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Advances in Design and Control SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines.

Editor-in-Chief Ralph C. Smith, North Carolina State University

Editorial Board Athanasios C. Antoulas, Rice University Siva Banda, Air Force Research Laboratory Belinda A. Batten, Oregon State University John Betts, The Boeing Company (retired) Stephen L. Campbell, North Carolina State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Arthur J. Krener, University of California, Davis Kirsten Morris, University of Waterloo Richard Murray, California Institute of Technology Ekkehard Sachs, University of Trier

Series Volumes Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical Approaches Speyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and Control Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue- Based Approach ¸ Adaptive Control Tutorial Ioannou, Petros and Fidan, Barıs, Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems Huang, J., Nonlinear Output Regulation: Theory and Applications Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems Gunzburger, Max D., Perspectives in Flow Control and Optimization Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J. William and James, Matthew R., Extending H1 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

Shapes and Geometries

Metrics, Analysis, Differential Calculus, and Optimization Second Edition

M. C. Delfour Université de Montréal Montréal, Québec Canada

J.-P. Zolésio

National Center for Scientific Research (CNRS) and National Institute for Research in Computer Science and Control (INRIA) Sophia Antipolis France

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2011 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. The research of the first author was supported by the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the National Sciences and Engineering Research Council of Canada; and the FQRNT program of the Ministère de l’Éducation du Québec. Library of Congress Cataloging-in-Publication Data Delfour, Michel C., 1943Shapes and geometries : metrics, analysis, differential calculus, and optimization / M. C. Delfour, J.-P. Zolésio. -- 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-898719-36-9 (hardcover : alk. paper) 1. Shape theory (Topology) I. Zolésio, J.-P. II. Title. QA612.7.D45 2011 514’.24--dc22 2010028846

is a registered trademark.

j

This book is dedicated to Alice, Jeanne, Jean, and Roger

Contents List of Figures Preface 1 Objectives and Scope of the Book . 2 Overview of the Second Edition . . 3 Intended Audience . . . . . . . . . 4 Acknowledgments . . . . . . . . . .

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1 Introduction: Examples, Background, and Perspectives 1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Geometry as a Variable . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of the Introductory Chapter . . . . . . . . . . . . . . 2 A Simple One-Dimensional Example . . . . . . . . . . . . . . . . . . 3 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Optimal Triangular Meshing . . . . . . . . . . . . . . . . . . . . . . . 6 Modeling Free Boundary Problems . . . . . . . . . . . . . . . . . . . 6.1 Free Interface between Two Materials . . . . . . . . . . . . . 6.2 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 7 Design of a Thermal Diﬀuser . . . . . . . . . . . . . . . . . . . . . . 7.1 Description of the Physical Problem . . . . . . . . . . . . . . 7.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 7.3 Reformulation of the Problem . . . . . . . . . . . . . . . . . . 7.4 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 7.5 Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Design of a Thermal Radiator . . . . . . . . . . . . . . . . . . . . . . 8.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 8.2 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 9 A Glimpse into Segmentation of Images . . . . . . . . . . . . . . . . 9.1 Automatic Image Processing . . . . . . . . . . . . . . . . . . 9.2 Image Smoothing/Filtering by Convolution and Edge Detectors 9.2.1 Construction of the Convolution of I . . . . . . . . 9.2.2 Space-Frequency Uncertainty Relationship . . . . . 9.2.3 Laplacian Detector . . . . . . . . . . . . . . . . . . . vii

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Objective Functions Deﬁned on the Whole Edge . . . . . . . 9.3.1 Eulerian Shape Semiderivative . . . . . . . . . . . . 9.3.2 From Local to Global Conditions on the Edge . . . 9.4 Snakes, Geodesic Active Contours, and Level Sets . . . . . . 9.4.1 Objective Functions Deﬁned on the Contours . . . . 9.4.2 Snakes and Geodesic Active Contours . . . . . . . . 9.4.3 Level Set Method . . . . . . . . . . . . . . . . . . . 9.4.4 Velocity Carried by the Normal . . . . . . . . . . . 9.4.5 Extension of the Level Set Equations . . . . . . . . 9.5 Objective Function Deﬁned on the Whole Image . . . . . . . 9.5.1 Tikhonov Regularization/Smoothing . . . . . . . . . 9.5.2 Objective Function of Mumford and Shah . . . . . . 9.5.3 Relaxation of the (N − 1)-Hausdorﬀ Measure . . . . 9.5.4 Relaxation to BV-, H s -, and SBV-Functions . . . . 9.5.5 Cracked Sets and Density Perimeter . . . . . . . . . Shapes and Geometries: Background and Perspectives . . . . . . . . 10.1 Parametrize Geometries by Functions or Functions by Geometries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Shape Analysis in Mechanics and Mathematics . . . . . . . . 10.3 Characteristic Functions: Surface Measure and Geometric Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Distance Functions: Smoothness, Normal, and Curvatures . . 10.5 Shape Optimization: Compliance Analysis and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Shape Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Shape Calculus and Tangential Diﬀerential Calculus . . . . . 10.8 Shape Analysis in This Book . . . . . . . . . . . . . . . . . . Shapes and Geometries: Second Edition . . . . . . . . . . . . . . . . 11.1 Geometries Parametrized by Functions . . . . . . . . . . . . . 11.2 Functions Parametrized by Geometries . . . . . . . . . . . . . 11.3 Shape Continuity and Optimization . . . . . . . . . . . . . . 11.4 Derivatives, Shape and Tangential Diﬀerential Calculuses, and Derivatives under State Constraints . . . . . . . . . . . . . .

2 Classical Descriptions of Geometries and Their Properties 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Abelian Group Structures on Subsets of a Fixed Holdall D . 2.2.1 First Abelian Group Structure on (P(D), ) . . . . 2.2.2 Second Abelian Group Structure on (P(D), ) . . . 2.3 Connected Space, Path-Connected Space, and Geodesic Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bouligand’s Contingent Cone, Dual Cone, and Normal Cone 2.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Sets Sets 5.1 5.2 5.3 5.4

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2.5.2 The Space W0m,p (Ω) . . . . . . . . . . . . . . . . . . 61 2.5.3 Embedding of H01 (Ω) into H01 (D) . . . . . . . . . . 62 2.5.4 Projection Operator . . . . . . . . . . . . . . . . . . 63 Spaces of Continuous and Diﬀerentiable Functions . . . . . . 63 2.6.1 Continuous and C k Functions . . . . . . . . . . . . 63 2.6.2 H¨ older (C 0, ) and Lipschitz (C 0,1 ) Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6.3 Embedding Theorem . . . . . . . . . . . . . . . . . 65 2.6.4 Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to Path-Connected Domains via the Geodesic Distance 66 Locally Described by an Homeomorphism or a Diﬀeomorphism 67 Sets of Classes C k and C k, . . . . . . . . . . . . . . . . . . . 67 Boundary Integral, Canonical Density, and Hausdorﬀ Measures 70 3.2.1 Boundary Integral for Sets of Class C 1 . . . . . . . 70 3.2.2 Integral on Submanifolds . . . . . . . . . . . . . . . 71 3.2.3 Hausdorﬀ Measures . . . . . . . . . . . . . . . . . . 72 Fundamental Forms and Principal Curvatures . . . . . . . . . 73 Globally Described by the Level Sets of a Function . . . . . . . 75 Locally Described by the Epigraph of a Function . . . . . . . . 78 Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs and the Space H of Dominating Functions . . . . . . . . . . . 79 olderian/Lipschitzian Sets . . . . 87 Local C k, -Epigraphs and H¨ Local C k, -Epigraphs and Sets of Class C k, . . . . . . . . . . 89 Locally Lipschitzian Sets: Some Examples and Properties . . 92 5.4.1 Examples and Continuous Linear Extensions . . . . 92 5.4.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . 93 5.4.3 Boundary Measure and Integral for Lipschitzian Sets 94 5.4.4 Geodesic Distance in a Domain and in Its Boundary 97 5.4.5 Nonhomogeneous Neumann and Dirichlet Problems 100 Locally Described by a Geometric Property . . . . . . . . . . . 101 Deﬁnitions and Main Results . . . . . . . . . . . . . . . . . . 102 Equivalence of Geometric Segment and C 0 Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Equivalence of the Uniform Fat Segment and the Equi-C 0 Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . 109 Uniform Cone/Cusp Properties and H¨ olderian/Lipschitzian Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.1 Uniform Cone Property and Lipschitzian Sets . . . 114 6.4.2 Uniform Cusp Property and H¨ olderian Sets . . . . . 115 Hausdorﬀ Measure and Dimension of the Boundary . . . . . 116

3 Courant Metrics on Images of a Set 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generic Constructions of Micheletti . . . . . . . . . . . . . 2.1 Space F(Θ) of Transformations of RN . . . . . . . 2.2 Diﬀeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

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Contents Closed Subgroups G . . . . . . . . . . . . . . . . . . . . . . . Courant Metric on the Quotient Group F(Θ)/G . . . . . . . Assumptions for B k (RN , RN ), C k (RN , RN ), and C0k (RN , RN ) 2.5.1 Checking the Assumptions . . . . . . . . . . . . . . 2.5.2 Perturbations of the Identity and Tangent Space . . 2.6 Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) . . . . . . 2.6.1 Checking the Assumptions . . . . . . . . . . . . . . 2.6.2 Perturbations of the Identity and Tangent Space . . Generalization to All Homeomorphisms and C k -Diﬀeomorphisms . .

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4 Transformations Generated by Velocities 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Metrics on Transformations Generated by Velocities . . . . . . . 2.1 Subgroup GΘ of Transformations Generated by Velocities 2.2 Complete Metrics on GΘ and Geodesics . . . . . . . . . . 2.3 Constructions of Azencott and Trouv´e . . . . . . . . . . . 3 Semiderivatives via Transformations Generated by Velocities . . 3.1 Shape Function . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gateaux and Hadamard Semiderivatives . . . . . . . . . . 3.3 Examples of Families of Transformations of Domains . . . 3.3.1 C ∞ -Domains . . . . . . . . . . . . . . . . . . . . 3.3.2 C k -Domains . . . . . . . . . . . . . . . . . . . . 3.3.3 Cartesian Graphs . . . . . . . . . . . . . . . . . 3.3.4 Polar Coordinates and Star-Shaped Domains . . 3.3.5 Level Sets . . . . . . . . . . . . . . . . . . . . . . 4 Unconstrained Families of Domains . . . . . . . . . . . . . . . . . 4.1 Equivalence between Velocities and Transformations . . . 4.2 Perturbations of the Identity . . . . . . . . . . . . . . . . 4.3 Equivalence for Special Families of Velocities . . . . . . . 5 Constrained Families of Domains . . . . . . . . . . . . . . . . . . 5.1 Equivalence between Velocities and Transformations . . . 5.2 Transformation of Condition (V2D ) into a Linear Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Continuity of Shape Functions along Velocity Flows . . . . . . . 5 Metrics via Characteristic Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Abelian Group Structure on Measurable Characteristic Functions 2.1 Group Structure on Xµ (RN ) . . . . . . . . . . . . . . . . 2.2 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Complete Metric for Characteristic Functions in Lp -Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lebesgue Measurable Characteristic Functions . . . . . . . . . . 3.1 Strong Topologies and C ∞ -Approximations . . . . . . . . 3.2 Weak Topologies and Microstructures . . . . . . . . . . . 3.3 Nice or Measure Theoretic Representative . . . . . . . . .

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xi 3.4 The Family of Convex Sets . . . . . . . . . . . . . . . . . . . 3.5 Sobolev Spaces for Measurable Domains . . . . . . . . . . . . Some Compliance Problems with Two Materials . . . . . . . . . . . 4.1 Transmission Problem and Compliance . . . . . . . . . . . . 4.2 The Original Problem of C´ea and Malanowski . . . . . . . . . 4.3 Relaxation and Homogenization . . . . . . . . . . . . . . . . Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . Caccioppoli or Finite Perimeter Sets . . . . . . . . . . . . . . . . . . 6.1 Finite Perimeter Sets . . . . . . . . . . . . . . . . . . . . . . . 6.2 Decomposition of the Integral along Level Sets . . . . . . . . 6.3 Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces . . . . . . . . . . . . . . . . . . . 6.4 Compactness and Uniform Cone Property . . . . . . . . . . . Existence for the Bernoulli Free Boundary Problem . . . . . . . . . . 7.1 An Example: Elementary Modeling of the Water Wave . . . 7.2 Existence for a Class of Free Boundary Problems . . . . . . . 7.3 Weak Solutions of Some Generic Free Boundary Problems . . 7.3.1 Problem without Constraint . . . . . . . . . . . . . 7.3.2 Constraint on the Measure of the Domain Ω . . . . 7.4 Weak Existence with Surface Tension . . . . . . . . . . . . .

6 Metrics via Distance Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniform Metric Topologies . . . . . . . . . . . . . . . . . . . . . . . 2.1 Family of Distance Functions Cd (D) . . . . . . . . . . . . . . 2.2 Pomp´eiu–Hausdorﬀ Metric on Cd (D) . . . . . . . . . . . . . . 2.3 Uniform Complementary Metric Topology and Cdc (D) . . . . c 2.4 Families Cdc (E; D) and Cd,loc (E; D) . . . . . . . . . . . . . . . 3 Projection, Skeleton, Crack, and Diﬀerentiability . . . . . . . . . . . 4 W 1,p -Metric Topology and Characteristic Functions . . . . . . . . . 4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 4.2 Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . . 5 Sets of Bounded and Locally Bounded Curvature . . . . . . . . . . . 5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Reach and Federer’s Sets of Positive Reach . . . . . . . . . . . . . . 6.1 Deﬁnitions and Main Properties . . . . . . . . . . . . . . . . 6.2 C k -Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A Compact Family of Sets with Uniform Positive Reach . . . 7 Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Characterization of Convex Sets . . . . . . . . . . . . . . . . . . . . 8.1 Convex Sets and Properties of dA . . . . . . . . . . . . . . . . 8.2 Semiconvexity and BV Character of dA . . . . . . . . . . . . 8.3 Closed Convex Hull of A and Fenchel Transform of dA . . . . 8.4 Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and c (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cd,loc

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Compactness Theorems for Sets of Bounded Curvature . . . . . . . . 324 9.1 Global Conditions in D . . . . . . . . . . . . . . . . . . . . . 325 9.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 327

7 Metrics via Oriented Distance Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Family of Oriented Distance Functions Cb (D) . . . . . . 2.2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . 3 Projection, Skeleton, Crack, and Diﬀerentiability . . . . . . . . . . . 4 W 1,p (D)-Metric Topology and the Family Cb0 (D) . . . . . . . . . . . 4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 4.2 Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . . 5 Boundary of Bounded and Locally Bounded Curvature . . . . . . . . 5.1 Examples and Limit of Tubular Norms as h Goes to Zero . . 6 Approximation by Dilated Sets/Tubular Neighborhoods . . . . . . . 7 Federer’s Sets of Positive Reach . . . . . . . . . . . . . . . . . . . . . 7.1 Approximation by Dilated Sets/Tubular Neighborhoods . . . 7.2 Boundaries with Positive Reach . . . . . . . . . . . . . . . . . 8 Boundary Smoothness and Smoothness of bA . . . . . . . . . . . . . 9 Sobolev or W m,p Domains . . . . . . . . . . . . . . . . . . . . . . . . 10 Characterization of Convex and Semiconvex Sets . . . . . . . . . . . 10.1 Convex Sets and Convexity of bA . . . . . . . . . . . . . . . . 10.2 Families of Convex Sets Cb (D), Cb (E; D), and Cb,loc (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 BV Character of bA and Semiconvex Sets . . . . . . . . . . . 11 Compactness and Sets of Bounded Curvature . . . . . . . . . . . . . 11.1 Global Conditions on D . . . . . . . . . . . . . . . . . . . . . 11.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 12 Finite Density Perimeter and Compactness . . . . . . . . . . . . . . 13 Compactness and Uniform Fat Segment Property . . . . . . . . . . . 13.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Equivalent Conditions on the Local Graph Functions . . . . . 14 Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 De Giorgi Perimeter of Caccioppoli Sets . . . . . . . . . . . . 14.2 Finite Density Perimeter . . . . . . . . . . . . . . . . . . . . . 15 The Families of Cracked Sets . . . . . . . . . . . . . . . . . . . . . . 16 A Variation of the Image Segmentation Problem of Mumford and Shah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 16.2 Cracked Sets without the Perimeter . . . . . . . . . . . . . . 16.2.1 Technical Lemmas . . . . . . . . . . . . . . . . . . . 16.2.2 Another Compactness Theorem . . . . . . . . . . . 16.2.3 Proof of Theorem 16.1 . . . . . . . . . . . . . . . . . 16.3 Existence of a Cracked Set with Minimum Density Perimeter

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xiii 16.4

Uniform Bound or Penalization Term in the Objective Function on the Density Perimeter . . . . . . . . . . . . . . . 407

8 Shape Continuity and Optimization 1 Introduction and Generic Examples . . . . . . . . . . . . . . . . . . . 1.1 First Generic Example . . . . . . . . . . . . . . . . . . . . . . 1.2 Second Generic Example . . . . . . . . . . . . . . . . . . . . . 1.3 Third Generic Example . . . . . . . . . . . . . . . . . . . . . 1.4 Fourth Generic Example . . . . . . . . . . . . . . . . . . . . . 2 Upper Semicontinuity and Maximization of the First Eigenvalue . . 3 Continuity of the Transmission Problem . . . . . . . . . . . . . . . . 4 Continuity of the Homogeneous Dirichlet Boundary Value Problem . 4.1 Classical, Relaxed, and Overrelaxed Problems . . . . . . . . . 4.2 Classical Dirichlet Boundary Value Problem . . . . . . . . . . 4.3 Overrelaxed Dirichlet Boundary Value Problem . . . . . . . . 4.3.1 Approximation by Transmission Problems . . . . . . 4.3.2 Continuity with Respect to X(D) in the Lp (D)-Topology . . . . . . . . . . . . . . . . . . . . 4.4 Relaxed Dirichlet Boundary Value Problem . . . . . . . . . . 5 Continuity of the Homogeneous Neumann Boundary Value Problem 6 Elements of Capacity Theory . . . . . . . . . . . . . . . . . . . . . . 6.1 Deﬁnition and Basic Properties . . . . . . . . . . . . . . . . . 6.2 Quasi-continuous Representative and H 1 -Functions . . . . . . 6.3 Transport of Sets of Zero Capacity . . . . . . . . . . . . . . . 7 Crack-Free Sets and Some Applications . . . . . . . . . . . . . . . . 7.1 Deﬁnitions and Properties . . . . . . . . . . . . . . . . . . . . 7.2 Continuity and Optimization over L(D, r, O, λ) . . . . . . . . 7.2.1 Continuity of the Classical Homogeneous Dirichlet Boundary Condition . . . . . . . . . . . . . . . . . . 7.2.2 Minimization/Maximization of the First Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . 8 Continuity under Capacity Constraints . . . . . . . . . . . . . . . . . 9 Compact Families Oc,r (D) and Lc,r (O, D) . . . . . . . . . . . . . . . 9.1 Compact Family Oc,r (D) . . . . . . . . . . . . . . . . . . . . 9.2 Compact Family Lc,r (O, D) and Thick Set Property . . . . . 9.3 Maximizing the Eigenvalue λA (Ω) . . . . . . . . . . . . . . . 9.4 State Constrained Minimization Problems . . . . . . . . . . . 9.5 Examples with a Constraint on the Gradient . . . . . . . . .

409 409 411 411 411 412 412 417 418 418 421 423 423

9 Shape and Tangential Diﬀerential Calculuses 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Diﬀerentiation in Topological Vector Spaces 2.1 Deﬁnitions of Semiderivatives and Derivatives . 2.2 Derivatives in Normed Vector Spaces . . . . . . 2.3 Locally Lipschitz Functions . . . . . . . . . . . 2.4 Chain Rule for Semiderivatives . . . . . . . . .

457 457 458 458 461 465 465

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

424 425 426 429 429 431 432 434 434 437 437 438 440 447 447 450 452 453 454

xiv

Contents

3

4

5

6

2.5 Semiderivatives of Convex Functions . . . . . . . . . . . 2.6 Hadamard Semiderivative and Velocity Method . . . . . First-Order Shape Semiderivatives and Derivatives . . . . . . . 3.1 Eulerian and Hadamard Semiderivatives . . . . . . . . . 3.2 Hadamard Semidiﬀerentiability and Courant Metric Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Perturbations of the Identity and Gateaux and Fr´echet Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Shape Gradient and Structure Theorem . . . . . . . . . Elements of Shape Calculus . . . . . . . . . . . . . . . . . . . . 4.1 Basic Formula for Domain Integrals . . . . . . . . . . . 4.2 Basic Formula for Boundary Integrals . . . . . . . . . . 4.3 Examples of Shape Derivatives . . . . . . . . . . . . . . 4.3.1 Volume of Ω and Surface Area of Γ . . . . . . 4.3.2 H 1 (Ω)-Norm . . . . . . . . . . . . . . . . . . . 4.3.3 Normal Derivative . . . . . . . . . . . . . . . . Elements of Tangential Calculus . . . . . . . . . . . . . . . . . 5.1 Intrinsic Deﬁnition of the Tangential Gradient . . . . . 5.2 First-Order Derivatives . . . . . . . . . . . . . . . . . . 5.3 Second-Order Derivatives . . . . . . . . . . . . . . . . . 5.4 A Few Useful Formulae and the Chain Rule . . . . . . . 5.5 The Stokes and Green Formulae . . . . . . . . . . . . . 5.6 Relation between Tangential and Covariant Derivatives 5.7 Back to the Example of Section 4.3.3 . . . . . . . . . . . Second-Order Semiderivative and Shape Hessian . . . . . . . . 6.1 Second-Order Derivative of the Domain Integral . . . . 6.2 Basic Formula for Domain Integrals . . . . . . . . . . . 6.3 Nonautonomous Case . . . . . . . . . . . . . . . . . . . 6.4 Autonomous Case . . . . . . . . . . . . . . . . . . . . . 6.5 Decomposition of d2 J(Ω; V (0), W (0)) . . . . . . . . . .

. . . .

. . . .

. . . .

467 469 471 471

. . . 476 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

10 Shape Gradients under a State Equation Constraint 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Min Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 An Illustrative Example and a Shape Variational Principle 2.2 Function Space Parametrization . . . . . . . . . . . . . . . 2.3 Diﬀerentiability of a Minimum with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Application of the Theorem . . . . . . . . . . . . . . . . . . 2.5 Domain and Boundary Integral Expressions of the Shape Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Transport of H0k (Ω) by W k,∞ -Transformations of RN . . . 4.2 Laplacian and Bi-Laplacian . . . . . . . . . . . . . . . . . . 4.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

476 479 482 482 484 486 486 487 488 491 492 495 496 497 498 498 501 501 502 504 505 510 515

. . . .

519 519 521 521 522

. 523 . 526 . . . . . .

530 532 535 536 537 546

Contents 5

6

xv Saddle 5.1 5.2 5.3 5.4

Point Formulation and Function Space Parametrization . . An Illustrative Example . . . . . . . . . . . . . . . . . . . . Saddle Point Formulation . . . . . . . . . . . . . . . . . . . Function Space Parametrization . . . . . . . . . . . . . . . Diﬀerentiability of a Saddle Point with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application of the Theorem . . . . . . . . . . . . . . . . . . 5.6 Domain and Boundary Expressions for the Shape Gradient Multipliers and Function Space Embedding . . . . . . . . . . . . . 6.1 The Nonhomogeneous Dirichlet Problem . . . . . . . . . . . 6.2 A Saddle Point Formulation of the State Equation . . . . . 6.3 Saddle Point Expression of the Objective Function . . . . . 6.4 Veriﬁcation of the Assumptions of Theorem 5.1 . . . . . . .

. . . .

551 551 552 553

. . . . . . . .

555 559 561 562 562 563 564 566

Elements of Bibliography

571

Index of Notation

615

Index

619

List of Figures 1.1 1.2 1.3 1.4 1.5 1.6

Graph of J(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Column of height one and cross section area A under the load . Triangulation and basis function associated with node Mi . . . . . Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . . Heat spreading scheme for high-power solid-state devices. . . . . (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω; . . . . . . . . . . . . . . . . . . . . . (C) Surface D generating Ω. 1.7 Volume Ω and its cross section. . . . . . . . . . . . . . . . . . . . and its generating surface A. . . . . . . . . . . . . . . 1.8 Volume Ω 1.9 Image I of objects and their segmentation in the frame D. . . . . 1.10 Image I containing black curves or cracks in the frame D. . . . . 1.11 Example of a two-dimensional strongly cracked set. . . . . . . . . 1.12 Example of a surface with facets associated with a ball. . . . . .

. . . . .

. . . . .

4 5 8 11 14

. . . . . . .

. . . . . . .

15 19 20 22 22 35 37 68 79 91 92

2.8

Diﬀeomorphism gx from U (x) to B. . . . . . . . . . . . . . . . . . . Local epigraph representation (N = 2). . . . . . . . . . . . . . . . . . Domain Ω0 and its image T (Ω0 ) spiraling around the origin. . . . . . Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin. . . . Examples of arbitrary and axially symmetrical O around the direction d = Ax (0, eN ). . . . . . . . . . . . . . . . . . . . . . . . . . The cone x + Ax C(λ, ω) in the direction Ax eN . . . . . . . . . . . . . Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α , h(θ) = θ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. . .

4.1

Transport of Ω by the velocity ﬁeld V . . . . . . . . . . . . . . . . . . 171

2.1 2.2 2.3 2.4 2.5 2.6 2.7

5.1 5.2 5.3 5.4 5.5 5.6

. . . . . . . . . . . . . Smiling sun Ω and expressionless sun Ω. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 . . . . . . . . . . . . . . Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . The function f (x, y) = 56 (1 − |x| − |y|)6 . . . . . . . . . . . . . Optimal distribution and isotherms with k1 = 2 (black) and k2 (white) for the problem of section 4.1. . . . . . . . . . . . . . . Optimal distribution and isotherms with k1 = 2 (black) and k2 (white) for the problem of C´ea and Malanowski. . . . . . . . . xvii

. . . . . . . . . . . . =1 . . . =1 . . .

109 114 118 119

220 227 228 234 235 239

xviii

List of Figures

5.7

The staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1 6.2 6.3 6.4 6.5

Skeletons Sk (A), Sk (A), and Sk (∂A) = Sk (A) ∪ Sk (A). Nonuniqueness of the exterior normal. . . . . . . . . . . . . Vertical stripes of Example 4.1. . . . . . . . . . . . . . . . . ∇dA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . Set of critical points of A. . . . . . . . . . . . . . . . . . . .

. . . . .

280 286 293 301 318

7.1 7.2

∇bA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . . W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R2 with uniformly bounded density perimeter to a set with empty interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of a two-dimensional strongly cracked set. . . . . . . . . . . The two-dimensional strongly cracked set of Figure 7.3 in an open frame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The two open components Ω1 and Ω2 of the open domain Ω for N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

356

7.3 7.4 7.5

. . . . .

. . . . .

. . . . .

. . . . .

387 396 400 406

Preface

1

Objectives and Scope of the Book

The objective of this book is to give a comprehensive presentation of mathematical constructions and tools that can be used to study problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. In that context, a good analytical framework and good modeling techniques must be able to handle the occurrence of singular behaviors whenever they are compatible with the mechanics or the physics of the problems at hand. In some optimization problems, the natural intuitive notion of a geometric domain undergoes mutations into relaxed entities such as microstructures. So the objects under consideration need not be smooth open domains, or even sets, as long as they still makes sense mathematically. This book covers the basic mathematical ideas, constructions, and methods that come from diﬀerent ﬁelds of mathematical activities and areas of applications that have often evolved in parallel directions. The scope of research is frighteningly broad because it touches on areas that include classical geometry, modern partial differential equations, geometric measure theory, topological groups, and constrained optimization, with applications to classical mechanics of continuous media such as ﬂuid mechanics, elasticity theory, fracture theory, modern theories of optimal design, optimal location and shape of geometric objects, free and moving boundary problems, and image processing. Innovative modeling or new issues raised in some applications force a new look at the fundamentals of well-established mathematical areas such as geometry, to relax basic notions of volume, perimeter, and curvature or boundary value problems, and to ﬁnd suitable relaxations of solutions. In that spirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notion of volume to the one of measure on an equivalence class of measurable sets in 1907. He was followed in that endeavor in the early 1950s by the celebrated work of E. De Giorgi, who used the relaxed notion of perimeter deﬁned on the class of Caccioppoli sets to solve Plateau’s problem of minimal surfaces. The material that is pertinent to the study of geometric objects and the entities and functions that are deﬁned on them would necessitate an encyclopedic investment to bring together the basic theories and their ﬁelds of applications. This objective is obviously beyond the scope of a single book and two authors. The xix

xx

Preface

coverage of this book is more modest. Yet, it contains most of the important fundamentals at this stage of evolution of this expanding ﬁeld. Even if shape analysis and optimization have undergone considerable and important developments on the theoretical and numerical fronts, there are still cultural barriers between areas of applications and between theories. The whole ﬁeld is extremely active, and the best is yet to come with fundamental structures and tools beginning to emerge. It is hoped that this book will help to build new bridges and stimulate cross-fertilization of ideas and methods.

2

Overview of the Second Edition

The second edition is almost a new book. All chapters from the ﬁrst edition have been updated and, in most cases, considerably enriched with new material. Many chapters or parts of chapters have been completely rewritten following the developments in the ﬁeld over the past 10 years. The book went from 9 to 10 chapters with a more elaborate sectioning of each chapter in order to produce a much more detailed table of contents. This makes it easier to ﬁnd speciﬁc material. A series of illustrative generic examples has been added right at the beginning of the introductory Chapter 1 to motivate the reader and illustrate the basic dilemma: parametrize geometries by functions or functions by geometries? This is followed by the big picture: a section on background and perspectives and a more detailed presentation of the second edition. The former Chapter 2 has been split into Chapter 2 on the classical descriptions and properties of domains and sets and a new Chapter 3, where the important material on Courant metrics and the generic constructions of A. M. Micheletti have been reorganized and expanded. Basic deﬁnitions and material have been added and regrouped at the beginning of Chapter 2: Abelian group structure on subsets of a set, connected and path-connected spaces, function spaces, tangent and dual cones, and geodesic distance. The coverage of domains that verify some segment property and have a local epigraph representation has been considerably expanded, and Lipschitzian (graph) domains are now dealt with as a special case. The new Chapter 3 on domains and submanifolds that are the image of a ﬁxed set considerably expands the material of the ﬁrst edition by bringing up the general assumptions behind the generic constructions of A. M. Micheletti that lead to the Courant metrics on the quotient space of families of transformations by subgroups of isometries such as identities, rotations, translations, or ﬂips. The general results apply to a broad range of groups of transformations of the Euclidean space and to arbitrary closed subgroups. New complete metrics on the whole spaces of homeomorphisms and C k -diﬀeomorphisms are also introduced to extend classical results for transformations of compact manifolds to general unbounded closed sets and open sets that are crack-free. This material is central in classical mechanics and physics and in modern applications such as imaging and detection. The former Chapter 7 on transformations versus ﬂows of velocities has been moved right after the Courant metrics as Chapter 4 and considerably expanded. It now specializes the results of Chapter 3 to spaces of transformations that are

2. Overview of the Second Edition

xxi

generated by the ﬂow of a velocity ﬁeld over a generic time interval. One important motivation is to introduce a notion of semiderivatives as well as a tractable criterion for continuity with respect to Courant metrics. Another motivation for the velocity point of view is the general framework of R. Azencott and A. Trouv´e starting in 1994 with applications in imaging. They construct complete metrics in relation with geodesic paths in spaces of diﬀeomorphisms generated by a velocity ﬁeld. The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and 5 on distance and oriented distance functions have become Chapters 5, 6, and 7. Those chapters have been renamed Metrics Generated by . . . in order to emphasize one of the main thrusts of the book: the construction of complete metrics on shapes and geometries.1 Those chapters emphasize the function analytic description of sets and domains: construction of metric topologies and characterization of compact families of sets or submanifolds in the Euclidean space. In that context, we are now dealing with equivalence classes of sets that may or may not have an invariant open or closed representative in the class. For instance, they include Lebesgue measurable sets and Federer’s sets of positive reach. Many of the classical properties of sets can be recovered from the smoothness or function analytic properties of those functions. The former Chapter 6 on optimization of shape functions has been completely rewritten and expanded as Chapter 8 on shape continuity and optimization. With meaningful metric topologies, we can now speak of continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc., compact families of sets, and existence of optimal geometries. The chapter concentrates on continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the objective function is deﬁned as an inﬁmum over a family of functions over a ﬁxed domain or set such as the eigenvalue problems. We ﬁrst characterize the continuity of the transmission problem and the upper semicontinuity of the ﬁrst eigenvalue of the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of deﬁnition since they require diﬀerent constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial diﬀerential equations. An introduction is also given to the concepts and results from capacity theory from which very general families of sets stable with respect to boundary conditions can be constructed. Note that some material has been moved from one chapter to another. For instance, section 7 on the continuity of the Dirichlet boundary problem in the former Chapter 3 has been merged with the content of the former Chapter 4 in the new Chapter 8. The former Chapters 8 and 9 have become Chapters 9 and 10. They are devoted to a modern version of the shape calculus, an introduction to the tangential diﬀerential calculus, and the shape derivatives under a state equation constraint. In Chapters 3, 5, 6, and 7, we have constructed complete metric spaces of geometries. Those spaces are nonlinear and nonconvex. However, several of them have a group 1 This is in line with current trends in the literature such as in the work of the 2009 Abel Prize ´moli and G. winner M. Gromov [1] and its applications in imaging by G. Sapiro [1] and F. Me Sapiro [1] to identify objects up to an isometry.

xxii

Preface

structure and, in some cases, it is possible to construct C 1 -paths in the group from velocity ﬁelds. This leads to the notion of Eulerian semiderivative that is somehow the analogue of a derivative on a smooth manifold. In fact, two types of semiderivatives are of interest: the weaker Gateaux style semiderivative and the stronger Hadamard style semiderivative. In the latter case, the classical chain rule is still available even for nondiﬀerentiable functions. In order to prepare the ground for shape derivatives, an enriched self-contained review of the pertinent material on semiderivatives and derivatives in topological vector spaces is provided. The important Chapter 10 concentrates on two generic examples often encountered in shape optimization. The ﬁrst one is associated with the so-called compliance problems, where the shape functional is itself the minimum of a domain-dependent energy functional. The special feature of such functionals is that the adjoint state coincides with the state. This obviously leads to considerable simpliﬁcations in the analysis. In that case, it will be shown that theorems on the diﬀerentiability of the minimum of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the minimizer is not unique. The second one will deal with shape functionals that can be expressed as the saddle point of some appropriate Lagrangian. As in the ﬁrst example, theorems on the diﬀerentiability of the saddle point of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the solution of the saddle point equations is not unique. Avoiding the diﬀerentiation of the state equation with respect to the domain is particularly advantageous in shape problems.

3

Intended Audience

The targeted audience is applied mathematicians and advanced engineers and scientists, but the book is also suitable for a broader audience of mathematicians as a relatively well-structured initiation to shape analysis and calculus techniques. Some of the chapters are fairly self-contained and of independent interest. They can be used as lecture notes for a mini-course. The material at the beginning of each chapter is accessible to a broad audience, while the latter sections may sometimes require more mathematical maturity. Thus the book can be used as a graduate text as well as a reference book. It complements existing books that emphasize speciﬁc mechanical or engineering applications or numerical methods. It can be considered ´sio [9], Introduction a companion to the book of J. Sokolowski and J.-P. Zole to Shape Optimization, published in 1992. Earlier versions of parts of this book have been used as lecture notes in graduate courses at the Universit´e de Montr´eal in 1986–1987, 1993–1994, 1995–1996, and 1997–1998 and at international meetings, workshops, or schools: S´eminaire de Math´ematiques Sup´erieures on Shape Optimization and Free Boundaries (Montr´eal, Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis (K´enitra, Morocco, December 1993), course of the COMETT MATARI European Program on Shape Optimization and Mutational Equations (Sophia-Antipolis, France, September 27 to October 1, 1993), CRM Summer School on Boundaries,

4. Acknowledgments

xxiii

Interfaces and Transitions (Banﬀ, Canada, August 6–18, 1995), and CIME course on Optimal Design (Troia, Portugal, June 1998).

4

Acknowledgments

The ﬁrst author is pleased to acknowledge the support of the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the constant support of the National Sciences and Engineering Research Council of Canada; ´ and the FQRNT program of the Minist`ere de l’Education du Qu´ebec. Many thanks also to Louise Letendre and Andr´e Montpetit of the Centre de Recherches Math´ematiques, who provided their technical support, experience, and talent over the extended period of gestation of this book. Michel Delfour Jean-Paul Zol´esio August 13, 2009

Chapter 1

Introduction: Examples, Background, and Perspectives 1 1.1

Orientation Geometry as a Variable

The central object of this book1 is the geometry as a variable. As in the theory of functions of real variables, we need a diﬀerential calculus, spaces of geometries, evolution equations, and other familiar concepts in analysis when the variable is no longer a scalar, a vector, or a function, but is a geometric domain. This is motivated by many important problems in science and engineering that involve the geometry as a modeling, design, or control variable. In general the geometric objects we shall consider will not be parametrized or structured. Yet we are not starting from scratch, and several building blocks are already available from many ﬁelds: geometric measure theory, physics of continuous media, free boundary problems, the parametrization of geometries by functions, the set derivative as the inverse of the integral, the parametrization of functions by geometries, the Pomp´eiu–Hausdorﬀ metric, and so on. As is often the case in mathematics, spaces of geometries and notions of derivatives with respect to the geometry are built from well-established elements of functional analysis and diﬀerential calculus. There are many ways to structure families of geometries. For instance, a domain can be made variable by considering 1 The numbering of equations, theorems, lemmas, corollaries, deﬁnitions, examples, and remarks is by chapter. When a reference to another chapter is necessary it is always followed by the words in Chapter and the number of the chapter. For instance, “equation (2.18) in Chapter 9.” The text of theorems, lemmas, and corollaries is slanted; the text of deﬁnitions, examples, and remarks is normal shape and ended by a square . This makes it possible to aesthetically emphasize certain words especially in deﬁnitions. The bibliography is by author in alphabetical order. For each author or group of coauthors, there is a numbering in square brackets starting with [1]. A ´ [3] and a reference to an reference to an item by a single author is of the form J. Dieudonne item with several coauthors S. Agmon, A. Douglis, and L. Nirenberg [2]. Boxed formulae or statements are used in some chapters for two distinct purposes. First, they emphasize certain important deﬁnitions, results, or identities; second, in long proofs of some theorems, lemmas, or corollaries, they isolate key intermediary results which will be necessary to more easily follow the subsequent steps of the proof.

1

2

Chapter 1. Introduction: Examples, Background, and Perspectives

the images of a ﬁxed domain by a family of diﬀeomorphisms that belong to some function space over a ﬁxed domain. This naturally occurs in physics and mechanics, where the deformations of a continuous body or medium are smooth, or in the numerical analysis of optimal design problems when working on a ﬁxed grid. This construction naturally leads to a group structure induced by the composition of the diﬀeomorphisms. The underlying spaces are no longer topological vector spaces but groups that can be endowed with a nice complete metric space structure by introducing the Courant metric. The practitioner might or might not want to use the underlying mathematical structure associated with his or her constructions, but it is there and it contains information that might guide the theory and inﬂuence the choice of the numerical methods used in the solution of the problem at hand. The parametrization of a ﬁxed domain by a ﬁxed family of diﬀeomorphisms obviously limits the family of variable domains. The topology of the images is similar to the topology of the ﬁxed domain. Singularities that were not already present there cannot be created in the images. Other constructions make it possible to considerably enlarge the family of variable geometries and possibly open the doors to pathological geometries that are no longer open sets with a nice boundary. Instead of parametrizing the domains by functions or diﬀeomorphisms, certain families of functions can be parametrized by sets. A single function completely speciﬁes a set or at least an equivalence class of sets. This includes the distance functions and the characteristic function, but also the support function from convex analysis. Perhaps the best known example of that construction is the Pomp´eiu–Hausdorﬀ metric topology. This is a very weak topology that does not preserve the volume of a set. When the volume, the perimeter, or the curvatures are important, such functions must be able to yield relaxed deﬁnitions of volume, perimeter, or curvatures. The characteristic function that preserves the volume has many applications. It played a fundamental role in the integration theory of Henri Lebesgue at the beginning of the 20th century. It was also used in the 1950s by E. De Giorgi to deﬁne a relaxed notion of perimeter in the theory of minimal surfaces. Another technique that has been used successfully in free or moving boundary problems, such as motion by mean curvature, shock waves, or detonation theory, is the use of level sets of a function to describe a free or moving boundary. Such functions are often the solution of a system of partial diﬀerential equations. This is another way to build new tools from functional analysis. The choice of families of function parametrized sets or of families of set parametrized functions, or other appropriate constructions, is obviously problem dependent, much like the choice of function spaces of solutions in the theory of partial diﬀerential equations or optimization problems. This is one aspect of the geometry as a variable. Another aspect is to build the equivalent of a diﬀerential calculus and the computational and analytical tools that are essential in the characterization and computation of geometries. Again, we are not starting from scratch and many building blocks are already available, but many questions and issues remain open. This book aims at covering a small but fundamental part of that program. We had to make diﬃcult choices and refer the reader to appropriate books and references for background material such as geometric measure theory and specialized topics such as homogenization theory and microstructures which are available in excellent

2. A Simple One-Dimensional Example

3

books in English. It was unfortunately not possible to include references to the considerable literature on numerical methods, free and moving boundary problems, and optimization.

1.2

Outline of the Introductory Chapter

We ﬁrst give a series of generic examples where the shape or the geometry is the modeling, control, or optimization variable. They will be used in the subsequent chapters to illustrate the many ways such problems can be formulated. The ﬁrst example is the celebrated problem of the optimal shape of a column formulated by Lagrange in 1770 to prevent buckling. The extremization of the eigenvalues has also received considerable attention in the engineering literature. The free interface between two regions with diﬀerent physical or mechanical properties is another generic problem that can lead in some cases to a mixing or a microstructure. Two typical problems arising from applications to condition the thermal environment of satellites are described in sections 7 and 8. The ﬁrst one is the design of a thermal diﬀuser of minimal weight subject to an inequality constraint on the output thermal power ﬂux. The second one is the design of a thermal radiator to eﬀectively radiate large amounts of thermal power to space. The geometry is a volume of revolution around an axis that is completely speciﬁed by its height and the function which speciﬁes its lateral boundary. Finally, we give a glimpse at image segmentation, which is an example of shape/geometric identiﬁcation problems. Many chapters of this book are of direct interest to imaging sciences. Section 10 presents some background and perspectives. A fundamental issue is to ﬁnd tractable and preferably analytical representations of a geometry as a variable that are compatible with the problems at hand. The generic examples suggest two types of representations: the ones where the geometry is parametrized by functions and the ones where a family of functions is parametrized by the geometry. As is always the case, the choice is very much problem dependent. In the ﬁrst case, the topology of the variable sets is ﬁxed; in the second case the families of sets are much larger and topological changes are included. The book presents the two points of view. Finally, section 11 sketches the material in the second edition of the book.

2

A Simple One-Dimensional Example

A general feature of minimization problems with respect to a shape or a geometry subject to a state equation constraint is that they are generally not convex and that, when they have a solution, it is generally not unique. This is illustrated in ´a [2]: minimize the objective function the following simple example from J. Ce a def 2 |ya (x) − 1| dx, J(a) = 0

where a ≥ 0 and ya is the solution of the boundary value problem (state equation) d2 ya def (x) = −2 in Ωa = (0, a), dx2

dya (0) = 0, dx

ya (a) = 0.

(2.1)

4

Chapter 1. Introduction: Examples, Background, and Perspectives

Here the one-dimensional geometric domain Ωa = ]0, a[ is the minimizing variable. We recognize the classical structure of a control problem, except that the minimizing variable is no longer under the integral sign but in the limits of the integral sign. One consequence of this diﬀerence is that even the simplest problems will usually not be convex or convexiﬁables. They will require a special analysis. In this example it is easy to check that the solution of the state equation is ya (x) = a2 − x2

and J(a) =

8 5 4 3 a − a + a. 15 3

The graph of J, shown in Figure 1.1, is not the graph of a convex function. Its global minimum in a0 = 0, local maximum in a1 , and local minimum in a2 , 3 3 1 1 def def a1 = 1 − √ , a2 = 1+ √ , 4 4 3 3 are all diﬀerent.

a1

a2

Figure 1.1. Graph of J(a). To avoid a trivial solution, a strictly positive lower bound must be put on a. A unique minimizing solution is obtained for a ≥ a1 where the gradient of J is zero. For 0 < a < a2 , the minimum will occur at the preset lower or upper bound on a.

3

Buckling of Columns

The next example illustrates the fact that even simple problems can be nondiﬀerentiable with respect to the geometry. This is generic of all eigenvalue problems when the eigenvalue is not simple. One of the early optimal design problems was formulated by J. L. Lagrange [1] in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by the Danish mathematician and astronomer T. Clausen [1] in 1849. It consists in ﬁnding the best proﬁle of a vertical column of ﬁxed volume to prevent buckling.

3. Buckling of Columns

5

It turns out that this problem is in fact a hidden maximization of an eigenvalue. Many incorrect solutions had been published until 1992. This problem and other problems related to columns have been revisited in a series of papers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, but a complete theoretical and numerical solution for the buckling of a column was given only in 1992 by S. J. Cox and M. L. Overton [1]. The diﬃculty was that the eigenvalue is not simple and hence not diﬀerentiable with respect to the geometry. Consider a normalized column of unit height and unit volume (see Figure 1.2). Denote by the magnitude of the normalized axial load and by u the resulting transverse displacement. Assume that the potential energy is the sum of the bending and elongation energies 1 1 EI |u |2 dx − |u |2 dx, 0

0

where I is the second moment of area of the column’s cross section and E is its Young’s modulus. For suﬃciently small load the minimum of this potential energy with respect to all admissible u is zero. Euler’s buckling load λ of the column is the largest for which this minimum is zero. This is equivalent to ﬁnding the following minimum: 1 EI |u |2 dx def 0 λ = inf , (3.1) 1 |2 dx 0=u∈V |u 0 where V = H02 (0, 1) corresponds to the clamped case, but other types of boundary conditions can be contemplated. This is an eigenvalue problem with a special Rayleigh quotient. Assume that E is constant and that the second moment of area I(x) of the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

1

x

normalized load

cross section area A(x)

0

Figure 1.2. Column of height one and cross section area A under the load .

6

Chapter 1. Introduction: Examples, Background, and Perspectives

cross-sectional area A(x),

I(x) = c A(x)

1

A(x) dx = 1.

and 0

Normalizing λ by cE and taking into account the engineering constraints ∃0 < A0 < A1 , ∀x ∈ [0, 1],

0 < A0 ≤ A(x) ≤ A1 ,

we ﬁnally get

1

A |u |2 dx , 1 |2 dx 0=u∈V A∈A |u 0 1 def 2 A(x) dx = 1 . A = A ∈ L (0, 1) : A0 ≤ A ≤ A1 and sup λ(A),

def

λ(A) =

inf

0

(3.2) (3.3)

0

4

Eigenvalue Problems

Let D be a bounded open Lipschitzian domain in RN and A ∈ L∞ (D; L(RN , RN )) be a matrix function deﬁned on D such that ∗

A=A

and αI ≤ A ≤ βI

(4.1)

for some coercivity and continuity constants 0 < α ≤ β and ∗A is the transpose of A. Consider the minimization or the maximization of the ﬁrst eigenvalue sup λA (Ω) A∇ϕ · ∇ϕ dx Ω∈A(D) def A Ω inf 1 , (4.2) λ (Ω) = A |ϕ|2 dx 0=ϕ∈H0 (Ω) inf λ (Ω) Ω Ω∈A(D) where A(D) is a family of admissible open subsets of D (cf., for instance, sections 2, 7, and 9 of Chapter 8). In the vectorial case, consider the following linear elasticity problem: ﬁnd U ∈ H01 (Ω)3 such that Cε(U )·· ε(W ) dx = F · W dx (4.3) ∀W ∈ H01 (Ω)3 , Ω

Ω

for some distributed loading F ∈ L2 (Ω)3 and a constitutive law C which is a bilinear symmetric transformation of def def

σij τij Sym3 = τ ∈ L(R3 ; R3 ) : ∗ τ = τ , σ·· τ = 1≤i,j≤3

(L(R3 ; R3 ) is the space of all linear transformations of R3 or 3 × 3-matrices) under the following assumption. Assumption 4.1. The constitutive law is a transformation C ∈ Sym3 for which there exists a constant α > 0 such that Cτ ·· τ ≥ α τ ·· τ for all τ ∈ Sym3 .

5. Optimal Triangular Meshing

7

For instance, for the Lam´e constants µ > 0 and λ ≥ 0, the special constitutive law Cτ = 2µ τ + λ tr τ I satisﬁes Assumption 4.1 with α = 2µ. The associated bilinear form is def Cε(U )·· ε(W ) dx, aΩ (U, W ) = Ω

where U is a vector function, D(U ) is the Jacobian matrix of U , and def

ε(U ) =

1 (D(U ) + ∗D(U )) 2

is the strain tensor. The ﬁrst eigenvalue is the minimum of the Rayleigh quotient aΩ (U, U ) 1 3 (Ω) , U = 0 . : ∀U ∈ H λ(Ω) = inf 0 |U |2 dx Ω A typical problem is to ﬁnd the sensitivity of the ﬁrst eigenvalue with respect to the shape of the domain Ω. In 1907, J. Hadamard [1] used displacements along the normal to the boundary Γ of a C ∞ -domain to compute the derivative of the ﬁrst eigenvalue of the clamped plate. As in the case of the column, this problem is not diﬀerentiable with respect to the geometry when the eigenvalue is not simple.

5

Optimal Triangular Meshing

The shape calculus that will be developed in Chapters 9 and 10 for problems governed by partial diﬀerential equations (the continuous model ) will be readily applicable to their discrete model as in the ﬁnite element discretization of elliptic boundary value problems. However, some care has to be exerted in the choice of the formula for the gradient, since the solution of a ﬁnite element discretization problem is usually less smooth than the solution of its continuous counterpart. Most shape objective functionals will have two basic formulas for their shape gradient: a boundary expression and a volume expression. The boundary expression is always nicer and more compact but can be applied only when the solution of the underlying partial diﬀerential equation is smooth and in most cases smoother than the ﬁnite element solution. This leads to serious computational errors. The right formula to use is the less attractive volume expression that requires only the same smoothness as the ﬁnite element solution. Numerous computational experiments conﬁrm that fact (cf., for instance, E. J. Haug and J. S. Arora [1] or E. J. Haug, K. K. Choi, and V. Komkov [1]). With the volume expression, the gradient of the objective function with respect to internal and boundary nodes can be readily obtained by plugging in the right velocity ﬁeld. A large class of linear elliptic boundary value problems can be expressed as the minimum of a quadratic function over some Hilbert space. For instance, let Ω be a bounded open domain in RN with a smooth boundary Γ. The solution u of the boundary value problem −∆u = f in Ω,

u = 0 on Γ

8

Chapter 1. Introduction: Examples, Background, and Perspectives

is the minimizing element in the Sobolev space H01 (Ω) of the energy functional def E(v, Ω) = |∇v|2 − 2f v dx, Ω def |∇u|2 dx. J(Ω) = inf1 E(v, Ω) = E(u, Ω) = − v∈H0 (Ω)

Ω

The elements of this problem are a Hilbert space V , a continuous symmetrical coercive bilinear form on V , and a continuous linear form on V . With this notation ∃u ∈ V,

E(u) = inf E(v), v∈V

def

E(v) = a(v, v) − 2 (v)

and u is the unique solution of the variational equation ∃u ∈ U, ∀v ∈ V,

a(u, v) = (v).

In the ﬁnite element approximation of the solution u, a ﬁnite-dimensional subspace Vh of V is used for some small mesh parameter h. The solution of the approximate problem is given by ∃uh ∈ Vh ,

E(uh ) = inf E(vh ), vh ∈Vh

⇒ ∃uh ∈ Uh , ∀vh ∈ Vh ,

a(uh , vh ) = (vh ).

It is easy to show that the error can be expressed as follows: a(u − uh , u − uh ) = u − uh 2V = 2 [E(uh ) − E(u)] . Assume that Ω is a polygonal domain in RN . In the ﬁnite element method, the ¯ domain is partitioned into a set τh of small triangles by introducing nodes in Ω def

M = {Mi ∈ Ω : 1 ≤ i ≤ p}

def

def

∂M = {Mi ∈ ∂Ω : p + 1 ≤ i ≤ p + q}

and M = M ∪ ∂M

for some integers p ≥ N + 1 and q ≥ 1 (see Figure 1.3). Therefore the triangularization τh = τh (M ), the solution space Vh = Vh (M ), and the solution uh = uh (M ) are functions of the positions of the nodes of the set M . Assuming that the total 0

0

0 Mi

1 0

0 0

Figure 1.3. Triangulation and basis function associated with node Mi .

5. Optimal Triangular Meshing

9

number of nodes is ﬁxed, consider the following optimal triangularization problem: def

inf j(M ), M

j(M ) = E(uh (τh (M )), Ω) =

inf

vh ∈Vh (M )

E(vh , Ω),

|∇(u − uh )|2 dx = 2 E(uh , Ω(τh (M ))) − E(u, Ω) Ω

= 2 J(Ω(τh (M ))) − J(Ω) , def |∇uh |2 dx. J(Ω(τh (M ))) = inf E(vh , Ω(τh (M ))) = E(uh , Ω(τh (M ))) = − u −

uh 2V

=

vh ∈Vh

Ω

The objective is to compute the partial derivative of j(M ) with respect to the th component (Mi ) of the node Mi : ∂j (M ). ∂(Mi ) This partial derivative can be computed by using the velocity method for the special ´sio [3]) velocity ﬁeld (cf. M. C. Delfour, G. Payre, and J.-P. Zole Vi (x) = bMi (x) e , where bMi ∈ Vh is the (piecewise P 1 ) basis function associated with the node Mi : bMi (Mj ) = δij for all i, j. In that method each point X of the plane is moved according to the solution of the vector diﬀerential equation dx (t) = V (x(t)), dt

x(0) = X.

def

This yields a transformation X → Tt (X) = x(t; X) : R2 → R2 of the plane, and it is natural to introduce the following notion of semiderivative: def

dJ(Ω; V ) = lim

t0

J(Tt (Ω)) − J(Ω) . t

For t ≥ 0 small, the velocity ﬁeld must be chosen in such a way that triangles are moved onto triangles and the point Mi is moved in the direction e : Mi → Mit = Mi + t e . This is achieved by choosing the following velocity ﬁeld: Vi (t, x) = bMit (x) e , where bMit is the piecewise P 1 basis function associated with node Mit : bMit (Mj ) = δij for all i, j. This yields the family of transformations Tt (x) = x + t bMi (x) e which moves the node Mi to Mi + t e and hence ∂j (M ) = dJ(Ω; Vi ). ∂(Mi )

10

Chapter 1. Introduction: Examples, Background, and Perspectives Going back to our original example, introduce the shape functional def 2 |∇u| dx, E(Ω, v) = |∇v|2 − 2 f v dx. J(Ω) = inf1 E(Ω, v) = − v∈H0 (Ω)

Ω

Ω

In Chapter 9, we shall show that we have the following boundary and volume expressions for the derivative of J(Ω):

dJ(Ω; V ) =

2 ∂u dJ(Ω; V ) = − V · n dΓ, Γ ∂n A (0)∇u · ∇u − 2 [div V (0)f + ∇f · V (0)] u dx,

Ω

A (0) = div V (0) I − ∗DV (0) − DV (0). For a P 1 -approximation def

¯ : v|K ∈ P 1 (K), ∀K ∈ τh Vh = v ∈ C 0 (Ω) and the trace of the normal derivative on Γ is not deﬁned. Thus, it is necessary to use the volume expression. For the velocity ﬁeld Vi DVi = e ∗∇bMi , div DVi = e · ∇bMi , A (0) = e · ∇bMi I − e ∗∇bMi − ∇bMi ∗ e . Since ∂j (M ) = dJ(Ω; Vi ), ∂(Mi ) we ﬁnally obtain the formula for the derivative of the function j(M ) with respect to node Mi in the direction e : ∂j (M ) = [ e · ∇bMi I − e ∗∇bMi − ∇bMi ∗ e ] ∇; uh · ∇uh ∂(Mi ) Ω − 2 [ e · ∇bMi f + ∇f · e bMi ] uh dx. Since the support of bMi consists of the triangles having Mi as a vertex, the gradient with respect to the nodes can be constructed piece by piece by visiting each node.

6

Modeling Free Boundary Problems

The ﬁrst step towards the solution of a shape optimization is the mathematical modeling of the problem. Physical phenomena are often modeled on relatively smooth or nice geometries. Adding an objective functional to the model will usually push the system towards rougher geometries or even microstructures. For instance, in the optimal design of plates the optimization of the proﬁle of a plate led to highly oscillating proﬁles that looked like a comb with abrupt variations ranging from zero

6. Modeling Free Boundary Problems

11

to maximum thickness. The phenomenon began to be understood in 1975 with the paper of N. Olhoff [1] for circular plates with the introduction of the mechanical notion of stiﬀeners. The optimal plate was a virtual plate, a microstructure, that is a homogenized geometry. Another example is the Plateau problem of minimal surfaces that experimentally exhibits surfaces with singularities. In both cases, it is mathematically natural to replace the geometry by a characteristic function, a function that is equal to 1 on the set and 0 outside the set. Instead of optimizing over a restricted family of geometries, the problem is relaxed to the optimization over a set of measurable characteristic functions that contains a much larger family of geometries, including the ones with boundary singularities and/or an arbitrary number of holes.

6.1

Free Interface between Two Materials

´a and K. Malanowski [1] Consider the optimal design problem studied by J. Ce in 1970, where the optimization variable is the distribution of two materials with diﬀerent physical characteristics within a ﬁxed domain D. It cannot a priori be assumed that the two regions are separated by a smooth interface and that each region is connected. This problem will be covered in more details in section 4 of Chapter 5. Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assume for the moment that the domain D is partitioned into two subdomains Ω1 and Ω2 separated by a smooth interface ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 1.4. Domain Ω1 (resp., Ω2 ) is made up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y be the solution of the transmission problem − k1 y = f in Ω1 y = 0 on ∂D

and − k2 y = f in Ω2 , ∂y ∂y + k2 = 0 on Ω1 ∩ Ω2 , and k1 ∂n1 ∂n2

(6.1)

where n1 (resp., n2 ) is the unit outward normal to Ω1 (resp., Ω2 ) and f is a given function in L2 (D). Assume that k1 > k2 . The objective is to maximize the equivalent of the compliance f y dx (6.2) J(Ω1 ) = − D

Ω1

Ω2

Figure 1.4. Fixed domain D and its partition into Ω1 and Ω2 .

12

Chapter 1. Introduction: Examples, Background, and Perspectives

over all domains Ω1 in D subject to the following constraint on the volume of material k1 which occupies the part Ω1 of D: m(Ω1 ) ≤ α,

0 < α < m(D)

(6.3)

for some constant α. If χ denotes the characteristic function of the domain Ω1 , / Ω1 , χ(x) = 1 if x ∈ Ω1 and 0 if x ∈ the compliance J(χ) = J(Ω1 ) can be expressed as the inﬁmum over the Sobolev space H01 (D) of an energy functional deﬁned on the ﬁxed set D: J(χ) = def

min

ϕ∈H01 (D)

E(χ, ϕ),

(k1 χ + k2 (1 − χ)) |∇ϕ|2 − 2 χf ϕ dx.

E(χ, ϕ) =

(6.4) (6.5)

D

J(χ) can be minimized or maximized over some appropriate family of characteristic functions or with respect to their relaxation to functions between 0 and 1 that would correspond to microstructures. As in the eigenvalue problem, the objective function is an inﬁmum, but here the inﬁmum is over a space that does not depend on the function χ that speciﬁes the geometric domain. This will be handled by the special techniques of Chapter 10 for the diﬀerentiation of the minimum of a functional.

6.2

Minimal Surfaces

The celebrated Plateau’s problem, named after the Belgian physicist and professor J. A. F. Plateau [1] (1801–1883), who did experimental observations on the geometry of soap ﬁlms around 1873, also provides a nice example where the geometry is a variable. It consists in ﬁnding the surface of least area among those bounded by a given curve. One of the diﬃculties in studying the minimal surface problem is the description of such surfaces in the usual language of diﬀerential geometry. For instance, the set of possible singularities is not known. Measure theoretic methods such as k-currents (k-dim surfaces) were used by E. R. Reifenberg [1, 2, 3, 4] around 1960, H. Federer and W. H. Fleming [1] in 1960 (normals and integral currents), F. J. Almgren, Jr. [1] in 1965 (varifolds), and H. Federer [5] in 1969. In the early 1950s, E. De Giorgi [1, 2, 3] and R. Caccioppoli [1] considered a hypersurface in the N -dimensional Euclidean space RN as the boundary of a set. In order to obtain a boundary measure, they restricted their attention to sets whose characteristic function is of bounded variation. Their key property is an associated natural notion of perimeter that extends the classical surface measure of the boundary of a smooth set to the larger family of Caccioppoli sets named after the celebrated Neapolitan mathematician Renato Caccioppoli.2 2 In 1992 his tormented personality was remembered in a ﬁlm directed by Mario Martone, The Death of a Neapolitan Mathematician (Morte di un matematico napoletano).

7. Design of a Thermal Diﬀuser

13

Caccioppoli sets occur in many shape optimization problems (or free boundary problems), where a surface tension is present on the (free) boundary, such as in the free interface water/soil in a dam (C. Baiocchi, V. Comincioli, E. Magenes, and G. A. Pozzi [1]) in 1973 and in the free boundary of a water wave (M. Souli ´sio [1, 2, 3, 4, 5]) in 1988. More details will be given in Chapter 5. and J.-P. Zole

7

Design of a Thermal Diﬀuser

Shape optimization problems are everywhere in engineering, physics, and medicine. We choose two illustrative examples that were proposed by the Canadian Space Program in the 1980s. The ﬁrst one is the design of a thermal diﬀuser to condition the thermal environment of electronic devices in communication satellites; the second one is the design of a thermal radiator that will be described in the next section. There are more and more design and control problems coming from medicine. For instance, the design of endoprotheses such as valves, stents, and coils in blood vessels or left ventricular assistance devices (cardiac pumps) in interventional cardiology helps to improve the health of patients and minimize the consequences and costs of therapeutical interventions by going to mini-invasive procedures.

7.1

Description of the Physical Problem

This problem arises in connection with the use of high-power solid-state devices (HPSSD) in communication satellites (cf. M. C. Delfour, G. Payre, and J.´sio [1]). An HPSSD dissipates a large amount of thermal power (typ. P. Zole > 50 W) over a relatively small mounting surface (typ. 1.25 cm2 ). Yet, its junction temperature is required to be kept moderately low (typ. 110◦ C). The thermal resistance from the junction to the mounting surface is known for any particular HPSSD (typ. 1◦ C/W), so that the mounting surface is required to be kept at a lower temperature than the junction (typ. 60◦ C). In a space application the thermal power must ultimately be dissipated to the environment by the mechanism of radiation. However, to radiate large amounts of thermal power at moderately low temperatures, correspondingly large radiating areas are required. Thus we have the requirement to eﬃciently spread the high thermal power ﬂux (TPF) at the HPSSD source (typ. 40 W/cm2 ) to a low TPF at the radiator (typ. 0.04 W/cm2 ) so that the source temperature is maintained at an acceptably low level (typ. < 60◦ C) at the mounting surface. The eﬃcient spreading task is best accomplished using heatpipes, but the snag in the scheme is that heatpipes can accept only a limited maximum TPF from a source (typ. max 4 W/cm2 ). Hence we are led to the requirement for a thermal diﬀuser. This device is inserted between the HPSSD and the heatpipes and reduces the TPF at the source (typ. > 40 W/cm2 ) to a level acceptable to the heatpipes (typ. > 4 W/cm2 ). The heatpipes then suﬃciently spread the heat over large space radiators, reducing the TPF from a level at the diﬀuser (typ. 4 W/cm2 ) to that at the radiator (typ. 0.04 W/cm2 ). This scheme of heat spreading is depicted in Figure 1.5. It is the design of the thermal diﬀuser which is the problem at hand. We may assume that the HPSSD presents a uniform thermal power ﬂux to the diﬀuser

14

Chapter 1. Introduction: Examples, Background, and Perspectives schematic drawing not to scale

radiator to space heatpipes

heatpipe saddle

thermal diﬀuser

high-power solid-state diﬀuser

Figure 1.5. Heat spreading scheme for high-power solid-state devices. at the HPSSD/diﬀuser interface. Heatpipes are essentially isothermalizing devices, and we may assume that the diﬀuser/heatpipe saddle interface is indeed isothermal. Any other surfaces of the diﬀuser may be treated as adiabatic.

7.2

Statement of the Problem

Assume that the thermal diﬀuser is a volume Ω symmetrical about the z-axis (cf. Figure 1.6 (A)) whose boundary surface is made up of three regular pieces: the mounting surface Σ1 (a disk perpendicular to the z-axis with center in (r, z) = (0, 0)), the lateral adiabatic surface Σ2 , and the interface Σ3 between the diffuser and the heatpipe saddle (a disk perpendicular to the z-axis with center in (r, z) = (0, L)). The temperature distribution over this volume Ω is the solution of the stationary heat equation k∆T = 0 (∆T , the Laplacian of T ) with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω): k

∂T = qin on Σ1 , ∂n

k

∂T = 0 on Σ2 , ∂n

T = T3 (constant) on Σ3 ,

(7.1)

where n always denotes the outward unit normal to the boundary surface Σ and ∂T /∂n is the normal derivative to the boundary surface Σ, ∂T = ∇T · n (∇T = the gradient of T ). ∂n

(7.2)

The parameters appearing in (7.1) are k = thermal conductivity (typ. 1.8W/cm×◦ C), qin = uniform inward thermal power ﬂux at the source (positive constant).

7. Design of a Thermal Diﬀuser

15

The radius R0 of the mounting surface Σ1 is ﬁxed so that the boundary surface Σ1 is already given in the design problem. For practical considerations, we assume that the diﬀuser is solid without interior hollows or cutouts. The class of shapes for the diﬀuser is characterized by the design parameter L > 0 and the positive function R(z), 0 < z ≤ L, with R(0) = R0 > 0. They are volumes of revolution Ω about the z-axis generated by the surface A between the z-axis and the function R(z) (cf. Figure 1.6 (B)), that is, def

Ω = (x, y, z) : 0 < z < L, x2 + y 2 < R(z)2 .

(7.3)

So the shape of Ω is completely speciﬁed by the length L > 0 and the function R(z) > 0 on the interval [0, L]. z

z Σ3

L

ζ

L A R(z)

S3

1 S4

D

Σ2 0

0

Σ1

S2 y

0

0

y

R0

(A)

0

0

(B)

1 S1

ρ (C)

Figure 1.6. (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω; (C) Surface D generating Ω. Assuming that the diﬀuser is made up of a homogeneous material of uniform density (no hollow) the design objective is to minimize the volume def

J(Ω) =

L

R(z)2 dz

dx = π Ω

(7.4)

0

subject to a uniform constraint on the outward thermal power ﬂux at the interface Σ3 between the diﬀuser and the heatpipe saddle: sup −k

p∈Σ3

∂T ∂T (p) ≤ qout or k + qout ≥ 0 on Σ3 , ∂z ∂n

(7.5)

where qout is a speciﬁed positive constant. It is readily seen that the minimization problem (7.4) subject to the constraint (7.5) (where T is the solution of the heat equation with the boundary conditions (7.1)) is independent of the ﬁxed temperature T3 on the boundary Σ3 . In other words the optimal shape Ω, if it exists, is independent of T3 . As a result, from now on we set T3 equal to 0.

16

7.3

Chapter 1. Introduction: Examples, Background, and Perspectives

Reformulation of the Problem

In a shape optimization problem the formulation is important from both the theoretical and the numerical viewpoints. In particular condition (7.5) is diﬃcult to numerically handle since it involves the pointwise evaluation of the normal derivative on the piece of boundary Σ3 . This problem can be reformulated as the minimization of T on Σ3 , where T is now the solution of a variational inequality. Consider the following minimization problem over the subspace of functions that are positive or zero on Σ3 : def

(7.6) V + (Ω) = v ∈ H 1 (Ω) : v|Σ3 ≥ 0 , 1 inf |∇v|2 dx − qin v dΣ + qout v dΣ. (7.7) v∈V + (Ω) Ω 2 Σ1 Σ3 H 1 (Ω) is the usual Sobolev space on the domain Ω, and the inequality on Σ3 has to be interpreted quasi-everywhere in the capacity sense. Leaving aside those technicalities, the minimizing solution of (7.7) is characterized by ∂T ∂T = qin on Σ1 , k = 0 on Σ2 , −k ∆T = 0 in Ω, k ∂n ∂n ∂T ∂T + qout ≥ 0, T k + qout = 0 on Σ3 . T ≥ 0, k ∂n ∂n

(7.8)

The former constraint (7.5) is veriﬁed and replaced by the new constraint T = 0 on Σ3 .

(7.9)

If there exists a nonempty domain Ω of the form (7.3) such that T = 0 on Σ3 , the problem is feasible. In this formulation the pointwise constraint on the normal derivative of the temperature on Σ3 has been replaced by a pointwise constraint on the less demanding trace of the temperature on Σ3 . Yet, we now have to solve a variational inequality instead of a variational equation for the temperature T .

7.4

Scaling of the Problem

In the above formulations the shape parameter L and the shape function R are not independent of each other since the function R is deﬁned on the interval [0, L]. This motivates the following changes of variables and the introduction of the dimensionless temperature y: x y z , y → ξ2 = , z → ζ = , 0 ≤ ζ ≤ 1, x → ξ1 = R0 R0 L R(Lζ) ˜ = L , R(ζ) ˜ L = , R0 R0 k y(ξ1 , ξ2 , ζ) = T (R0 ξ1 , R0 ξ2 , Lζ), Lqin def ˜ 2 . D = (ξ1 , ξ2 , ζ) : 0 < ζ < 1, ξ 2 + ξ 2 < R(z) 1

2

7. Design of a Thermal Diﬀuser

17

˜ now appears as a coeﬃcient in the partial diﬀerential equation The parameter L 2 ∂ y ∂2y ∂2y 2 ˜ L + 2 + 2 = 0 in D 2 ∂ξ1 ∂ξ2 ∂ζ with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D: ∂y = 1 on S1 , ∂νA

∂y = 0 on S2 , ∂νA

y = 0 on S3 ,

(7.10)

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface S, ∂y ∂y ∂y ∂y 2 ˜ + ν3 . = L ν1 + ν2 ∂νA ∂ξ1 ∂ξ2 ∂ζ Finally, the optimal design problem depends only on the ratio q = qout /qin through the constraint ∂y qout + ≥ 0 on S3 . ∂νA qin ˜ > 0 and the function R ˜ > 0 now deﬁned The design variables are the parameter L on the ﬁxed interval [0, 1].

7.5

Design Problem

The fact that this speciﬁc design problem can be reduced to ﬁnding a parameter and a function gives the false or unfounded impression that it can now be solved by standard mathematical programming and numerical methods. Early work on such problems revealed a diﬀerent reality, such as oscillating boundaries and convergence towards nonphysical designs. Clearly, the geometry refused to be handled by standard methods without a better understanding of the underlying physics and its inception in the modeling of the geometric variable. At the theoretical level, the existence of solution requires a concept of continuity with respect to the geometry of the solution of either the heat equation with an inequality constraint on the TPF or the variational inequality with an equality constraint on the temperature. The other element is the lower semicontinuity of the objective functional that is not too problematic for the volume functional as long as the chosen topology on the geometry preserves the continuity of the volume functional. For instance, the classical Hausdorﬀ metric topology does not preserve the volume. In the context of ﬂuid mechanics (cf., for instance, O. Pironneau [1]), it means that a drag minimizing sequence of sets with constant volume may converge to a set with twice the volume (cf. Example 4.1 in Chapter 6). A wine making industry exploiting the convergence in the Hausdorﬀ metric topology could yield miraculous proﬁts. Other serious issues are, for instance, the lack of diﬀerentiability of the solution of a variational inequality at the continuous level that will inadvertently aﬀect the diﬀerentiability or the evolution of a gradient method at the discrete level. We shall

18

Chapter 1. Introduction: Examples, Background, and Perspectives

see that there is not only one topology for shapes but a whole range that selectively preserve some but not all of the geometrical features. Again the right choice is problem dependent, much like the choice of the right Sobolev space in the theory of partial diﬀerential equations.

8

Design of a Thermal Radiator

Current trends indicate that future communications satellites and spacecrafts will grow ever larger, consume ever more electrical power, and dissipate larger amounts of thermal energy. Various techniques and devices can be deployed to condition the thermal environment for payload boxes within a spacecraft, but it is desirable to employ those which oﬀer good performance for low cost, low weight, and high reliability. A thermal radiator (or radiating ﬁn) which accepts a given TPF from a payload box and radiates it directly to space can oﬀer good performance and high reliability at low cost. However, without careful design, such a radiator can be unnecessarily bulky and heavy. It is the mass-optimized design of the thermal radiator ´which is the problem at hand (cf. M. C. Delfour, G. Payre, and J.-P. Zole sio [2]). We may assume that the payload box presents a uniform TPF (typ. 0.1 to 1.0 W/cm2 ) into the radiator at the box/radiator interface. The radiating surface is a second surface mirror which consists of a sheet of glass whose inner surface has silver coating. We may assume that the TPF out of the radiator/space interface is governed by the T 4 radiation law, although we must account also for a constant TPF (typ. 0.01 W/cm2 ) into this interface from the sun. Any other surfaces of the radiator may be treated as adiabatic. Two constraints restrict freedom in the design of the thermal radiator: (i) the maximum temperature at the box/radiator interface is not to exceed some constant (typ. 50◦ ); and (ii) no part of the radiator is to be thinner than some constant (typ. 1 mm).

8.1

Statement of the Problem

Assume that the radiator is a volume Ω symmetrical about the z-axis (cf. Figure 1.7) whose boundary surface is made up of three regular pieces: the contact surface Σ1 (a disk perpendicular to the z-axis with center at the point (r, z) = (0, 0)), the lateral adiabatic surface Σ2 , and the radiating surface Σ3 (a disk perpendicular to the z-axis with center at (r, z) = (0, L)). More precisely

Σ1 = (x, y, z) : z = 0 and x2 + y 2 ≤ R02 ,

Σ2 = (x, y, z) : x2 + y 2 = R(z)2 , 0 ≤ z ≤ L , (8.1)

2 2 2 Σ3 = (x, y, z) : z = L and x + y ≤ R(L) , where the radius R0 > 0 (typ. 10 cm), the length L > 0, and the function R : [0, L] → R,

R(0) = R0 ,

are given (R, the ﬁeld of real numbers).

R(z) > 0, 0 ≤ z ≤ L,

(8.2)

8. Design of a Thermal Radiator

19

z

z

Σ3

L Σ2 y

0

0

R0 Σ1

r

Figure 1.7. Volume Ω and its cross section. The temperature distribution (in Kelvin degrees) over the volume Ω is the solution of the stationary heat equation ∆T = 0

(the Laplacian of T )

(8.3)

with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω): k

∂T = qin on Σ1 , ∂n

k

∂T = 0 on Σ2 , ∂n

k

∂T = −σεT |T |3 + qs on Σ3 , ∂n

(8.4)

where n denotes the outward normal to the boundary surface Σ, ∂T /∂n is the normal derivative on the boundary surface Σ, and ∂T = ∇T · n (∇T = the gradient of T ). ∂n The parameters appearing in (8.1)–(8.4) are k = thermal conductivity (typ. 1.8 W/cm×◦ C), qin = uniform inward thermal power ﬂux at the source (typ. 0.1 to 1.0 W/cm2 ), σ = Boltzmann’s constant (5.67 × 10−8 W/m2 K4 ), ε = surface emissivity (typ. 0.8), qs = solar inward thermal power ﬂux (0.01 W/cm2 ). The optimal design problem consists in minimizing the volume L def R(z)2 dz J(Ω) = π

(8.5)

0

over all length L > 0 and shape function R subject to the constraint T (x, y, z) ≤ Tf

(typ. 50◦ C),

∀(x, y, z) ∈ Σ1 .

In this analysis we shall drop the requirement (ii) in the introduction.

(8.6)

20

8.2

Chapter 1. Introduction: Examples, Background, and Perspectives

Scaling of the Problem

As in the case of the diﬀuser, it is convenient to introduce the following dimensionless coordinates and temperature y (see Figure 1.8): x → ξ1 =

x , R0

y → ξ2 =

y , R0

z → ζ =

z , L

0 ≤ ζ ≤ 1,

R(Lζ) ˜ = L , R(ζ) ˜ L = , R0 R0 1/3 σεR0 T (R0 ξ1 , R0 ξ2 , Lζ), y(ξ1 , ξ2 , ζ) = k def ˜ 2 . D = (ξ1 , ξ2 , ζ) : 0 < ζ < 1, ξ 2 + ξ 2 < R(z) 1

2

˜ now appears as a coeﬃcient in the partial diﬀerential equation The parameter L 2 ∂ y ∂2y ∂ 2y 2 ˜ L + = 0 in D + ∂ξ12 ∂ξ22 ∂ζ 2 with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D: 1/3 qin ∂y σεR0 ˜ on S1 , R0 =L ∂νA k k ∂y =0 ∂νA

(8.7)

on S2 ,

∂y 3 ˜ ˜ + Ly|y| =L ∂νA

σεR0 k

1/3

qin qs R0 k qin

on S3 ,

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface Σ, ∂y ˜ 2 ν1 ∂y + ν2 ∂y + ν3 ∂y . =L ∂νA ∂ξ1 ∂ξ2 ∂ζ

ζ

ζ

˜3 Σ

1

˜2 Σ

S3

1 A

r 1 ˜1 Σ

0

S2 r

0

1 S1

and its generating surface A. Figure 1.8. Volume Ω

9. A Glimpse into Segmentation of Images

9

21

A Glimpse into Segmentation of Images

The study of the problem of linguistic or visual perceptions was initiated by several pioneering authors, such as H. Blum [1] in 1967, D. Marr and E. Hildreth [1] in 1980, and D. Marr [1] in 1982. It involves specialists of psychology, artiﬁcial intelligence, and experimentalists such as D. H. Hubel and T. N. Wiesel [1] in 1962 and F. W. Campbell and J. G. Robson [1] in 1968. In the ﬁrst part of this section, we revisit the pioneering work of D. Marr and E. Hildreth [1] in 1980 on the smoothing of the image by convolution with a suﬃciently diﬀerentiable normalized function as a function of the scaling parameter. We extend the space-frequency uncertainty principle to N -dimensional images. It is the analogue of the Heisenberg uncertainty principle of quantum mechanics. We revisit the Laplacian ﬁlter and we generalize the linearity assumption of D. Marr and E. Hildreth [1] from linear to curved contours. In the second part, we show how shape analysis methods and the shape and tangential calculuses can be applied to objective functionals deﬁned on the whole contour of an image. We anticipate Chapters 9 and 10 on shape derivatives by the velocity method and show how they can be applied to snakes, active geodesic contours, and level sets (cf., for instance, the book of S. Osher and N. Paragios [1]). It shows that the Eulerian shape semiderivative is the basic ingredient behind those representations, including the case of the oriented distance function. In all cases, the evolution equation for the continuous gradient descent method is shown to have the same structure. For more material along the same lines, the reader is referred to M. Dehaes [1] and M. Dehaes and M. C. Delfour [1].

9.1

Automatic Image Processing

The ﬁrst level of image processing is the detection of the contours or the boundaries of the objects in the image. For an ideal image I : D → R deﬁned in an open two-dimensional frame D with values in an interval of greys continuously ranging from white to black (see Figure 1.9), the edges of an object correspond to the loci of discontinuity of the image I (cf. D. Marr and E. Hildreth [1]), also called “step edges” by D. Marr [1]. As can be seen from Figure 1.9, the loci of discontinuity may only reveal part of an object hidden by another one and a subsequent and diﬀerent level of processing is required. A more diﬃcult case is the detection of black curves or cracks in a white frame (cf. Figure 1.10), where the function I becomes a measure supported by the curves rather than a function. In practice, the frame D of the image is divided into periodically spaced cells P (square, hexagon, diamond) with a quantized value or pixel from 256 grey levels. For small squares a piecewise linear continuous interpolation or higher degree C 1 -interpolation can be used to remove the discontinuities at the intercell boundaries. In addition, observations and measurements introduce noise or perturbations and the interpolated image I needs to be further smoothed or ﬁltered. When the characteristics of the noise are known, an appropriate ﬁlter can do the job (see, for instance, the use of low pass ﬁlters in A. Rosenfeld and M. Thurston [1], A. P. Witkin [1], and A. L. Yuille and T. Poggio [1]).

22

Chapter 1. Introduction: Examples, Background, and Perspectives

frame D levels of grey I

open set Ω

Figure 1.9. Image I of objects and their segmentation in the frame D.

frame D

Figure 1.10. Image I containing black curves or cracks in the frame D. Given an ideal grey level image I deﬁned in a ﬁxed bounded open frame D we want to identify the edges or boundaries of the objects contained in the image as shown in Figure 1.9. The edge identiﬁcation or segmentation problem is a lowlevel processing of a real image which should also involve higher-level processings or tasks (cf. M. Kass, M. Witkin, and D. Terzopoulos [1]). Intuitively the edges coincide with the loci of discontinuities of the image function I. At such points the norm |∇I| of the gradient ∇I is inﬁnite. Roughly speaking the segmentation of an ideal image I consists in ﬁnding the loci of discontinuity of the function I. The idea is now to ﬁrst smooth I by convolution with a suﬃciently diﬀerentiable function. This operation will be followed and/or combined with the use of an edge detector as the zero crossings of the Laplacian. In the literature the term ﬁlter often applies to both the ﬁlter and the detector. In this section we shall make the distinction between the two operations.

9.2

Image Smoothing/Filtering by Convolution and Edge Detectors

In this section we revisit the work of D. Marr and E. Hildreth [1] in 1980 on the smoothing of the image I by convolution with a suﬃciently diﬀerentiable normalized function ρ as a function of the scaling parameter ε > 0. In the second part of this

9. A Glimpse into Segmentation of Images

23

section we revisit the Laplacian ﬁlter and we generalize the linearity assumption of D. Marr and E. Hildreth [1] from linear to curved contours. 9.2.1

Construction of the Convolution of I

Let ρ : RN → R, N ≥ 1, be a suﬃciently smooth function such that ρ(x) dx = 1. ρ≥0 and RN

Associate with the image I, ρ, and a scaling parameter ε > 0 the normalized convolution x−y 1 def Iε (x) = (I ∗ ρε )(x) = N dy, (9.1) I(y) ρ ε ε RN where for x ∈ RN def

ρε (x) =

1 x ρ εN ε

and RN

ρε (x) dx = 1.

(9.2)

The function ρε plays the role of a probability density and ρε (x) dx of a probability measure. Under appropriate conditions Iε converges to the original image I as ε → 0. For a suﬃciently small ε > 0 the loci of discontinuity of the function I are transformed into loci of strong variation of the gradient of the convolution Iε . When ρ has compact support, the convolution acts locally around each point in a neighborhood whose size is of order ε. A popular choice for ρ is the Gaussian with integral normalized to one in RN deﬁned by 2 1 1 def e− 2 |x| and GN (x) dx = 1 GN (x) = √ ( 2π)N RN and the normalized Gaussian of variance ε GN ε (x) =

1 x 2 1 1 √ e− 2 | ε | N ε ( 2π)N

and RN

GN ε (x) dx = 1.

(9.3)

As noted in L. Alvarez, P.-L. Lions, and J.-M. Morel [1] in dimension N = 2, the function u(t) = G2√2t ∗ I is the solution of the parabolic equation ∂u (t, x) = ∆u(t, x), ∂t 9.2.2

u(0, x) = I(x).

Space-Frequency Uncertainty Relationship

N It is interesting to compute the Fourier transform F(GN ε ) of Gε to make explicit the N relationship between the mean square deviations of Gε and its Fourier transform N F(GN ε ). For ω ∈ R , deﬁne the Fourier transform of a function f by 1 def f (x) e−i ω·x . (9.4) F(f )(ω) = √ ( 2π)N RN

24

Chapter 1. Introduction: Examples, Background, and Perspectives

In applications the integral of the square of f often corresponds to an energy. We shall refer to the L2 -norm of f as the energy norm and the function f 2 (x) as the energy density. The Fourier transform (9.4) of the normalized Gaussian (9.3) is 2 1 1 1 def −i ω·x √ )(ω) = GN dx = √ e− 2 |εω| . (9.5) F(GN ε ε (x) e N N ( 2π) ( 2π) RN D. Marr and E. Hildreth [1] notice that there is an uncertainty relationship between the mean square deviation with respect to the energy density f (x)2 and the mean square deviation with respect to the energy density F(f )(ω)2 of its Fourier transform in dimension 1 using a result√of R. Bracewell [1, pp. 160–161], where he uses the constant 1/(2π) instead of 1/ 2π in the deﬁnition of the Fourier transform that yields 1/(4π) instead of 1/2 as a lower bound to the product of the two mean square deviations. This uncertainty relationship generalizes to dimension N . First deﬁne the notions of centro¨ıd and variance with respect to the energy density f (x)2 . Given a function f ∈ L1 (RN ) ∩ L2 (RN ) and x ∈ RN , deﬁne the centro¨ıd as 2 N x f (x) dx x = R f (x)2 dx RN def

(9.6)

and the variance as def

∆x2 = x − x2 =

|x|2 f (x)2 dx − |x|2 . 2 dx f (x) RN

RN

Theorem 9.1 (Uncertainty relationship). Given N ≥ 1 and f ∈ W 1,1 (RN ) ∩ W 1,2 (RN ) such that −ix f ∈ L1 (RN ) ∩ L2 (RN ), ∆x ∆ω ≥

N . 2

(9.7)

For f = GN ε 1 x 2 1 1 N ε2 √ , e− 2 | ε | ⇒ ∆x2 = N ε ( 2π)N 2 2 1 1 N fˆ(ω) = F(GN e− 2 |εω| ⇒ ∆ω2 = ε )(ω) = √ 2 ε2 ( 2π)N

f (x) = GN ε (x) =

⇒ ∆x∆ω =

N 2

N for GN ε and F(Gε ).

(9.8) (9.9) (9.10)

So the normalized Gaussian ﬁlter is indeed an optimal ﬁlter since it achieves the lower bound in all dimensions as stated by D. Marr and E. Hildreth [1] in dimension 1 in 1980. In the context of quantum mechanics, this relationship is the analogue of the Heisenberg uncertainty principle.

9. A Glimpse into Segmentation of Images 9.2.3

25

Laplacian Detector

One way to detect the edges of a regular object is to start from the convolutionsmoothed image Iε . Given a direction v, |v| = 1, and a point x ∈ R2 an edge point will correspond to a local minimum or maximum of the directional derivative def f (t) = ∇Iε (x + tv) · v with respect to t. Denote by tˆ such a point. Then a necessary condition for an extremal is D2 Iε (x + tˆv)v · v = 0. The point x ˆ = x + tˆv is a zero crossing following the terminology of D. Marr and E. Hildreth [1] of the second-order directional derivative in the direction v. Thus we are looking for the pairs (ˆ x, vˆ) verifying the necessary condition D2 Iε (ˆ x)ˆ v · vˆ = 0

(9.11)

and, more precisely, lines or curves C such that ∀x ∈ C, ∃v(x), |v(x)| = 1,

D2 Iε (x)v(x) · v(x) = 0.

(9.12)

This condition is necessary in order that, in each point x ∈ C, there exists a direction v(x) such that ∇Iε (x) · v is extremal. In order to limit the search to points x rather than to pairs (x, v), the Laplacian detector was introduced in D. Marr and E. Hildreth [1] under two assumptions: the linear variation of the intensity along the edges C of the object and the condition of zero crossing of the second-order derivative in the direction normal to C. Assumption 9.1 (Linear variation condition). The intensity Iε in a neighborhood of lines parallel to C is locally linear (aﬃne). Assumption 9.2. The zero-crossing condition is veriﬁed in all points of C in the direction of the normal n at the point, that is, D2 Iε n · n = 0 on C.

(9.13)

Under those two assumptions and for a line C, it is easy to show that the points of C verify the necessary condition ∆Iε = 0 on C.

(9.14)

Such conditions can be investigated for edges C that are the boundary Γ of a smooth domain Ω ⊂ RN of class C 2 by using the tangential calculus developed in section 5.2 of Chapter 9 and M. C. Delfour [7, 8] for objects in RN , N ≥ 1, and not just in dimension N = 2. Indeed we want to ﬁnd points x ∈ Γ such that the function ∇Iε (x)·n(x) is an extremal of the function f (t) = ∇Iε (x+t ∇bΩ (x))·∇bΩ (x) in t = 0. This yields the local necessary condition D2 Iε ∇bΩ · ∇bΩ = D2 Iε n · n = 0 on Γ of Assumption 9.2. As for Assumption 9.1, its analogue is the following.

26

Chapter 1. Introduction: Examples, Background, and Perspectives

Assumption 9.3. The restriction to the curve C of the gradient of the intensity Iε is a constant vector c, that is, ∇Iε = c,

c is a constant vector on C.

(9.15)

Indeed the Laplacian of Iε on C can be decomposed as follows (cf. section 5.2 of Chapter 9): ∆Iε = div Γ ∇Iε + D2 Iε n · n, where div Γ ∇Iε is the tangential divergence of ∇Iε that can be deﬁned as follows: div Γ ∇Iε = div (∇Iε ◦ pC )|C and pC is the projection onto C. Hence, under Assumption 9.3, div Γ (∇Iε ) = 0 on C since div Γ (∇Iε ) = div (∇Iε ◦ pΓ )|Γ = div (c)|Γ = 0, and, in view of (9.13) of Assumption 9.2, D2 Iε n · n = 0 on C. Therefore ∆Iε = div Γ (∇Iε ) + D2 Iε n · n = 0 on C, and we obtain the necessary condition ∆Iε = 0 on C.

9.3

(9.16)

Objective Functions Deﬁned on the Whole Edge

With the pioneering work of M. Kass, M. Witkin, and D. Terzopoulos [1] in 1988 we go from a local necessary condition at a point of the edge to a global necessary condition by introducing objective functionals deﬁned on the entire edge of an object. Here many computations and analytical studies can be simpliﬁed by adopting the point of view that a closed curve in the plane is the boundary of a set and using the whole machinery developed for shape and geometric analysis and the tangential and shape calculuses in Chapter 9. 9.3.1

Eulerian Shape Semiderivative

In this section we brieﬂy summarize the main elements of the velocity method. Deﬁnition 9.1. Given D, ∅ = D ⊂ RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D is the holdall or the universe. A shape functional is a map J : A → E from an admissible family A in P(D) with values in a topological vector space E (usually R).

9. A Glimpse into Segmentation of Images

27

Given a velocity ﬁeld V : [0, τ ] × RN → RN (the notation V (t)(x) = V (t, x) will often be used), consider the transformations def

T (t, X) = x(t, X),

t ≥ 0, X ∈ RN ,

(9.17)

where x(t, X) = x(t) is deﬁned as the ﬂow of the diﬀerential equation dx (t) = V (t, x(t)), t ≥ 0, dt

x(0, X) = X

(9.18)

(here the notation x → Tt (x) = T (t, x) : RN → RN will be used). The Eulerian shape semiderivative of J in Ω in the direction V is deﬁned as def

dJ(Ω; V ) = lim

t0

J(Ωt ) − J(Ω) t

(9.19)

(when the limit exists in E), where Ωt = Tt (Ω) = {Tt (x) : x ∈ Ω}. Under appropriate assumptions on the family {V (t)}, the transformations {Tt } are homeomorphisms that transport the boundary Γ of Ω onto the boundary Γt of Ωt and the interior Ω onto the interior of Ωt . 9.3.2

From Local to Global Conditions on the Edge

For simplicity, drop the subscript ε of the convolution-smoothed image Iε and assume that the edge of the object is the boundary Γ of an open domain Ω of class C 2 . Following V. Caselles, R. Kimmel, and G. Sapiro [1], it is important to choose objective functionals that are intrinsically deﬁned and do not depend on an arbitrary parametrization of the boundary. For instance, given a frame D = ]0, a[ × ]0, b[ and a smoothed image I : D → R, ﬁnd an extremum of the objective functional ∂I def dΓ, (9.20) E(Ω) = Γ ∂n where the integrand is the normal derivative of I. Using the velocity method the Eulerian shape directional semiderivative is given by the expression (cf. Chapter 9) ∂I ∂ ∂I + V · n dΓ, (9.21) dE(Ω; V ) = H ∂n ∂n ∂n Γ where H = ∆bΩ is the mean curvature and n = ∇bΩ is the outward unit normal. Proceeding in a formal way a necessary condition would be ∂I ∂ ∂I + = 0 on Γ H ∂n ∂n ∂n ⇒ ∆bΩ ∇I · ∇bΩ + ∇(∇I · ∇bΩ ) · ∇bΩ = 0 ⇒ ∆bΩ ∇I · ∇bΩ + D2 I∇bΩ · ∇bΩ + D2 bΩ ∇I · ∇bΩ = 0 ⇒ D2I n · n + H

∂I = 0 on Γ. ∂n

(9.22)

28

Chapter 1. Introduction: Examples, Background, and Perspectives

This global condition is to be compared with the local condition (9.13). It can also be expressed in terms of the Laplacian and the tangential Laplacian of I as ∆I − ∆Γ I = 0 on Γ,

(9.23)

which can be compared with the local condition (9.16), ∆I = 0. This arises from the decomposition of the Laplacian of I with respect to Γ using the following identity for a smooth vector function U : div U = div Γ U + DU n · n,

(9.24)

where the tangential divergence of U is deﬁned as div Γ U = div (U ◦ pΓ )|Γ and pΓ is the projection onto Γ. Applying this to U = ∇I and recalling the deﬁnition of the tangential Laplacian ∆I = div Γ ∇I + D 2 I n · n = ∆Γ I + H

∂I + D2 I n · n, ∂n

where the tangential gradient ∇Γ I and the Laplace–Beltrami operator ∆Γ I are deﬁned as ∇Γ I = ∇(I ◦ pΓ )|Γ

9.4 9.4.1

and

∆Γ I = div Γ (∇Γ I).

Snakes, Geodesic Active Contours, and Level Sets Objective Functions Deﬁned on the Contours

In the literature the objective or energy functional is generally made up of two terms: one (image energy) that depends on the image and one (internal energy) that speciﬁes the smoothness of Γ. A general form of objective functional is def E(Ω) = g(I) dΓ, g(I) is a function of I. (9.25) Γ

The directional semiderivative with respect to a velocity ﬁeld V is given by ∂ Hg(I) + dE(Ω; V ) = g(I) n · V dΓ. (9.26) ∂n Γ This gradient will make the snakes move and will activate the contours. 9.4.2

Snakes and Geodesic Active Contours

If a gradient descent method is used to minimize (9.25) starting from an initial curve C 0 = C, the iterative process is equivalent to following the evolution C t (boundary of the smooth domain Ωt ) of the closed curve C given by the equation ∂C t = − [Ht gI + (∇gI · nt )] nt on C t , ∂t

(9.27)

9. A Glimpse into Segmentation of Images

29

where Ht is the mean curvature, nt is the unit exterior normal, and the right-hand side of the equation is formally the “derivative” of (9.25) given by (9.26). Equation (9.27) is referred to as the geodesic ﬂow. For gI = 1 it is the motion by mean curvature ∂C t = −Ht nt on C t . ∂t 9.4.3

(9.28)

Level Set Method

The idea to represent the contours C t by the zero-level set of a function ϕt (x) = ϕ(t, x) for a function ϕ : [0, τ ] × RN → R by setting

C t = x ∈ RN : ϕ(t, x) = 0 = ϕ−1 t {0} and to replace (9.27) by an equation for ϕ is due to S. Osher and J. A. Sethian [1] in 1988. This approach seems to have been simultaneously introduced in image ´, T. Coll, and F. Dibos [1] in 1993 under processing by V. Caselles, F. Catte the name “geometric partial diﬀerential equations” with, in addition to the mean curvature term, a “transport” term and by R. Malladi, J. A. Sethian, and B. C. Vemuri [1] in 1995 under the name “level set approach” combined with the notion of “extension velocity.” Let (t, x) → ϕ(t, x) : [0, τ ] × RN → R be a smooth function and Ω be a subset N of R of boundary Γ = Ω ∩ Ω such that

int Ω = x ∈ RN : ϕ(0, x) < 0 and Γ = x ∈ RN : ϕ(0, x) = 0 . (9.29) Let V : [0, τ ]×RN → RN be a suﬃciently smooth velocity ﬁeld so that the transformations {Tt } are diﬀeomorphisms. Moreover, assume that the images Ωt = Tt (Ω) verify the following properties: for all t ∈ [0, τ ]

int Ωt = x ∈ RN : ϕ(t, x) < 0 and Γt = x ∈ RN : ϕ(t, x) = 0 . (9.30) Assuming that the function ϕt (x) = ϕ(t, x) is at least of class C 1 and that ∇ϕt = 0 on ϕ−1 t {0}, the total derivative with respect to t of ϕ(t, Tt (x)) for x ∈ Γ yields d ∂ ϕ(t, Tt (x)) + ∇ϕ(t, Tt (x)) · Tt (x) = 0. ∂t dt

(9.31)

By substituting the velocity ﬁeld in (9.18), we get ∂ ϕ(t, Tt (x)) + ∇ϕ(t, Tt (x)) · V (t, Tt (x)) = 0, ∀ Tt (x) ∈ Γt ∂t ∂ ϕt + ∇ϕt · V (t) = 0 on Γt , t ∈ [0, τ ]. ⇒ ∂t

(9.32)

This last equation, the level set evolution equation, is veriﬁed only on the boundaries or fronts Γt , 0 ≤ t ≤ τ . Can we ﬁnd a representative ϕ in the equivalence class

−1 [ϕ]Ω,V = ϕ : ϕ−1 t {0} = Γt and ϕt {< 0} = int Ωt , ∀t ∈ [0, τ ]

30

Chapter 1. Introduction: Examples, Background, and Perspectives

of functions ϕ that verify conditions (9.29) and (9.30) in order to extend (9.32) from Γt to the whole RN or at least almost everywhere in RN ? Another possibility is to consider the larger equivalence class

[ϕ]Γ,V = ϕ : ϕ−1 t {0} = Γt , ∀t ∈ [0, τ ] . 9.4.4

Velocity Carried by the Normal

In D. Adalsteinsson and J. A. Sethian [1], J. Gomes and O. Faugeras [1], and R. Malladi, J. A. Sethian, and B. C. Vemuri [1], the front moves under the eﬀect of a velocity ﬁeld carried by the normal with a scalar velocity that depends on the curvatures of the level set or the front. So we are led to consider velocity ﬁelds V of the form V (t)|Γt = v(t) nt ,

x ∈ Γt ,

(9.33)

for a scalar velocity (t, x) → v(t)(x) : [0, τ ] × RN → R. By using the expression ∇ϕt /|∇ϕt | of the exterior normal nt as a function of ∇ϕt , (9.32) now becomes ∂ ϕt + v(t) |∇ϕt | = 0 on Γt . ∂t

(9.34)

For instance, consider the example of the minimization of the total length of the curve C of (9.27) for the metric g(I) dC as introduced by V. Caselles, R. Kimmel, and G. Sapiro [1]. By using the computation (9.26) of the shape semiderivative with respect to the velocity ﬁeld V of the objective functional (9.25), a natural direction of descent is given by

v(t)nt ,

∂ v(t) = − Ht g(I) + g(I) ∂nt

on Γt .

(9.35)

The normal nt and the mean curvature Ht can be expressed as a function of ∇ϕt : ∇ϕt ∇ϕt ∇ϕt and Ht = div Γt nt = div Γt = div nt = . (9.36) |∇ϕt | |∇ϕt | |∇ϕt | Γt By substituting in (9.34), we get the following evolution equation: ∂ ∇ϕt ∂ϕt g(I) ∇ϕt · − Ht g(I) + = 0 on Γt ∂t ∂nt |∇ϕt |

(9.37)

and ∂ϕt − Ht g(I) + ∂ g(I) |∇ϕt | = 0 ∂t ∂nt ϕ = ϕ0 0

on Γt , on Γ0 .

(9.38)

9. A Glimpse into Segmentation of Images

31

By substituting expressions (9.36) for nt and Ht in terms of ∇ϕt , we ﬁnally get ∇ϕt ∂ϕt − div ∇ϕt g(I) + ∇g(I) · |∇ϕt | = 0 ∂t |∇ϕt | |∇ϕt | ϕ = ϕ0 0

on Γt ,

(9.39)

on Γ0 .

The negative sign arises from the fact that we have chosen the outward rather than the inward normal. The main references to the existence and uniqueness theorems related to (9.39) can be found in Y. G. Chen, Y. Giga, and S. Goto [1]. 9.4.5

Extension of the Level Set Equations

Equation (9.39) on the fronts Γt is not convenient from both the theoretical and the numerical viewpoints. So it would be desirable to be able to extend (9.39) in a small tubular neighborhood of thickness h def

Uh (Γt ) = x ∈ RN : dΓt (x) < h

(9.40)

of the front Γt for a small h > 0 (dΓt , the distance function to Γt ). In theory, the velocity ﬁeld associated with Γt is given by ∇ϕt ∇ϕt ∇ϕt (9.41) V (t) = − div g(I) + ∇g(I) · on Γt . |∇ϕt | |∇ϕt | |∇ϕt | Given the identities (9.36), the expressions of nt and Ht extend to a neighborhood of Γt and possibly to RN if ∇ϕt is suﬃciently smooth and ∇ϕt = 0 on Γt . Equation (9.39) can be extended to RN at the price of violating the assumptions (9.29)–(9.30), either by loss of smoothness of ϕt or by allowing its gradient to be zero: ∇ϕt ∂ϕt − div ∇ϕt g(I) + ∇g(I) · |∇ϕt | = 0 ∂t |∇ϕt | |∇ϕt | ϕ = ϕ0 0

in RN ,

(9.42)

in RN .

´, T. Coll, and F. Dibos [1] In fact, the starting point of V. Caselles, F. Catte was the following geometric partial diﬀerential equation: ∂ϕt − div ∇ϕt g(I) + ν g(I) |∇ϕt | = 0 in RN , ∂t |∇ϕt | (9.43) ϕ = ϕ0 in RN 0

for a constant ν > 0 and the function g(I) = 1/(1+|∇I|2 ). They prove the existence, in dimension 2, of a viscosity solution unique for initial data ϕ0 ∈ C([0, 1] × [0, 1]) ∩ W 1,∞ ([0, 1] × [0, 1]) and g ∈ W 1,∞ (R2 ).

32

9.5 9.5.1

Chapter 1. Introduction: Examples, Background, and Perspectives

Objective Function Deﬁned on the Whole Image Tikhonov Regularization/Smoothing

The convolution is only one approach to smoothing an image I ∈ L2 (D) deﬁned in a frame D. Another way is to use the Tikhonov regularization: given ε > 0, ﬁnd a function Iε ∈ H 1 (D) that minimizes the objective functional def 1 ε|∇ϕ|2 + |ϕ − I|2 dx, Eε (ϕ) = 2 D where the value of ε can be suitably adjusted to get the desired degree of smoothness. The minimizing function Iε ∈ H 1 (D) is solution of the following variational equation: ∀ϕ ∈ H 1 (D),

ε ∇Iε · ∇ϕ + (Iε − I) ϕ dx = 0.

(9.44)

D

The price to pay now is that the computation is made on the whole image. This can be partly compensated by combining in a single operation or level of processing the smoothing of the image with the detection of the edges. To do that D. Mumford and J. Shah [1] in 1985 and [2] in 1989 introduced a new objective functional that received a lot of attention from the mathematical and engineering communities. 9.5.2

Objective Function of Mumford and Shah

Given a grey level ideal image in a ﬁxed bounded open frame D we are looking for an open subset Ω of D such that its boundary reveals the edges of the two-dimensional objects contained in the image of Figure 1.9. Deﬁnition 9.2. Let D be a bounded open subset of RN with Lipschitzian boundary. (i) An image in the frame D is speciﬁed by a function I ∈ L2 (D). (ii) We say that {Ωj }j∈J is an open partition of D if {Ωj }j∈J is a family of disjoint connected open subsets of D such that mN (∪j∈J Ωj ) = mN (D)

and

mN (∂ ∪j∈J Ωj ) = 0,

where mN is the N -dimensional Lebesgue measure. Denote by P(D) the family of all such open partitions of D. The idea behind the formulation of D. Mumford and J. Shah [1] in 1985 is to do a Tikhonov regularization Iε,j ∈ H 1 (Ωj ) of I on each element Ωj of the partition and then minimize the sum of the local minimizations over Ωj over all open

9. A Glimpse into Segmentation of Images

33

partitions of the image D: ﬁnd an open partition P = {Ωj }j∈J in P(D) solution of the minimization problem inf inf1 ε|∇ϕj |2 + |ϕj − I|2 dx (9.45) P ∈P(D)

j∈J

ϕj ∈H (Ωj )

Ωj

for some ﬁxed constant ε > 0. Observe that without the condition mN (∪j∈J Ωj ) = mN (D) the empty set would be a solution of the problem. The question of existence requires a more speciﬁc family of open partitions or a penalization term that controls the “length” of the interfaces in some sense: inf inf1 ε|∇ϕj |2 + |ϕj − I|2 dx + c HN −1 (∂ ∪j∈J Ωj ) (9.46) P ∈P(D)

j∈J

ϕj ∈H (Ωj )

Ωj

for some c > 0, where HN −1 is the Hausdorﬀ (N − 1)-dimensional measure. Equivalently, we are looking for Ω = ∪j∈J Ωj open in D such that χΩ = χD a.e. in D (χΩ and χD , characteristic functions) solution of the following minimization problem: inf inf1 ε|∇ϕ|2 + |ϕ − I|2 dx + c HN −1 (∂Ω). (9.47) Ω open ⊂D ϕ∈H (Ω) χΩ =χD a.e.

Ω

In general, the Hausdorﬀ measure is not a lower semicontinuous functional. So either the (N − 1)-Hausdorﬀ measure HN −1 is relaxed to a lower semicontinuous notion of perimeter or the minimization problem is reformulated with respect to a more suitable family of open domains and/or a space of functions larger than H 1 (Ω). Another way of looking at the problem would be to minimize the number J of connected subsets of the open partition, but this seems more diﬃcult to formalize. 9.5.3

Relaxation of the (N − 1)-Hausdorﬀ Measure

The choice of a relaxation of the (N − 1)-Hausdorﬀ measure HN −1 is critical. Here the ﬁnite perimeter of Caccioppoli (cf. section 6 in Chapter 5) reduces to the perimeter of D since the characteristic function of Ω is almost everywhere equal to the characteristic function of D. However, the relaxation of HN −1 (∂Ω) to the ´sio [8] (N − 1)-dimensional upper Minkowski content by D. Bucur and J.-P. Zole is much more interesting and in view of its associated compactness theorem (cf. section 12 in Chapter 7) it yields a unique solution to the problem. 9.5.4

Relaxation to BV-, H s -, and SBV-Functions

The other avenue to explore is to reformulate the minimization problem with respect to a space of functions larger than H 1 (Ω). One possibility is to use functions with possible jumps, such as functions of bounded variations. In dimension 1, the BV-functions can be decomposed into an

34

Chapter 1. Introduction: Examples, Background, and Perspectives

absolutely continuous function in W 1,1 (0, 1) plus a jump part at a countable number of points of discontinuity. Such functions have their analogue in dimension N . With this in mind one could consider the penalized objective function def JBV (ϕ) = |ϕ − I|2 dx + εϕ2BV(D) , (9.48) D

which is similar to the Tikhonov regularization when rewritten in the form def JH 1 (ϕ) = |ϕ − I|2 dx + εϕ2H 1 (D) = (1 + ε)|ϕ − I|2 dx + ε|∇ϕ|2 dx, D

D

where we square the penalization term to make it diﬀerentiable. Unfortunately, the square of the BV-norm is not diﬀerentiable. Going back to Figure 1.9, assume that the triangle T has an intensity of 1, the circle C has an intensity of 2/3, and the remaining part of the square S has an intensity of 1/3. The remainder of the image is set equal to 0. Then the image functional is precisely 2 1 I(x) = χT (x) + χC + χS\(T ∪C) . 3 3 We know from Theorem 6.9 in section 6.3 of Chapter 5 that the characteristic function of a Lipschitzian set is a BV-function. Therefore, if we minimize the objective functional (9.48) over all ϕ ∈ BV(D), we get exactly ϕˆ = I. If we insist on formulating the minimization problem over a Hilbert space as for the Tikhonov regularization, the norm of the penalization term can be chosen in the space H s (D) for some s ∈ (0, 1]: def JH s (ϕ) = |ϕ − I|2 dx + ε ϕ2H s (D) . D

For 0 < s < 1/2, the norm is given by the expression def

ϕH s (D) =

dx D

dy D

|ϕ(y) − ϕ(x)|2 |y − x|N +2s

and the relaxation is very close to the one in BV(D) since BV(D)∩L∞ (D) ⊂ H s (D), 0 < s < 1/2 (cf. Theorem 6.9 (ii) in Chapter 5). However, as seducing as those relaxations can be, it is not clear that the objectives of the original formulation are preserved. Take the BV-formulation. It is implicitly assumed that there are clean jumps across the interfaces. This is true in dimension 1, but not in dimension strictly greater than 1 as explained in L. Ambrosio [1], where he points out that the distributional gradient of a BV-function which is a vector measure has three parts: an absolutely continuous part, a jump part, and a nasty Cantor part. The space BV(D) contains pathological functions of Cantor–Vitali type that are continuous with an approximate diﬀerential 0 almost everywhere, and this class of functions is dense in L2 (D), making the inﬁmum of

9. A Glimpse into Segmentation of Images

35

JBV over BV(D) equal to zero without giving information about the segmentation. To get around this diﬃculty, he introduces the smaller space SBV(D) of functions whose distributional gradient does not have a Cantor part. He considers the mathematical framework for the minimization of the objective functional def |ϕ − I|2 + ε|∇ϕ|2 dx + c HN −1 (K) J(ϕ, K) = D\K

with respect to all closed sets K ⊂ Ω and ϕ ∈ H 1 (Ω\K), where I ∈ L∞ (D). This problem makes sense and has a solution in SBV(D) when K is replaced by the set Sϕ of points where the function ϕ ∈ SBV(D) has a jump, that is, def

|ϕ − I|2 + ε|∇ϕ|2 dx + c HN −1 (Sϕ ),

J(ϕ) =

D

that turns out to be well-deﬁned on SBV(D). 9.5.5

Cracked Sets and Density Perimeter

The alternate avenue to explore is to reformulate the minimization problem with respect to a more suitable family of open domains (cf. section 15 of Chapter 7). An additional reason to do that would be to remove the term on the length of the interface. When a penalization on the length of the segmentation is included, long slender objects are not “seen” by the numerical algorithms since they have too large a perimeter. In order to retain a segmentation of piecewise H 1 -functions ´sio [38] introduced the without a perimeter term, M. C. Delfour and J.-P. Zole family of cracked sets (see Figure 1.11) that yields a compactness theorem in the W 1,p -topology associated with the oriented distance function (cf. Chapter 7).

Figure 1.11. Example of a two-dimensional strongly cracked set. The originality of this approach is that it does not require a penalization term on the length of the segmentation and that, within the set of solutions, there exists one with minimum density perimeter as deﬁned by D. Bucur and ´sio [8]. J.-P. Zole

36

Chapter 1. Introduction: Examples, Background, and Perspectives

Theorem 9.2. Let D be a bounded open subset of RN and α > 0 and h > 0 be real numbers.3 Consider the families Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1, def , F(D, h, α) = Ω ⊂ D : dΓ (x + td) ≥ α such that inf (9.49) 0 0 there exists a compact subset K of Ω such that, for all x ∈ Ω ∩ K, |∂ α f (x)| ≤ ε. Denote by C0k (Ω) the space of all such functions. Clearly C0k (Ω) ⊂ C k (Ω) ⊂ Bk (Ω) ⊂ C k (Ω). Endowed with the norm (2.16), C0k (Ω), C k (Ω), and Bk (Ω) are Banach spaces. Finally % % % def def def C k (Ω), C0∞ (Ω) = C0k (Ω), and B(Ω) = Bk (Ω). C ∞ (Ω) = k≥0

k≥0

k≥0

function f : Ω → R is uniformly continuous if for each ε > 0 there exists δ > 0 such that for all x and y in Ω such that |x − y| < δ we have |f (x) − f (y)| < ε. 3A

2. Notation and Deﬁnitions

65

When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by C0k (Ω)m or C0k (Ω, Rm ), C k (Ω)m or C k (Ω, Rm ), B k (Ω)m or Bk (Ω, Rm ), C k (Ω)m or C k (Ω, Rm ), etc. We quote the following classical compactness theorem. Theorem 2.4 (Ascoli–Arzel`a theorem). Let Ω be a bounded open subset of RN . A subset K of C(Ω) is precompact in C(Ω) provided the following two conditions hold: (i) there exists a constant M such that for all f ∈ K and x ∈ Ω, |f (x)| ≤ M ; (ii) for every ε > 0 there exists δ > 0 such that if f ∈ K, x, y in Ω, and |x−y| < δ, then |f (x) − f (y)| < ε. 2.6.2

H¨ older (C 0, ) and Lipschitz (C 0,1 ) Continuous Functions

Given λ, 0 < λ ≤ 1, a function f is (0, λ)-H¨ older continuous in Ω if ∃c > 0, ∀x, y ∈ Ω,

|f (y) − f (x)| ≤ c |x − y|λ .

When λ = 1, we also say that f is Lipschitz or Lipschitz continuous. Similarly for k ≥ 1, f is (k, λ)-H¨ older continuous in Ω if |∂ α f (y) − ∂ α f (x)| ≤ c |x − y|λ .

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω,

Denote by C k,λ (Ω) the space of all (k.λ)-H¨older continuous functions on Ω. Deﬁne for k ≥ 0 the subspaces4 & ∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω def (2.20) C k,λ (Ω) = f ∈ C k (Ω) : α |∂ f (y) − ∂ α f (x)| ≤ c |x − y|λ of C k (Ω). By deﬁnition for each α, 0 ≤ |α| ≤ k, ∂ α f has a unique, bounded, continuous extension to Ω. Endowed with the norm |∂ α f (y) − ∂ α f (x)| def f C k,λ (Ω) = max f C k (Ω) , max sup (2.21) |x − y|λ 0≤|α|≤k x,y∈Ω x=y

C k,λ (Ω) is a Banach space. Finally denote by C0k,λ (Ω) the space C k,λ (Ω) ∩ C0k (Ω). 2.6.3

Embedding Theorem

In general C k+1 (Ω) ⊂ C k,1 (Ω), but the inclusion is true for a large class of domains, including convex domains.5 We ﬁrst quote the following embedding theorem. 4 The notation C k,λ (Ω) should not be confused with the notation C k,λ (Ω) for (k, λ)-H¨ older continuous functions in Ω without the uniform boundedness assumption in Ω. In particular, C k,λ (RN ) is contained in but not equal to C k,λ (RN ). 5 For instance the convexity can be relaxed to the more general condition: there exists M > 0 such that for all x and y in Ω, there exists a path γx,y in Ω (that is, a C 1 injective map from the

interval [0, 1] into Ω with γ(0) = x, γ(1) = y, and

1 0

γ (t) dt ≤ M |x − y|).

66

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 2.5 (R. A. Adams [1]). Let k ≥ 0 be an integer and 0 < ν < λ ≤ 1 be real numbers. Then the following embeddings exist: C k+1 (Ω) → C k (Ω),

(2.22)

(Ω) → C (Ω),

(2.23)

C C

k,λ

k,λ

k

(Ω) → C

k,ν

(Ω).

(2.24)

If Ω is bounded, then the embeddings (2.23) and (2.24) are compact. If Ω is convex, we have the further embeddings C k+1 (Ω) → C k,1 (Ω), C

k+1

(Ω) → C

k,ν

(Ω).

(2.25) (2.26)

If Ω is convex and bounded, then embeddings (2.22) and (2.26) are compact. As a consequence of the second part of the theorem, the deﬁnition of C k,λ (Ω) simpliﬁes when Ω is convex: def

C k,λ (Ω) = f ∈ Bk (Ω) : ∀α, |α| = k, ∃c > 0, ∀x, y ∈ Ω, |∂ α f (y) − ∂ α f (x)| ≤ c |x − y|λ ,

(2.27)

and its norm is equivalent to the norm def

|∂ α f (y) − ∂ α f (x)| . |x − y|λ x,y∈Ω

f C k,λ (Ω) = f C k (Ω) + max sup |α|=k

(2.28)

x=y

When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by C k,λ (Ω)m or C k,λ (Ω, Rm ). 2.6.4

Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to Path-Connected Domains via the Geodesic Distance

By Rademacher’s theorem (cf., for instance, L. C. Evans and R. F. Gariepy [1]) k+1,∞ C k,1 (Ω) ⊂ Wloc (Ω)

and, when Ω is convex, C k,1 (Ω) = W k+1,∞ (Ω). This last convexity assumption can be relaxed as follows. From Rademacher’s ´zis [1, p. 154]) theorem and the inequality (cf. H. Bre ∀u ∈ W 1,∞ (Ω) and ∀x, y ∈ Ω, we have the following theorem.

|u(x) − u(y)| ≤ ∇uL∞ (Ω) distΩ (x, y)

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

67

Theorem 2.6. Let Ω be a bounded, path-connected, open subset of RN such that ∃c, ∀x, y ∈ Ω,

distΩ (x, y) ≤ c |x − y|.

(2.29)

Then we have W 1,∞ (Ω) = C 0,1 (Ω) algebraically and topologically, and there exists a constant c such that ∀u ∈ W 1,∞ (Ω) and ∀x, y ∈ Ω,

|u(x) − u(y)| ≤ c ∇uL∞ (Ω) |x − y|.

Hence for all integers k ≥ 0, W k+1,∞ (Ω) = C k,1 (Ω) algebraically and topologically. This is true for Lipschitzian domains (D. Gilbarg and N. S. Trudinger [1]). Corollary 1. Let Ω be a bounded, open, path-connected, and locally Lipschitzian6 subset of RN . Then property (2.29) and Theorem 2.6 are veriﬁed. Proof. By Theorem 5.8 later in section 5.4 of this chapter.

3

Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

Open domains in the Euclidean space RN are classically described by introducing at each point of their boundary a local diﬀeomorphism (that is, deﬁned in a neighborhood of the point) that locally ﬂattens the boundary. The smoothness of the boundary is determined by the smoothness of the local diﬀeomorphism. After introducing the main deﬁnition in section 3.1, we brieﬂy recall the deﬁnition of an (embedded) submanifold of codimension greater than or equal to 1 in section 3.2 in order to make sense of the associated boundary integral and, more generally, the integral over smooth submanifolds of arbitrary dimension in RN . The deﬁnition of the integral can be generalized by introducing the Hausdorﬀ measures that extend the integration theory from smooth submanifolds to arbitrary Hausdorﬀ measurable subsets of RN , thus making the writing of the integral completely independent of the choice of the local diﬀeomorphisms. Section 3.3 completes the section by deﬁning the fundamental forms and curvatures for a smooth submanifold of RN of codimension 1.

3.1

Sets of Classes C k and C k,

Let {e1 , . . . , eN } be the standard unit orthonormal basis in RN . We use the notation ζ = (ζ , ζN ) for a point ζ = (ζ1 , . . . , ζN ) in RN , where ζ = (ζ1 , . . . , ζN −1 ). Denote by B the open unit ball in RN and deﬁne the sets def

B0 = {ζ ∈ B : ζN = 0} , def

B+ = {ζ ∈ B : ζN > 0} ,

def

B− = {ζ ∈ B : ζN < 0} .

(3.1) (3.2)

The main elements entering Deﬁnition 3.1 are illustrated in Figure 2.1 for N = 2. 6 Cf. Deﬁnition 5.2 of section 5 and Theorem 5.3, which says that a locally Lipschitzian domain is equi-Lipschitzian when ∂Ω is bounded.

68

Chapter 2. Classical Descriptions of Geometries and Their Properties ζN gx B+ ζ

set Ω B0 ∂Ω

x U (x)

hx = gx−1

hx (x) = (0, 0) B−

ball B

Figure 2.1. Diﬀeomorphism gx from U (x) to B. Deﬁnition 3.1. Let Ω be a subset of RN such that ∂Ω = ∅, 0 ≤ k be an integer or +∞, and 0 ≤ ≤ 1 be a real number. (i)

- Ω is said to be locally of class C k at x ∈ ∂Ω if there exist (a) a neighborhood U (x) of x, and (b) a bijective map gx : U (x) → B with the following properties: def gx ∈ C k U (x); B , hx = gx−1 ∈ C k B; U (x) , int Ω ∩ U (x) = hx (B+ ), def Γx = ∂Ω ∩ U (x) = hx (B0 ), B0 = gx (Γx ).

(3.3) (3.4) (3.5)

- Given 0 < < 1, Ω is said to be locally (k, )-H¨ olderian at x ∈∂Ω if ∈ C k, U (x), B with conditions (a) and (b) are satisﬁed with a map g x −1 k, B, U (x) . inverse hx = gx ∈ C - Ω is said to be locally k-Lipschitzian at x ∈ ∂Ω if conditions (a) and−1(b) k,1 are satisﬁed with a map g U (x), B with inverse hx = gx ∈ ∈ C x C k,1 B, U (x) . (ii)

- Ω is said to be locally of class C k if, for each x ∈ ∂Ω, Ω is locally of class C k at x. - Given 0 < < 1, Ω is said to be locally (k, )-H¨ olderian if, for each x ∈ ∂Ω, Ω is locally (k, )-H¨olderian at x. - Ω is said to be locally k-Lipschitzian if, for each x ∈ ∂Ω, Ω is locally k-Lipschitzian at x.

Remark 3.1. By deﬁnition int Ω = ∅ and int Ω = ∅. The above deﬁnitions are usually given for an open set Ω called a domain. This terminology naturally arises in partial diﬀerential equations, where this open set is indeed the domain in which the solution

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

69

of the partial diﬀerential equation is deﬁned.7 At this point it is not necessary to assume that the set Ω is open. The family of diﬀeomorphisms {gx : x ∈ ∂Ω} characterizes the equivalence class of sets [Ω] = {A ⊂ RN : int A = int Ω and ∂A = ∂Ω} with the same interior and boundary. The sets int Ω and ∂Ω are invariants for the equivalence class [Ω] of sets of class C k, . The notation Γ for ∂Ω and the standard terminology domain for the unique (open) set int Ω associated with the class [Ω] will be used. In what follows, a C k, -mapping gx with C k, -inverse will be called a C k, -diﬀeomorphism. Consider the solution y = y(Ω) of the Dirichlet problem on the bounded open domain Ω: −∆y = f in Ω,

y = g on ∂Ω.

(3.6)

When Ω is of class C ∞ , the solution y(Ω) can be sought in any Sobolev space H m (Ω), m ≥ 1, by choosing suﬃciently smooth data f and g and appropriate compatibility conditions. However, when Ω is only of class C k , 1 ≤ k < ∞, m cannot be made arbitrary large by choosing smoother data f and g and appropriate compatibility conditions. The reader is referred to R. Dautray and J.-L. Lions [1, Chap. VII, sect. 3, pp. 1271–1304] for classical smoothness results of solutions to elliptic problems in domains of class C k . It is important to understand that we shall consider shape problems involving the solution of a boundary value problem over domains with minimal smoothness. Since we know that, at least for elliptic problems, the smoothness of the solution depends not only on the smoothness of the data but also on the smoothness of the domain. This issue will be of paramount importance to make sure that the shape problems are well posed. Remark 3.2. We shall see in section 5 that domains that are locally the epigraph of a C k, , k ≥ 1 (resp., Lipschitzian), function are of class C k, (resp., C 0,1 ), but domains which are of class C 0,1 are not necessarily locally the epigraph of a Lipschitzian function (cf. Examples 5.1 and 5.2). Nevertheless globally C 0,1 -mappings with a C 0,1 inverse are important since they transport Lp -functions onto Lp -functions and W 1,p -functions ˇas [1, Lems. 3.1 and 3.2, pp. 65–66]). onto W 1,p -functions (cf. J. Nec For sets of class C 1 , the unit exterior normal to the boundary Γ = ∂Ω can be characterized through the Jacobian matrices of gx and hx . By deﬁnition of B0 , {e1 , . . . , eN −1 } ⊂ B0 and the tangent space Ty Γ, Γ = ∂Ω, at y to Γx is the vector space spanned by the N − 1 vectors {Dhx (ζ , 0)ei : 1 ≤ i ≤ N − 1} ,

(ζ , 0) = gx (y) ∈ B0 ,

(3.7)

7 Classically for k ≥ 1 a bounded open domain Ω in RN is said to be of class C k if its boundary is a C k -submanifold of RN of codimension 1 and Ω is located on one side of its boundary Γ = ∂Ω (cf. S. Agmon [1, Def. 9.2, p. 178]).

70

Chapter 2. Classical Descriptions of Geometries and Their Properties

where Dhx (ζ) is the Jacobian matrix of hx at the point ζ: def

(Dhx )m = ∂m (hx ) . So from (3.7) a normal vector ﬁeld to Γx at y ∈ Γx is given by mx (y) = − ∗(Dhx )−1 (ζ , 0) eN = − ∗Dgx (y) eN ,

hx (ζ , 0) = y,

(3.8)

since −mx (y) · Dhx (ζ , 0) ei = eN · ei = δiN ,

1 ≤ i ≤ N.

Thus the outward unit normal ﬁeld n(y) at y ∈ Γx is given by (Dhx )−1 (ζ , 0) eN , | ∗(Dhx )−1 (ζ , 0) eN | ∗ (Dhx )−1 (h−1 x (y)) eN . n(y) = − ∗ −1 | (Dhx ) (h−1 x (y)) eN |

∀hx (ζ , 0) = y ∈ Γx , ∀y ∈ Γx ,

n(y) = −

∗

(3.9)

It can be veriﬁed that n is uniquely deﬁned on Γ by checking that for y ∈ Γx ∩ Γx , n is uniquely deﬁned by (3.9).

3.2 3.2.1

Boundary Integral, Canonical Density, and Hausdorﬀ Measures Boundary Integral for Sets of Class C 1

The family of neighborhoods U (x) associated with all the points x of Γ is an open cover of Γ. If Γ = ∂Ω is assumed to be compact, then there exists a ﬁnite open subcover: that is, there exists a ﬁnite sequence of points {xj : 1 ≤ j ≤ m} of Γ such that Γ ⊂ U1 ∪ · · · ∪ Um , where Uj = U (xj ). For simplicity, index all the previous symbols by j instead of xj . The boundary integration on Γ is obtained by using a partition of unity {rj : 1 ≤ j ≤ m} for the family of open neighborhoods {Uj : 1 ≤ j ≤ m} of Γ: r ∈ D(Uj ), 0 ≤ rj (x) ≤ 1, j m (3.10) rj (x) = 1 in a neighborhood U of Γ, j=1

such that U ⊂ ∪m j=1 Uj , where D(Uj ) is the set of all inﬁnitely continuously diﬀerentiable functions with compact support in Uj . If f ∈ C(Γ), then (f rj ) ◦ hj ∈ C(B0 ),

1 ≤ j ≤ m.

Deﬁne the boundary integral of f rj on Γj as def f rj dΓ = (f rj ) ◦ hj (ζ , 0) ωj (ζ ) dζ , Γj

B0

Γj = U (xj ) ∩ Γ,

(3.11)

(3.12)

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

71

where ωj = ωxj and ωx is the density term ωx (ζ ) = |mx (hx (ζ , 0))| | det Dhx (ζ , 0)|, ∗

mx (y) = − (Dhx )

−1

(h−1 x (y)) eN

= − Dgx (y) eN .

From this deﬁne the boundary integral of f on Γ as m def f dΓ = f rj dΓ. Γ

j=1

(3.13)

∗

(3.14)

Γj

The expression on the right-hand side of (3.12) results from the parametrization of the boundary Γj by B0 through the diﬀeomorphism hj . 3.2.2

Integral on Submanifolds

In diﬀerential geometry there is a general procedure to deﬁne the canonical density for a d-dimensional submanifold V in RN parametrized by a C k -mapping, k ≥ 1 (cf., for instance, M. Berger and B. Gostiaux [1, Def. 2.1.1, p. 48 (56) and Prop. 6.62, p. 214 (239)]). Deﬁnition 3.2. Let ∅ = S ⊂ RN , let k ≥ 1 and 1 ≤ d ≤ N be integers, and let 0 ≤ ≤ 1 be a real number. (i) Given x ∈ S, S is said to be locally a C k - (resp., C k, -) submanifold of dimension d at x in RN if there exists an open subset U (x) of RN containing k k, x and a C - (resp., C -) diﬀeomorphism gx from U (x) onto its open image gx U (x) , such that gx U (x) ∩ S = gx U (x) ∩ Rd , where Rd = (x1 , . . . , xd , 0, . . . , 0) ∈ RN : ∀(x1 , . . . , xd ) ∈ Rd . (ii) S is said to be a C k - (resp., C k, -) submanifold of dimension d in RN if, for each x ∈ S, it is locally a C k - (resp., C k, -) submanifold of dimension d at x in RN . For k ≥ 1 the canonical density ωx on the submanifold S at a point y ∈ U (x) ∩ S is given by ' ωx = | det B|, where the d × d matrix B is given by (B)ij = (Dhx ei ) · (Dhx ej ), 1 ≤ i, j ≤ d,

hx = gx−1 on gx (U (x)).

In the special case d = N − 1, it is easy to verify that ∗ C C ∗C c ∗ , B = ∗C C, Dhx Dhx = ∗ ∗ cC cc

72

Chapter 2. Classical Descriptions of Geometries and Their Properties

where C is the (N × (N − 1))-matrix and c is the N -vector deﬁned by Cij = {Dhx }ij ,

1 ≤ i ≤ N, 1 ≤ j ≤ N − 1,

c = Dhx eN .

Denote by M (A) the matrix of cofactors associated with a matrix A: M (A)ij is equal to the determinant of the matrix obtained after deleting the ith row and the jth column times (−1)i+j . Then M (A) = (det A) ∗ A−1 , M ( ∗ A) = ∗ M (A), and for two invertible matrices A1 and A2 , M (A1 A2 ) = M (A1 )M (A2 ). As a result det B = M ( ∗Dhx Dhx )N N = eN · M ( ∗Dhx Dhx )eN , where M ( ∗Dhx Dhx )N N is the N N -cofactor of the matrix ∗Dhx Dhx . Then M ( ∗Dhx Dhx )N N = eN · M ( ∗Dhx Dhx )eN = eN · M ( ∗Dhx ) M (Dhx )eN = |M (Dhx )eN |2 . In view of the previous considerations and (3.13) √ det B = |M (Dhx )eN | = | det Dhx | | ∗(Dhx )−1 eN | = ωx . 3.2.3

Hausdorﬀ Measures

Deﬁnition 3.2 gives the classical construction of a d-dimensional surface measure on the boundary of a C 1 -domain. In 1918, F. Hausdorff [1] introduced a ddimensional measure in RN which gives the same surface measure for smooth submanifolds but is deﬁned on all measurable subsets of RN . When d = N , it is equal to the Lebesgue measure. To complete the discussion we quote the deﬁnition from F. Morgan [1, p. 8] or L. C. Evans and R. F. Gariepy [1, p. 60]. Deﬁnition 3.3. For any subset S of RN , deﬁne the diameter of S as def

diam (S) = sup{|x − y| : x, y ∈ S}. Let α(d) denote the Lebesgue measure of the unit ball in Rd . The d-dimensional Hausdorﬀ measure Hd (A) of a subset A of RN is deﬁned by the following process. For δ small, cover A eﬃciently by countably many sets Sj with diam (Sj ) ≤ δ, add up all the terms d

α(d) (diam (Sj )/2) , and take the limit as δ → 0: def

Hd (A) = lim

δ0

inf

A⊂∪Sj diam (Sj )≤δ

j

α(d)

d diam (Sj ) , 2

where the inﬁmum is taken over all countable covers {Sj } of A whose members have diameter at most δ.

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

73

For 0 ≤ d < ∞, Hd is a Borel regular measure, but not a Radon measure for d < N since it is not necessarily ﬁnite on each compact subset of RN . The Hausdorﬀ dimension of a set A ⊂ RN is deﬁned as def

Hdim (A) = inf{0 ≤ s < ∞ : Hs (A) = 0}.

(3.15)

By deﬁnition Hdim (A) ≤ N and ∀k > Hdim (A),

Hk (A) = 0.

If a submanifold S of dimension d, 1 ≤ d < N , of Deﬁnitions 3.2 and 3.3 is characterized by a single C 1 -diﬀeomorphism g, that is, h = g −1 ,

g(S) = Rd and S = h(Rd ), then for any Lebesgue-measurable set E ⊂ Rd ω dx = Hd (h(E)). E

This is a generalization to submanifolds of codimension greater than 1 of formula (3.12). The deﬁnition of the Hausdorﬀ measure and the Hausdorﬀ dimension extend from integers d to reals s, 0 ≤ s ≤ ∞, by modifying Deﬁnition 3.3 as follows. Deﬁnition 3.4. For any real s, 0 ≤ s ≤ ∞, the s-dimensional Hausdorﬀ measure Hs (A) of a subset A of RN is deﬁned by the following process. For δ small, cover A eﬃciently by countably many sets Sj with diam (Sj ) ≤ δ, and add all the terms α(s) (diam (Sj )/2)d , where def

α(s) =

π s/2 , Γ(s/2 + 1)

Take the limit as δ → 0: def

Hs (A) = lim

δ0

inf

def

∞

Γ(t) =

A⊂∪Sj diam (Sj )≤δ

e−x xt−1 .

(3.16)

0

j

α(s)

d diam (Sj ) , 2

where the inﬁmum is taken over all countable covers {Sj } of A whose members have diameter at most δ. The Hausdorﬀ dimension is deﬁned by the same formula (3.15).

3.3

Fundamental Forms and Principal Curvatures

Consider a set Ω locally of class C 2 in RN . Its boundary Γ = ∂Ω is an (N − 1)dimensional submanifold of RN of class C 2 . At each point x ∈ Γ there is a C 2 diﬀeomorphism gx from a neighborhood U (x) of x onto B. Denoting its inverse by

74

Chapter 2. Classical Descriptions of Geometries and Their Properties

hx = gx−1 , the covariant basis at a point y ∈ U (x) ∩ Γ is deﬁned as def

aα (y) =

∂hx (ζ , 0), ∂ζα

α = 1, . . . , N − 1,

hx (ζ , 0) = y,

and aN is chosen as the inward unit normal : def

aN (y) =

∗

−1 (Dhx (h−1 eN x (y))) . −1 | ∗(Dhx (hx (y)))−1 eN |

The standard convention that the Greek indices range from 1 to N − 1 and the Roman indices from 1 to N will be followed together with Einstein’s rule of summation over repeated indices. The associated contravariant basis {ai } = {ai (y)} is deﬁned from the covariant one {ai } = {ai (y)} as ai · aj = δij , where δij is the Kronecker index function. The ﬁrst, second, and third fundamental forms a, b, and c are deﬁned as def

aαβ = aα · aβ ,

def

bαβ = −aα · aN,β ,

def

cαβ = bλα bλβ ,

where aN,β =

∂aN , ∂ζβ

bλα = aλ · aµ bµα .

The above deﬁnitions extend to sets of class C 1,1 for which hx ∈ C 1,1 (B) and hence hx ∈ C 1,1 (B0 ). So by Rademacher’s theorem in dimension N − 1 (cf., for instance, L. C. Evans and R. F. Gariepy [1]), hx ∈ W 2,∞ (B0 ) and the deﬁnitions of the second and third fundamental forms still make sense HN −1 almost everywhere on Γ. The eigenvalues of bαβ are the (N − 1) principal curvatures κi , 1 ≤ i ≤ N − 1, of the submanifold Γ. The mean curvature H and the Gauss curvature K are deﬁned as N −1 N −1 ( 1 def def H = κα and K = κα . N − 1 α=1 α=1 The choice of the inner normal for aN is necessary to make the principal curvatures of the sphere (boundary of the ball) positive. The factor 1/(N − 1) is used to make the mean curvature of the unit sphere equal to 1 in all dimensions. The reader should keep in mind the long-standing diﬀerences in usage between geometry and partial diﬀerential equations, where the outer unit normal is used in the integration-byparts formulae for the Euclidean space RN . For integration by parts on submanifolds of RN the sum of the principal curvatures will naturally occur rather than the mean curvature. It will be convenient to redeﬁne H as the sum of the principal curvatures and introduce the notation H for the classical mean curvature: N −1 1 κα , H = κα . H = N − 1 α=1 α=1 def

N −1

def

(3.17)

4. Sets Globally Described by the Level Sets of a Function

4

75

Sets Globally Described by the Level Sets of a Function

From the deﬁnition of a set of class C k, , k ≥ 1 and 0 ≤ ≤ 1, the set Ω can also be locally described by the level sets of the C k, -function def

fx (y) = gx (y) · eN ,

(4.1)

since by deﬁnition int Ω ∩ U (x) = {y ∈ U (x) : fx (y) > 0}, ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0}. The boundary ∂Ω is the zero level set of fx and the gradient ∇fx (y) = ∗Dgx (y)eN = 0 is normal to that level set. Thus the exterior normal to Ω is given by n(y) = −

∗ ∗ ∇fx (y) Dgx (y)eN (Dhx (gx (y)))−1 eN =− ∗ =− ∗ . |∇fx (y)| | Dgx (y)eN | | (Dhx (gx (y)))−1 eN |

From this local construction of the functions {fx : x ∈ ∂Ω}, a global function on RN can be constructed to characterize Ω. Theorem 4.1. Given k ≥ 1, 0 ≤ ≤ 1, and a set Ω in RN of class C k, with compact boundary, there exists a Lipschitz continuous f : RN → R such that int Ω = {y ∈ RN : f (y) > 0}

and

∂Ω = {y ∈ RN : f (y) = 0}

(4.2)

and a neighborhood W of ∂Ω such that f ∈ C k, (W ) and ∇f = 0 on W and n = −

∇f , |∇f |

(4.3)

where n is the outward unit normal to Ω on ∂Ω. Proof. Construction of the function f . Fix x ∈ ∂Ω and consider the function fx deﬁned by (4.1). By continuity of ∇fx , let V (x) ⊂ U (x) be a neighborhood of x such that 1 ∀y ∈ V (x), |∇fx (y) − ∇fx (x)| ≤ |∇fx (x)|. 2 Furthermore, let W (x) be a bounded open neighborhood of x such that W (x) ⊂ V (x). For ∂Ω compact there exists a ﬁnite subcover {Wj = W (xj ) : 1 ≤ j ≤ m} of ∂Ω for a ﬁnite sequence {xj : 1 ≤ j ≤ m} ⊂ ∂Ω. Index all the previous symbols by j instead of xj and deﬁne W = ∪m j=1 Wj . Let {rj : 1 ≤ j ≤ m} be a partition of unity for {Vj = V (xj )} such that r ∈ D(Vj ), 0 ≤ rj (x) ≤ 1, 1 ≤ j ≤ m, j m (4.4) rj (x) = 1 in W , j=1

76

Chapter 2. Classical Descriptions of Geometries and Their Properties

where, by deﬁnition, W is an open neighborhood of ∂Ω such that W ⊂ V = ∪m j=1 Vj and D(Vj ) is the set of all inﬁnitely continuously diﬀerentiable functions with compact support in Vj . Deﬁne the function f : RN → R: def

f (y) = dW ∪Ωc − dW ∪Ω +

m j=1

rj (y)

fj (y) , |∇fj (xj )|

where dA (x) = inf{|y − x| : y ∈ A} is the distance function from a point x to a nonempty set A of RN . The function dA is Lipschitz continuous with Lipschitz constant equal to 1. Since for all j, fj ∈ C k, (Vj ), rj ∈ D(Vj ), and W j ⊂ Vj , rj fj ∈ C k, (Vj ) with compact support in Vj . Therefore since k ≥ 1, rj fj is Lipschitz continuous in RN . So, by deﬁnition, f is Lipschitz continuous on RN as the ﬁnite sum of Lipschitz continuous functions on RN . Properties (4.2). Introduce the index set def

J(y) = {j : 1 ≤ j ≤ m, rj (y) > 0} for y ∈ V . For all y ∈ W , J(y) = ∅, dW ∪Ωc = 0 = dW ∪Ω , and f (y) =

rj (y)

j∈J(y)

fj (y) . |∇fj (xj )|

(4.5)

For y ∈ ∂Ω = W ∩ ∂Ω, fj (y) = 0 for all j ∈ J(y) and hence f (y) = 0; for y ∈ int Ω ∩ W , fj (y) > 0 and rj (y) > 0 for all j ∈ J(y) and hence f (y) > 0; for y ∈ int Ωc ∩ W , fj (y) < 0 and)rj (y) > 0 for all j ∈ J(y) and hence f (y) < 0. For y ∈ Ω\W , fj ≥ 0, rj ≥ 0, m rj fj ≥ 0, dW ∪Ωc > 0, dW ∪Ω = 0, and f > 0; )mj=1 c for y ∈ Ω \W , fj ≤ 0, rj ≥ 0, j=1 rj fj ≤ 0, dW ∪Ωc = 0, dW ∪Ω > 0, and f < 0. So we have proven that ∂Ω ⊂ f −1 (0), int Ω ⊂ {f > 0}, and int Ωc ⊂ {f < 0}. Hence f has the properties (4.2). Properties (4.3). Recall that on W the function f is given by expression (4.5). It belongs to C k, (W ), and, a fortiori, to C 1 (W ), as the ﬁnite sum of C k, -functions on W since k ≥ 1. The gradient of f in W is given by ∇f (y) =

∇rj (y)

j∈J(y)

fj (y) ∇fj (y) + rj (y) . |∇fj (xj )| |∇fj (xj )|

It remains to show that it is nonzero on ∂Ω. On ∂Ω ∩ Vj , fj = 0 and ∀j ∈ J(y),

∇fj (y) = −n(y). |∇fj (y)|

Therefore in W , ∇f can be rewritten in the form |∇fj (y)| rj (y) ∇f (y) = −n(y) |∇fj (xj )| j∈J(y) and |∇fj (y)| rj (y) 1 − ∇f (y) + n(y) = n(y) . |∇fj (xj )| j∈J(y)

4. Sets Globally Described by the Level Sets of a Function Finally, by construction of W ,

|∇f (y)|≥ |n(y)| − |n(y)| ≥1−

j∈J(y)

rj (y)

j∈J(y)

77

|∇fj (y)| rj (y) 1 − |∇fj (xj )|

1 1 = 2 2

and ∇f = 0 in W . This proves properties (4.3). This theorem has a converse. Theorem 4.2. Associate with a continuous function f : RN → R the set def

Ω = {y ∈ RN : f (y) > 0}.

(4.6)

f −1 (0) = {y ∈ RN : f (y) = 0} = ∅

(4.7)

Assume that def

and that there exists a neighborhood V of f −1 (0) such that f ∈ C k, (V ) for some k ≥ 1 and 0 ≤ ≤ 1 and that ∇f = 0 in f −1 (0). Then Ω is a set of class C k, , int Ω = Ω

and

∂Ω = f −1 (0).

(4.8)

Proof. By continuity of f , Ω is open, int Ω = Ω, Ω is closed, Ω ⊂ {y ∈ RN : f (y) ≥ 0},

Ω = Ω = {y ∈ RN : f (y) ≤ 0},

and ∂Ω ⊂ f −1 (0). Conversely for each x ∈ ∂Ω, deﬁne the function def

g(t) = f (x + t∇f (x)). There exists δ > 0 such that for all |t| < δ, x + t∇f (x) ∈ V and the function g is C 1 in (−δ, δ). Hence since g(0) = 0 and g (t) = ∇f (x + t∇f (x)) · ∇f (x)

t

∇f (x + s∇f (x)) · ∇f (x) ds.

f (x + t∇f (x)) = 0

By continuity of ∇f in V and the fact that ∇f (x) = 0, there exists δ , 0 < δ < δ, such that ∀s, 0 ≤ |s| ≤ δ ,

∇f (x + s∇f (x)) · ∇f (x) ≥

1 |∇f (x)|2 > 0. 2

Hence for all t, 0 < t ≤ δ , f (x + t∇f (x)) > 0. So any point in f −1 (0) can be approximated by a sequence {xn = x + tn ∇f (x) : n ≥ 1, 0 < tn ≤ δ } in Ω, tn → 0, and f −1 (0) ⊂ Ω. Similarly, using a sequence of negative tn ’s, f −1 (0) ⊂ Ω and hence f −1 (0) ⊂ ∂Ω. This proves (4.8).

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Chapter 2. Classical Descriptions of Geometries and Their Properties

Fix x ∈ ∂Ω = f −1 (0). Since ∇f (x) = 0, deﬁne the unit vector eN (x) = ∇f (x)/|∇f (x)|. Associate with eN (x) unit vectors e1 (x), . . . , eN −1 (x), which form an orthonormal basis in RN with eN (x). Deﬁne the map gx : V → RN as f (y) def N −1 gx (y) = {(y − x) · eα (x)}α=1 , ⇒ gx ∈ C k, (V ; RN ). |∇f (x)| The transpose of the Jacobian matrix of gx is given by ∗

Dgx (y) = (e1 (x), . . . , eN −1 (x), ∇f (y)/|∇f (x)|)

and Dgx (x) = I, the identity matrix in the {ei (x)} reference system. By the inverse mapping theorem gx has a C k, inverse hx in some neighborhood U (x) of x in V . Therefore, Ω ∩ U (x) = {y ∈ U (x) : f (x) > 0} = int Ω ∩ U (x), {y ∈ U (x) : f (x) = 0} = ∂Ω ∩ U (x), and the set Ω is of class C k, . We complete this section with the important theorem of Sard. ´ [3, Vol. III, sect. 16.23, p. 167]). Let X and Y Theorem 4.3 (J. Dieudonne be two diﬀerential manifolds, f : X → Y be a C ∞ -mapping, and E be the set of critical points of f . Then f (E) is negligible in Y , and Y − f (E) is dense in Y . Combining this theorem with Theorem 4.1, this means that, for almost all t in the range of a C ∞ -function f : RN → R, the set

y ∈ RN : f (x) > t is of class C ∞ in the sense of Deﬁnition 3.1. The following theorem extends and completes Sard’s theorem. Theorem 4.4 (H. Federer [5, Thm. 3.4.3, p. 316]). If m > ν ≥ 0 and k ≥ 1 are integers, A is an open subset of Rm , B ⊂ A, Y is a normed vector space, and f : A → Y is a map of class k,

dim Im Df (x) ≤ ν for x ∈ B,

then Hν+(m−ν)/k (f (B)) = 0.

5

Sets Locally Described by the Epigraph of a Function

After diﬀeomorphisms and level sets, the local epigraph description provides a third point of view and another way to characterize the smoothness of a set or a domain. For instance, Lipschitzian domains, which are characterized by the property that their boundary is locally the epigraph of a Lipschitzian function, play a central role in the theory of Sobolev spaces and partial diﬀerential equations. They can equivalently be characterized by the geometric uniform cone property. We shall see that sets that are locally the epigraph of C k, -functions are equivalent to sets that are locally of class C k, for k ≥ 1.

5. Sets Locally Described by the Epigraph of a Function

5.1

79

Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs and the Space H of Dominating Functions

Let eN in RN be a unit reference vector and introduce the following notation: H = {eN }⊥ , def

∀ζ ∈ RN ,

ζ = PH (ζ), def

def

ζN = eN · ζ

(5.1)

for the associated reference hyperplane H orthogonal to eN , the orthogonal projection PH onto H, and the normal component ζN of ζ. The vector ζ is equal to ζ + ζN eN and will often be denoted by (ζ , ζN ). In practice the vector eN is chosen as eN = (0, . . . , 0, 1), but the actual form of the unit vector eN is not important. A direction in RN is speciﬁed by a unit vector d ∈ RN , |d| = 1. Alternatively, it can be speciﬁed by AeN for some matrix A in the orthogonal subgroup of N × N matrices O(N) = {A : ∗A A = A ∗A = I} , def

(5.2)

where ∗A is the transposed matrix of A. Conversely, for any unit vector d in RN , |d| = 1, there exists8 A ∈ O(N) such that d = AeN . For all x ∈ RN and A ∈ O(N), it is easy to check that |Ax| = |x| = |A−1 x| for all x ∈ RN . The main elements of Deﬁnition 5.1 are illustrated in Figure 2.2 for N = 2. ζN graph of ax

set Ω Ax eN ∂Ω

ζ x

0 Vx ⊂ H = {eN }⊥ U(x)

Figure 2.2. Local epigraph representation (N = 2). Deﬁnition 5.1. Let eN be a unit vector in RN , H be the hyperplane {eN }⊥ , and Ω be a subset of RN with nonempty boundary ∂Ω. (i) Ω is said to be locally a C 0 epigraph if for each x ∈ ∂Ω there exist (a) an open neighborhood U(x) of x; (b) a matrix Ax ∈ O(N); the orthonormal basis with respect to d by adding N −1 unit vectors d1 , d2 , . . . , dN −1 and construct the matrix whose columns are the vectors A = [d1 d2 . . . dN −1 d] for which AeN = d. 8 Complete

80

Chapter 2. Classical Descriptions of Geometries and Their Properties (c) a bounded open neighborhood Vx of 0 in H such that U(x) ⊂ {y ∈ RN : PH (A−1 x (y − x)) ∈ Vx };

(5.3)

(d) and a function ax ∈ C 0 (Vx ) such that ax (0) = 0 and & ζ ∈ Vx , U(x) ∩ ∂Ω = U(x) ∩ x + Ax (ζ + ζN eN ) : ζN = ax (ζ ) & ζ ∈ Vx U(x) ∩ int Ω = U(x) ∩ x + Ax (ζ + ζN eN ) : . ζN > ax (ζ )

(5.4)

(5.5)

(ii) Ω is said to be a C 0 epigraph if it is locally a C 0 epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

(iii) Ω is said to be an equi-C 0 epigraph if it is a C 0 epigraph and the family of functions {ax : x ∈ ∂Ω} is uniformly bounded and equicontinuous: ∃c > 0 such that ∀x ∈ ∂Ω, ∀ξ ∈ V, ∀ε > 0, ∃δ > 0 such that ∀x ∈ ∂Ω, ∀y, ∀z ∈ V such that |z − y| < δ,

|ax (ξ )| ≤ c, (5.6) |ax (z) − ax (y)| < ε.

Remark 5.1. Conditions (5.3), (5.4), and (5.5) are equivalent to the following three conditions: & ζ ∈ Vx U(x) ∩ ∂Ω ⊂ x + Ax (ζ + ζN eN ) : , (5.7) ζN = ax (ζ ) & ζ ∈ Vx , (5.8) U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : ζN > ax (ζ ) & ζ ∈ V x U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : . (5.9) ζN < ax (ζ ) In particular, Ω is locally a C 0 epigraph if and only if Ω is locally a C 0 epigraph. As a result, conditions (5.3), (5.4), and (5.5) are also equivalent to conditions (5.3), (5.4), and the following condition: & ζ ∈ Vx (5.10) U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : ζN < ax (ζ ) in place of condition (5.5).

5. Sets Locally Described by the Epigraph of a Function

81

Remark 5.2. Note that from (5.3) PH (A−1 x (U(x) − x)) ⊂ Vx and that this yields the condition PH (U ) ⊂ V in part (iii) of the deﬁnition. Remark 5.3. It is always possible to redeﬁne U(x) − x or Vx to be open balls or open hypercubes centered in 0. For instance, in the case of balls, for Vx = BH (0, ρ) ⊂ Vx , choose the associated neighborhood U (x) = U(x) ∩ {y ∈ RN : PH (A−1 x (y − x)) ∈ BH (0, ρ)}. Similarly, for U (x) = B(x, r) ⊂ U(x) choose the associated neighborhood Vx = −1 PH (A−1 x (B(x, r)−x) = BH (0, r) ⊂ Vx , since, from (5.3), Vx ⊃ PH (Ax (U(x)−x)) ⊃ −1 PH (Ax B(x, r) − x)) = BH (0, r). In both cases properties (5.3) to (5.5) are veriﬁed for the new neighborhoods. Remark 5.4. Sets Ω ⊂ RN that are locally C 0 epigraphs have a boundary ∂Ω with zero N -dimensional Lebesgue measure. The three cases considered in Deﬁnition 5.1 diﬀer only when the boundary ∂Ω is unbounded. But we ﬁrst introduce some notation. Notation 5.1. (i) Under condition (5.3), a point y in U(x) ⊂ RN is represented by (ζ , ζN ) ∈ Vx ×R, where ζ = PH (A−1 x (y − x)) ∈ Vx ⊂ H, def

ζN = A−1 x (y − x) · eN (x). def

We can also identify H with RN −1 = {(z , 0) ∈ RN : z ∈ RN −1 }. (ii) The graph, epigraph, and hypograph of ax : Vx → R will be denoted as follows: def

A0x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ζN = ax (ζ )} , def

A+ x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ∀ζN > ax (ζ )} , def

A− x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ∀ζN < ax (ζ )} . def

The equi-C 0 epigraph case is important since we shall see in Theorem 5.2 that the three cases of Deﬁnition 5.1 are equivalent when the boundary ∂Ω is compact. We now show that, for an arbitrary boundary, the equicontinuity of the graph functions {ax : x ∈ ∂Ω} can be expressed in terms of the continuity in 0 of a dominating function h. For this purpose, deﬁne the space of dominating functions as follows: def

H = {h : [0, ∞[ → [0, ∞[ : h(0) = 0 and h is continuous in 0} .

(5.11)

By continuity in 0, h is locally bounded in 0: ∀c > 0, ∃ρ > 0 such that ∀θ ∈ [0, ρ],

|h(θ)| ≤ c.

The only part of the function h that will really matter is in a bounded neighborhood of zero. Its extension outside to [0, ∞[ can be relatively arbitrary.

82

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 5.1. Let Ω be a C 0 -epigraph. The following conditions are equivalent: (i) Ω is an equi-C 0 -epigraph. (ii) There exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ζ , ξ ∈ BH (0, ρ) such that |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |). (5.12)

(iii) There exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ξ ∈ BH (0, ρ),

ax (ξ ) ≤ h(|ξ |).

(5.13)

Inequalities (5.12) and (5.13) for h ∈ H are also veriﬁed with lim sup h that also belongs to H. Proof. Since the ax ’s are uniformly continuous, inequalities (5.12) and (5.13) with h ∈ H imply the same inequalities with lim sup h that also belongs to H by taking the pointwise lim sup of both sides of the inequalities. (ii) ⇒ (i). For h ∈ H, there exists ρ1 > 0 such that 0 < ρ1 ≤ ρ/2 and |h(θ)| ≤ 1 for all θ ∈ [0, ρ1 ]. By construction BH (0, ρ1 ) ⊂ V and from inequality (5.12) ∀ζ , ξ ∈ BH (0, ρ1 ), |ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |), ∀ ξ ∈ BH (0, ρ1 ), |ax (ξ )| ≤ h(|ξ |) ≤ 1 and the family {ax : BH (0, ρ1 ) → R : x ∈ ∂Ω} is equicontinuous and uniformly bounded. (i) ⇒ (ii). By Deﬁnition 5.1 (iii), there exist a neighborhood V of 0 in H and a neighborhood U of 0 in RN such that for all x ∈ ∂Ω Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U,

and the family {ax ∈ C 0 (V ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous. Choose r > 0 such that B(0, 3r) ⊂ U . From property (5.3) in Deﬁnition 5.1

⇒ BH (0, 3r) ⊂ V. U(x) ⊂ y ∈ RN : PH A−1 x (y − x) ∈ V Deﬁne the modulus of continuity ∀θ ∈ [0, 3r],

h(θ) = sup sup

sup

ζ ∈H x∈∂Ω ξ ∈V |ζ |=1 ξ +θζ ∈V

|ax (θζ + ξ ) − ax (ξ )|.

(5.14)

The function h(θ), as the sup of the family θ → |ax (θζ + ξ ) − ax (ξ )| of continuous functions with respect to (ζ , ξ ), is lower semicontinuous with respect to θ and bounded in [0, 3r] since the ax ’s are uniformly bounded. By construction h(0) = 0 and by equicontinuity of the ax , for all ε > 0, there exists δ > 0 such that for all θ ∈ [0, 3r], |θ| < δ, ∀x ∈ ∂Ω, ∀ζ ∈ H, |ζ | = 1,

|ax (θζ + ξ ) − ax (ξ )| < ε

⇒ |h(θ)| < ε,

5. Sets Locally Described by the Epigraph of a Function

83

where h is continuous in 0. In view of Remark 5.3, choose ρ = 3r and the new neighborhoods V = BH (0, ρ) and U = {ξ ∈ B(0, ρ) : PH ξ ∈ V } = B(0, ρ) and extend h by the constant h(ρ) to [ρ, ∞[ . Therefore h ∈ H and, by deﬁnition of h, for all x ∈ ∂Ω and for all ξ and ζ in BH (0, ρ) such that |ξ − ζ | < ρ ξ −ζ |ax (ξ ) − ax (ζ )| = ax ζ + |ξ − ζ | − a (ζ ) x ≤ h(|ξ − ζ |) |ξ − ζ | ⇒ ∀x ∈ ∂Ω, ∀ξ , ζ ∈ BH (0, ρ), |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |).

(ii) ⇒ (iii) Choose ζ = 0 in inequality (5.12): ∀ξ ∈ BH (0, ρ) such that |ξ | < ρ,

ax (ξ ) ≤ |ax (ξ )| ≤ h(|ξ |).

(iii) ⇒ (ii). In view of Remark 5.3, choose the new neighborhoods V = BH (0, ρ) and U = U ∩ {y ∈ RN : PH (y) ∈ BH (0, ρ)} and the restriction of the functions {ax : V → R : ∀x ∈ ∂Ω} to V . Choose r > 0 such that B(0, 4r) ⊂ U (hence 4r ≤ ρ) and consider the region O = {ζ + ζN eN : ζ ∈ BH (0, 2r) and h(|ζ |) < ζN < r} . def

For any x ∈ ∂Ω and ζ ∈ O, consider the point yζ = x + Ax (ζ + ζN eN ). It is readily ' 2 seen that |yζ − x| = |ζ| < (2r) + r 2 < 3r and that yζ ∈ U(x). In addition y ζ ∈ A+ x . From (5.13) ax (ζ ) ≤ h(|ζ |) for all ζ ∈ V = BH (0, ρ) and hence ax (ζ ) ≤ + h(|ζ |) < ζN for all ζ ∈ O (2r < ρ). As a result yζ ∈ U(x) ∩ Ax = U(x) ∩ int Ω ⊂ int Ω and ∀x ∈ ∂Ω,

x + Ax O ⊂ int Ω.

(5.15)

Given ζ and ξ in BH (0, 2r), consider the two points yζ = x + Ax (ζ + ax (ζ )eN ) ∈ A0x = U(x) ∩ ∂Ω, def

yξ = x + Ax (ξ + ax (ξ )eN ) ∈ A0x = U(x) ∩ ∂Ω. def

Since yξ and yζ belong to U(x) ∩ ∂Ω, yξ does not belong to yζ + Ax O ⊂ int Ω or / Ax O. Hence equivalently yξ − yζ ∈ ξ − ζ + (ax (ξ ) − ax (ζ ))eN = A−1 / O. x (yξ − yζ ) ∈ For |ξ − ζ | < 2r, this is equivalent to saying that one of the following inequalities is veriﬁed: h(|ξ − ζ |) ≥ Ax eN · (yξ − yζ ) = ax (ξ ) − ax (ζ ) or r ≤ Ax eN · (yξ − yζ ) = ax (ξ ) − ax (ζ ) by deﬁnition of ax and the fact that ξ − ζ ∈ BH (0, 2r) ⊂ BH (0, ρ) = V . But the second inequality contradicts the continuity of ax . Hence the ﬁrst inequality holds and, by interchanging the roles of ξ and ζ , we get inequality (5.12) with a new ρ equal to 2r.

84

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 5.2. When the boundary ∂Ω is compact, the three properties of Deﬁnition 5.1 are equivalent: locally C 0 epigraph

⇐⇒

C 0 epigraph

⇐⇒

equi-C 0 epigraph.

Under such conditions, there exist neighborhoods V of 0 in H and U of 0 in RN such that PH U ⊂ V and the neighborhoods Vx of 0 in H and the neighborhoods U(x) of x in RN can be chosen of the form Vx = V

and

∃Ax ∈ O(N) such that U(x) = x + Ax U.

(5.16)

Moreover, the family of functions {ax : V → R : x ∈ ∂Ω} can be chosen uniformly bounded and equicontinuous with respect to x ∈ ∂Ω. In addition, there exist ρ > 0 such that BH (0, ρ) ⊂ V and a function h ∈ H such that ∀x ∈ ∂Ω, ∀ξ , ζ ∈ BH (0, ρ), |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |). (5.17)

Proof. It is suﬃcient to show that a locally C 0 epigraph is an equi-C 0 epigraph. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx ) ⊂ U(x). Since ∂Ω is compact, m there exist a ﬁnite sequence {xi }m i=1 of points of ∂Ω and a ﬁnite subcover {Bi }i=1 , Bi = B(xi , Ri ), Ri = rxi . (i) We ﬁrst claim that ∃R > 0, ∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi . We proceed by contradiction. If this is not true, then for each n ≥ 1, ∃xn ∈ ∂Ω, ∀i, 1 ≤ i ≤ m, B xn , 1/n ⊂ Bi , / Bi . ⇒ ∀i, 1 ≤ i ≤ m, ∃yin ∈ B xn , 1/n such that yin ∈ Since ∂Ω is compact, there exist a subsequence of {xn } and x ∈ ∂Ω such that xnk → x as k goes to inﬁnity. Hence for each i, 1 ≤ i ≤ m, |yink − x| ≤ |yink − xnk | + |xnk − x| ≤

1 + |xnk − x| → 0 nk

and yink → x as k goes to inﬁnity. For each i, the set Bi is closed. Thus x ∈ Bi and m * / Bi . yink ∈ Bi → x ∈ Bi ⇒ ∃ x ∈ ∂Ω such that x ∈ i=1

But this contradicts the fact that {Bi }m i=1 is an open cover of ∂Ω. This property is obviously also satisﬁed for all neighborhoods N of 0 such that ∅ = N ⊂ B(0, R). (ii) Construction of the neighborhoods Vx and U (x) and the maps Ax and ax . From part (i) for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi = B(xi , Ri ) ⊂ U(xi ) ⇒

PH (A−1 xi (B(x, R)

− xi )) ⊂ PH (A−1 xi (U(xi ) − xi )) ⊂ Vxi

⇒ PH (A−1 xi (B(x − xi , R))) ⊂ Vxi .

5. Sets Locally Described by the Epigraph of a Function

85

Since x ∈ ∂Ω ∩ U(xi ), it is of the form x = xi + Axi (ξi + axi (ξi )eN ),

ξi = PH (A−1 xi (x − xi ))

−1 −1 ⇒ PH (A−1 xi (B(x − xi , R))) = PH (Axi (x − xi )) + PH (Axi (B(0, R)))

= ξi + BH (0, R)

⇒ ξi + BH (0, R) ⊂ Vxi .

(5.18)

Choose the neighborhoods def

def

U = B(0, R) and V = BH (0, R). Associate with x ∈ ∂Ω the following new neighborhoods and functions: Vx = V, def

Ax = Axi , def

U(x) = x + Ax U, def

ζ → ax (ζ ) = axi (ζ + ξi ) − axi (ξi ) : V → R . def

It is readily seen from identity (5.18) that ax is well-deﬁned, ax (0) = 0, and ax ∈ C 0 (V ). Since the family {axi : 1 ≤ i ≤ m} is ﬁnite, ax C 0 (V ) ≤ 2 axi C 0 (Vx

i

)

≤ 2 max axi C 0 (Vx ) < ∞ 1≤i≤m

i

and the family {ax ; x ∈ ∂Ω} is uniformly bounded. Since each axi is uniformly continuous, for all ε > 0 there exists δi > 0 such that ∀ζ1 , ζ2 ∈ Vxi such that |ζ1 − ζ2 | < δi ,

|axi (ζ1 ) − axi (ζ2 )| < ε.

And since the family {axi : 1 ≤ i ≤ m} is ﬁnite pick δ = min1≤i≤m δi to get the following: for all ε > 0, there exists δ > 0 such that for all i ∈ {1, . . . , m}, ∀ζ1 , ζ2 ∈ Vxi such that |ζ1 − ζ2 | < δ,

|axi (ζ1 ) − axi (ζ2 )| < ε.

Therefore, for all ε > 0, there exists δ > 0 such that ∀x ∈ ∂Ω, ∀ζ1 , ζ2 ∈ V such that |ζ1 − ζ2 | < δ,

|ax (ζ1 ) − ax (ζ2 )| < ε

(5.19)

and the family {ax ; x ∈ ∂Ω} is uniformly equicontinuous. By construction, properties (a) and (b) of Deﬁnition 5.1 are veriﬁed for U (x) and Ax . By construction, properties (a) and (b) are veriﬁed for U (x) and Ax . Always by construction in x ∈ ∂Ω U (x) = x + Ax B(0, R) = x + B(0, R) = B(x, R), PH ((Ax )−1 (U (x) − x)) = PH (B(0, R)) = BH (0, R) = V = Vx

⇒ U (x) ⊂ y ∈ RN : PH ((Ax )−1 (y − x)) ∈ Vx

86

Chapter 2. Classical Descriptions of Geometries and Their Properties

and condition (c) of Deﬁnition 5.1 is veriﬁed. For condition (d) of Deﬁnition 5.1 in x ∈ ∂Ω, let xi be the point associated with x such that B(x, R) ⊂ U(xi ). Since U (x) = B(x, R), U (x) ∩ ∂Ω ⊂ U(xi ) ∩ ∂Ω = {xi + Axi (ζ + axi (ζ )eN ) : ∀ζ ∈ Vxi } ∩ ∂Ω and for all y ∈ U (x) ∩ ∂Ω there exists ξ ∈ Vxi such that y = xi + Axi (ξ + axi (ξ )eN ) = x + Axi (ξ + axi (ξ )eN ) − (x − xi ). But x also has a representation with respect to xi x = xi + Axi (ξi + axi (ξi )eN ),

⇒ y = x + Axi (ξ −

ξi

ξi ∈ Vxi

+ (axi (ξ ) − axi (ξi ))eN )

that can be rewritten as y = x + Axi (ξ − ξi + (axi ((ξ − ξi ) + ξi ) − axi (ξi ))eN ) ⇒ ζ = PH (A−1 xi (y − x)) = ξ − ξi ∈ V def

−1 (U (x) − x)) ⊂ Vx = V . Finally, since by (c) PH (A−1 xi (U (x) − x)) = PH ((Ax ) there exists ζ ∈ V such that

y = x + Axi (ζ + (axi (ζ + ξi ) − axi (ξi ))eN ) = x + Axi (ζ + ax (ζ )eN ) and property (5.4) is veriﬁed. The constructions and the proof of property (5.5) are similar: U (x) ∩ int Ω ⊂ U(xi ) ∩ int Ω = U(xi ) ∩ xi + Axi (ζ + ζN eN ) : ∀ζ ∈ Vxi , ∀ζN > axi (ζ ) and for all y ∈ U (x)∩int Ω there exists (ξ , ξN ) ∈ Vxi ×R such that ξN > axi (ξ ) and y = xi + Axi (ξ + ξN eN ) = x + Axi (ξ + ξN eN ) − (x − xi ) = x + Axi (ξ − ξi + (ξN − axi (ξi ))eN ) ⇒ ζ = PH (A−1 xi (y − x)) = ξ − ξi ∈ V, def

ζN = (A−1 xi (y − x)) · eN = ξN − axi (ξi ) def

−1 (U (x) − x)) ⊂ Vx = V . Finally, since by (c) PH (A−1 xi (U (x) − x)) = PH ((Ax ) there exists (ζ , ζN ) ∈ V × R such that

y = x + Axi (ζ + ax (ζ )eN ), ζN = ξN − axi (ξi ) = ξN − axi (ξ ) + axi (ξ ) − axi (ξi ) = ξN − axi (ξ ) + axi (ζ + ξi ) − axi (ξi ) = ξN − axi (ξ ) + ax (ζ ) > ax (ζ )

since ξN > axi (ξ ) and property (5.5) in (d) is veriﬁed. (iii) The other properties and (5.17) follow from Theorem 5.1.

5. Sets Locally Described by the Epigraph of a Function

5.2

87

Local C k, -Epigraphs and H¨ olderian/Lipschitzian Sets

We extend the three deﬁnitions of C 0 epigraphs to C k, epigraphs. C 0,1 is the important family of Lipschitzian sets and C 0, , 0 < < 1, of H¨ olderian sets. Deﬁnition 5.2. Let Ω be a subset of RN such that ∂Ω = ∅. Let k ≥ 0, 0 ≤ ≤ 1. (i) Ω is said to be locally a C k, epigraph if for each x ∈ ∂Ω there exist (a) an open neighborhood U(x) of x; (b) a matrix Ax ∈ O(N); (c) a bounded open neighborhood Vx of 0 in H such that U(x) ⊂ {y ∈ RN : PH (A−1 x (y − x)) ∈ Vx };

(5.20)

(d) and a function ax ∈ C k, (Vx ) such that ax (0) = 0 and U(x) ∩ ∂Ω = U(x) ∩

x + Ax (ζ + ζN eN ) :

U(x) ∩ int Ω = U(x) ∩

&

ζ ∈ Vx

,

ζN = ax (ζ )

x + Ax (ζ + ζN eN ) :

(5.21)

&

ζ ∈ Vx ζN > ax (ζ )

,

(5.22)

where ζ = (ζ1 , . . . , ζN −1 ) ∈ RN −1 . We shall say that Ω is locally Lipschitzian in the C 0,1 case and locally H¨ olderian of index in the C 0, , 0 < < 1, case. (ii) Ω is said to be a C k, epigraph if it is locally a C k, epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x)−x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

def

Vx = V

def

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

olderian of index We shall say that Ω is Lipschitzian in the C 0,1 case and H¨ in the C 0, , 0 < < 1, case. (iii) Given (k, ) = (0, 0), Ω is said to be an equi-C k. epigraph if it is a C k, epigraph and ∃c > 0, ∀x ∈ ∂Ω,

ax C k, (V ) ≤ c,

(5.23)

where f C k, (V ) is the norm on the space C k, (V ) as deﬁned in (2.21): def

f C k, (V ) = f C k (V ) + max

sup

0≤|α|≤k x,y∈V x=y

def

C k,0 (V ) = C k (V )

and

|∂ α f (y) − ∂ α f (x)| , |y − x| def

if 0 < ≤ 1,

f C k,0 (V ) = f C k (V ) .

88

Chapter 2. Classical Descriptions of Geometries and Their Properties Ω is said to be an equi-C 0,0 epigraph if it is an equi-C 0 epigraph in the sense of Deﬁnition 5.1 (iii). We shall say that Ω is equi-Lipschitzian in the C 0,1 case and equi-H¨ olderian of index in the C 0, , 0 < < 1, case.

Remark 5.5. In Deﬁnition 5.1 (iii) for (k, ) = (0, 0) it follows from condition (5.23) that the functions {ax : x ∈ ∂Ω} are uniformly equicontinuous on V . In the case (k, ) = (0, 0) this is no longer true. So we have to include it by using Deﬁnition 5.1 (iii). Theorem 5.2 readily extends to the C k, case. Theorem 5.3. When the boundary ∂Ω is compact, the three types of sets of Deﬁnition 5.2 are equi-C k, epigraphs in the sense of Deﬁnitions 5.2 (iii) and 5.1 (iii). Proof. The proof is the same as the one of Theorem 5.2 for the C 0 case. The equiC k, property is obtained by choosing in the proof of Theorem 5.2 the common constant as c = max1≤i≤m cxi , where cxi is the constant associated with axi on Vxi , axi C k (Vxi ) ≤ cxi . Sets Ω that are locally a C k, epigraph are Lebesgue measurable in RN , their “volume” is locally ﬁnite, and their boundary has zero “volume.” Theorem 5.4. Let k ≥ 0, let 0 ≤ ≤ 1, and let Ω be locally a C k, epigraph. (i) The complement Ω is also locally a C k, epigraph, and int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω, int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω. Moreover, for all x ∈ ∂Ω m U(x) ∩ ∂Ω = 0

⇒ m ∂Ω = 0,

(5.24)

(5.25)

where m is the N -dimensional Lebesgue measure. (ii) If Ω = ∅ is open, then Ω = int Ω. Proof. (i) The fact that the complement of Ω is locally a C k, epigraph follows from Remark 5.1. To complete the proof, it is suﬃcient to establish the ﬁrst two properties of the ﬁrst line of (5.24). The second line is the ﬁrst line applied to Ω. Since U(x) is a neighborhood of x, there exists r > 0 such that B(x, 3r) ⊂ U(x). Since ax ∈ C 0 (Vx ), ax (0) = 0, and Vx is a neighborhood of 0, there exists ρ, 0 < ρ < r, such that BH (0, ρ) ⊂ Vx and for all ζ , |ζ | < ρ, |ax (ζ )| ≤ r. Given ζ ∈ BH (0, ρ), the point yζ = x + A(ζ + ax (ζ )eN ) ∈ ∂Ω ∩ U(x). Given a real α, |α| < r, consider the point zα = x + A(ζ + (ax (ζ ) + α)eN ). By construction |zα − x| = |A(ζ + (ax (ζ ) + α)eN )| = |ζ + (ax (ζ ) + α)eN | < 3r and zα ∈ U(x). Hence the segment (yζ , yζ + rAx eN ) ⊂ A+ ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅. Also the segment (yζ , yζ − rAx eN ) ⊂ A− ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅.

5. Sets Locally Described by the Epigraph of a Function

89

Finally, associate with yζ ∈ ∂Ω and the sequence αn = r/n the points zαn ∈ int Ω. As n goes to inﬁnity, zn → yζ and ∂Ω ⊂ int Ω and hence Ω = int Ω. From the following Lemma 5.1 we get the third identity in the ﬁrst line of (5.24). (ii) From part (i) ∂(int Ω) = ∂Ω and by Lemma 5.1 (ii) int Ω = Ω. Since Ω is open, Ω = int Ω = Ω and int Ω = Ω = Ω. Lemma 5.1. ⇐⇒

(i) ∂(int Ω) = ∂Ω (ii) ∂(int Ω) = ∂Ω

int Ω = Ω.

⇐⇒

int Ω = Ω.

Proof. It is suﬃcient to prove (i). (⇒) By deﬁnition of the closure, Ω = int Ω ∪ ∂Ω = int Ω ∪ ∂(int Ω) = int Ω. Conversely (⇐) By deﬁnition ∂Ω = Ω ∩ Ω, Ω ∩ Ω = int Ω ∩ Ω = int Ω ∩ (int Ω) = int Ω ∩ (int Ω) = ∂(int Ω).

5.3

Local C k, -Epigraphs and Sets of Class C k,

Sets or domains which are locally the epigraph of a C k, -function, k ≥ 0, 0 ≤ ≤ 1 (resp., locally Lipschitzian), are sets of class C k, (resp., C 0,1 ). However, we shall see from Examples 5.1 and 5.2 that a domain of class C 0,1 is generally not locally the epigraph of a Lipschitzian function. Theorem 5.5. Let , 0 ≤ ≤ 1, be a real number. (i) If Ω is locally a C k, epigraph for k ≥ 0, then it is locally of class C k, . (ii) If Ω is locally of class C k, for k ≥ 1, then it is locally a C k, epigraph. Proof. (i) For each point x ∈ ∂Ω, let U(x), Vx , and ax be the associated neighborhoods and the C k, -function. Deﬁne the C k, -mappings for ξ = (ξ , ξN ) ∈ Vx × R hx (ξ) = x + Ax (ξ + [ξN + ax (ξ )] eN ), def

−1 −1 gx (y) = (PH A−1 x (y − x), eN · Ax (y − x) − ax (PH Ax (y − x))). def

It is easy to verify that hx is the inverse of gx , hx (gx (y)) = y in U(x) and gx (hx (ξ)) = ξ in gx (U(x)), −1 and that since for y ∈ U(x) ∩ ∂Ω, eN · A−1 x (y − x) = ax (PH Ax (y − x)

gx (U(x) ∩ ∂Ω) ∈ H = {(ξ , ξN ) ∈ RN : ξN = 0}, −1 and that since for y ∈ U(x) ∩ int Ω, eN · A−1 x (y − x) > ax (PH Ax (y − x)

gx (U(x) ∩ int Ω) ⊂ {(ξ , ξN ) ∈ RN : ξN > 0}. The set Ω is of class C k, .

(5.26) (5.27)

90

Chapter 2. Classical Descriptions of Geometries and Their Properties

(ii) In the discussion preceding Theorem 4.1, we showed that a set of class C k, is locally the level set of a C k, -function. From the deﬁnition of a set of class C k, , k ≥ 1 and 0 ≤ ≤ 1, the set Ω can be locally described by the level sets of the C k, -function def

fx (y) = gx (y) · eN , since by deﬁnition int Ω ∩ U (x) = {y ∈ U (x) : fx (y) > 0}, ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0}. The boundary ∂Ω is the zero level set of fx and the gradient ∇fx (y) = ∗Dgx (y)eN = 0 is normal to that level set. The exterior normal to Ω is given by n(y) = −

∗ ∗ Dgx (y)eN (Dhx (gx (y)))−1 eN ∇fx (y) =− ∗ =− ∗ . |∇fx (y)| | Dgx (y)eN | | (Dhx (gx (y)))−1 eN |

Since fx is C 1 and ∇fx (x) = 0, choose a smaller neighborhood of x, still denoted by U (x), such that ∀y ∈ U (x),

∇fx (y) · ∇fx (x) > 0.

To construct the graph function around the point x ∈ Γ = ∂Ω, choose Ax ∈ O(N) such that Ax eN = −n(x) and H = {eN }⊥ . Consider the C k, -function, k ≥ 1, λ(ζ , ζN ) = fx (x + Ax (ζ + ζN eN )) . def

By construction, λ(0, 0) = 0 and 0 = ∇λ(0, 0) · (0, ξN ) = ∇fx (x) · (0, ξN ) = ξN |∇fx (x)|

⇒ ξN = 0.

Therefore, by the implicit function theorem, there exist a neighborhood Vx ⊂ B0 of (0, 0) and a function ax ∈ C k. (Vx ) such that λ(0, 0) = 0 and ∀ζ ∈ Vx ,

λ(ζ , ax (ζ )) = fx (x + Ax (ζ + ax (ζ )eN ) = 0.

By construction, x + Ax (ζ + ax (ζ )eN ) ∈ Γ ∩ U (x). Choose U(x) = U (x) ∩ {x + Ax (ζ + ζN eN ) : ζ ∈ Vx and ζN ∈ R}. def

By construction, Vx and U(x) satisfy condition (5.3). Moreover, from (3.4)–(3.5) int Ω ∩ U (x) = hx (B+ ) and ∂Ω ∩ U (x) = hx (B0 ) int Ω ∩ U (x) = {y ∈ U (x) : f (y) > 0}, x ⇒ ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0},

5. Sets Locally Described by the Epigraph of a Function

91

and since U(x) ⊂ U (x) we get conditions (5.4) and (5.5): int Ω ∩ U(x) = {y ∈ U(x) : fx (y) > 0} = A+ ∩ U(x), ∂Ω ∩ U(x) = {y ∈ U(x) : fx (y) = 0} = A0 ∩ U(x). Recall that, by construction of U(x), for all (ζ , ζN ) ∈ A−1 x (U(x) − {x}), ζN > ax (ζ ) ⇐⇒ λ(ζ , ζN ) = fx (x + Ax (ζ + ζN eN )) > 0, since ∇fx (x) ∂λ (ζ , ζN ) = ∇fx (x + Ax (ζ + ζN en )) · Ax eN = ∇fx (x + Ax ζ) · >0 ∂ζN |∇fx (x)| −1 in the neighborhood A−1 x (U(x) − {x}) ⊂ Ax (U (x) − {x}) of (0, 0).

Example 5.1 (R. Adams, N. Aronszajn, and K. T. Smith [1]). Consider the open convex (Lipschitzian) set Ω0 = {ρ eiθ : 0 < ρ < 1, 0 < θ < π/2} and its image Ω = T (Ω0 ) by the C 0,1 -diﬀeomorphism (see Figure 2.3) T (ρ eiθ ) = ρ ei(θ−log ρ) ,

T −1 (ρ eiθ ) = ρ ei(θ+log ρ) .

It is readily seen that, as ρ goes to zero, the image of the two pieces of the boundary of Ω0 corresponding to θ = 0 and θ = π/2 begin to spiral around the origin. As a result Ω is not locally the epigraph of a function at the origin.

1

1

0

0 0 (a) quarter disk Ω0

1

0

1

(b) T (Ω0 ) spirals around 0

Figure 2.3. Domain Ω0 and its image T (Ω0 ) spiraling around the origin. Example 5.2. This second example can be found in F. Murat and J. Simon [1], where it is attributed to M. Zerner. Consider the Lipschitzian function λ deﬁned on [0, 1] as follows: λ(0) = 0, and on each interval [1/3n+1 , 1/3n ] 2 1 1 ≤ s ≤ n+1 , 2 s − , n+1 n+1 3 3 3 def λ(s) = 1 2 1 −2 s − , ≤ s ≤ n, n n+1 3 3 3

92

Chapter 2. Classical Descriptions of Geometries and Their Properties

where n ranges over all integers n ≥ 0. Associate with λ and a real δ > 0 the set def

Ω = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, |x2 − λ(x1 )| < δ x1 } (see Figure 2.4). The set Ω is the image of the triangle def

Ω0 = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, |x2 | < δ x1 } through the C 0,1 -homeomorphism def

T (x1 , x2 ) = (x1 , x2 + λ(x1 )),

T −1 (y1 , y2 ) = (y1 , y2 − λ(y1 )).

Since the triangle Ω0 is Lipschitzian, its image is a set of class C 0,1 . But Ω is not locally Lipschitzian in (0, 0) since Ω zigzags like a lightning bolt as it gets closer to the origin. Thus, however small the neighborhood around (0, 0), a direction eN (0, 0) cannot be found to make the domain locally the epigraph of a function.

1

1

0

0 0

1 (a) triangle Ω0

0

1 (b) T (Ω0 ) zigzags to 0

Figure 2.4. Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin.

5.4

Locally Lipschitzian Sets: Some Examples and Properties

The important subfamily of sets that are locally a Lipschitzian epigraph (that is, locally a C 0,1 epigraph) enjoys most of the properties of smooth domains. Convex sets and domains that are locally the epigraph of a C 1 - or smoother function are locally Lipschitzian. In a bounded path-connected Lipschitzian set, the geodesic distance between any two points is uniformly bounded. Their boundary has zero volume and locally ﬁnite boundary measure (cf. Theorem 5.7 below). Sobolev spaces deﬁned on Ω have linear extension to RN . 5.4.1

Examples and Continuous Linear Extensions

According to this deﬁnition the whole space RN and the closed unit ball with its center or a small crack removed are not locally Lipschitzian since in the ﬁrst case

5. Sets Locally Described by the Epigraph of a Function

93

the boundary is empty and in the second case the conditions cannot be satisﬁed at the center or along the crack, which is a part of the boundary. Similarly the set ∞ *

def

Ω =

Ωn ,

def

Ωn =

n=1

1 1 y ∈ RN : y − n < n+2 2 2

is not locally Lipschitzian since the conditions of Deﬁnition 5.2 (i) are not satisﬁed in 0 ∈ ∂Ω. However, the set def

Ω =

∞ * n=1

Ωn ,

def

Ωn =

y ∈ R : |y − n| < N

1

2n+2

is locally Lipschitzian, but not Lipschitzian or equi-Lipschitzian in the sense of Deﬁnition 5.2. One of the important properties of a bounded (open) Lipschitzian domain Ω is the existence of a continuous linear extension of functions of the Sobolev space H k (Ω) from Ω to functions in H k (RN ) deﬁned on RN , that is, (5.28) E : H k (Ω) → H k (RN ) and ∀ϕ ∈ H k (Ω), (Eϕ) Ω = ϕ (cf. the Calder´ on extension theorem in R. A. Adams [1, p. 83, Thm. 4.32, p. 91] and ˇas [1, Thm. 3.10, p. 80]). This property is also important in the existence of J. Nec optimal domains when it is uniform for a given family. For instance, this will occur for the family of domains satisfying the uniform cone property of section 6.4.1. 5.4.2

Convex Sets

Theorem 5.6. Any convex subset Ω of RN such that Ω = RN and int Ω = ∅ is locally Lipschitzian, and for each x ∈ ∂Ω the neighborhood Vx and the function ax can be chosen convex. Proof. (i) Construction of the function ax . Start with the unit vector eN and the hyperplane H orthogonal to eN . Pick any point x0 in the interior of Ω and choose ε > 0 such that B(x0 , 2ε) ⊂ int Ω. Since Ω = RN , ∂Ω = ∅ and we can associate with each x ∈ ∂Ω the direction d(x) = (x0 − x)/|x0 − x| and the matrix Ax ∈ O(N) such that d(x) = Ax eN . Note that the hyperplane Ax H is orthogonal to the vector / Ω and the minimum d(x) = Ax eN . By convexity, the point x− 0 = x − (x0 − x) ∈ distance δ from x− to Ω is such that 0 < δ < |x − x |. As a result there exists ρ, 0 0 0 < ρ < ε, such that for all ζ ∈ H, |ζ | < ρ, the line Lζ = {x + Ax (ζ + ζN eN ) : |ζN | ≤ |x − x0 |} def

from the point x0 + Ax ζ to x− 0 + Ax ζ in the direction Ax eN contains a point of Ω and a point of its complement. Hence there exists ζˆN , |ζˆN | ≤ |x − x0 |, such that yˆ = x + Ax (ζ + ζˆN eN ) ∈ ∂Ω ∩ Lζ minimizes ζN = Ax eN · (y − x) over all y ∈ Ω ∩ Lζ . If yˆ1 and yˆ2 are two distinct minimizing points such that ζˆ1N = ζˆ2N ,

94

Chapter 2. Classical Descriptions of Geometries and Their Properties

then yˆ1 = x + Ax (ζ + ζˆ1N eN ) = x + Ax (ζ + ζˆ2N eN ) = yˆ2 . As a result, for all ζ ∈ H, |ζ | < ρ, the function ax (ζ ) =

def

inf

Ax eN · (y − x)

y∈Ω∩Lζ

is well-deﬁned and ﬁnite and there exists a unique ζˆN , |ζˆN | < |x − x0 |, such that ax (ζ ) = ζˆN and yˆ = x + Ax (ζ + ζˆN eN ) is the unique minimizer. Choose the neighborhoods def

U(x) =

y∈R : N

&

|PH A−1 x (y − x)| < ρ |A−1 x (y − x) · eN | < |x − x0 |

,

def

Vx = {z ∈ H : |z| < ρ} .

Then condition (5.20) of Deﬁnition 5.2 is satisﬁed and it is easy to verify that the function ax satisﬁes conditions (5.21) and (5.22). (ii) Convexity of the function ax . By construction the neighborhoods Vx and U(x) are convex. The set U(x) ∩ Ω is convex. Then recall from (5.21)–(5.22) that U(x) ∩ {x + Ax (ζ + ζN eN ) : ζ ∈ Vx , ζN ≥ ax (ζ )} = U(x) ∩ Ω. Thus the set on the left-hand side is convex. In particular, ∀y 1 , y 2 ∈ Ω ∩ U(x), ∀α ∈ [0, 1],

αy 1 + (1 − α)y 2 ∈ Ω ∩ U(x).

In particular, for any two y 1 and y 2 in ∂Ω ∩ U(x),

j y j = x + Ax (ζ j + ζN eN ),

j ζN = ax (ζ j ), j = 1, 2,

and y α = αy 1 + (1 − α)y 2 = x + Ax [αζ 1 + (1 − α)ζ 2 ] ∈ Ω ∩ U(x) 1 2 + (1 − α)ζN ≥ ax αζ 1 + (1 − α)ζ 2 ⇒ αζN ⇒ αax (ζ 1 ) + (1 − α)ax (ζ 2 ) ≥ ax αζ 1 + (1 − α)ζ 2 and ax is convex. (iii) The function ax is Lipschitzian. It is suﬃcient to show that ax is bounded in Vx . Then we conclude from I. Ekeland and R. Temam [1, Lem. 3.1, p. 11] that the function is continuous and convex and hence Lipschitz continuous in Vx . By construction the minimizing point yˆ belongs to U(x), which is bounded. 5.4.3

Boundary Measure and Integral for Lipschitzian Sets

The boundary measure on ∂Ω, which was deﬁned in section 3.2 from the local C 1 -diﬀeomorphism, can also be deﬁned from the local graph representation of the boundary and extended to the Lipschitzian case. Assuming that ∂Ω is compact, there is a ﬁnite subcover of open neighborhoods U(x) for ∂Ω that can be represented by a ﬁnite family of Lipschitzian graphs. Let

5. Sets Locally Described by the Epigraph of a Function

95

def

{Uj }m j=1 , Uj = U(xj ), be a ﬁnite open cover of ∂Ω corresponding to some ﬁnite m sequence {xj }m j=1 of points of ∂Ω. Denote by eN , H, {Vj }j=1 , Vj = Vxj , and {aj }, aj = axj , the associated elements of Deﬁnition 5.2. Introduce the notation Γ = ∂Ω,

Γj = Γ ∩ Uj , 1 ≤ j ≤ m,

(5.29)

and the map hj = hxj and its inverse gj = gxj as deﬁned in (5.26)–(5.27): hx (ξ) = x + Ax (ξ + [ξN + ax (ξ )] eN ), def

−1 −1 gx (y) = (PH A−1 x (y − x), eN · Ax (y − x) − ax (PH Ax (y − x))). def

By Rademacher’s theorem the Lipschitzian function ax is diﬀerentiable almost everywhere in Vx and belongs to W 1,∞ Vx (cf., for instance, L. C. Evans and R. F. Gariepy [1]). Since hj is deﬁned from the Lipschitzian function aj , Dhj is deﬁned almost everywhere in Vx ×R, but also HN −1 almost everywhere on Vx ×{0}. Thus the canonical density for C 1 -domains still makes sense and is given by the same formula (3.13): ωj (ζ ) = ωxj (ζ ) = | det Dhj (ζ , 0)| | ∗(Dhj )−1 (ζ , 0) eN |. def

(5.30)

It is easy to verify that for almost all ξ ∈ Vj Dhj (ξ , ξN ) = Aj

... 0 .. . .. , det Dh (ξ , ξ ) = 1, .. j N . . .. . 0 . . . ∂N −1 aj (ξ ) 1

1

0 .. .

0 .. . 0 ∂1 aj (ξ )∂2 aj (ξ )

...

where ∂i aj is the partial derivative of aj with respect to the ith component of ζ = (ζ1 , . . . , ζN −1 ) ∈ Vj . The matrix Dhj is invertible and its coeﬃcients belong to L∞ and ∗(Aj )−1 = Aj . By direct computation

1

0 ∗ (Dhj )−1 (ξ , ξN ) = Aj ... . ..

0 .. .

...

−∂1 aj (ξ )

0 .. .

.

0 .. .

...

0

, −∂N −1 aj (ξ ) 1

= |(Aj )−1 ∗(Dhj )−1 (ζ , 0) eN | =

1 + |∇aj (ζ )|2 .

0

..

0

−∂2 aj (ξ ) .. .

(5.31)

(5.32)

and ﬁnally ωj (ζ ) = | ∗(Dhj )−1 (ζ , 0) eN |

96

Chapter 2. Classical Descriptions of Geometries and Their Properties

Let {r1 , . . . , rm } be a partition of unity for the Uj ’s, that is, r ∈ D(Uj ), 0 ≤ rj (x) ≤ 1, j m rj (x) = 1 in a neighborhood U of Γ

(5.33)

j=1

0 such that U ⊂ ∪m j=1 Uj . For any function f in C (Γ) the functions fj and fj ◦ hj deﬁned as

fj (y) = f (y)rj (y), fj ◦ hj (ζ , 0) = fj (x + Aj (ξ + aj (ξ ) eN ),

y ∈ Γj , ζ ∈ Vj ,

(5.34)

respectively, belong to C 0 (Γj ) and C 0 (Vj ). The integral of f on Γ is then deﬁned as

def

f dΓ = Γ

m j=1

fj dΓj , Γj

def

fj (hj (ζ , 0)) ωj (ζ ) dζ .

fj dΓ = Γj

(5.35)

Vj

Since ωj ∈ L∞ (Vj ), 1 ≤ j ≤ m, this integral is also well-deﬁned for all f in L1 (Γ); that is, the function fj ◦ hj ωj belongs to L1 (Vj ) for all j, 1 ≤ j ≤ m. As in section 2, the tangent plane to Γj at y = hj (ζ , 0) is deﬁned by Dhj (ζ , 0) Aj H, where Aj H is the hyperplane orthogonal to Aj eN . An outward normal ﬁeld to Γj is given by ∂1 aj (ζ ) .. . mx (ζ) = − ∗(Dhj )−1 (ζ , ζN )eN = Aj ∂N −1 aj (ζ ) −1 and the unit outward normal to Γj at y = hj (ζ , 0) ∈ Γj is given by ∂1 aj (ζ ) .. 1 . Aj n(y) = n(hj (ζ , 0)) = ' . ∂N −1 aj (ζ ) 1 + |∇aj (ζ )|2 −1

(5.36)

The matrix Aj can be removed by redeﬁning the functions ax on the space Aj H orthogonal to Aj eN . The formulae also hold for C k, -domains, k ≥ 1. We have seen that sets that are locally a C k, epigraph, k ≥ 0, have nice properties. For instance, their boundary has zero “volume.” However, they generally have no locally ﬁnite “surface measure” (cf. M. C. Delfour, N. Doyon, and J.´sio [1, 2] for speciﬁc examples) as we shall see in section 6.5. Fortunately, P. Zole Lipschitzian sets have locally ﬁnite “surface measure.”

5. Sets Locally Described by the Epigraph of a Function Theorem 5.7. Let Ω ⊂ RN be locally Lipschitzian. Then ' HN −1 U(x) ∩ ∂Ω ≤ 1 + c2x mN −1 Vx ,

97

(5.37)

where mN −1 and HN −1 are the respective (N − 1)-dimensional Lebesgue measure and (N − 1)-dimensional Hausdorﬀ measure, and cx > 0 is the Lipschitz constant of ax in Vx . If, in addition, ∂Ω is compact, then HN −1 (∂Ω) < ∞.

(5.38)

Proof. From Theorem 5.4 and from formula (5.32),

1/2 1/2 dζ ≤ 1 + c2x mN −1 Vx . 1 + |∇ax (ζ )|2 HN −1 U(x) ∩ ∂Ω = V (x)

When the boundary of Ω is compact, then we can ﬁnd a ﬁnite subcover of open neighborhoods {U(xj )}m j=1 . From the previous estimate, HN −1 (∂Ω) is bounded by a ﬁnite sum of bounded terms. 5.4.4

Geodesic Distance in a Domain and in Its Boundary

We can now justify property (2.29) in Theorem 2.6. Theorem 5.8. Let Ω ⊂ RN be bounded, open, and locally Lipschitzian. (i) If Ω is path-connected, then ∃cΩ , ∀x, y ∈ Ω,

distΩ (x, y) ≤ cΩ |x − y|.

(5.39)

dist∂Ω (x, y) ≤ c∂Ω |x − y|.

(5.40)

(ii) If ∂Ω is path-connected, then ∃c∂Ω , ∀x, y ∈ ∂Ω,

Remark 5.6. The boundary of a path-connected, bounded, open, and locally Lipschitzian set is generally not path-connected. As an example consider the set {x ∈ R2 : 1 < |x| < 2} whose boundary is made up of the two disjoint circles of radii 1 and 2. Remark 5.7. An open subset of RN with compact path-connected boundary is generally not pathconnected as can be seen from a domain in R2 made up of two tangent squares. Proof. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx ) ⊂ U(x). Since ∂Ω is compact, there exist a ﬁnite sequence {xi }m i=1 of points of ∂Ω and a ﬁnite , B = B(x , R ), R = r . Moreover, from part (i) of the proof subcover {Bi }m i i i i x i i=1 of Theorem 5.2, ∃R > 0, ∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi .

98

Chapter 2. Classical Descriptions of Geometries and Their Properties

Always from part (i) of the proof of Theorem 5.2, the neighborhoods U and V can be chosen as def

U = B(0, R)

and

def

V = BH (0, R),

and to each point x ∈ ∂Ω we can associate the following new neighborhoods and functions: Vx = V, def

Ax = Axi , def

U(x) = x + Ax U, def

ζ → ax (ζ ) = axi (ζ + ξi ) − axi (ξi ) : V → R . def

Since V = BH (0, R) is path-connected and for each ζ ∈ V the map ζN → x + Ax [ζ + ζN eN ] is continuous, Lζ = B(x, R) ∩ {x + Ax [ζ + ζN eN ] : ζN ∈ R} def

is a (path-connected) line segment. This is nice, but we need to further reduce the size of the neighborhoods U (x) and Vx to make the piece of boundary ∂Ω ∩ U (x) and hence Ω∩U (x) path-connected. Since the family of functions {ax (ζ ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous, there exists c > 0 such that ∀x ∈ ∂Ω,

∀ζ1 , ζ2 ∈ V,

|ax (ζ2 ) − ax (ζ1 )| ≤ c |ζ2 − ζ1 |.

Choose ρ, 0 < ρ ≤ R/(4c + 4). Then ∀ζ ∈ BH (0, ρ), |ax (ζ )| ≤ R/4 ⇒ |Ax (ζ + ax (ζ )eN )| < R ⇒ {x + Ax [ζ + ax (ζ )eN ] : ζ ∈ BH (0, ρ)} ⊂ ∂Ω ∩ B(x, R).

∀x ∈ ∂Ω,

Since BH (0, ρ) is path-connected and the map ζ → x + Ax [ζ + ax (ζ )eN ] is continuous, the image of BH (0, ρ) is path-connected. Redeﬁne V as BH (0, ρ) and U as B(0, R) ∩ {y ∈ RN : PH (y) ∈ BH (0, ρ)} and take the restrictions of the functions ax to BH (0, ρ). From this construction, for all x ∈ ∂Ω the new sets U(x ) ∩ ∂Ω and U(x ) ∩ int Ω are path-connected. Using the same notation for the ﬁnite sequence xi ∈ ∂Ω, let {U (xi ) : 1 ≤ i ≤ k} be the new ﬁnite subcovering of ∂Ω. For each x ∈ int Ω there exists 0 < rx < R/2 such that B(x, rx ) ⊂ int Ω. Consider the new set ΩR = Ω\ ∪ki=1 U (xi ). Since ΩR is compact, there exists a ﬁnite open covering of ΩR by the sets {B(xi , ri ) : k + 1 ≤ i ≤ m}, xi ∈ ΩR , and ri = rxi . By construction B(xi , ri ) ⊂ int Ω. (i) We ﬁrst show that distΩ (x, y) is uniformly bounded for any two points of Ω. Since Ω is covered by a ﬁnite number of balls, it is suﬃcient to show that the geodesic distance between any two points x and y in each ball is bounded. For k + 1 ≤ i ≤ m and x, y ∈ B(xi , ri ), distΩ (x, y) = |x − y| < R. For 1 ≤ i ≤ k, and x, y ∈ U (xi ), there is a graph representation x = xi + Ai (ζ + ζN eN ) and y = xi + Ai (ξ + ξN eN ). Choose as ﬁrst path the line from x = xi + Ai (ζ + ζN eN ) to xi + Ai (ζ + ai (ζ )eN ), follow the boundary along the second path {λζ +(1−λ)ξ , ai (λζ +(1−λ)ξ ) : λ ∈ [0, 1]} from xi +Ai (ζ +ai (ζ )eN )

5. Sets Locally Described by the Epigraph of a Function

99

to xi +Ai (ξ +ai (ξ )eN ), and ﬁnally as third path the line from xi +Ai (ξ +ai (ξ )eN ) to y = xi + Ai (ξ + ξN eN ). Its length is bounded by 1' |ζ − ξ |2 + |∇ai (λζ + (1 − λ)ξ ) · (ζ − ξ )|2 dλ + 2R, 2R + 0

√ which is bounded by 2R + 1 + c2 2ρ + 2R. Finally we proceed by contradiction. By deﬁnition of the geodesic distance distΩ (x, y)/|x − y| ≥ 1. If (5.39) is not true, there exist sequences {xn } and {yn } in Ω such that distΩ (xn , yn )/|xn − yn | goes to +∞, and hence, by boundedness of distΩ (xn , yn ), |xn − yn | goes to zero. Since Ω is bounded there exist subsequences, still denoted by {xn } and {yn }, and a point x ∈ Ω such that xn → x and yn → x. If x ∈ int Ω, there exists r > 0 such that B(x, r ) ⊂ int Ω and N such that for all n > N , xn , yn ∈ B(x, r ), where distΩ (xn , yn ) = |xn − yn | → 0 and we get a contradiction. If x ∈ ∂Ω, PH (B(x, ρ) − x) = BH (0, ρ) = V . There exists N such that, for all n > N , xn , yn ∈ B(0, ρ) and there exist (ζn , ζnN ), and (ξn , ξnN ), ζn , ξn ∈ BH (0, ρ), ξnN ≥ ax (ξn ), and ζnN ≥ ax (ζn ) such that xn = x + Ax [ξn + ξnN eN ] and yn = x + Ax [ζn + ζnN eN ]. The path {p(λ); 0 ≤ λ ≤ 1} deﬁned as λζn + (1 − λ)ξn p(λ) = x + Ax ax (λζn + (1 − λ)ξn ) + λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )] stays in B(x, R) ∩ Ω. Indeed λζn + (1 − λ)ξn ∈ BH (0, ρ), eN · A−1 x (p(λ) − x) = ax (λζn + (1 − λ)ξn ) + λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )] ≥ ax (λζn + (1 − λ)ξn ) , and p(λ) ∈ B(x, R). This last property follows from the estimate ' |p(λ) − x| ≤ |λζn + (1 − λ)ξn |2 + |ax (λζn + (1 − λ)ξn ) |2 + |λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )]| ' ≤ 1 + c2 |λζn + (1 − λ)ξn | + λ |ζnN − ax (ζn )| + (1 − λ) |ξnN − ax (ξn )| ' ' ≤ 1 + c2 ρ + λ 2ρ + (1 − λ) 2ρ = (2 + 1 + c2 ) ρ < R. Since the path lies in B(x, R) ∩ Ω, its length can be estimated by computing ζn − ξn p (λ) = Ax ∇ax (λζn + (1 − λ)ξn ) · (ζn − ξn ) + [ζnN − ξnN ] + ax (ξn ) − ax (ζn ) ⇒ |p (λ)| ≤ |ζn − ξn | + 2c|ζn − ξn | ≤ (1 + 2c) |yn − xn | 1 ⇒ distΩ (xn , yn ) ≤ |p (λ)| dλ ≤ (1 + 2c) |yn − xn | ≤ (1 + 2c) 2ρ < R 0

and we again get a contradiction. (ii) The case where ∂Ω is path-connected follows by the same constructions and proof as in part (i).

100 5.4.5

Chapter 2. Classical Descriptions of Geometries and Their Properties Nonhomogeneous Neumann and Dirichlet Problems

With the help of the previous deﬁnition of boundary measure in section 5.4.3, we can now make sense of the nonhomogeneous Neumann and Dirichlet problems (for the Laplace equation) in Lipschitzian domains. Assuming that the functions aj belong to W 1,∞ (Vj ), it can be shown that the classical Stokes divergence theorem holds for such domains. Given a bounded smooth domain D in RN and a Lipschitzian domain Ω in D, then ∀ ϕ ∈ C 1 (D, RN ), div ϕ

dx = ϕ

· n dΓ, (5.41) Ω

Γ

where the outward unit normal ﬁeld n is deﬁned by (5.36) for almost all ∈ Γj (with y = hx (ζ , 0), ζ ∈ Vj ) as (∂1 aj (ζ ), . . . , ∂N −1 aj (ζ ), −1) ' . 1 + |∇aj (ζ )|2

∗

n(y) = Aj

The trace of a function g in W 1,1 (D) on Γ is deﬁned through a W 1,∞ (D)N -extension N of the normal n as def gϕ dΓ = div (gϕN ) dx (5.42) ∀ϕ ∈ D(RN ), Γ

Ω

(cf. S. Agmon, A. Douglis, and L. Nirenberg [1, 2]). The right-hand side is well-deﬁned in the usual sense so that by the Stokes divergence theorem we obtain the trace g Γ deﬁned on Γ. This trace is uniquely deﬁned, denoted by γΓ g, and (5.43) γΓ ∈ L W 1,1 (Ω), L1 (Γ) . Let Ω be a Lipschitzian domain in D. Given f ∈ L2 (D) and g ∈ H 1 (D), consider the following problem: ∃y ∈ H 1 (Ω) such that ∀ϕ ∈ H 1 (Ω), ∇y · ∇ϕ dx = f ϕ dx + g ϕ dΓ. Ω

If the condition

Ω

(5.44)

Γ

f dx + Ω

g dΓ = 0

(5.45)

Γ

is satisﬁed, then by the Lax–Milgram theorem, problem (5.44)–(5.45) has a unique solution in H 1 (Ω)/ R: any two solutions of problem (5.44) can diﬀer only by a constant. For instance, we can associate with the solution y = y(Ω) in H 1 (Ω)/ R of (5.44) the following objective function: def def 2 J(Ω) = c Ω, y(Ω) , c(Ω, ϕ) = (|∇ϕ(x)| − G) dx (5.46) Ω

for some G in L2 (D).

6. Sets Locally Described by a Geometric Property

101

For the nonhomogeneous Dirichlet problem let Ω be a Lipschitzian domain in D and consider the weak form of the problem ∃y ∈ H 1 (Ω) such that ∀ϕ ∈ H 2 (Ω) ∩ H01 (Ω), ∂ϕ y ϕ dx = − g f ϕ dx − dΓ + ∂n Ω Γ Ω

(5.47)

for data (f, g) in L2 (D) × H 1 (D). If this problem has a solution y, then by Green’s theorem ∂ϕ ∂ϕ dΓ = − g dΓ + ∇y · ∇ϕ dx − y f ϕ dx (5.48) ∂n Ω Γ Γ ∂n Ω and −y = f ∈ L2 (Ω),

1

y|Γ = g ∈ H 2 (Γ).

Here, for instance, we can associate with y the objective function def def 1 |ϕ − G|2 dx, G ∈ L2 (D). J(Ω) = c Ω, y(Ω) , c(Ω, ϕ) = 2 Ω

6

(5.49)

(5.50)

Sets Locally Described by a Geometric Property

A large class of sets Ω can be characterized by geometric segment properties. The basic segment property is equivalent to the property that the set is locally a C 0 epigraph (cf. section 6.2). For instance, it is suﬃcient to get the density of C k (Ω) in the Sobolev space W m,p (Ω) for any m ≥ 1 and k ≥ m. In this section, we establish the equivalence of the segment, the uniform segment, and the more recent9 uniform fat segment properties with the respective locally C 0 , the C 0 , and the equi-C 0 epigraph properties (cf. sections 6.2, 6.3, and 6.4). Under the uniform fat segment property the local functions all have the same modulus of continuity speciﬁed by the continuity of the dominating function at the origin (cf. section 6.3). For sets with a compact boundary the three segment properties are equivalent (cf. section 6.1). However, for sets with an unbounded boundary the uniform segment property is generally too meager to make the local epigraphs uniformly bounded and equicontinuous. Section 6.4 specializes the uniform fat segment property to the uniform cusp and cone properties, which imply that ∂Ω is, respectively, an equi-H¨olderian and equi-Lipschitzian epigraph. Section 6.5 discusses the existence of a locally ﬁnite boundary measure. We have already seen in section 5.4.3 that Lipschitzian sets have a locally ﬁnite boundary or surface (that is Hausdorﬀ HN −1 ) measure. So we can make sense of boundary conditions associated with a partial diﬀerential equation on the domain. In general, this is not true for H¨ olderian domains. To illustrate that point, we construct examples of H¨ olderian sets for which the Hausdorﬀ dimension of the boundary is strictly greater than N − 1 and hence HN −1 (∂Ω) = +∞. 9 This was introduced starting with the uniform cusp property in M. C. Delfour and J.-P. Zo´sio [37] in 2001 and further reﬁned and generalized in the sequence of papers by M. C. Delfour, le ´sio [1, 2] in 2005 and M. C. Delfour and J.-P. Zole ´sio [43] in 2007. N. Doyon, and J.-P. Zole

102

6.1

Chapter 2. Classical Descriptions of Geometries and Their Properties

Deﬁnitions and Main Results

In this section we use the notation introduced at the beginning of section 5: eN is a unit vector in RN and H = {eN }⊥ . The open segment between two distinct points x and y of RN will be denoted by def

(x, y) = {x + t(y − x) : ∀t, 0 < t < 1}. Deﬁnition 6.1. Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is said to satisfy the segment property if ∀x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃Ax ∈ O(N), such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λAx eN ) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform segment property if ∃r > 0, ∃λ > 0 such that ∀x ∈ ∂Ω, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λAx eN ) ⊂ int Ω.

(iii) Ω is said to satisfy the uniform fat segment property if there exist r > 0, λ > 0, and an open region O of RN containing the segment (0, λeN ) and not 0 such that for all x ∈ ∂Ω, ∃Ax ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax O ⊂ int Ω.

(6.1)

Of course, when Ω is an open domain, Ω = int Ω, and we are back to the standard deﬁnition. Yet, as in Deﬁnition 3.1, those deﬁnitions involve only ∂Ω, Ω, and int Ω and the properties remain true for all sets in the class

[Ω]b = A ⊂ RN : A = Ω and ∂A = ∂Ω . A set having the segment property must have an (N −1)-dimensional boundary and cannot simultaneously lie on both sides of any given part of its boundary. In fact, a domain satisfying the segment property is locally a C 0 epigraph in the sense of Deﬁnition 5.1, as we shall see in section 6.2. We ﬁrst give a few general properties. Theorem 6.1. Let Ω be a subset of RN such that ∂Ω = ∅. (i) If ∀x ∈ ∂Ω, ∃λ > 0, ∃Ax ∈ O(N),

(x, x + λAx eN ) ∈ int Ω,

then int Ω = ∅ and int Ω = Ω

and

∂(int Ω) = ∂Ω.

(6.2)

6. Sets Locally Described by a Geometric Property

103

(ii) Ω satisﬁes the segment property if and only if Ω satisﬁes the segment property. In particular int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω, int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω. Proof. (i) Pick any point x ∈ ∂Ω. By deﬁnition, for all µ ∈ ]0, 1[ , xµ = x+µ λ AeN ∈ int Ω and int Ω = ∅. As a result there exists a sequence µn > 0 → 0 such that int Ω xµn → x

⇒ ∂Ω ⊂ int Ω ⊂ Ω

⇒ Ω ⊂ int Ω ⊂ Ω.

The second identity follows from Lemma 5.1 (i). (ii) Assume that Ω satisﬁes the segment property. For any x ∈ ∂Ω, there exist r > 0, λ > 0, and A ∈ O(N) such that for all y ∈ B(x, r) ∩ Ω, (y, y + λAeN ) ⊂ int Ω. We claim that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λ(−A)eN ) ⊂ int Ω.

Indeed, if there exist z ∈ B(x, r) ∩ Ω and α ∈ (0, λ) such that y − αAeN ∈ Ω, then y = (y − αAeN ) + αAeN ∈ int Ω, which leads to a contradiction. In the other direction we interchange Ω and Ω and repeat the proof. Finally, the two sets of properties follow from part (i) applied to Ω and Ω. We summarize the equivalences between the epigraph and segment properties that will be detailed in Theorems 6.5, 6.6, and 6.7 with the associated constructions and additional motivation. Theorem 6.2. Let Ω be a subset of RN such that ∂Ω = ∅. (i) (a) Ω is locally a C 0 epigraph if and only if Ω has the segment property; (b) Ω is a C 0 epigraph if and only if Ω has the uniform segment property; (c) Ω is an equi-C 0 epigraph if and only if Ω has the uniform fat segment property. (ii) If, in addition, ∂Ω is compact, the six properties are equivalent and there exists a dominating function h ∈ H that satisﬁes the properties of Theorems 5.1 and 5.2. Proof. (i) From Theorems 6.5, 6.6, and 6.7. (ii) When ∂Ω is compact, from part (i) and the equivalences in Theorem 5.2. We quote the following density results from R. A. Adams [1, Thm. 3.18, p. 54]. Theorem 6.3. If Ω has the segment property, then the set

f |int Ω : ∀f ∈ C0∞ (RN ) of restrictions of functions of C0∞ (RN ) to int Ω is dense in W m,p (int Ω) for 1 ≤ p < ∞ and m ≥ 1. In particular, C k (int Ω) is dense in W m,p (int Ω) for any m ≥ 1 and k ≥ m.

104

6.2

Chapter 2. Classical Descriptions of Geometries and Their Properties

Equivalence of Geometric Segment and C 0 Epigraph Properties

As for sets that are locally a C 0 epigraph in Deﬁnition 5.1, the three cases of Deﬁnition 6.1 diﬀer only when ∂Ω is unbounded. We ﬁrst deal with the ﬁrst two and will come back to the third one in section 6.3. Theorem 6.4. Let ∂Ω = ∅ be compact. Then the segment property and the uniform segment property of Deﬁnition 6.1 are equivalent. Proof. It is suﬃcient to show that the segment property implies the uniform segment property. Since ∂Ω is compact there exists a ﬁnite open subcover {Bi }m i=1 , Bi = of points of ∂Ω. From part (i) of B(xi , rxi ), of ∂Ω for some ﬁnite sequence {xi }m i=1 the proof of Theorem 5.2 ∃r > 0, ∀x ∈ ∂Ω,

∃i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi .

Deﬁne λ = min1≤i≤m λi > 0. Therefore for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi and hence, by choosing Ax = Axi ∈ O(N), we get ∀y ∈ Ω ∩ Bi ,

(y, y + λAx eN ) ⊂ (y, y + λi Axi eN ) ⊂ int Ω.

By restricting the above property to B(x, r) ⊂ Bi , ∀y ∈ Ω ∩ B(x, r),

(y, y + λAx eN ) ⊂ (y, y + λi Ax eN ) ⊂ int Ω.

Since r and λ are independent of x ∈ ∂Ω, Ω has the uniform segment property. Theorem 6.5. Let Ω be a subset of RN such that ∂Ω = ∅. (i) If Ω satisﬁes the segment property, then Ω is locally a C 0 epigraph. For x ∈ ∂Ω, let B(x, rx ), λx , and Ax be the associated open ball, the height, and the rotation matrix. Then there exists ρx , def

0 < ρx ≤ rxλ = min {rx , λx /2} ,

(6.3)

which is the largest radius such that

BH (0, ρx ) ⊂ PH (A−1 x (y − x)) : ∀y ∈ B(x, rxλ ) ∩ ∂Ω . The neighborhoods of Deﬁnition 5.1 can be chosen as def

Vx = BH (0, ρx ) and

def U(x) = B(x, rxλ ) ∩ y ∈ RN : PH (A−1 x (y − x)) ∈ Vx ,

(6.4)

where10 BH (0, ρx ) denotes the open ball of radius ρx in the hyperplane H. For each ζ ∈ Vx , there exists a unique yζ ∈ ∂Ω∩U(x) such that PH A−1 x (yζ −x) = ζ and the function ζ → ax (ζ ) = (yζ − x) · (Ax eN ) : Vx → R def

10 Note

that U(x) − {x} = Ax [B(0, rxλ ) ∩ {ζ : PH ζ ∈ Vx }].

6. Sets Locally Described by a Geometric Property

105

is well-deﬁned, bounded, ∀ζ ∈ Vx ,

|ax (ζ )| < rxλ ,

(6.5)

and uniformly continuous in Vx , that is, ax ∈ C 0 (Vx ). (ii) Conversely, if Ω is locally a C 0 epigraph, both Ω and Ω satisfy the segment property. Proof. (i) By assumption for each x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃A ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λAeN ) ⊂ int Ω.

Consider the set def

W = PH A−1 (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω ,

rλ = min{r, λ/2}.

Therefore ∀ζ ∈ W,

∃y ∈ B(x, r) ∩ ∂Ω,

PH A−1 (y − x) = ζ .

If there exist y1 , y2 ∈ B(x, rλ )∩∂Ω such that PH A−1 (y1 −x) = PH A−1 (y2 −x) = ζ , then y2 ∈ (y1 , y1 + AeN · (y2 − y1 ) AeN ) ⊂ (y1 , y1 + λAeN ) ∈ int Ω since |y2 − y1 | < 2rλ < λ and we get a contradiction. Therefore, the map ζ → a(ζ ) = AeN · (y − x) = eN · A−1 (y − x) : W → R def

is well-deﬁned and a(0) = 0. (W is open and 0 ∈ W ). By deﬁnition, 0 ∈ W ⊂ BH (0, rλ ). Given any point ξ ∈ W , the segment property is satisﬁed at the point yξ = x + A(ξ + a(ξ )eN ) ∈ ∂Ω ∩ B(x, rλ ), def

and (yξ , yξ + λAeN ) ⊂ int Ω and (yξ , yξ − λAeN ) ⊂ int Ω by Theorem 6.1 (ii): (yξ , yξ + λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ B(x, rλ ), (yξ , yξ − λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ ∩B(x, rλ ). Therefore, there exists a neighborhood BH (ξ , R) of ξ such that BH (ξ , R) ⊂ BH (0, rλ ) and for all ζ ∈ BH (ξ , R), Lζ ∩ int Ω ∩ B(x, rλ ) = ∅, where Lζ = x + A(ζ + R eN ) is the line through x + Aζ in the direction AeN . If ξ is not an interior point of W , there exists a sequence {ζn } ⊂ BH (ξ , R), ζn → ξ , such that for all n, (Lζn ∩ B(x, rλ )) ∩ ∂Ω = ∅. Therefore, either Lζn ∩ B(x, rλ ) ⊂ int Ω or Lζn ∩ B(x, rλ ) ⊂ int Ω. Since for points ζn ∈ BH (ξ , R) we already know that Lζn ∩ B(x, rλ ) ∩ int Ω = ∅, then Lζn ∩ B(x, rλ ) ⊂ int Ω. Choose def 1 α = a(ξ ) − rλ2 − |ξ |2 2

106

Chapter 2. Classical Descriptions of Geometries and Their Properties

and the sequence zn = x + A(ζn + αeN ) → z = yξ − def

def

1 2

a(ξ ) +

rλ2 − |ξ |2 AeN .

We get the following estimate: |z − x|

N − α NΩ,i (ε) ≤

m i=1

NΩ,i (ε) ≤ m

1 εN −α

(rλ + ε)

⇒ NΩ (ε)εβ ≤ εβ−N +α m (rλ + ε) ⇒ ∀β > N − α,

N −1

N −1

α √ N − 1 + ε1−α c

α √ N − 1 + ε1−α c

Hβ (∂Ω) = 0.

By deﬁnition, the Hausdorﬀ dimension of ∂Ω is less than or equal to N − α.

118

Chapter 2. Classical Descriptions of Geometries and Their Properties

It is possible to construct examples of sets verifying the uniform cusp property for which the Hausdorﬀ dimension of the boundary is strictly greater than N − 1 and hence HN −1 (∂Ω) = +∞. Example 6.1. This following two-dimensional example of an open domain with compact boundary satisfying the uniform cusp condition for the function h(θ) = θ α , 0 < α < 1, can easily be generalized to an N -dimensional example. Consider the open domain def

Ω = {(x, y) : −1 < x ≤ 0 and 0 < y < 2} ∩ {(x, y) : 0 < x < 1 and f (x) < y < 2} ∩ {(x, y) : 1 ≤ x < 2 and 0 < y < 2} in R2 (see Figure 2.7), where f : [0, 1] → R is deﬁned as follows: def

f (x) = dC (x)α ,

0 ≤ x ≤ 1,

and C is the Cantor set on the interval [0, 1] and dC (x) is the distance function from the point x to the set C (dC is uniformly Lipschitzian of constant 1). This function is equal to 0 on C. Any point in [0, 1]\C belongs to one of the intervals of length 3−k , k ≥ 1, which has been deleted from [0, 1] in the sequential construction of the Cantor set. Therefore the distance function dC (x) is equal to the distance function to the two end points of that interval. In view of this special structure it can be shown that ∀x, y ∈ [0, 1],

|dC (y)α − dC (x)α | ≤ |y − x|α .

Denote by Γ the piece of the boundary ∂Ω speciﬁed by the function f = dC . On Γ the uniform cusp condition is veriﬁed with ρ = 1/6, λ = (1/6)α , and h(θ) = θ α (see

2 y

1

domain Ω

C(λ, h, ρ, e2 )

f (x) 0

−1

x 0

1

2

Figure 2.7. Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α , h(θ) = θα .

6. Sets Locally Described by a Geometric Property

0

1/9

2/9

1/3

119

2/3

7/9

8/9

1

Figure 2.8. f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. Figure 2.8). Clearly the number NΩ (ε) of hypercubes of dimension N and side ε required to cover ∂Ω is greater than the number NΓ (ε) of hypercubes of dimension N and side ε required to cover Γ. The construction of the Cantor set is done by sequentially deleting intervals. At step k = 0 the interval (1/3, 2/3) of width 3−1 is removed. At step k a total of 2k intervals of width 3−(k+1) are removed. Thus if we pick ε = 3−(k+1) , the interval [0, 1] can be covered with exactly 3(k+1) intervals. Here we are interested in ﬁnding a lower bound to the total number of squares of side ε necessary to cover Γ. For this purpose we keep only the 2k intervals removed at step k. Vertically it takes α −1 −(k+1) α 2−1 3−(k+1) 2 3 ≥ − 1. 3−(k+1) 3−(k+1) Then we have for β ≥ 0 0 NΩ (ε) ≥ NΓ (ε)≥ 2k

2−1 3−(k+1) 3−(k+1)

1

α −1

k ≥ 2k−α 3(k+1)(1−α) − 2k = 3(1−α) 2 2−α 3(1−α) − 2k and NΩ (ε)(3

−k 1+β

)

−k(1+β)

≥3

≥ 3

−(α+β)

3

(1−α)

k 2

2

k 2

2

−α (1−α)

3

−α (1−α)

3

−

−2

2 3(1+β)

k

k .

The second term goes to zero as k goes to inﬁnity. The ﬁrst term goes to inﬁnity as k goes to inﬁnity if 3−(α+β) 2 > 1, that is, 0 < α+β < ln 2/ ln 3. Under this condition, H1+β (∂Ω) = H1+β (Γ) = +∞ for all 0 < α < ln 2/ ln 3 and all 0 ≤ β < ln 2/ ln 3 − α.

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Chapter 2. Classical Descriptions of Geometries and Their Properties

Therefore given 0 < α < ln 2/ ln 3 ∀β, 0 ≤ β + α < ln 2/ ln 3,

H1+β (∂Ω) = +∞

and the Hausdorﬀ dimension of ∂Ω is strictly greater than 1. Given 0 < α < 1, it is possible to construct an optimal example of a set verifying the uniform cusp property for which the Hausdorﬀ dimension of the boundary is exactly N − α and hence HN −1 (∂Ω) = +∞. Example 6.2 (Optimal example of a set that veriﬁes the uniform cusp property with h(θ) = |θ|α , 0 < α < 1, and whose boundary has Hausdorﬀ dimension exactly equal to N − α.). For that purpose, we need a generalization of the Cantor set. Denote by C1 the Cantor set. Recall that each x, 0 ≤ x ≤ 1, can be written uniquely (if we make a certain convention) as ∞ aj (3, x) x= , 3j j=1 where aj (3, x) can be regarded as the jth digit of x written in basis 3. From this the Cantor set is characterized as follows: x ∈ C1 ⇐⇒ ∀j, aj (3, x) = 1. Similarly for an arbitrary integer k ≥ 1, each x ∈ [0, 1] can be uniquely written in the form ∞ aj (2k + 1, x) x= (2k + 1)j j=1 and we can deﬁne the set Ck as x ∈ Ck ⇐⇒ ∀j, aj (2k + 1, x) = k. In a certain sense, if k1 > k2 , Ck1 contains more points than Ck2 . We now use these sets to construct the family of set Dk as follows: x ∈ D1 ⇐⇒ 2x ∈ C1 and for k > 1, x ∈ Dk ⇐⇒ 2k+1 (x − 2k ) ∈ Ck . 2 Note that if k1 = k2 , Dk1 Dk2 = ∅ since the Dk ’s contain only points from the def interval [1 − 2k−1 , 1 − 2k ]. Consider now the set D = ∪∞ k=1 Dk and go back to def

Example 6.1 with the function f replaced by the function f (x) = dD (x)α . Again it can be shown that for all x, y ∈ [0, 1], |dD (y)α − dD (x)α | ≤ |y − x|α . Note that on the interval [1 − 2k−1 , 1 − 2k ] we have dD (x)α = dDk (x)α . Denote by Γ the piece of boundary ∂Ω speciﬁed by the function f = dD and by Γk the part of boundary ∂Ω speciﬁed by the function f = dD (= dDk ) on the interval [1 − 2k−1 , 1 − 2k ]. Once again on Γ the uniform cusp property is veriﬁed

6. Sets Locally Described by a Geometric Property

121

with ρ = 1/6, λ = (1/6)α , and h(θ) = θα . Clearly the number NΩ (ε) of hypercubes of dimension N and side ε required to cover ∂Ω is greater than the number NΓk (ε) of hypercubes of dimension N and side ε required to cover Γk . The construction of the set Ck is also done sequentially by deleting intervals. At step j = 0 the interval ]k/(2k + 1), (k + 1)/(2k + 1)[ of width (2k + 1)−1 is removed. At step j a total of 2j intervals of width (2j + 1)−(j+1) are removed. If we consider the intervals that remain at step j, a total of 2j+1 nonempty disjoint intervals of width (k/(2k + 1))j+1 remain in the set Ck . Each of these intervals contains a gap of length (k/(2k + 1))j+1 1/(2k + 1) created at step j + 1. If we construct the set Dk in the same way, at step j a total of 2j nonempty disjoint intervals of width (k/(2k + 1))j+1 1/2k remain in the set Dk . Each of these intervals contains a gap of length (k/(2k + 1))j+1 1/(2k (2k + 1)). Pick j+1 1 k ε= k 2 2k + 1 and look for a lower bound on the number of squares of side ε necessary to cover Γk . For this purpose, consider only the 2j+1 nonempty disjoint intervals remaining at step j. As they each contain a gap of length j+1 1 k 2k + 1 2k (2k + 1) vertically it takes 1α 0 j+1 j+1 k 1 2k + 1 k 2 2k + 1 2k+1 (2k + 1) k 1α 0 j+1 j+1 1 2k + 1 k k 2 −1 ≥ 2k + 1 2k+1 (2k + 1) k ε-cubes. Then we have for β ≥ 0 00 1α 1 j+1 j+1 k 1 2k + 1 j+1 k NΩ (ε) ≥ NΓ (ε) ≥ 2 2 −1 2k + 1 2k+1 (2k + 1) k j+1 2(2k + 1)kα 2k ≥ − 2j+1 k(2k + 1)α 2α(k+1) (2k + 1)α j+1 2k(1−α)−α − 2j+1 = 2(2k + 1)1−α k α−1 (2k + 1)α and hence ε1+β NΩ (ε) 0 j+1 11+β k(1−α)−α 1 k 1−α α−1 j+1 2 j+1 ≥ k 2(2k + 1) − 2 2k 2k + 1 (2k + 1)α 0 0 α+β 1j+1 j+1 11+β 2k(1−α)−α 1 k k j+1 2 −2 . ≥ 2k + 1 2k 2k + 1 2k(1+β) (2k + 1)α

122

Chapter 2. Classical Descriptions of Geometries and Their Properties

The second term goes to zero as j goes to inﬁnity. The ﬁrst term goes to inﬁnity as j goes to inﬁnity if ∀k,

k 2k + 1

α+β 2 > 1,

that is ∀k,

0 0 such that for all x such that |x| > ρ0 , |g(x)| < ε. But f also belongs to C00 (RN , RN ) and

126

Chapter 3. Courant Metrics on Images of a Set

there exists ρ ≥ 2ρ0 such that for all x such that |x| > ρ, |f (x)| < ρ0 . In particular, |x + f (x)| ≥ |x| − |f (x)| > ρ − ρ0 ≥ ρ0 and hence |g(x + f (x))| < ε. In conclusion for all ε > 0, there exists ρ > 0 such that for all x such that |x| > ρ, |g(x + f (x))| < ε and g ◦ (I + f ) ∈ C00 (RN , RN ). (4) Θ = C 0,1 (RN , RN ). This is the space of bounded Lipschitz continuous functions on RN that is contained in C 0 (RN , RN ). For f, g ∈ C 0,1 (RN , RN ), f ◦ (I + g)C 0 = f C 0 < ∞. Since g is Lipschitz with Lipschitz constant c(g), I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is also Lipschitz with constant c(f ) (1 + c(g)). Proof of Theorem 2.1. I ∈ F(Θ) for f = 0. If I + f ∈ F(Θ), (I + f )−1 is bijective, (I + f )−1 − I ∈ Θ, (F −1 )−1 = F = I + f , and (F −1 )−1 − I = f ∈ Θ. Hence (I + f )−1 ∈ F(Θ) and I = (I + f ) ◦ (I + f )−1 = (I + f )−1 + f ◦ (I + f )−1 ⇒ (I + f )−1 − I = −f ◦ (I + f )−1 . The last property to check is the composition (I + f ) ◦ (I + g). It is bijective as the composition of two bijective transformations. Moreover, by linearity of Θ and Assumption 2.1 (I + f ) ◦ (I + g) − I = g + f ◦ (I + g) ∈ Θ.

(2.4)

From (2.4) and the invertibility of the composition [(I + f ) ◦ (I + g)]−1 − I = −[g + f ◦ (I + g)] ◦ (I + g)−1 ◦ (I + f )−1 ∈ Θ by using Assumption 2.1 twice. Conversely, if (2.4) is true for all f ∈ Θ and I + g ∈ F(Θ), by linearity of Θ, Assumption 2.1 is veriﬁed. Associate with F ∈ F(Θ) the following function of I and F : def

d(I, F ) =

inf

n

F =F1 ◦···◦Fn Fi ∈F (Θ) i=1

Fi − IΘ + Fi−1 − IΘ ,

(2.5)

where the inﬁmum is taken over all ﬁnite factorizations of F in F(Θ) of the form F = F1 ◦ · · · ◦ Fn ,

Fi ∈ F(Θ).

In particular d(I, F ) = d(I, F −1 ). Extend this function to all F and G in F(Θ): d(F, G) = d(I, G ◦ F −1 ). def

By deﬁnition, d is right-invariant since for all F , G, and H in F(Θ) d(F, G) = d(F ◦ H, G ◦ H).

(2.6)

2. Generic Constructions of Micheletti

127

To show that d is a metric5 on F(Θ) necessitates a second assumption. Assumption 2.2. (Θ, · ) is a normed vector space of mappings from RN into RN and for each r > 0 there exists a continuous function c0 (r) such that ∀{fi }ni=1 ⊂ Θ such that

n

fi < α < r;

(2.7)

(I + f1 ) ◦ · · · ◦ (I + fn ) − I < α c0 (r).

(2.8)

i=1

then

Theorem 2.2. Under Assumptions 2.1 and 2.2, d is a right-invariant metric on F(Θ). Example 2.2. (1), (2), (3) Since the norm on the three spaces B0 (RN , RN ), C 0 (RN , RN ), and C00 (RN , RN ) is the same and Assumption 2.2 involves only the norm, it is suﬃcient to check it for B 0 (RN , RN ). Consider {fi }ni=1 ⊂ B0 (RN , RN ) such that n fi C 0 < α < r. (2.9) i=1

Using the convention (I + fn ) ◦ (I + fn ) = (I + fn ) in the summation (I + f1 ) ◦ · · · ◦ (I + fn ) − I = (I + fn ) − I +

n−1

(I + fi ) ◦ · · · ◦ (I + fn ) − (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

= fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α < r

⇒ c0 (r) = r.

i=1

(4) Θ = C 0,1 (RN , RN ). Consider a sequence {fi }ni=1 ⊂ C 0,1 (RN , RN ) such that n

max {fi C 0 , c(fi )} < α < r.

(2.10)

i=1 5A

function d : X × X → R is said to be a metric on X if (cf. J. Dugundji [1])

(i) d(F.G) ≥ 0, for all F, G, (ii) d(F, G) = 0 ⇐⇒ F = G, (iii) d(F, G) = d(G, F ), for all F, G, (iv) d(F, H) ≤ d(F, G) + d(G, H), for all F, G, H. A metric d on a group (F , ◦) is said to be right-invariant if for all H ∈ F , d(F ◦H, G◦H) = d(F, G).

128

Chapter 3. Courant Metrics on Images of a Set

From the previous example (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α.

i=1

In addition, (I + f1 ) ◦ · · · ◦ (I + fn ) − I = fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ c((I + f1 ) ◦ · · · ◦ (I + fn ) − I) ≤ c(fn ) +

n−1

c(fi ) (1 + c(fi+1 )) . . . (1 + c(f1 ))

i=1

≤ c(fn ) + ≤

n

n−1

)i+1 c(fi ) e

i=1

)n

c(fi ) e[

i=1

k=1

c(fi )]

c(fk )

< α eα

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0,1 < α max{eα , 1} < α er and we can choose c0 (r) = er . Proof of Theorem 2.2. Properties (i) and (iii) are veriﬁed by deﬁnition. For the triangle inequality (iv), F ◦ H −1 = (F ◦ G−1 ) ◦ (G ◦ H −1 ) is a ﬁnite factorization of F ◦ H −1 . Consider arbitrary ﬁnite factorizations of F ◦ G−1 and G ◦ H −1 : F ◦ G−1 = (I + α1 ) ◦ · · · ◦ (I + αm ), G◦H

−1

= (I + β1 ) ◦ · · · ◦ (I + βn ),

I + αi ∈ F(Θ), I + βj ∈ F(Θ).

This yields a new ﬁnite factorization of F ◦ H −1 . By deﬁnition of d, d(I, F ◦ H −1 ) ≤

m

αi + αi ◦ (I + αi )−1 +

i=1

n

βj + βj ◦ (I + βj )−1 ,

i=j

and by taking inﬁma over each factorization, d(F, H) = d(I, F ◦ H −1 ) ≤ d(I, F ◦ G−1 ) + d(I, G ◦ H −1 ) = d(F, G) + d(G, H) using the symmetry property (iii). To verify (ii) it is suﬃcient to show that F = I ⇐⇒ d(I, F ) = 0. Clearly I ◦ I is a factorization of F = I and 0 ≤ d(I, F ) ≤ 0. Conversely, if d(I, F ) = 0, for ε, 0 < ε < 1, there exists a ﬁnite factorization F = F1 ◦ · · · ◦ Fn such that n

Fi − I + Fi−1 − I < ε < 1.

i=1

By Assumption 2.2, F − I + F −1 − I < ε c0 (1), which implies that F − I = 0 and F −1 − I = 0. This completes the proof.

2. Generic Constructions of Micheletti

129

An important aspect of the previous construction was to build in the triangle inequality to make d a (right-invariant) metric. However, there are other ways to introduce a topology on F(Θ). The interest of this speciﬁc choice will become apparent later on. For instance, the original Courant metric of A. M. Micheletti [1] in 1972 was constructed from the following slightly diﬀerent right-invariant metric on F(Θ): given H ∈ F(Θ), consider ﬁnite factorizations H1 ◦ · · · ◦ Hm , Hi ∈ F(Θ), 1 ≤ i ≤ m, of H and ﬁnite factorizations G1 ◦ · · · ◦ Gn , Gj ∈ F(Θ), 1 ≤ j ≤ n, of H −1 , and deﬁne m n def d1 (I, H) = inf Hi − I + inf Gi − I, (2.11) H=H1 ◦···◦Hm i=1 m≥1

H −1 =G1 ◦···◦Gn i=1 n≥1

where the inﬁma are taken over all ﬁnite factorizations of H and H −1 in F(Θ) and ∀F, G ∈ F(Θ),

d1 (G, F ) = d1 (I, F ◦ G−1 ). def

(2.12)

Another example is the topology generated by the right-invariant semimetric 6 ∀H ∈ F(Θ),

d0 (I, H) = H − I + H −1 − I, def

∀F, G ∈ F(Θ),

d0 (G, F ) = d0 (I, F ◦ G−1 ). def

(2.13)

Given r > 0 and F ∈ F(Θ), let Br (F ) = {G ∈ F(Θ) : d0 (F, G) < r}. Let τ0 be the weakest topology on F(Θ) such that the family {Br (F ) : 0 ≤ r < ∞} is the base for τ0 . By deﬁnition d1 (F, G) ≤ d(F, G) ≤ d0 (F, G) and the injections (F, d1 ) → (F, d) → (F, d0 ) are continuous. Theorem 2.3. Under Assumptions 2.1 and 2.2, the topologies generated by d1 , d, and d0 are equivalent. Proof. Since d0 , d, and d1 are right-invariant on the group F(Θ), it is suﬃcient to prove the equivalence for d0 and d1 around I. For all ε > 0 with δ = ε ∀H ∈ F(Θ), d0 (I, H) < δ,

d1 (I, H) < ε,

since d1 (I, H) ≤ d0 (I, H). Conversely, by Assumption 2.2, for any δ, 0 < δ < 1, d1 (I, H) < δ < 1

⇒ d0 (I − H) = H − I + H −1 − I < 2δ c(1).

So for any ε > 0, pick δ = min{1, ε/c(1)}/2. Hence ∀ε > 0, ∃δ > 0,

∀H ∈ F(Θ), d1 (I, H) < δ,

d0 (I, H) < ε.

This concludes the proof of the equivalences. 6 Given a space X, a function d : X × X → R is said to be a semimetric if (i) d(F.G) ≥ 0, for all F, G,

(ii) d(F, G) = 0 ⇐⇒ F = G, (iii) d(F, G) = d(G, F ), for all F, G. This notion goes back to Fr´echet and Menger.

130

Chapter 3. Courant Metrics on Images of a Set F. Murat and J. Simon [1] in 1976 considered as candidates for Θ the spaces W k+1,c (RN , RN ) def = f ∈ W k,∞ (RN , RN ) : ∀ 0 ≤ |α| ≤ k + 1, ∂ α f ∈ C(RN , RN )

and W k+1,∞ (RN , RN ), k ≥ 0. The ﬁrst space W k+1,c (RN , RN ) is equivalent to the space C k+1 (RN , RN ), k ≥ 0, algebraically and topologically. The second space W k+1,∞ (RN , RN ) coincides with C k,1 (RN , RN ), k ≥ 0. For the spaces C k+1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, they obtained the following pseudo triangle inequality for the semimetric d0 : d0 (F1 ,F3 ) ≤d0 (F1 , F2 ) + d0 (F2 , F3 ) + d0 (F1 , F2 ) d0 (F2 , F3 ) P [d0 (F1 , F2 ) + d0 (F2 , F3 )] ,

(2.14) ∀F1 , F2 , F3 ,

where P : R+ → R+ is a continuous increasing function. With that additional property, they called d0 a pseudodistance 7 and showed that they could construct a metric from it. Lemma 2.1. Assume that the semimetric d0 satisﬁes the pseudo triangle inequality (2.14). Then, for all α, 0 < α < 1, there exists a constant ηα > 0 such that the (α) function d0 : F(Θ) × F(Θ) → R+ deﬁned as def

(α)

d0 (F1 , F2 ) = inf{d0 (F1 , F2 ), ηα }α is a metric on E. The Micheletti construction with the metric d1 was given in 2001 for the spaces C k (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, in M. C. Delfour and J.-P. Zo´sio [37] so that pseudodistances can be completely bypassed. The metrics d and le d1 constructed from the semimetric d0 under the general Assumptions 2.1 and 2.2 are both more general and interesting since they are of geodesic type by the use of inﬁma over ﬁnite factorizations (I + θ1 ) ◦ · · · ◦ (I + θn ) of I + θ when the semimetric d0 is interpreted as an energy n

d0 (I + θi , I) =

i=1

n

θi + − θi ◦ (I + θi )−1

(2.15)

i=1

7 This terminology is not standard. In the literature a pseudometric or pseudodistance is usually reserved for a function d : X × X → R satisfying

(i) d(F.G) ≥ 0,

for all F, G,

(ii’) d(F, G) = 0 ⇐= F = G, (iii) d(F, G) = d(G, F ),

for all F, G.

(iv) d(F, H) ≤ d(F, G) + d(G, H),

for all F, G, H.

Condition (ii) for a metric is weakened to condition (ii’) for a pseudodistance.

2. Generic Constructions of Micheletti

131

over families of piecewise constant trajectories T : [0, 1] → F(Θ) starting from I at time 0 to I + θ at time 1 in the group F(Θ) by assigning times ti , 0 = t0 < t1 < · · · < tn < tn+1 = 1, at each jump: I, 0 ≤ t < t1 , (I + θ1 ), t 1 < t < t2 , t 2 < t < t3 , (I + θ2 ) ◦ (I + θ1 ), def ... T (t) = (2.16) (I + θi ) ◦ · · · ◦ (I + θ1 ), ti < t < ti+1 , ... (I + θn ) ◦ · · · ◦ (I + θ1 ), tn < t ≤ 1. The jump in the group F(Θ) at time ti is given by − −1 [T (t)]ti = T (t+ −I i ) ◦ T (ti ) def

= [(I + θi ) ◦ · · · ◦ (I + θ1 )] ◦ [(I + θi−1 ) ◦ · · · ◦ (I + θ1 )]−1 − I = θi as an element of Θ that could be viewed as the tangent space to F(Θ) ⊂ I + Θ. The choice of d in a geodesic context is more interesting than the choice of d1 that would involve a double inﬁma in the energy (2.15). At this stage, in view of the equivalence Theorem 2.3, working with the metric d or d1 is completely equivalent. Assumption 2.3. (Θ, · ) is a normed vector space of mappings from RN into RN and there exists a continuous function c1 such that for all I + f ∈ F(Θ) and g ∈ Θ g ◦ (I + f ) ≤ g c1 (f ).

(2.17)

The next assumption is a kind of uniform continuity. Assumption 2.4. (Θ, · ) is a normed vector space of mappings from RN into RN . For each g ∈ Θ, for all ε > 0, there exists δ > 0 such that ∀γ ∈ Θ such that γ < δ,

g ◦ (I + γ) − g < ε.

(2.18)

Theorem 2.4. Under Assumptions 2.1 to 2.4, F(Θ) is a topological (metric) group. Example 2.3 (Checking Assumption 2.3). (1), (2), (3) Since the norm on the three spaces B 0 (RN , RN ), C 0 (RN , RN ), and C00 (RN , RN ) is the same and Assumption 2.2 involves only the norm, it is suﬃcient to check it for B 0 (RN , RN ). Consider g ∈ B0 (RN , RN ) and I +f ∈ F(B 0 (RN , RN )). Since I + f is a bijection g ◦ (I + f )C 0 ≤ gC 0 and the constant function c1 (r) = 1 satisﬁes Assumption 2.3.

(2.19)

132

Chapter 3. Courant Metrics on Images of a Set

(4) Θ = C 0,1 (RN , RN ). Consider g ∈ C 0,1 (RN , RN ) and I + f ∈ F(C 0,1 (RN , RN )). From (1) g ◦ (I + f )C 0 ≤ gC 0 and since all the functions are Lipschitz c(g ◦ (I + f )) ≤ c(g) (1 + c(f )) ⇒ g ◦ (I + f )C 0,1 ≤ gC 0,1 (1 + c(f )) ≤ gC 0,1 (1 + f C 0,1 ) and the continuous function c1 (r) = 1 + r satisﬁes Assumption 2.3. Assumption 2.4 is a little trickier. It involves not only norms but also a “uniform continuity” of the function g that is not veriﬁed for g ∈ B0 (RN , RN ) and for the Lipschitz part c(g) of the norm of C 0,1 (RN , RN ). Example 2.4 (Checking Assumption 2.4). (1) Θ = B0 (RN , RN ). In that space there exists some g that is not uniformly continuous. For that g there exists ε > 0 such that for all n > 0, there exists xn , yn , |yn − xn | < 1/n such that |g(yn ) − g(xn )| ≥ ε. Deﬁne the function γn (x) = yn − xn . Therefore, there exists ε > 0 such that for all n with γn C 0 < 1/n, g ◦ (I + γn ) − gC 0 ≥ |g(xn + γn (xn )) − g(xn )| = |g(xn + yn − xn ) − g(xn )| ≥ ε and Assumption 2.4 is not veriﬁed. However, B0 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : B 0 (RN , RN ) → R is continuous. (2) and (3) Θ = C 0 (RN , RN ). By the uniform continuity of g ∈ C 0 (RN , RN ), for all ε > 0 there exists δ > 0 such that ∀x, y ∈ RN ,

|g(y) − g(x)| < ε/2.

For γC 0 < δ ∀x ∈ RN , ∀x ∈ RN ,

|(I + γ)(x) − x| = |γ(x)| ≤ γC 0 < δ,

|g(x + γ(x)) − g(x)| < ε/2

⇒ g ◦ (I + γ) − gC 0 ≤ ε/2 < ε

and Assumption 2.4 is veriﬁed. In addition, C 0 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : C 0 (RN , RN ) → R is continuous. (4) Θ = C 0,1 (RN , RN ). Since g ∈ C 0,1 (RN , RN ) ∀x, y ∈ RN ,

|g(y) − g(x)| < c(g) |y − x|.

Therefore for def

γC 0,1 = max{γC 0 , c(γ)} < δ, γC 0 < δ and we get g ◦ (I + γ) − gC 0 ≤ c(g) γC 0 < c(g) δ.

2. Generic Constructions of Micheletti

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Unfortunately, for the part c(g ◦ (I + γ)) of the C 0,1 -norm, for all x = y ∈ RN |g(y + γ(y)) − g(y) − (g(x + γ(x)) − g(x))| ≤ c(g) + c(g) c(I + γ), |y − x| c(g ◦ (I + γ) − g) ≤ c(g) [1 + c(γ)] < c(g) [1 + δ] and we cannot satisfy Assumption 2.4. However, C 0,1 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : C 0,1 (RN , RN ) → R is continuous. Remark 2.1. Since the spaces C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) have the same norm and Assumptions 2.2 and 2.3 involve only the norms, they will always be veriﬁed for C0k (RN , RN ) and C k (RN , RN ) if they are veriﬁed for B k (RN , RN ), k ≥ 0. Solely Assumption 2.1 will require special attention. The fact that Assumption 2.4 is not always veriﬁed will not prevent us from proving that the space (F(Θ), d) is complete. The completeness will be proved in Theorem 2.6 under Assumptions 2.1 to 2.3, where Assumption 2.4 will be replaced by the condition Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous. The following theorem will be proved in Theorems 2.11, 2.12, and 2.13 of section 2.5 and Theorems 2.15 and 2.16 of section 2.6. Theorem 2.5. Let k ≥ 0 be an integer. (i) C0k (RN , RN ) and C k (RN , RN ) satisfy Assumptions 2.1 to 2.4, C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ C 0 (RN , RN ), and, for all x ∈ RN , the mapping f → f (x) : C k (RN , RN ) → RN is continuous. (ii) B k (RN , RN ) and C k,1 (RN , RN ) verify Assumptions 2.1 to 2.3, C k,1 (RN , RN ) ⊂ B k (RN , RN ) ⊂ C 0 (RN , RN ), and, for all x ∈ RN , the mapping f → f (x) : B k (RN , RN ) → RN is continuous. Proof of Theorem 2.4. From J. L. Kelley [1], it is suﬃcient to prove that ∀F ∈ F(Θ), ∀H ∈ F(Θ) such that d(I, H) → 0,

d(I, F −1 ◦ H ◦ F ) → 0.

Rewrite F −1 ◦ H ◦ F as follows: F −1 ◦ H ◦ F − I = F −1 ◦ [I + H − I] ◦ F − F −1 ◦ F

= F −1 ◦ (I + γ) − F −1 ◦ F

= (F −1 − I) ◦ (I + γ) − (F −1 − I) ◦ F + γ ◦ F, γ = H − I and g = (F −1 − I) ◦ (I + γ) − (F −1 − I). def

def

By Assumption 2.3, γ ◦ F ≤ γ c1 (F − I),

g ◦ F ≤ g c1 (F − I);

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Chapter 3. Courant Metrics on Images of a Set

by Assumption 2.4, for all ε > 0, there exists δ, 0 < δ < min{1, ε/(4c0 (1) c1 (F − I))}, such that γ = H − I < δ c0 (1)

⇒

g =(F −1 − I) ◦ (I + γ) − (F −1 − I) < ε/(4 c1 (F − I)).

Combining the above properties ∀H such that d(I, H) < δ, ⇒ F

−1

H − I < δ c0 (1)

◦ H ◦ F − I ≤ γ c1 (F − I) + ε/2 ≤ δ c0 (1) c1 (F − I) + ε/4 ≤ ε/2.

By Assumption 2.2, for all H such that d(I, H) < δ < 1, we have H − I < δ c0 (1) and H −1 − I < δ c0 (1). Finally, for all ε > 0, there exists δ > 0 such that d(I, H) < δ implies F −1 ◦ H ◦ F − I ≤ ε/2. By a similar argument, for all ε > 0, there exists δ > 0 such that d(I, H) < δ implies F −1 ◦ H −1 ◦ F − I ≤ ε/2. Therefore, for all ε > 0 there exists δ0 = min{δ, δ } > 0 such that for all H, d(I, H) < δ0 , d(I, F −1 ◦ H ◦ F ) ≤ F −1 ◦ H ◦ F − I + F −1 ◦ H −1 ◦ F − I < ε and F(Θ) is a topological group for the metric d. Theorem 2.6. (i) Let Assumptions 2.1 to 2.3 be veriﬁed for a Banach space Θ contained in C 0 (RN , RN ) such that, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous. Then (F(Θ), d) is a complete metric space. (ii) Let Assumptions 2.1 to 2.4 be veriﬁed for a Banach space Θ. Then F(Θ) is a complete topological (metric) group. Remark 2.2. F(C 0,1 (RN , RN )) is a complete metric space since the assumptions of Theorem 2.6 are veriﬁed, but it is not a topological group since Assumption 2.4 is not veriﬁed. Proof of Theorem 2.6. (i) Let {Fn } be a Cauchy sequence in F(Θ). (a) Boundedness of {Fn − I} and {Fn−1 − I} in Θ. For each ε, 0 < ε < 1, there exists N such that, for all m, n > N , d(Fm , Fn ) < ε. By the triangle inequality |d(I, Fm ) − d(I, Fn )| ≤ d(Fm , Fn ) < ε, {d(I, Fn } is Cauchy, and, a posteriori, is bounded by some constant L. For each n, n ) such that there exists a factorization Fn = (I + f1n ) ◦ · · · ◦ (I + fν(n)

ν(n)

0≤

fin + fin ◦ (I + f1n )−1 − d(I, Fn ) < ε < 1

i=1

ν(n)

⇒

i=1

fin + fin ◦ (I + fin )−1 < L + ε < L + 1.

2. Generic Constructions of Micheletti

135

By Assumption 2.2 Fn − I + Fn−1 − I < 2(L + ε) c0 (L + 1) and the sequences {Fn − I} and {Fn−1 − I} are bounded in Θ. (b) Convergence of {Fn − I} and {Fn−1 − I} in Θ. For any m, n −1 Fn − Fm = (Fn ◦ Fm − I) ◦ Fm

and by Assumption 2.3 −1 −1 Fm − Fn = (Fn ◦ Fm − I) ◦ Fm ≤ Fn ◦ Fm − I c1 (Fm − I) −1 ⇒ Fm − Fn ≤ c+ 1 Fn ◦ Fm − I,

def

c+ 1 = sup c1 (Fm − I) < ∞ m

since {Fm − I} is bounded. For all ε, 0 < ε < c0 (1) c+ 1 , there exists N such that ∀n, m, > N,

−1 d(I, Fn ◦ Fm ) = d(Fm , Fn ) < ε/(c0 (1) c+ 1 ) < 1.

By Assumption 2.2 + −1 Fn ◦ Fm − I < c0 (1)ε/(c0 (1) c+ 1 ) = ε/c1

⇒ ∀n, m, > N,

−1 Fm − I − (Fn − I) ≤ c+ 1 Fn ◦ Fm − I < ε

and {Fn − I} is a Cauchy sequence in Θ that converges to some F − I ∈ Θ since Θ is a Banach space. Similarly, we get that {Fn−1 − I} is a Cauchy sequence in Θ that converges to some G − I ∈ Θ. (c) F is bijective, G = F −1 , and F ∈ F(Θ). For all n, Fn , Fn−1 , F , and G are continuous since Θ ⊂ C 0 (RN , RN ). By continuity of f → f (x) : Θ → RN , f → x + f (x) : Θ → RN is continuous and [Fn−1 (x) − x| + x → [G(x) − x] + x. By the same argument for Fn , x = Fn (Fn−1 (x)) → F (G(x)) = (F ◦ G)(x) and for all x ∈ RN , (F ◦ G)(x) = x. Similarly, starting from Fn−1 ◦ Fn = I, we get (G ◦ F )(x) = x. Hence, G ◦ F = I = F ◦ G. So F is bijective, F − I, F −1 − I ∈ Θ, and F ∈ F(Θ). (d) Convergence of Fn → F in F(Θ). To check that Fn → F in F(Θ) recall that by Assumption 2.3 d(Fn , F ) = d(I, F ◦ Fn−1 ) ≤I − F ◦ Fn−1 + I − Fn ◦ F −1 = (Fn − F ) ◦ Fn−1 + (F − Fn ) ◦ F −1 ≤ c1 (Fn−1 − I)Fn − F + c1 (F −1 − I)F − Fn ≤ max c1 (Fn−1 − I) + c1 (F −1 − I) F − Fn , n

which goes to zero as Fn − I goes to F − I in Θ. (ii) The proof (ii) diﬀers from the proof of (i) only in part (c). Consider G ◦ F − I = G ◦ F − G ◦ Fn + (G − Fn−1 ) ◦ Fn .

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Chapter 3. Courant Metrics on Images of a Set

Since {Fn − I} is bounded in Θ and Fn−1 → G, the term (G − Fn−1 ) ◦ Fn goes to 0 as n → ∞ by Assumption 2.3. The second terms can be rewritten as G ◦ F − G ◦ Fn = F − Fn + (G − I) ◦ F − (G − I) ◦ Fn = F − Fn + g ◦ F − g ◦ Fn , where g = G − I ∈ Θ. The term F − Fn goes to zero. As for the second terms

g ◦ F − g ◦ Fn = g ◦ [I + (F − Fn ) ◦ Fn−1 ] − g ◦ Fn . Let gn = g ◦ [I + (F − Fn ) ◦ Fn−1 ] − g and let γn = (F − Fn ) ◦ Fn−1 . By boundedness of {Fn − I} and {Fn−1 − I} in Θ and Assumption 2.3, γn = (F − Fn ) ◦ Fn−1 ≤ F − Fn c1 (Fn−1 − I) ≤ c+ 1 F − Fn , gn ◦ Fn ≤ gn c1 (Fn − I) ≤ c+ 1 gn , −1 c+ 1 = sup{c1 (Fn − I), c1 (Fn − I)} < ∞. def

n

By Assumption 2.4, for all ε > 0 there exists δ > 0 such that γn < δ c+ 1

⇒ gn = g ◦ (I + γn ) − g < ε/c+ 1.

By combining for all ε > 0, there exists δ > 0 such that F − Fn < δ

+ ⇒ γn ≤ c+ 1 F − Fn < c1 δ + ⇒ gn ◦ Fn < c+ 1 ε/c1 = ε ⇒ F − Fn < δ ⇒ g ◦ F − g ◦ Fn < ε.

As a result G ◦ F − I is bounded by an expression that goes to zero as n → ∞ and G ◦ F = I. By an analogous argument, we can prove that F ◦ G − I = 0, that is, F −1 = G. So F is bijective, F − I, F −1 − I ∈ Θ. Hence F ∈ F(Θ).

2.2

Diﬀeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

Consider as other candidates for Θ the spaces k N N ∞ k N N B(RN , RN ) = ∩∞ k=0 B (R , R ) = ∩k=0 C (R , R ),

(2.20)

k N N C0∞ (RN , RN ) = ∩∞ k=0 C0 (R , R )

(2.21)

def

def

and the resulting groups F(B(RN , RN ))

I + f : f ∈ B(RN , RN ), (I + f ) bijective and (I + f )−1 − I ∈ B(RN , RN ) and F(C0∞ (RN , RN ))

I + f : f ∈ C0∞ (RN , RN ), (I + f ) bijective and (I + f )−1 − I ∈ C0∞ (RN , RN )

2. Generic Constructions of Micheletti

137

that satisfy Assumption 2.1 but not the other assumptions since B(RN , RN ) and C0∞ (RN , RN ) are Fr´echet but not Banach spaces. To get around this, observe that k N N ∞ k N N F(B(RN , RN )) = ∩∞ k=0 F(B (R , R )) = ∩k=0 F(C (R , R )), k N N F(C0∞ (RN , RN )) = ∩∞ k=0 F(C0 (R , R )),

where (F(C k (RN , RN )), dk ) and (F(C0k (RN , RN )), dk ) are topological (metric) groups with the same metrics {dk } and the following monotony property: dk (F, G) ≤ dk+1 (F, G) resulting from the monotony of the norms f C k = max f C i ≤ 0≤i≤k

max f C i = f C k+1 .

0≤i≤k+1

Theorem 2.7. The spaces F(B(RN , RN )) and F(C0∞ (RN , RN )) are complete topological (metric) groups for the distance def

d∞ (F, G) =

∞ 1 dk (F, G) , 2k 1 + dk (F, G)

(2.22)

k=0

where dk is the metric associated with C k (RN , RN ) and C0k (RN , RN ).8 This theorem is a consequence of the following general lemma. Lemma 2.2. Assume that (Fk , dk ), k ≥ 0, is a family of groups each complete with respect to their metric dk and that ∀k ≥ 0,

Fk+1 ⊂ Fk

and

dk (F, G) ≤ dk+1 (F, G), ∀F, G ∈ ∩∞ k=0 Fk .

(2.23)

Then the intersection F∞ = ∩∞ k=0 Fk is a group that is complete with respect to the metric d∞ deﬁned in (2.22). If, in addition, each (Fk , dk ) is a topological (metric) group, then F∞ is a topological (metric) group. Proof. By deﬁnition, F∞ is clearly a group. (i) d∞ is a metric on F∞ . d∞ (F, G) = 0 implies that dk (F, G) = 0 for all k and F = G and conversely. Also d∞ (F, G) = d∞ (G, F ). The function x/(1 + x) is monotone strictly increasing with x ≥ 0. Therefore, dk (F, H) ≤ dk (F, G)+dk (G, H) implies dk (F, H) dk (F, G) dk (G, H) ≤ + 1 + dk (F, H) 1 + dk (F, G) + dk (G, H) 1 + dk (F, G) + dk (G, H) dk (G, H) dk (F, G) + , ≤ 1 + dk (F, G) 1 + dk (G, H) ∞ ∞ ∞ 1 dk (F, H) 1 dk (F, G) 1 dk (G, H) ≤ + 2k 1 + dk (F, H) 2k 1 + dk (F, G) 2k 1 + dk (G, H) k=0

k=0

k=0

and d∞ (F, H) ≤ d∞ (F, G) + d∞ (G, H). 8 In view of (2.20), it is important to note that F (B(RN , RN )) is a topological group by using the family (F (C k (RN , RN )), dk ) in Lemma 2.2 and not the family (F (B k (RN , RN )), dk ).

138

Chapter 3. Courant Metrics on Images of a Set

(ii) (F∞ .d∞ ) is complete. Let {Fn } be a Cauchy sequence in F∞ ; that is, for all ε, 0 < ε < 1/2, there exists N0 such that for n, m > N0 , d∞ (Fn , Fm ) < ε. In particular ∞

1 dk (Fn , Fm ) d0 (Fn , Fm ) ≤ N1 , d∞ (Fn , Fm ) < ε/22 . Then ∞

1 dk (Fn , Fm ) 1 1 d1 (Fn , Fm ) ≤ < 2 ε ⇒ d1 (Fn , Fm ) < ε/(1 − ε) < 2ε 2 1 + d1 (Fn , Fm ) 2k 1 + dk (Fn , Fm ) 2 k=0

and {Fn } is a Cauchy sequence in F1 that converges to some G1 in F1 . By continuous injection of F1 into F0 , G1 = F ∈ F0 ∩ F1 . Proceeding in that way, we get that there exists F ∈ F∞ = ∩∞ k=0 Fk such that Fn → F in Fk for all k ≥ 0. Therefore, F∞ is complete.

2.3

Closed Subgroups G

Given an arbitrary nonempty subset Ω0 of RN , consider the family def

X (Ω0 ) = F (Ω0 ) ⊂ RN : ∀F ∈ F(Θ)

(2.24)

of images of Ω0 by the elements of F(Θ). By introducing the subgroup def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 } ,

(2.25)

[F ] → F (Ω0 ) : F(Θ)/G(Ω0 ) → X (Ω0 )

(2.26)

we obtain a bijection

between the set of images X (Ω0 ) and the quotient space F(Θ)/G(Ω0 ). Using this bijection, the topological structure of X (Ω0 ) will be identiﬁed with the topological structure of the quotient space. Lemma 2.3. Let Assumptions 2.1 to 2.3 hold with Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , let the mapping f → f (x) : Θ → RN be continuous.9 If the nonempty subset Ω0 of RN is closed or if Ω0 = int Ω0 ,10 the family def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

(2.27)

is a closed subgroup of F(Θ). 9 This 10 Such

means that F(Θ) ⊂ I + Θ ⊂ Hom(RN , RN ). a set will be referred to as a crack-free set in Deﬁnition 7.1 (ii) of Chapter 8. Indeed,

by deﬁnition Ω is crack-free if Ω = Ω. If, in addition, Ω is open, then Ω = Ω = Ω and Ω = Ω = int Ω.

2. Generic Constructions of Micheletti

139

Proof. G(Ω0 ) is clearly a subgroup of F(Θ). It is suﬃcient to show that for any sequence {Fn } in G(Ω0 ) that converges to F in F(Θ), then F ∈ G(Ω0 ). The other properties are straightforward. If {Fn } converges to F in F(Θ), then by Assumption 2.3 we have Fn ◦ F −1 − I + F ◦ Fn−1 − I → 0. From F ◦ Fn−1 − I → 0, for each x ∈ Ω0 the sequence {F (Fn−1 (x))} ⊂ F (Ω0 ) converges to x ∈ Ω0 and hence Ω0 ⊂ F (Ω0 ). From the convergence Fn ◦F −1 −I → 0 for each x ∈ F (Ω0 ), F −1 (x) ∈ Ω0 and the sequence {Fn (F −1 (x))} ⊂ Ω0 converges to x ∈ F (Ω0 ) and hence F (Ω0 ) ⊂ Ω0 . Therefore, Ω0 ⊂ F (Ω0 ) ⊂ Ω0 and Ω0 = F (Ω0 ). Since F is a homeomorphism, for all A, F (A) = F (A) and F (A) = F (A). As a result int F (A) = F (int A). In view of the identity Ω0 = F (Ω0 ), F (Ω0 ) = F (Ω0 ) = Ω0

and

int Ω0 = int F (Ω0 ) = F (int Ω0 ).

From this, if Ω0 is closed, Ω0 = Ω0 and F (Ω0 ) = Ω0 . If Ω0 = int Ω0 , we also get F (Ω0 ) = Ω0 . In both cases F ∈ G(Ω0 ). Remark 2.3. A. M. Micheletti [1] assumes that Ω0 is a bounded connected open domain of class C 3 in order to make all the images F (Ω0 ) bounded connected open domains of class C 3 with Θ = C03 (RN , RN ). Open domains of class C k , k ≥ 1, are locally C k epigraphs and domains that are locally C 0 epigraphs are crack-free11 by Theorem 5.4 (ii) of Chapter 2. Remark 2.4. F. Murat and J. Simon [1] were mainly interested in families of images of locally Lipschitzian (epigraph) domains. Recall that F(C 1 (RN , RN )) transports locally Lipschitzian (epigraph) domains onto locally Lipschitzian (epigraph) domains (cf. A. Bendali [1] and A. Djadane [1]), but that F(C 0,1 (RN , RN )) does not, as shown in Examples 5.1 and 5.2 of section 5.3 in Chapter 2. Of special interest are the closed submanifolds of RN of codimension greater than or equal to 1. For instance, we can choose Ω0 = S n , 2 ≤ n < N , the n-sphere of radius 1 centered in 0. We get a family of closed curves in R3 for n = 2 and of closed surfaces in R3 for n = 3. Similarly, by choosing Ω0 = B1n (0), the closed unit ball of dimension n, we get a family of curves for n = 1 and a family of surfaces for n = 2. The choice of [0, 1]n yields a family of surfaces with corners for n = 2. Explicit expressions can be obtained for normals, curvatures, and density measures as in Chapter 2. Example 2.5 (Closed surfaces in R3 ). For Ω0 = S 2 in R3 , f ∈ C 1 (R3 , R3 ), and F = I + f , the tangent ﬁeld at the point ξ ∈ S 2 is Tξ S 2 = {ξ}⊥ and the unit normal is ξ. At a point x = F (ξ) of the image 11 Cf.

Deﬁnition 7.1 (ii) of Chapter 8.

140

Chapter 3. Courant Metrics on Images of a Set

F (S 2 ), the tangent space and the unit normal are given by TF (ξ) F (S 2 ) = DF (ξ){ξ}⊥ and n(F (x)) =

∗

DF (ξ)−1 ξ . | ∗DF (ξ)−1 ξ|

Example 2.6 (Curves in RN ). Let eN be a unit vector in RN , let H = {eN }⊥ , and let def

Ω0 = {ζN eN : ζN ∈ [0, 1]} ⊂ RN . For f ∈ C 1 (RN , RN ), and F = I + f , the tangent ﬁeld at the point ξ ∈ Ω0 is Tξ Ω0 = R eN and the normal space (Tξ Ω0 )⊥ is H. At a point x = F (ξ) of the curve F (Ω0 ), the tangent and normal spaces are given by TF (ξ) F (Ω0 ) = R DF (ξ)eN

TF (ξ) F (Ω0 )⊥ = {DF (ξ)eN }⊥ .

and

In practice, one might want to use other subgroups, such as all the translations by some vector a ∈ RN : def

Fa (x) = a + x, ⇒

Fa−1 (x)

x ∈ RN ,

− a + x = F−a (x)

⇒ Fa − I = a ∈ Θ ⇒

Fa−1

− I = −a ∈ Θ

(2.28) (2.29)

for all Θ that contain the constant functions such as Bk (RN , RN ), C k (RN , RN ), and C k,1 (RN , RN ), but not C0k (RN , RN ). Therefore def

Gτ = Fa : ∀a ∈ RN

(2.30)

is a subgroup of F(Θ). In that case, it is also a closed subgroup of F(Θ): Fa ◦ Fb (x) = a + b + x = Fa+b (x),

F0 = I

and (Gτ , ◦) is isomorphic to the one-dimensional closed abelian group (RN , +). This means that all the topologies on Gτ are equivalent and (Gτ , ◦) is closed in F(Θ).

2.4

Courant Metric on the Quotient Group F (Θ)/G

When F(Θ) is a topological right-invariant metric group and G is a closed subgroup of F(Θ), the quotient metric def

dG (F ◦ G, H ◦ G) =

˜ inf d(F ◦ G, H ◦ G)

˜ G,G∈G

(2.31)

induces a complete metric topology on the quotient group F(Θ)/G.12 The quotient metric dG(Ω0 ) with Θ = C0k (RN , RN ) was referred to as the Courant metric by 12 Cf.,

for instance, D. Montgomery and L. Zippin [1 sect. 1.22, p. 34, sect. 1.23, p. 36])

(i) If G is a topological group whose open sets at e have a countable basis, then G is metrizable and, moreover, there exists a metric which is right-invariant. (ii) If G is a closed subgroup of a metric group F, then F/G is metrizable.

2. Generic Constructions of Micheletti

141

A. M. Micheletti [1]. In that case F(Θ) is a topological (metric) group. However, we have seen that for Θ = C 0,1 (RN , RN ) and B0 (RN , RN ), F(Θ) is not a topological group. Yet, the completeness of (F (Θ), d) and the right-invariance of d are suﬃcient to recover the result for an arbitrary closed subgroup G of F. It is important to have the weakest possible assumptions on G to secure the widest range of applications. Theorem 2.8. Assume that (F, d) is a group with right-invariant metric d that is complete for the topology generated by d. For any closed subgroup G of F, the function dG : F × F → R, def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G), G∈G

(2.32)

is a right-invariant metric on F(Θ)/G and the space (F/G, dG ) is complete. The topology induced by dG coincides with the quotient topology of F/G. Proof. (i) F/G is clearly a group with unit element G under the composition law def

[F ◦ G] ◦ [H ◦ G] = [F ◦ H] ◦ G. (ii) (dG is a metric). Since d is right-invariant, the deﬁnition (2.31) is equivalent to def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G). G∈G

(2.33)

Since d is symmetrical, so is dG . For the triangle inequality let g1 , g2 ∈ G: dG (F ◦ G, H ◦ G) ≤ d(F, H ◦ g1 ◦ g2 ) ≤ d(F, G ◦ g2 ) + d(G ◦ g2 , H ◦ g1 ◦ g2 ) ≤ d(F, G ◦ g2 ) + d(G, H ◦ g1 ) by right-invariance. The triangle inequality for dG follows by taking the inﬁma over g1 , g2 ∈ G. Since F/G is a group, it remains to show that dG (I ◦ G, H ◦ G) = 0 if and only if H ∈ G. Clearly dG (I ◦G, I ◦G) ≤ d(I, I) = 0. Conversely, let dG (I ◦G, H ◦G) = 0. For all n > 0, there exists gn ∈ G such that d(gn−1 , H) = d(I, H ◦ gn ) < 1/n. Therefore gn−1 → H in F(Θ). But since G is closed limn→∞ gn−1 ∈ G and H ∈ G. In particular, the canonical mapping F → π(F ) = F ◦ G : F(Θ) → F(Θ)/G is continuous since dG (I ◦ G, F ◦ G) ≤ d(I, F ) for all F ∈ F(Θ). (iii) (Completeness). Consider a Cauchy sequence {Fn ◦ G} in F/G. It is suﬃcient to show that there exists a subsequence which converges to some F ◦ G ∈ F(Θ)/G. Construct the subsequence {Fν ◦ G} such that dG (Fν ◦ G, Fν+1 ◦ G) < 1/(2ν ). Then there exists a sequence {Hν } in F such that (i) Hν ∈ Fν ◦ G and (ii) d(Hν , Hν+1 ) < 1/(2ν ). We proceed by induction. By deﬁnition of dG , if dG (F1 ◦ G, F2 ◦G) < 1/2, there exist H1 ∈ F1 ◦G and H2 ∈ F2 ◦G such that d(H1 , H2 ) < 1/2. Similarly, if Hν ∈ Fν ◦ G, it follows that there exists Hν+1 with the required property. Because dG (Fν ◦ G, Fν+1 ◦ G) < 1/2ν , there exist G1 and G2 in G such that d(Fν ◦ G1 , Fν+1 ◦ G2 ) < 1/2ν . If G3 is such that Fν ◦ G1 = Hν ◦ G3 , then we can

142

Chapter 3. Courant Metrics on Images of a Set

choose Hν+1 = Fν+1 ◦ G2 ◦ G−1 3 since d(Hν , Fν+1 ◦ G2 ◦ G−1 3 ) = d(Hν ◦ G3 , Fν+1 ◦ G2 ) = d(Fν ◦ G1 , Fν+1 ◦ G2 ). It is easy to check that {Hν } is a Cauchy sequence in F. Indeed ∀i < j,

d(Hi , Hj ) ≤

j−1

d(Hn , Hn+1 ) ≤

n=i

1 1 1 + · · · + j ≤ i−1 . 2i 2 2

By completeness of F, {Hν } converges to some H in F. From (ii) the canonical map π : F → F/G is continuous. So the sequence {π(Hν ) = Fν ◦ G} converges to π(H) = H ◦ G in F/G. R (iv) Associate with G the equivalence relation F ∼ H if F ◦ H −1 ∈ G. By deﬁnition, the quotient topology on F/R is the ﬁnest topology T on F/R for which the canonical map F → π(F ) : F → F/R is continuous. But, by deﬁnition, F/R = F/G and, from part (ii), π is also continuous for the right-invariant quotient metric. Therefore, the identity map i : (F/R, T ) → (F/G, dG ) is continuous. The topology T in [I] is deﬁned by the fundamental system of neighborhoods {π(F ) : d(F, I) < ε}. But dG (π(F ), π(I)) ≤ d(F, I) < ε and i({π(F ) : d(F, I) < ε}) ⊂ {π(F ) : dG ([F ], [I]) < ε} that is a neighborhood of [I] in the (F/G, dG ) topology. Therefore i−1 is continuous and the two topologies coincide. We now have all the ingredients to conclude. Theorem 2.9. Let Θ be any of the spaces Bk (RN , RN ), C k (RN , RN ), C0k (RN , RN ), C k,1 (RN , RN ), k ≥ 0, or C ∞ (RN , RN ), C0∞ (RN , RN ). (i) The group (F(Θ), d) is a complete right-invariant metric space. For Θ equal to C k (RN , RN ) or C0k (RN , RN ), 0 ≤ k ≤ ∞, (F(Θ), d) is also a topological group. (ii) For any closed subgroup G of F(Θ), the function dG : F(Θ) × F(Θ) → R, def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G), G∈G

(2.34)

is a right-invariant metric on F(Θ)/G and the space (F(Θ)/G, dG ) is complete. The topology induced by dG coincides with the quotient topology of F(Θ)/G. (iii) Let Ω0 , ∅ = Ω0 ⊂ RN , be closed or let Ω0 = int Ω0 . Then def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

(2.35)

is a closed subgroup of F(Θ). Finally, it is interesting to recall that A. M. Micheletti [1] used the quotient metric to prove the following theorem.

2. Generic Constructions of Micheletti

143

Theorem 2.10. Fix k = 3, the Courant metric of F(C03 (RN , RN ))/G(Ω0 ), and a bounded open connected domain Ω0 of class C 3 in RN . Let X (Ω0 ) be the family of images of Ω0 by elements of F(C03 (RN , RN ))/G(Ω0 ). The subset of all bounded open domains Ω in X (Ω0 ) such that the spectrum of the Laplace operator −∆ on Ω with homogeneous Dirichlet conditions on the boundary ∂Ω does not have all its eigenvalues simple is of the ﬁrst category.13 This theorem says that, up to an arbitrarily small perturbation of the domain, the eigenvalues of the Laplace operator of a C 3 -domain can be made simple.

2.5

Assumptions for Bk (RN , RN ), C k (RN , RN ), and C0k (RN , RN )

2.5.1

Checking the Assumptions

By deﬁnition, for all integers k ≥ 0 C0k (RN , RN )) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) ⊂ B0 (RN , RN ) ⊂ C 0 (RN , RN ) ∀x ∈ RN ,

f → f (x) : B0 (RN , RN ) → RN

with continuous injections between the spaces and continuity of the evaluation map. The spaces are all Banach spaces endowed with the same norm:14 def

f Ck =

max ∂ α f C .

0≤|α|≤k

As a consequence, the evaluation map f → f (x) is continuous for all the spaces. The following results are quoted directly from A. M. Micheletti [1] and apply to all spaces Θ ⊂ C k (RN , RN ). Denote by S(x)[y1 ][y2 ], . . . , [yk ] at the point x ∈ RN a k-linear form with arguments y1 , . . . , yk . Lemma 2.4. Assume that F and G are two mappings from RN to RN such that F is k-times diﬀerentiable in an open neighborhood of x and G is k-times diﬀerentiable in an open neighborhood of F (x). Then the kth derivative of G ◦ F in x is the sum of a ﬁnite number of k-linear applications on RN of the form . / (h1 , h2 , . . . , hk ) → G() (F (x)) F (λ1 ) (x)[h1 ][h1 ] . . . [hλ1 ] / . (2.36) . . . F (λ ) (x)[hk−λ +1 ] . . . [hk ] , where = 1, . . . , k and λ1 + · · · + λ = k. 13 A set B is said to be nowhere dense if its closure has no interior or, alternatively, if B is dense. A set is said to be of the ﬁrst category if it is the countable union of nowhere dense sets (cf. J. Dugundji [1, Def. 10.4, p. 250]). 14 Here

RN is endowed with the norm |x| =

)N

i=1

x2i

1/2

.

144

Chapter 3. Courant Metrics on Images of a Set

Proof. We proceed by induction on k. The result is trivially true for k = 1. Assuming that it is true for k − 1, we prove that it is also true for k. This is an obvious consequence of the observation that for a mapping from RN into L((RN )k−1 , RN ) of the form x → G() (F (x))[F (λ1 ) (x)] . . . [F (λ ) (x)], where = 1, . . . , k − 1 and λ1 + · · · + λ = k − 1, the diﬀerential in x is G(+1) (F (x))[F (1) ][F (λ1 ) (x)] . . . [F (λ ) (x)] + G() (F (x))[F (λ1 +1) (x)] . . . [F (λ ) (x)] + · · · + G() (F (x))[F (λ1 ) (x)] . . . [F (λ +1) (x)], (2.37) and the result is true for k. Lemma 2.5. Given g and f in C k (RN , RN ), let ψ = g ◦ (I + f ). Then for each x ∈ RN |ψ(x)| = |g(x + f (x))|, |ψ

(1)

(x)| ≤ |g (1) (x + f (x))| [1 + |f (1) (x)|],

|ψ (i) (x)| ≤ |g(1) (x + f (x))| |f (i) (x)| +

i

|g (j) (x + f (x))| aj (|f (1) (x)|, . . . , |f (i−1) (x)|)

j=2

for i = 2, . . . , k, where aj is a polynomial with positive coeﬃcients. Proof. This is an obvious consequence of Lemma 2.4. Theorem 2.11. Assumptions 2.1 and 2.3 are veriﬁed for the spaces C0k (RN , RN ), C k (RN , RN ), and B k (RN , RN ), k ≥ 0. Proof. (i) Assumption 2.3. From Lemma 2.5 it readily follows that f ◦ (I + g) ∈ Bk (RN , RN ) for all f ∈ B k (RN , RN ) and I + g in F(B k (RN , RN )). Moreover, ψC = f C , ψ

(1)

C ≤ f (1) C (1 + g (1) C ),

ψ (i) C ≤ f (1) C g (i) C +

i

f (j) C aj (g (1) C , . . . , g (i−1) C )

j=2

for i = 2, . . . , k, where aj is a polynomial. Hence ψC k = max ψ i C ≤ f C k c1 (gC k ) i=0,...,k

for some polynomial function c1 that depends only on k.

2. Generic Constructions of Micheletti

145

(ii) Assumption 2.1. The case of Bk (RN , RN ) is a consequence of (i). For C (RN , RN ) and each i, ψ(i) is uniformly continuous as the ﬁnite sum and product of compositions of uniformly continuous functions by Lemma 2.4. As for C0k (RN , RN ), for all I + g ∈ F(C0k (RN , RN )) and f ∈ C0k (RN , RN ), h = f ◦ (I + g) ∈ C k (RN , RN ) from the previous case. It remains to show that h ∈ C0k (RN , RN ). This amounts to showing that |h(x)| and |h(i) (x)| go to zero as |x| → ∞, i = 1, . . . , k. We again proceed by induction on k. Since |g(x)| → 0 as |x| → ∞, |x + g(x)| ≥ |x| − |g(x)| → ∞ as |x| → ∞ and |h(x)| = |f (x + g(x))| → 0 as |x| → ∞ and Assumption 2.1 is true for k = 0. Always, from Lemma 2.5 and the identity h(x) = f (x + g(x)), k

|h(1) (x)| ≤ |f (1) (x)| [1 + |g (1) (x)|] → 0 as |x| → 0 and Assumption 2.1 is true for k = 1. For k ≥ 2 and 2 ≤ i ≤ k, again, from Lemma 2.5 |h(i) (x)| ≤ |f (1) (x + g(x))| g (i) C +

i

|f (j) (x + g(x))| aj (g (1) C , . . . , g (i−1) C )

j=2

for i = 2, . . . , k. Since for each j, |f (j) (x)| → 0 as |x| → ∞, by the same argument, |f (j) (x + g(x))| → 0 as |x| → ∞ and hence the whole right-hand side goes to 0. Theorem 2.12. Assumption 2.2 is veriﬁed for the spaces C0k (RN , RN ), C k (RN , RN ), and Bk (RN , RN ), k ≥ 0. It is a consequence of the following lemma of A. M. Micheletti [1]. Lemma 2.6. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0 with the following property: if the sequence f1 , . . . , fn in C r (RN , RN ) is such that n fi C r < α, 0 < α < s, (2.38) i=1

then for the map F = (I + fn ) ◦ · · · ◦ (I + f1 ), F − IC r ≤ α c(r, s).

(2.39)

Proof. We again proceed by induction on r. For simplicity deﬁne def

Fi = (I + fi ) ◦ · · · ◦ (I + f1 ),

Fn = F.

By the deﬁnition of F we have F − I = f1 + f2 ◦ (I + f1 ) + · · · + fn ◦ (I + fn−1 ) ◦ · · · ◦ (I + f1 ) = f1 + f2 ◦ F1 + · · · + fn ◦ Fn−1 .

(2.40)

146

Chapter 3. Courant Metrics on Images of a Set

)n Then F − IC 0 ≤ i=1 fi C 0 ≤ α and c(0, s) = 1 for all integers s > 0. From (2.40) and Lemma 2.5 we further get sup |(F − I)(1) (x)| ≤ f1 C 1 + f2 C 1 [1 + f1 C 1 ] x

≤

n i=1

fi C 1

n (

+ · · · + fn C 1 [1 + fn−1 C 1 ] . . . [1 + f1 C 1 ] (2.41) 0 n 1 )n (1 + fi C 1 ) ≤ fi C 1 e( i=1 fi C 1 ) ≤ α eα ≤ α es

i=1

i=1

and F − IC 1 ≤ α eα . Hence c(1, s) = es for all integers s > 0. We now show that if the result is true for r − 1, it is true for r. We have to evaluate |(Fn − I)(r) (x)|. It is obvious that Fn − I = (I + fn ) ◦ Fn−1 − I = Fn−1 − I + fn ◦ Fn−1 and hence (Fn − I)(r) (x) = (Fn−1 − I)(r) (x) + (fn ◦ Fn−1 )(r) (x).

(2.42)

For r ≥ 2 from Lemma 2.5 we have |(fn ◦ Fn−1 )(r) (x)| ≤ |fn(1) (Fn−1 (x))| |(Fn−1 − I)(r) (x)| r |fn(j) (Fn−1 (x))| aj (|(Fn−1 − I)(1) (x)|, . . . , |(Fn−1 − I)(r−1) (x)|). + j=2

By the induction assumption |(Fn−1 − I)(i) (x)| ≤ (Fn−1 − I)C r−1 ≤ α c(r − 1, s), i = 1, . . . , r − 1, and the fact that aj is a polynomial dependent on r for j = 2, . . . , r, there exists a constant L(r, s) such that |(fn ◦ Fn−1 )(r) (x)| r |fn(j) (Fn−1 (x))| ≤ |fn(1) (Fn−1 (x))| |(Fn−1 − I)(r) (x)| + L(r, s) j=2

(2.43)

≤ fn C r |(Fn−1 − I)(r) (x)| + (r − 1)L(r, s)fn C r . Deﬁne M (r, s) = (r − 1)L(r, s). From (2.42) and (2.43), we get |(Fn − I)(r) (x)| ≤ [1 + fn C r ] |(Fn−1 − I)(r) (x)| + M (r, s)fn C r . After repeating this procedure n − 1 times we have |(Fn − I)(r) (x)| ≤ [1 + fn C r ] . . . [1 + f2 C r ]f1 C r + [1 + fn C r ] . . . [1 + f2 C r ]M (r, s)f2 C r + · · · + M (r, s)fn C r ≤ max{M (r, s), 1} eα α. Henceforth c(r, s) = max {M (r, s), 1}es . Theorem 2.13. Assumption 2.4 is veriﬁed for C k (RN , RN ) and C0k (RN , RN ). It is a consequence of the following slightly modiﬁed versions of lemmas of A. M. Micheletti [1].

2. Generic Constructions of Micheletti

147

Lemma 2.7. Given f ∈ C k (RN , RN ) and γ ∈ C 0 (RN , RN ), for all ε > 0, there exists δ > 0 such that ∀γ, γC 0 < δ,

sup f (i) ◦ (I + γ) − f (i) C 0 < ε.

(2.44)

0≤i≤k

Proof. Since f and its derivatives f (i) all belong to C 0 (RN , RN ), they are uniformly continuous functions. Hence, for each ε > 0 there exists δ > 0 such that ∀x, y ∈ RN , |x − y| < δ,

sup |f (i) (y) − f (i) (x)| < ε. 0≤i≤k

In particular for γ ∈ C 0 (RN , RN ) such that γC 0 < δ |x + γ(x) − x| = |γ(x)| ≤ γC 0 < δ

⇒ sup |f (i) (x + γ(x)) − f (i) (x)| < ε 0≤i≤k

and we get the result. Lemma 2.8. For each g ∈ C k (RN , RN ), for all ε > 0, there exists δ > 0 such that ∀γ ∈ C k (RN , RN ) such that γC k < δ,

g ◦ (I + γ) − gC k < ε.

(2.45)

Proof. Again proceed by induction on k. For k = 0 the result is a consequence of Lemma 2.7. Then check that if it is true for k − 1 it is true for k. From Lemma 2.4 the kth derivative of g ◦(I +γ)−g is the sum of a ﬁnite number of k-linear mappings of the form g (r) (x + γ(x))[(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] − g (r) (x)[I (λ1 ) (x)] . . . [I (λr ) (x)] = g (r) (x + γ(x)) − g (r) (x) [(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] + g (r) (x)[(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] − g (r) (x)[I (λ1 ) (x)] . . . [I (λr ) (x)], where r = 1, . . . , k and λ1 + · · · + λr = k. Since |(I + γ)(λ1 ) (x)| |(I + γ)(λ2 ) (x)| . . . |(I + γ)(λr ) (x)| ≤ (γC k + 1)r , the norm of the mapping can be bounded above by the following expression: sup |g (r) (x + γ(x)) − g (r) (x)| (γC k + 1)r + γC k p(γC k , gC k ), x

(2.46)

where p is a polynomial. From this upper bound and Lemma 2.7 the result of the lemma is true for k. 2.5.2

Perturbations of the Identity and Tangent Space

By construction F(Θ) ⊂ I + Θ

148

Chapter 3. Courant Metrics on Images of a Set

and the tangent space in each point of the aﬃne space I + Θ is Θ. In general, I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there is a suﬃciently small ball B around 0 in Θ such that the so-called perturbations of the identity, I + B, are contained in F(Θ). In particular, for θ ∈ B, the family of transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C 1 -path in F(Θ) and the tangent space to each point of F(Θ) can be shown to be exactly Θ. Theorem 2.14. Let Θ be equal to C0k (RN , RN ), C k (RN , RN ), or B k (RN , RN ), k ≥ 1. (i) The map f → I + f : B(0, 1) ⊂ Θ → F(Θ) is continuous. (ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ. Remark 2.5. The generic framework of Micheletti is similar to an inﬁnite-dimensional Riemannian manifold. Proof. (i) For k ≥ 1 and f suﬃciently small, F is bijective and has a unique inverse. Given y ∈ RN , consider the mapping def

S(x) = y − f (x),

x ∈ RN .

Then for any x1 and x2 , S(x2 ) − S(x1 ) = −[f (x2 ) − f (x1 )] ⇒ |S(x2 ) − S(x1 )| ≤ |f (x2 ) − f (x1 )| ≤ f (1) C 0 |x2 − x1 |. For f (1) C 1 < 1, S is a contraction and, for each y, there exists a unique x such that y − f (x) = S(x) = x, [I + f ](x) = y. Therefore I + f is bijective. But in order to show that F ∈ F(B k (RN , RN )) we also need to prove that [I + f ]−1 ∈ B k (RN , RN )). The Jacobian matrix of F (x) is equal to I + f (1) (x) and ∀x ∈ RN ,

|F (1) (x) − I| ≤ F (1) − IC 0 = f (1) C 0 .

If there exists x ∈ RN such that F (1) (x) is not invertible, then for some 0 = y ∈ RN , F (1) (x)y = 0 and |y| = |F (1) (x)y − y| ≤ |F (1) (x) − I||y| ≤ f (1) C 0 |y|

⇒ 1 ≤ f (1) C 0 < 1,

which is a contradiction. Therefore, for all x ∈ RN , F (1) (x) is invertible. As a result the conditions of the implicit function theorem are met and from L. Schwartz [3, Vol. 1, Thm. 29, p. 294, Thm. 31, p. 299] we have that F −1 = [I + f ]−1 ∈ C k (RN , RN ). It remains to prove that F −1 = [I+f ]−1 ∈ Bk (RN , RN ) and the continuity. We again proceed by induction on k. Set y = F (x). By deﬁnition g(y) = F −1 (y) − y = x − F (x) = −f (x) and gC 0 = f C0 . Always, from Lemma 2.5 and the identity g(y) = −f (x) = −f (F −1 (y)) = −f (y + g(y)), |g (1) (y)| ≤ |f (1) (x)| [1 + |g(1) (y)|]

⇒ |g (1) (y)| [1 − |f (1) (x)|] ≤ |f (1) (x)|.

2. Generic Constructions of Micheletti

149

Since |f (1) (x)| ≤ f C0 < 1 0 ≤ |g (1) (y)| ≤

|f (1) (x)| 1 − f C0

⇒ g(1) C 0 ≤

|f (1) (x)| f (1) C0 ≤ . 1 − f C0 1 − f C0

(2.47)

For k ≥ 2, again from Lemma 2.5, we have )k 0 ≤ |g

(k)

j=2

(y)| ≤

)k j=2

≤

|f (j) (x)| aj (|g (1) (y)|, . . . , |g (k−1) (y)|) 1 − |f (1) (x)| |f

(j)

(x)| aj (|g (1) (y)|, . . . , |g (k−1) (y)|) 1 − f (1) C 0

(2.48) .

So there exists a constant ck such that F −1 − ICk ≤ ck F − IC k = ck f C k . Therefore, for k ≥ 1, ∀f ∈ Bk (RN , RN ) such that f (1) C 0 < 1,

F = I + f ∈ F(B k (RN , RN )).

Note that the condition is on f (1) C 0 and not f C k , but by deﬁnition f (1) C 0 ≤ f C 1 ≤ f C k and the condition f C k < 1 yields the same result. In particular, the map f → I + f : B(0, 1) ⊂ Bk (RN , RN ) → F ( B k (RN , RN )) is continuous. All this can now be specialized to the spaces C0k (RN , RN ) and C k (RN , RN ) by using the identity (I + f )−1 − I = −f ◦ (I + f )−1 . (ii) Since F(Θ) is contained in the aﬃne space I + Θ, any tangent element to F(Θ) will be contained in Θ. From part (i) for k ≥ 1 and for all θ ∈ Θ there exists τ > 0 such that for all t, 0 ≤ t ≤ τ , I + tΘ ∈ F(Θ). For any F ∈ F(Θ), t → (I + tθ) ◦ F : [0, τ ] → F(Θ) is a continuous path in F(Θ) and the limit of (I + tθ) − I [(I + tθ) ◦ F ] ◦ F −1 − I = =θ t t will be an element of the tangent space TF F(Θ) in F to F(Θ). Therefore for all F ∈ F(Θ), TF F(Θ) = Θ.

2.6 2.6.1

Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) Checking the Assumptions

By deﬁnition, for all integers k ≥ 0 C

k,1

N

def

(RN , R ) =

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω f ∈ C (RN , R ) : α |∂ f (y) − ∂ α f (x)| ≤ c |x − y| k

N

&

150

Chapter 3. Courant Metrics on Images of a Set

endowed with the same norm:15 def f C k,1 = max max ∂ α f C , ck (f ) = max {f C k , ck (f )} ,

(2.49)

0≤|α|≤k

def

|f (y) − f (x)| and ∀k ≥ 1, |y − x| y=x

c(f ) = sup

def

ck (f ) =

c(∂ α f ),

(2.50)

|α|=k

and the convention c0 (f ) = c(f ). C0k,1 (RN , RN ) is the space of functions f ∈ C k,1 (RN , RN ) such that f and all its derivatives go to zero at inﬁnity. They are all Banach spaces. By deﬁnition C0k,1 (RN , RN ) ⊂ C k,1 (RN , RN ) ⊂ B0 (RN , RN ) ⊂ C 0 (RN , RN ), ∀x ∈ RN ,

f → f (x) : B 0 (RN , RN ) → RN

with continuous injections between the spaces and continuity of the evaluation map. As a consequence, the evaluation map f → f (x) is continuous for all the spaces. We deal only with the case C k,1 (RN , RN ). The case C0k,1 (RN , RN ) is obtained from the case C k,1 (RN , RN ) as in the previous section. Theorem 2.15. Assumptions 2.1 and 2.3 are veriﬁed for the spaces C0k,1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0. Proof. The composition and multiplication of bounded and Lipschitz functions is bounded and Lipschitz. For f ∈ C 0,1 (RN , RN ) and I + g ∈ F(C 0,1 (RN , RN )), f ◦ (I + g)C 0 = f C 0 < ∞. Since g is Lipschitz with Lipschitz constant c(g), I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is also Lipschitz with constant c(f ◦ (I + g)) ≤ c(f ) (1 + c(g)). Hence f ◦ (I + g)C 0,1 ≤ f C 0,1 (1 + c(g)) ≤ f C 0,1 (1 + gC 0,1 ) and the continuous function c1 (r) = 1 + r satisﬁes Assumption 2.3. For k = 1, D(f ◦ (I + g)) = Df ◦ (I + g) (I + Dg) and D(f ◦ (I + g))C ≤ Df C (1 + DgC ), c(D(f ◦ (I + g))) ≤ c(Df ) (1 + c(g)) (1 + DgC ) + Df C (1 + c(Dg)) ⇒ D(f ◦ (I + g))C 0,1 ≤ Df C 0,1 (1 + DgC 0,1 ) (1 + c(g)) ≤ Df C 0,1 (1 + DgC 0,1 ) (1 + gC 0,1 ) ⇒ f ◦ (I + g)C 1,1 ≤ f C 1,1 (1 + DgC 0,1 ) (1 + gC 0,1 ) ≤ f C 1,1 (1 + gC 1,1 )2 . The general case is obtained by induction over k. Theorem 2.16. Assumption 2.2 is veriﬁed for C0k,1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0.

)N 2 1/2 15 N Here R

is endowed with the norm |x| =

i=1

xi

.

2. Generic Constructions of Micheletti

151

Proof. Consider a sequence {fi }ni=1 ⊂ C 0,1 (RN , RN ) such that n

max{fi C 0 , c(fi )} < α < r.

(2.51)

i=1

From Theorem 2.12 (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α.

i=1

In addition, (I + f1 ) ◦ · · · ◦ (I + fn ) − I = fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ c((I + f1 ) ◦ · · · ◦ (I + fn ) − I) ≤ c(fn ) + ≤ c(fn ) + ≤

n

n−1 i=1 n−1

c(fi ) (1 + c(fi+1 )) . . . (1 + c(f1 )) )i+1 c(fi ) e

k=1

c(fk )

i=1

)n

c(fi ) e[

i=1

c(fi )]

< α eα

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0,1 < α max{eα , 1} < α er and we can choose c0 (r) = er for k = 0. We use the same technique for k ≥ 1 but with heavier computations. 2.6.2

Perturbations of the Identity and Tangent Space

By construction F(Θ) ⊂ I + Θ and the tangent space in each point of the aﬃne space I + Θ is Θ. In general, I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there is a suﬃciently small ball B around 0 in Θ such that the so-called perturbations of the identity, I + B, are contained in F(Θ). In particular, for θ ∈ B, the family of transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C 1 -path in F(Θ) and the tangent space to each point of F(Θ) can be shown to be exactly Θ. Theorem 2.17. Let Θ be equal to C0k,1 (RN , RN ) or C k,1 (RN , RN ), k ≥ 0. (i) The map f → I + f : B(0, 1) ⊂ Θ → F(Θ) is continuous. (ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ. Remark 2.6. The generic framework of Micheletti is similar to an inﬁnite-dimensional Riemannian manifold.

152

Chapter 3. Courant Metrics on Images of a Set

Proof. (i) Consider perturbations of the identity of the form F = I + f , f ∈ C 0,1 (RN , RN ). Consider for any y ∈ RN the map def

x → S(x) = y − f (x) : RN → RN . For all x1 and x2 S(x2 ) − S(x1 ) = −[f (x2 ) − f (x1 )], |S(x2 ) − S(x1 )| = |f (x2 ) − f (x1 )| ≤ c(f ) |x2 − x1 |, and f is contracting for c(f ) < 1. Hence for each y ∈ RN there exists a unique x ∈ RN such that ⇒ y − f (x) = x

S(x) = x

⇒ F (x) = y def

and F is bijective. It remains to prove that g = F −1 − I ∈ C 0,1 (RN , RN ) to conclude that I + f ∈ F(C 0,1 (RN , RN )). For any y ∈ RN and x = F −1 (y) F −1 (y) − y = x − F (x), gC 0 = F −1 − IC 0 = sup |F −1 (y) − y| = sup |(I − F )(F −1 (y))| y∈RN

y∈RN

= sup |(I − F )(x)| = F − IC 0 < ∞ x∈RN

since F : RN → RN is bijective and gC 0 = f C 0 . For all y1 and y2 and xi = F −1 (yi ), |g(y2 ) − g(y1 )| = |F −1 (y2 ) − y2 − (F −1 (y1 ) − y1 )| ≤ |x2 − F (x2 ) − (x1 − F (x1 )| = |f (x2 ) − f (x1 )| ≤ c(f )|x2 − x1 | ≤ c(f )|F −1 (y2 ) − F −1 (y1 )| = c(f )|g(y2 ) − g(y1 ) + (y2 − y1 )| ⇒ |g(y2 ) − g(y1 )| ≤

c(f ) |y2 − y1 | 1 − c(f )

⇒ c(g) ≤

c(f ) < ∞. 1 − c(f )

Therefore ∀f ∈ C 0,1 (RN , RN ) such that c(f ) < 1,

F = I + f ∈ F(C 0,1 (RN , RN )).

Moreover since c(f ) ≤ f C 0,1 , the condition f C 0,1 < 1 yields c(f ) < 1 and I +f ∈ F(C 0,1 (RN , RN )). Furthermore, the map f → I + f : B(0, 1) ⊂ C 0,1 (RN , RN ) → F(C 0,1 (RN , RN )) is continuous. We use the same technique for k ≥ 1 plus the arguments of Theorem 2.14 (i). (ii) Same proof as Theorem 2.14 (ii).

3. Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

3

153

Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

With a variation of the generic constructions associated with the Banach space Θ of section 2, it is possible to construct a metric on the whole space of homeomorphisms Hom(RN , RN ) or on the whole space Diﬀ k (RN , RN ) of C k -diﬀeomorphisms of RN , k ≥ 1, or of an open subset D of RN . First, recall that, for an open subset D of RN , the topology of the vector space C k (D, RN ) can be speciﬁed by the family of seminorms def

∀K compact ⊂ D,

nK (F ) = F C k (K) .

(3.1)

This topology is equivalent to the one generated by the monotone increasing subfamily {Ki }i≥1 of compact sets 1 def Ki = x ∈ D : dD (x) ≥ and |x| ≤ i , i ≥ 1, (3.2) i where dD (x) = inf y∈D |y − x|. This leads to the construction of the metric def

δk (F, G) =

∞ 1 nKi (F − G) i 1 + n (F − G) 2 Ki i=1

(3.3)

that generates the same topology as the initial family of seminorms and makes C k (D, RN ) a Fr´echet space with metric δk . Given an open subset D of RN , consider the group Homk (D) def

= F ∈ C k (D, RN ) : F : D → D is bijective and F −1 ∈ C k (D, RN )

(3.4)

of transformations of D. For k = 0, Hom0 (D) = Hom(D) and for k ≥ 1, Homk (D) = Diﬀ k (D), where the condition F −1 ∈ C k (D, RN ) is redundant. One possible choice of a topology on Homk (D) is the topology induced by C k (D, RN ). This is the so-called weak topology on Homk (D) (cf., for instance, M. Hirsch [1, Chap. 2]). However, Homk (D) is not necessarily complete for that topology. In order to get completeness, we adapt the constructions of A. M. Micheletti [1]. Associate with F ∈ Homk (D) and each compact subset K ⊂ D the following function of I and F : def

qK (I, F ) =

inf

n

F =F1 ◦···◦Fn F ∈Homk (D) =1

nK (I, F ) + nK (I, F−1 ),

(3.5)

where the inﬁmum is taken over all ﬁnite factorizations F = F1 ◦ · · · ◦ Fn , Fi ∈ Homk (D), of F in Homk (D). By construction, qK (I, F −1 ) = qK (I, F ). Extend this deﬁnition to all pairs F and G in Homk (D): qK (F, G) = qK (I, G ◦ F −1 ). def

(3.6)

154

Chapter 3. Courant Metrics on Images of a Set

The function qK is a pseudometric 16 and the family {qK : K compact ⊂ D} deﬁnes a topology on Homk (D). This topology is metrizable for the metric def

dk (F, G) =

∞ 1 qKi (F, G) , 2i 1 + qKi (F, G) i=1

(3.7)

constructed from the monotone increasing subfamily {Ki }i≥1 of compact sets deﬁned in (3.2). By deﬁnition, dk (F, G) = dk (G, F ) and dk is symmetrical. It is right-invariant since for all F , G, and H in Homk (D)) ∀F, G, and H ∈ Homk (D), ∀F ∈ Homk (D),

dk (F, G) = dk (F ◦ H, G ◦ H), dk (I, F ) = dk (I, F −1 ).

Theorem 3.1. Homk (D), k ≥ 0, is a group under composition, dk is a rightinvariant metric on Homk (D), and (Homk (D), dk ) is a complete metric space. We need Lemmas 2.4 and 2.5 and the analogues of Lemma 2.6 and Theorem 2.11 for the spaces C k (K, RN ), K a compact subset of D, whose elements are bounded and uniformly continuous. Lemma 3.1. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0 with the following property: for any compact K ⊂ D and a sequence F1 , . . . , Fn in C r (K, RN ) is such that n

F − IC r (K) < α,

0 < α < s;

(3.8)

=1

then for the map F = Fn ◦ · · · ◦ F1 , F − IC r (K) ≤ α c(r, s).

(3.9)

Theorem 3.2. Given k ≥ 0, there exists a continuous function c1 such that for all F, G ∈ Homk (D) and any compact K ⊂ D F ◦ GC k (K) ≤ F C k (K) c1 (GC k (K) ). 16 A

(3.10)

map d : X × X → R on a set X is called a pseudometric (or ´ ecart, or gauge) whenever

(i) d(x, y) ≥ 0, for all x and y, (ii) x = y ⇒ d(x, y) = 0, (iii) d(x, y) = d(y, x), (iv) d(x, z) ≤ d(x, y) + d(y, z) (J. Dugundji [1, p. 198]). This notion is analogous to the one of seminorm for topological vector spaces, but for a group the terminology semimetric (cf. footnote 7) is used for a metric without the triangle inequality.

3. Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

155

Proof. From Lemma 2.5 it readily follows that ψ = F ◦ G ∈ Homk (D). Moreover, ψC 0 (K) = F C 0 (K) , ψ

(1)

C 0 (K) ≤ F (1) C 0 (K) G(1) C 0 (K) ,

ψ (i) C 0 (K) ≤ F (1) C 0 (K) G(i) C 0 (K) +

i

F (j) C 0 (K) aj (G(1) C 0 (K) , . . . , G(i−1) C 0 (K) )

j=2

for i = 2, . . . , k, where aj is a polynomial. Hence ψC k (K) = max ψ i C(K) ≤ F C k (K) c1 (GC k (K) ) i=0,...,k

for some polynomial function c1 that depends only on k. Proof of Theorem 3.1. (i) Homk (D) is a group. The composition of C k k diﬀeomorphisms is again a C -diﬀeomorphism. By deﬁnition, each element has an inverse and the neutral element is I. (ii) dk is a metric. dk (F, G) ≥ 0, dk (F, G) = dk (G, F ). For the triangle inequality F ◦ H −1 = (F ◦ G−1 ) ◦ (G ◦ H −1 ) is a ﬁnite factorization of F ◦ H −1 . Consider arbitrary ﬁnite factorizations of F ◦ G−1 and G ◦ H −1 : F ◦ G−1 = H1 ◦ · · · ◦ Hm ,

Hi ∈ C k (D, D),

G ◦ H −1 = L1 ◦ · · · ◦ Ln ,

Lj ∈ C k (D, D).

This yields a new ﬁnite factorization of F ◦ H −1 . By deﬁnition of qKi , qKi (I, F ◦ H

−1

)≤

m

nKi (I, H ) +

nKi (I, H−1 )

+

n

nKi (I, L ) + nKi (I, L−1 )

=1

=1

and by taking inﬁma over each factorization, qKi (F, H) = qKi (I, F ◦ H −1 ) ≤qKi (I, F ◦ G−1 ) + qKi (I, G ◦ H −1 ) = qKi (F, G) + qKi (G, H). By deﬁnition of dk , since {Ki } is a monotonically increasing sequence of compact sets and {qKi } is a monotonically increasing sequence of pseudometrics, we then get the triangle inequality dk (F, H) ≤ dk (F, G) + dk (G, H). Since Homk (D) is a group and dk is right-invariant, it remains to show that dk (I, F ) = 0 if and only if F = I. Clearly if F = I, qKi (I, F ) = 0 since the qKi ’s are pseudometrics and dk (I, F ) = 0. Conversely, if def

dk (I, F ) =

∞ 1 qKi (I, F ) = 0, i 1 + q (I, F ) 2 Ki i=1

156

Chapter 3. Courant Metrics on Images of a Set

then for all Ki , qKi (I, F ) = 0. For each Ki and all ε, 0 < ε < 1, there exists a ﬁnite factorization F = F1 ◦ · · · ◦ Fn such that n

nKi (F − I) + nKi (F−1 − I) < ε < 1

=1

⇒ nKi (F − I) ≤

nKi (F ◦ · · · ◦ Fn − F+1 ◦ · · · ◦ Fn ).

From Lemma 3.1 ∀ε > 0,

nKi (F − I) < ε c(k, 1)

and (F − I)|Ki = 0. Since this is true for all i, F − I = 0 on D and dk is a metric. (iii) Completeness. Let {Fn } be a Cauchy sequence in Homk (D). (a) Boundedness of {Fn } and {Fn−1 } in C k (D, RN ). For all ε, 0 < ε < 1, there exists N such that, for all m, n > N , dk (Fm , Fn ) < ε. By the triangle inequality |dk (I, Fm ) − dk (I, Fn )| ≤ dk (Fm , Fn ) < ε. For all i, {qKi (I, Fn } is Cauchy and, a posteriori, bounded by some constant L. For ν(n) each n, there exists a factorization Fn = Fn1 ◦ · · · ◦ Fn such that

ν(n)

0≤

Fni − IC k (Ki ) + (Fni )−1 − IC k (Ki ) − dk (I, Fn ) < ε < 1

i=1

ν(n)

⇒

Fni − IC k (Ki ) + (Fni )−1 − IC k (Ki ) < L + ε < L + 1.

i=1

By Lemma 3.1 Fn − IC k (Ki ) + Fn−1 − IC k (Ki ) < 2(L + ε) c(k, L + 1) and the sequences {Fn − I} and {Fn−1 − I} and, a fortiori, {Fn } and {Fn−1 } are bounded in C k (Ki , RN ). (b) Convergence of {Fn } and {Fn−1 } in C k (D, RN ). For any m, n −1 − I) ◦ Fm Fn − Fm = (Fn ◦ Fm

and by Theorem 3.2 −1 − I) ◦ Fm C k (Ki ) Fm − Fn C k (Ki ) = (Fn ◦ Fm −1 ≤ Fn ◦ Fm − IC k (Ki ) c1 (Fm C k (Ki ) ) −1 ⇒ Fm − Fn C k (Ki ) ≤ c+ 1 Fn ◦ Fm − IC k (Ki ) , def

c+ 1 = sup c1 (Fm C k (Ki ) ) < ∞ m

3. Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

157

since {Fm } is bounded in C k (Ki , RN ). For all ε, 0 < ε < c(k, 1) c+ 1 (where c(k, 1) is the constant of Lemma 3.1), ∃N, ∀n, m > N,

−1 dk (I, Fn ◦ Fm ) = dk (Fm , Fn ) < ε/(c(k, 1) c+ 1 ) < 1.

By Lemma 3.1 + −1 − IC k (Ki ) < c(k, 1)ε/(c(k, 1) c+ Fn ◦ Fm 1 ) = ε/c1

⇒ ∀n, m, > N,

−1 Fm − Fn )C k (Ki ) ≤ c+ 1 Fn ◦ Fm − IC k (Ki ) < ε

and, since this is true for all Ki , {Fn } is a Cauchy sequence in C k (D, RN ) that converges to some F ∈ C k (D, RN ) since C k (D, RN ) is a Fr´echet space for the family of seminorms {qKi }. Similarly, we get that {Fn−1 } is a Cauchy sequence in C k (D, RN ) that converges to some G ∈ C k (D, RN ). (c) F is bijective, G = F −1 , and F ∈ Homk (D). For all n, Fn , Fn−1 , F and G are continuous since C k (D, RN ) ⊂ C 0 (D, RN ). By continuity of F → F (x) : C k (D, RN ) → RN , Fn−1 (x) → G(x). By the same argument for Fn , x = Fn (Fn−1 (x)) → F (G(x)) = (F ◦ G)(x) and for all x ∈ RN , (F ◦ G)(x) = x. Similarly, starting from Fn−1 ◦ Fn = I, we get (G ◦ F )(x) = x. Hence, G ◦ F = I = F ◦ G. So F is bijective, F, F −1 ∈ C k (D, RN ), and F ∈ Homk (D). (d) Convergence of Fn → F in Homk (D). To check that Fn → F in Homk (D) recall that by Theorem 3.2 for each Ki qKi (Fn , F ) = qKi (I, F ◦ Fn−1 ) ≤ I − F ◦ Fn−1 C k (Ki ) + I − Fn ◦ F −1 C k (Ki ) = (Fn − F ) ◦ Fn−1 C k (Ki ) + (F − Fn ) ◦ F −1 C k (Ki ) ≤ c1 (Fn−1 C k (Ki ) )Fn − F C k (Ki ) + c1 (F −1 C k (Ki ) )F − Fn C k (Ki ) ≤ max c1 (Fn−1 C k (Ki ) ) + c1 (F −1 C k (Ki ) ) F − Fn C k (Ki ) n

that goes to zero as Fn goes to F in C k (D, RN ). This completes the proof. Remark 3.1. In section 2 it was possible to consider in F(Θ) a subgroup G of translations since Fa (x) = a + x

⇒ (Fa − I)(x) = a ∈ Θ

for spaces Θ that contain constants. Yet, since all of them were spaces of bounded functions in RN , it was not possible to include rotations or ﬂips. But this can be done now in Homk (RN ), and we can quotient out not only by the subgroup of translations but also by the subgroups of isometries or rotation that are important in image processing.

Chapter 4

Transformations Generated by Velocities 1

Introduction

In Chapter 3 we have constructed quotient groups of transformations F(Θ)/G and their associated complete Courant metrics. Such spaces are neither linear nor convex. In this chapter, we specialize the results of Chapter 3 to spaces of transformations that are generated at time t = 1 by the ﬂow of a velocity ﬁeld over a generic time interval [0, 1] with values in the tangent space Θ. The main motivation is to introduce a notion of semiderivatives in the direction θ ∈ Θ on such groups as well as a tractable criterion for continuity via C 1 or continuous paths in the quotient group endowed with the Courant metric. This point of view was adopted by J.´sio [2, 7] as early as 1973 and considerably expanded in his th`ese d’´etat in P. Zole 1979. One of his motivations was to solve a shape diﬀerential equation of the type A V (t) + G(Ωt (V )) = 0, t > 0, where G is the shape gradient of a functional and A a duality operator.1 At that time most people were using a simple perturbation of the identity to compute shape derivatives. The ﬁrst comprehensive book systematically promoting the velocity method was published in 1992 by J. Sokolowski and ´sio [9]. Structural theorems for the Eulerian shape derivative of smooth J.-P. Zole ´sio [7] and generalized to nonsmooth domains were ﬁrst given in 1979 by J.-P. Zole ´sio [14]. The velocity point domains in 1992 by M. C. Delfour and J.-P. Zole ´ [2], of view was also adopted in 1994 by R. Azencott [1], in 1995 by A. Trouve ´ and in 1998 by A. Trouve [3] and L. Younes [2] to construct complete metrics and geodesic paths in spaces of diﬀeomorphisms generated by a velocity ﬁeld with a broad spectrum of applications to imaging.2 The reader is referred to the forthcoming book of L. Younes [6] for a comprehensive exposition of this work and

1 Cf.

´sio [7] and the recent book by M. Moubachir and J.-P. Zole ´sio [1]. J.-P. Zole ﬁrst author would like to thank Robert Azencott, who pointed out his work and the contributions of his team during a visit in Houston early in 2010. Special thanks also to Alain Trouv´ e for an enlightening discussion shortly after in Paris. 2 The

159

160

Chapter 4. Transformations Generated by Velocities

to related papers such as the ones of P. W. Michor and D. Mumford [1, 2, 3] and L. Younes, P. W. Michor, I. Shah, and D. Mumford [1]. In view of the above motivations, this chapter begins with section 2, which specializes the results of Chapter 3 to transformations generated by velocity ﬁelds. It also explores the connections between the constructions of Azencott and Micheletti that implicitly use a notion of geodesic path with discontinuities. Section 3 motivates and adapts the deﬁnitions of Gateaux and Hadamard semiderivatives in topological vector spaces (cf. Chapter 9) to shape functionals deﬁned on shape spaces. The analogue of the Gateaux semiderivative for sets is obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative comes from the velocity (speed ) method. The ﬁrst notion does not extend to submanifolds and does not incorporate the chain rule for the semiderivative of the composition of functions (for instance, to get the semiderivative with respect to the parameters of an a priori parametrized geometry), while Hadamard does. In Chapter 9, ﬂows of velocity ﬁelds will be adopted as the natural framework for deﬁning shape semiderivatives. In section 3.3 the velocity and transformation viewpoints will be emphasized through a series of examples of commonly used families of transformations of sets. They include C k -domains, Cartesian graphs, polar coordinates, and level sets. The following two sections give technical results that will be used to characterize continuity and semidiﬀerentiability along local paths without restricting the analysis to the subgroup GΘ of F(Θ) of section 2. In section 4 we establish the equivalence between deformations obtained from a family of C 1 -paths and deformations obtained from the ﬂow of a velocity ﬁeld. Section 4.1 gives the equivalence under relatively general conditions. Section 4.2 shows that Lipschitzian perturbations of the identity operator can be generated by the ﬂow of a nonautonomous velocity ﬁeld. In section 4.3 the conditions of section 4.2 are sharpened for the special families of velocity ﬁelds in C0k (RN , RN ), C k (RN , RN ), and C k,1 (RN , RN ). The constrained case where the family of domains are subsets of a ﬁxed holdall is studied in section 5. In both sections 4 and 5 we show that, under appropriate conditions, starting from a family of transformations is locally equivalent to starting from a family of velocity ﬁelds. This key result bridges the two points of view. Section 6 builds on the results of section 4 to establish that the continuity of a shape function with respect to the Courant metric on F(Θ)/G is equivalent to its continuity along the ﬂows of all velocity ﬁelds V in C 0 ([0, τ ]; Θ) for Θ equal to C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0. This result is of both intrinsic and practical interest since it is generally easier to check the continuity along paths than directly with respect to the Courant metric. Finally, from the discussion in section 3, the technical results of section 4 will again be used in Chapter 9 to construct local C 1 -paths generated by velocity ﬁelds V in C 0 ([0, τ ]; Θ) to deﬁne the shape semiderivatives.

2. Metrics on Transformations Generated by Velocities

2

161

Metrics on Transformations Generated by Velocities

2.1

Subgroup GΘ of Transformations Generated by Velocities

From Chapter 3 recall the deﬁnition of the group def

F(Θ) = I + θ : θ ∈ Θ, (I + θ)−1 ∃, and (I + θ)−1 − I ∈ Θ for the Banach space Θ = C 0,1 (RN , RN ) with norm |θ(y) − θ(x)| def def , θΘ = max sup |θ(x)|, c(θ) , c(θ) = sup |y − x| N y=x x∈R

(2.1)

and the associated complete metric d on F(Θ). This section specializes to a subgroup of transformations I + θ of F(Θ) that are speciﬁed via a vector ﬁeld V on a generic interval [0, 1] with the following properties: (i) for all x ∈ RN , the function t → V (t, x) : [0, 1] → RN belongs to L1 (0, 1; RN ) and 1 1 V (t)C dt = sup |V (t, x)| dt < ∞; (2.2) 0

0 x∈RN

(ii) for almost all t ∈ [0, 1], the function x → V (t, x) : RN → RN is Lipschitzian and 1 1 |V (t, y) − V (t, x)| c(V (t)) dt = sup dt < ∞. (2.3) |y − x| 0 0 x=y In view of the assumptions on V , it is natural to introduce the norm 1 def V L1 (0,1;Θ) = V (t)Θ dt.

(2.4)

0

Associate with V ∈ L1 (0, 1; Θ) the ﬂow dTt (V ) = V (t) ◦ Tt (V ), dt

T0 (V ) = I.

(2.5)

In view of the assumptions on V , for each X ∈ RN , the diﬀerential equation x (t; X) = V (t, x(t; X)),

x(0; X) = X,

(2.6)

has a unique solution in W 1,1 (0, 1; RN ) ⊂ C([0, 1], RN ), the mapping X → x(·; X) : RN → C([0, 1], RN ) is continuous, and for all t ∈ [0, 1] the mapping X → Tt (V ) def (X) = x(t; X) : RN → RN is bijective. Deﬁne def

GΘ = T1 (V ) : ∀V ∈ L1 ([0, 1]; Θ) .

(2.7)

162

Chapter 4. Transformations Generated by Velocities

Theorem 2.1.

(i) The set GΘ with the composition ◦, ∀V1 , V2 ∈ L1 (0, 1; Θ),

T1 (V1 ) ◦ T1 (V2 ),

(2.8)

is a subgroup of F(Θ) and for all V ∈ L1 (0, 1; Θ), sup Tt (V ) − IΘ = sup θV (t)Θ ≤ V L1 exp2(1+V L1 ) , 0≤t≤1

(2.9)

0≤t≤1 def

where θV (t) = Tt (V ) − I and L1 stands for L1 (0, 1; Θ). In particular, for all t, Tt (V ) ∈ F(Θ) and t → Tt (V ) : [0, 1] → F(Θ) is a continuous path. (ii) For all V ∈ L1 (0, 1; Θ), (T1 (V ))−1 = T1 (V − ) for V − (t, x) = −V (1 − t, x), −θV ◦ (I + θV )−1 = (T1 (V ))−1 − I = T1 (V − ) − I = θV − , −

V L1 (0,1;Θ) = V L1 (0,1;Θ) ,

(2.10) (2.11)

and sup Tt (V )−1 − IΘ = sup θV− (t)Θ ≤ V L1 exp2(1+V L1 ) .

0≤t≤1

(2.12)

0≤t≤1

(iii) For all V ∈ L1 (0, 1; Θ), T1 (V ) − IΘ + (T1 (V ))−1 − IΘ ≤ 2 V L1 (0,1;Θ) exp2 (1+V L1 (0,1;Θ) ) (2.13) ⇒ d(T1 (V ), I) ≤ 2 V L1 (0,1;Θ) exp2 (1+V L1 (0,1;Θ) ) .

(2.14)

(iv) For all V, W ∈ L1 (0, 1; Θ), max sup θW (t) − θV (t)Θ , sup θW − (t) − θV − (t)Θ 0≤t≤1

≤V L1 exp

(1+V L1 )

0≤t≤1

(2.15) 1 + W L1 exp2(1+W L1 ) W (s) − V (s)L1 ,

where L1 stands for L1 (0, 1; Θ). Remark 2.1. Note that the condition (I − θ)−1 exists and (I − θ)−1 ∈ Θ in the deﬁnition of F(Θ) is automatically veriﬁed for the transformations I + θV generated by a velocity ﬁeld V ∈ L1 (0, 1; Θ). Proof. (i) Choosing V = 0, the identity I is in GΘ . To show that GΘ is closed under the composition, it is suﬃcient to construct a W ∈ L1 (0, 1; Θ) such that T1 (W ) = T1 (V1 ) ◦ T1 (V2 ). Given τ ∈ (0, 1) deﬁne the concatenation t 1 V , x , 0 ≤ t ≤ τ, τ 2 τ def (V1 ∗ V2 )(t, x) = (2.16) t−τ 1 V1 ,x , τ < t ≤ 1. 1−τ 1−τ

2. Metrics on Transformations Generated by Velocities

163

It is easy to check that W = V1 ∗ V2 ∈ L1 (0, 1; Θ) as the concatenation of the L1 -functions V1 and V2 so that the equation dTt (W ) = W (t) ◦ Tt (W ), dt

T0 (W ) = I

has a unique solution and T1 (W ) ∈ GΘ . On the interval [0, τ ]

s t/τ V2 (s , Ts τ (W )(X)) ds , Ts (W )(X) d = τ τ 0 0 t ⇒ Tt τ (W )(X) − X = V2 (s , Ts τ (W )(X)) ds , 0 ≤ t ≤ 1

Tt (W )(X) − X =

t

V2

s

0

⇒ Tt τ (W )(X) = Tt (V2 )(X)

and Tτ (W ) = T1 (V2 ).

On the interval [τ, 1]

t

Tt (W )(X) − T1 (V2 ) =

V1

τ t−τ 1−τ

=

s−τ , Ts (W )(X) 1−τ

d

s−τ 1−τ

V2 s , Tτ +s (1−τ ) (W )(X) ds

0 t

⇒ Tτ +t (1−τ ) t(W )(X) − T1 (V2 ) =

V2 s , Tτ +s (1−τ ) (W )(X) ds ,

0 ≤ t ≤ 1,

0

Tτ +t (1−τ ) (W )(X) = Tt (V1 ) ◦ T1 (V2 ) and T1 (W )(X) = T1 (V1 ) ◦ T1 (V2 ). To show that each T1 (V ) has an inverse in GΘ , consider the new function y(t; X) = [T1−t (V ) ◦ (T1 (V ))−1 ](X) = x(1 − t; (T1 (V ))−1 (X)), def

0 ≤ t ≤ 1,

for which y(0; X) = [T1 (V ) ◦ (T1 (V ))−1 ](X) = X and y(1; X) = (T1 (V ))−1 (X). Its equation is given by dy (t; X) = −V (1 − t, y(t; X)), dt def

y(0; X) = X,

and the associated vector ﬁeld in V − (t, x) = −V (1 − t, x) L1 (0, 1; Θ). Therefore GΘ is a group of homeomorphisms Hom(RN ). To show that GΘ ⊂ F(Θ), it is necessary to show T1 (V ) − I ∈ Θ. Given X ∈ RN , consider the functions

that clearly belongs to of RN , that is, GΘ ⊂ that for each V , θ = x(t) = Tt (V )(X) and

164

Chapter 4. Transformations Generated by Velocities

θ(t; X) = x(t) − X. By deﬁnition t V (s, X + θ(s; X)) ds, θ(t; X) = x(t) − X = 0 t t |V (s, X + θ(s; X)) − V (s, X)| ds + |V (s, X)| ds |θ(t; X)| ≤ 0 0 t t ≤ c(V (s)) |θ(s; X)| ds + |V (s, X)| ds, 0 0 t t c(V (s)) sup |θ(s; X)| ds + sup |V (s, X)| ds. sup |θ(t; X)| ≤ X

0

X

0

X

t Given 0 < α < 1 let gα (t) = exp( 0 c(V (s)) ds/α). Then it is easy to show that supX |θ(t; X)| 1 sup ≤ g (t) 1 − α α t∈[0,1] ⇒ sup θ(t)C ≤ t∈[0,1]

gα (1) 1−α

1

sup |V (s, X)| ds X

0

1

V (s)C ds. 0

To prove that θ is Lipschitzian, associate with X, Y ∈ RN , X = Y , the functions x(t) = Tt (V )(X), y(t) = Tt (V )(Y ), θ(t; X) = x(t) − X, and θ(t; Y ) = y(t) − Y . By deﬁnition t θ(t; X) − θ(t; Y ) = V (s, Y + θ(s; Y )) − V (s, X + θ(s; X)) ds, 0 t t c(V (s)) |θ(s; Y )) − θ(s; X)| ds + c(V (s)) |Y − X| ds |θ(t; X) − θ(t; Y )| ≤ 0 0 t t |θ(t; X) − θ(t; Y )| |θ(s; Y )) − θ(s; X)| ⇒ sup ≤ ds + c(V (s)) sup c(V (s)) ds |Y − X| |Y − X| X=Y X=Y 0 0 gα (1) 1 c(V (s)) ds. ⇒ sup c(θ(t)) ≤ 1−α 0 t∈[0,1] Combining the last inequality with the one of the ﬁrst part, for all α, 0 < α < 1, sup θ(t)Θ ≤ t∈[0,1]

gα (1) 1−α

1

V (t)Θ dt. 0

1 1 For V = 0, θ = 0; for V = 0, choose α = 0 V (t)Θ dt/( 0 V (t)Θ dt + 1). This yields (2.9). Since (I + θ)−1 = (T1 (V ))−1 = T1 (V − ) is of the same form for the velocity ﬁeld V − , (I + θ)−1 − I = (T1 (V ))−1 − I = T1 (V − ) − I ∈ Θ. Therefore GΘ is a subgroup of F(Θ). (ii) From the construction of the inverse in part (i) and the fact that (T1 (V ))−1 − I = −θV ◦ (I + θV )−1 . (iii) From inequality (2.9) applied to θV and θV − and identity (2.11).

2. Metrics on Transformations Generated by Velocities (iv) Let θV (t) = Tt (V ) − I and let θW (t) = Tt (W ) − I: t θW (t) − θV (t) = W (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds 0 t = (W (s) − V (s)) ◦ Ts (W ) ds 0 t V (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds, + 0 t W (s) − V (s)C ds θW (t) − θV (t)C ≤ 0 t + c(V (s)) θW (s) − θV (s)C ds 0

gα (1) 1−α

⇒ sup θW (t) − θV (t)C ≤ 0≤t≤1

for 0 < α < 1 and gα (t) = exp

t

1

W (s) − V (s)C ds 0

c(V (s) ds/α. Similarly,

0

t

θW (t) − θV (t) =

(W (s) − V (s)) ◦ Ts (W ) ds t V (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds, + 0 t c(θW (t) − θV (t)) ≤ c(W (s) − V (s)) (1 + c(θW (s))) ds 0 t c(V (s)) c(θW (s) − θV (s)) ds + 0

0

⇒ sup c(θW (t) − θV (t)) ≤ 0≤t≤1

gα (1) 1−α

1

c(W (s) − V (s)) (1 + c(θW (s))) ds. 0

From part (i) sup c(θW (s)) ≤ W L1 (0,1;Θ) exp2(1+W L1 (0,1;Θ) ) 0≤s≤1

and ﬁnally sup θW (t) − θV (t)Θ 0≤t≤1

≤

gα (1) (1 + W L1 (0,1;Θ) exp2(1+W L1 (0,1;Θ) ) ) W (s) − V (s)L1 (0,1;Θ) ds. 1−α

By choosing α = V L1 (0,1;Θ) /(1 + V L1 (0,1;Θ) ) sup θW (t) − θV (t)Θ 0≤t≤1

≤ V L1 exp(1+V L1 ) (1 + W L1 (0,1;Θ) exp2(1+W L1 ) ) W (s) − V (s)L1 ds.

165

166

2.2

Chapter 4. Transformations Generated by Velocities

Complete Metrics on GΘ and Geodesics

Since GΘ is a subgroup of the complete group F(Θ), its closure GΘ with respect to the metric d of F(Θ) is a closed subgroup whose elements can be “approximated” by elements constructed from the ﬂow of a velocity ﬁeld. Is (GΘ , d) complete? Can a velocity ﬁeld be associated with a limit element? It is very unlikely that (GΘ , d) can be complete unless we strengthen the metric d to make the associated Cauchy sequence of velocities {Vn } converge in L1 (0, 1; Θ). We have seen that for each V ∈ L1 (0, 1; Θ), we have a continuous path t → Tt (V ) in GΘ . Since the part of the metric on I +θV is deﬁned as an inﬁmum over all ﬁnite factorizations (I + θV1 ) ◦ · · · ◦ (I + θVn ) of I + θV and (I + θV − ) ◦ · · · ◦ (I + θVn− ) 1 of (I + θV )−1 , T1 (V ) = T1 (V1 ) ◦ T1 (V2 ) ◦ · · · ◦ T1 (Vn ), n inf θVi Θ + θV − Θ , d(I, T1 (V )) = T1 (V )= T1 (V1 )◦···◦T1 (Vn ) i=1

i

is there a velocity V ∗ ∈ L1 (0, 1; Θ) that achieves that inﬁmum? Do we have a geodesic path between I and T1 (V ) that would be achieved by V ∗ ∈ L1 (0, 1; Θ)? Given a velocity ﬁeld we have constructed a continuous path t → Tt (V ) : [0, 1] → F(Θ). At each t, there exists δ > 0 suﬃciently small such that the path t → Tt+s (V ) ◦ Tt−1 (V ) : [0, δ] → F(Θ) is diﬀerentiable in t in the group F(Θ): Tt+s (V ) ◦ Tt−1 (V ) − I 1 t+s V (r) ◦ Tr (V ) dr ◦ Tt−1 (V ) = s s t (2.17) −1 → V (t) ◦ Tt (V ) ◦ Tt (V ) = V (t) dF Tt ⇒ = V (t) a.e. in Θ. (2.18) dt With this deﬁnition a jump from I + θ1 to [I + θ2 ] ◦ [I + θ1 ] at time 1/2 becomes −1 (V ) − I = [I + θ2 ] ◦ [I + θ1 ] ◦ [I + θ1 ]−1 − I = I + θ2 − I = θ2 , T1/2+s (V ) ◦ T1/2

that is, a Dirac delta function θ2 in the tangent space Θ to F(Θ) at 1/2. Now the norm of the velocity V = θ2 δ1/2 in the space of measures becomes V M1 ((0,1);Θ) =

1

θ2 Θ δ1/2 dt = θ2 Θ . 0

If we have n such jumps, V M1 ((0,1);Θ) =

θi Θ .

1=1,...,n

The metric of Micheletti is the inﬁmum of this norm over all ﬁnite factorizations. A consequence of this analysis is that a factorization of an element I + θ of the form (I +θ1 )◦(I +θ2 )◦· · ·◦(I +θn ) is in fact a path t → Tt : [0, 1] → F(Θ) of bounded

2. Metrics on Transformations Generated by Velocities

167

variations in F(Θ) between I and I +θ by assigning times 0 < t1 < t2 < · · · < tn < 1 to each jump in the group F(Θ): 0 ≤ t < t1 , I, , t1 ≤ t < t2 , I + θ 1 def t2 ≤ t < t3 , Tt (V ) = (I + θ1 ) ◦ (I + θ2 ), ... (I + θ1 ) ◦ (I + θ2 ) ◦ · · · ◦ (I + θn ), tn ≤ t ≤ 1, n dF Tt def = V (t), V M1 ((0,1);Θ) = θi δ(ti ) ⇒ θi Θ , V (t) = dt 1=1,...,n i=1

where the time-derivative is taken in the group sense (2.17) and V is a bounded measure. With this interpretation in mind, for V ∈ L1 (0, 1; Θ) the metric of Micheletti should reduce to 3 1 13 3 dF Tt (v) 3 3 3 d(I, T1 (V )) = inf v(t)Θ dt = inf 3 dt 3 dt, v∈L1 (0,1;Θ) 0 v∈L1 (0,1;Θ) 0 Θ T1 (v)=T1 (V )

T1 (v)=T1 (V )

where the derivative is in the group F(Θ). As a result we could talk of a geodesic path in F(Θ) between I and T1 (V ). If this could be justiﬁed, the next question is whether GΘ is complete in (F(Θ), d) or complete with respect to the metric constructed from the function 1 def dG (I, T1 (V )) = inf v(t)Θ dt, (2.19) 1 v∈L (0,1;Θ) T1 (v)=T1 (V )

0

dG (T1 (V ), T1 (W )) = d(T1 (V ) ◦ T1 (W )−1 , I). def

(2.20)

By deﬁnition dG (T1 (V )−1 , I) = dG (T1 (V ), I),

dG (T1 (V ), T1 (W )) = dG (T1 (W ), T1 (V ))

and dG is right-invariant. The inﬁmum is necessary since the velocity taking I to T1 (V ) is not unique. Theorem 2.2. Let Θ be equal to C k,1 (RN , RN ), C k+1 (RN , RN ), C0k+1 (RN , RN ), or Bk+1 (RN , RN ), k ≥ 0. Then dG is a right-invariant metric on GΘ . Remark 2.2. The theorem is also true for the right-invariant metric dG (T1 (V ), I) = d(T1 (W ), T1 (V )) + def

inf

W ∈L1 (0,1;Θ) T1 (W )=T1 (V )

W L1 (0,1;Θ) ,

dG (T1 (V ), T1 (W )) = dG (T1 (V ) ◦ T1 (W )−1 , I). def

(2.21)

(2.22)

168

Chapter 4. Transformations Generated by Velocities

Proof. We give only the proof for C 0,1 (RN , RN ). It remains to show that the triangle inequality holds and that dG (T1 (W ), T1 (V )) = 0 if and only if T1 (W ) = T1 (V ). Given U, V, W in L1 (0, 1; Θ), T1 (W ) ◦ T1 (U )−1 = T1 (W ) ◦ T1 (V )−1 ◦ T1 (V ) ◦ T1 (U )−1 . For all v ∈ L1 (0, 1; Θ) such that T1 (v) = T1 (W ) ◦ T1 (V )−1 and for all u ∈ L1 (0, 1; Θ) such that T1 (v) = T1 (V ) ◦ T1 (U )−1 , the concatenation v ∗ u is such that T1 (v ∗ u) = T1 (W ) ◦ T1 (U )−1 . By deﬁnition dG (T1 (W ) ◦ T1 (U )−1 , I) ≤ v ∗ uL1 = vL1 + uL1 . By taking inﬁma with respect to v and u we get the triangle inequality dG (T1 (W ) ◦ T1 (U )−1 , I) ≤ dG (T1 (W ) ◦ T1 (V )−1 , I) + dG (T1 (V ) ◦ T1 (U )−1 , I). If T1 (V ) = I, then inf w∈L1 (0,1;Θ) wL1 (0,1;Θ) = 0 and dG (T1 (V ), I) = 0. ConT1 (w)=I

versely, if dG (T1 (V ), I) = 0, then inf v∈L1 (0,1;Θ) vL1 (0,1;Θ) = 0 and there exists a T1 (v)=T1 (V )

sequence vn ∈ L1 (0, 1; Θ) such that vn → 0 in L1 (0, 1; Θ) and T1 (vn ) = T1 (V ). By continuity of T1 (vn ) − I with respect to vn from Theorem 2.1 (iv), T1 (V ) − I = T1 (vn ) − I → T1 (0) − I = 0 and T1 (V ) = I. Getting the completeness with the metric dG and even dG is not obvious. For a Cauchy sequence {T1 (Vn )}, the L1 -convergence of the velocities {Vn } to some V would yield the convergence of {T1 (Vn )} to T1 (V ) and even the convergence of the geodesics, but the metrics do not seem suﬃciently strong to do that. To appreciate this point, we ﬁrst establish an inequality that really follows from the triangle inequality for dG . Associate with W a w such that T1 (w) = T1 (W ), with V a v such that T1 (v) = T1 (V ), and let u be such that T1 (u) = T1 (W )◦T1 (V )−1 . Then the concatenation w ∗ v− is such that T1 (v ∗ u) = T1 (W ). From this we get the inequalities inf T1 (w)=T1 (W )

wL1 ≤ v ∗ uL1 = vL1 + uL1

and by taking inﬁma we get the inequality inf T1 (w)=T1 (W )

wL1 ≤

inf

T1 (u)=T1 (W )◦T1 (V )−1

uL1 +

inf T1 (v)=T1 (V )

vL1

inf ⇒ inf wL1 − inf vL1 ≤ uL1 . T1 (w)=T1 (W ) T1 (v)=T1 (V ) T1 (u)=T1 (W )◦T1 (V )−1 Pick any Cauchy sequence {T1 (Vn )}. For any ε > 0, there exists N such that for all n, m > N 1 1 w − inf v uL1 < ε. inf inf L L ≤ −1 T1 (w)=T1 (Vm )

T1 (v)=T1 (Vn )

T1 (u)=T1 (Vm )◦T1 (Vn )

Therefore, there exists a sequence {vn } such that T1 (vn ) = T1 (Vn ) and 0 ≤ vn L1 −

inf T1 (v)=T1 (Vn )

vL1 < ε

⇒ |vm L1 − vn L1 | < 3ε. The sequence {vn L1 } is bounded in L1 (0, 1; Θ), but this does not seem suﬃcient to get the convergence of the vn ’s in L1 (0, 1; Θ).

2. Metrics on Transformations Generated by Velocities

169

One could think of many ways to strengthen the metric. For instance, one could introduce the minimization of the whole W 1,1 (0, 1; Θ)-norm of the path t → Tt (v) with respect to v from which the minimizing velocity v(t) = ∂Tt /∂t ◦ Tt−1 could be recovered, but the triangle inequality would be lost.

2.3

Constructions of Azencott and Trouv´ e

In 1994 R. Azencott [1] deﬁned the following program. Given a smooth manifold M , consider the time continuous curve T = (Tt )0≤t≤1 in Aut(M ) solutions of ∂Tt = V (t) ◦ Tt , ∂t

T0 = I,

(2.23)

for a continuous time family V (t) of vector ﬁelds in a Hilbert space H ⊂ Aut(M ) and deﬁne 1 def l(φ) with l(φ) = |V (t)|I dt. (2.24) d(I, φ) = inf V ∈C([0,1];H) T1 (V )=φ

0

A fully rigorous construction of this distance, in the context of inﬁnite-dimensional ´ [2] in 1995 and [3] in 1998. He shows Lie groups, was performed by A. Trouve that the subset of φ ∈ Aut(M ) for which d(I, φ) can be deﬁned is a subgroup A of invertible mappings. He extends this distance between two arbitrary mappings def φ and ψ by right-invariance: d(φ, ψ) = d(I, ψ ◦ φ−1 ). Hence, in this extended framework, the new problem should be to ﬁnd in A 1 1 φˆ = argmin |f˜ − f ◦ φ|2 dx + d(Id, φ)2 , 2 M 2 where M is a smooth manifold, the function f : M → R is a high dimensional representation template, and f˜ is a new observed image belonging to this family (cf. ´ [3]). This approach was further developed by the group of Azencott A. Trouve (cf., for instance, L. Younes [1]). ´ [3] in 1998: Quoting from the introduction of A. Trouve . . . Hence, for a large deformation φ, one can consider φ as the concatenation of small deformations φui . More precisely, if φ = Φn , where the Φk are recursively deﬁned by Φ0 = Id and φk+1 = φuk+1 ◦ Φk , the family Φ = (Φk )0≤k≤n deﬁnes a polygonal ) line in Aut(M ) whose length l(Φ) can be approximated by l(φ) = nk=1 |uk |e . At a naive level, one could deﬁne the distance d(Id, φ) as the inﬁmum of l(Φn ) for all family Φ such that Φn = φ. For a more rigorous setting, one should consider the time continuous curve Φ = (Φt )0≤t≤1 in Aut(M ). . . . In fact what he describes as a naive approach is the very precise construction of A. M. Micheletti [1] in 1972. He is certainly not the ﬁrst author who has overlooked that paper in Italian, where the central completeness result that is of interest to us was just a lemma in her analysis of the continuity of the ﬁrst eigenvalue of the Laplacian. We brought it up only in the 2001 edition of this book.

170

Chapter 4. Transformations Generated by Velocities

By introducing a Hilbert space H ⊂ Θ = C 0,1 (RN , RN ) and minimizing over L (0, 1; H) rather than L1 (0, 1; Θ), the bounded sequence {vn } lives in L2 (0, 1; H), where there exist a v ∈ L2 (0, 1; H) and a weakly converging subsequence to v. By using the complete metric of Micheletti, we had the existence of a F ∈ F(Θ) such that T1 (vn ) → F , but now we have a candidate v for which F = T1 (v). This is very much analogous to the Hilbert uniqueness method of J.-L. Lions [1] in the context of the controllability of partial diﬀerential equations (cf. also R. Glowinski and J.-L. Lions [1, 2]). 2

Remark 2.3. It is important to understand that ﬁnding a complete metric of Courant type for the various spaces of diﬀeomorphisms is a much less restrictive problem than constructing a metric from geodesics. The existence of a geodesic is not required for the Courant metrics.

3 3.1

Semiderivatives via Transformations Generated by Velocities Shape Function

All spaces F(Θ) and F(Θ)/G(Ω0 ) in Chapter 3 associated with the family of sets X (Ω0 ) = {F (Ω0 ) : ∀F ∈ F(Θ)}

(3.1)

are nonlinear and nonconvex and the elements of F(Θ)/G(Ω0 ) are equivalence classes of transformations. Deﬁning a diﬀerential with respect to such spaces is similar to deﬁning a diﬀerential on an inﬁnite-dimensional manifold. Fortunately, the tangent space is invariant and equal to Θ in every point of F(Θ). This considerably simpliﬁes the analysis. Deﬁnition 3.1. Given a nonempty subset D of RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D will be referred to as the underlying holdall or universe. A shape functional is a map J :A→E (3.2) from some admissible family A of sets in P(D) into a topological space E. For instance, A could be the set X (Ω). D can represent some physical or mechanical constraint, a submanifold of RN , or some mathematical constraint. In most cases, it can be chosen large and as smooth as necessary for the analysis. In the unconstrained case, D is equal to RN .

3.2

Gateaux and Hadamard Semiderivatives

Consider the solution (ﬂow) of the diﬀerential equation dx (t, X) = V (t, x(t, X)), t ≥ 0, dt

x(t, X) = X,

(3.3)

3. Semiderivatives via Transformations Generated by Velocities

171

V x(t)

X Ωt = Tt (Ω)

Ω

Figure 4.1. Transport of Ω by the velocity ﬁeld V . def

for velocity ﬁelds V (t)(x) = V (t, x) (cf. Figure 4.1). It generates the family of transformations {Tt : RN → RN : t ≥ 0} deﬁned as follows: def

X → Tt (X) = x(t; X) : RN → RN .

(3.4)

Given an initial set Ω ⊂ RN , associate with each t > 0 the new set def

Ωt = Tt (Ω) = {Tt (X) : ∀X ∈ Ω} .

(3.5)

This perturbation of the initial set is the basis of the velocity (speed ) method . The choice of the terminology “velocity” to describe this method is accurate but may become ambiguous in problems where the variables involved are themselves “physical velocities”: this situation is commonly encountered in continuum mechanics. In such cases it may be useful to distinguish between the “artiﬁcial velocity” and the “physical velocity.” This is at the origin of the terminology speed method, which has often been used in the literature. The latter terminology is convenient, but is not as accurate as the velocity method. We shall keep both terminologies and use the one that is most suitable in the context of the problem at hand. It is important to work with the weakest notion of semiderivative that preserves the basic elements of the classical diﬀerential calculus such as the chain rule for the diﬀerentiation of the composition of functions. It should be able to handle the norm of a function or functions deﬁned as an upper or lower envelope of a family of diﬀerentiable functions. Since the spaces of geometries are generally nonlinear and nonconvex, we also need a notion of semiderivative that yields diﬀerentials in the tangent space as for manifolds. We are looking for a very general notion of semidiﬀerential but not more! In that context the most suitable candidate is the Hadamard semiderivative3 that will be introduced later. Important nondiﬀerentiable functions are Hadamard semidiﬀerentiable and the chain rule is veriﬁed. 3 In his 1937 paper, M. Fre ´chet [1] extends to function spaces the Hadamard derivative and promotes it as more general than his own Fr´echet derivative since it does not require the space to have a metric in inﬁnite-dimensional spaces. In the same paper, he also introduces a relaxation of the Hadamard derivative that is almost a semiderivative. A precise deﬁnition of the Hadamard semiderivative explicitly appears in J.-P. Penot [1] in 1978 and in A. Bastiani [1] in 1964 as a well-known notion.

172

Chapter 4. Transformations Generated by Velocities

We proceed step by step. A ﬁrst candidate for the shape semiderivative of J at Ω in the direction V would be def

dJ(Ω; V ) = lim

t0

J(Ωt (V )) − J(Ω) t

(when the limit exists).

(3.6)

This very weak notion depends on the history of V for t > 0 and will be too weak for most of our purposes. For instance, its composition with another function would not verify the chain rule. To make it compatible with the usual notion of semiderivative in a direction θ of the tangent space Θ to F(Θ) in F (Ω), it must be strengthened as follows. Deﬁnition 3.2. Given θ ∈ Θ def

dJ(Ω; θ) =

J(Ωt (V )) − J(Ω) , t ([0,τ ];Θ)

lim

0

V ∈C V (0)=θ t0

(3.7)

where the limit depends only on θ and is independent of the choice of V for t > 0. Equivalently, the limit dJ(Ω; θ) is independent of the way we approach Ω for a ﬁxed direction θ. This is precisely the adaptation of the Hadamard semiderivative.4 Deﬁnitions based on a perturbation of the identity def

Tt (X) = x(t) = X + t θ(X),

t ≥ 0,

(3.8)

can be recast in the velocity framework by choosing the velocity ﬁeld V (t) = θ◦Tt−1 , that is, ∀x ∈ RN , ∀t ≥ 0,

V (t, x) = θ(Tt−1 (x)) = θ ◦ (I + tθ)−1 (x), def

(3.9)

which requires the existence of the inverse for suﬃciently small t. With that choice for each X, x(t) = Tt (X) is the solution of the diﬀerential equation dx (t) = V t, x(t) , dt

x(0) = X.

(3.10)

Under appropriate continuity and diﬀerentiability assumptions, ∀x ∈ RN , ∀x ∈ RN ,

def

V (0)(x) = V (0, x) = θ(x), def ∂V ˙ (t, x) V (0)(x) = = − [Dθ(x)] θ(x), ∂t t=0

(3.11) (3.12)

where Dθ(x) is the Jacobian matrix of θ at the point x. In compact notation, V (0) = θ

and V˙ (0) = − [Dθ] θ.

(3.13)

This last computation shows that at time 0, the points of the domain Ω are simultaneously aﬀected by the velocity ﬁeld V (0) = θ and the acceleration ﬁeld 4 Cf. footnote 2 in section 2 of Chapter 9 for the deﬁnitions and properties and the comparison of the various notions of derivatives and semiderivatives in Banach spaces.

3. Semiderivatives via Transformations Generated by Velocities

173

V˙ (0) = −[Dθ]θ. The same result will be obtained without acceleration term by replacing I + tθ by the transformation Tt generated by the solutions of the diﬀerential equation dx (t) = θ(x(t)), dt

x(0) = X,

def

Tt (X) = x(t),

for which V (0) = θ and V˙ (0) = 0. Under suitable assumptions the two methods will produce the same ﬁrst-order semiderivative. The deﬁnition of a shape semiderivative def

dJ(Ω; θ) = lim

t0

J([I + tθ]Ω) − J(Ω) t

(3.14)

by perturbation of the identity is similar to a Gateaux semiderivative that will not verify the chain rule. Moreover, second-order semiderivatives will diﬀer by an acceleration term that will appear in the expression obtained by the method of perturbation of the identity.

3.3

Examples of Families of Transformations of Domains

In section 3.2 we have formally introduced a notion of semiderivative of a shape functional via transformations generated by ﬂows of velocity ﬁelds and perturbations of the identity operator as special cases of the velocity method. Before proceeding with a more abstract treatment, we present several examples of deﬁnitions of shape semiderivatives that can be found in the literature. We consider special classes of domains (C ∞ , C k , Lipschitzian), Cartesian graphs, polar coordinates, and level sets that provide classical examples of parametrized and/or constrained deformations. In each case we construct the associated underlying family (not necessarily unique) of transformations {Tt : 0 ≤ t ≤ τ }. 3.3.1

C ∞ -Domains

Let Ω be an open domain of class C ∞ in RN . Recall from section 5 of Chapter 2 that in any point x ∈ Γ the unit outward normal is given by def

∀y ∈ Γx = U(x) ∩ Γ,

n(y) =

mx (y) , |mx (y)|

(3.15)

where mx (y) = − ∗(Dhx )−1 (h−1 x (y)) eN in U(x) ⇒ n=−

∗

(3.16)

−1

(Dhx ) eN ◦ h−1 x . | ∗(Dhx )−1 eN |

(3.17)

When Γ is compact, it is possible to ﬁnd a ﬁnite sequence of points {xj : 1 ≤ j ≤ J} in Γ such that def

Γ⊂U =

J * j=1

Uj ,

def

Uj = U(xj ).

174

Chapter 4. Transformations Generated by Velocities

As in the deﬁnition of the boundary integral, associate with {Uj } a partition of unity {rj }: rj ∈ D(Uj ),

0 ≤ rj ≤ 1,

J

rj = 1 in U0

j=1

for some neighborhood U0 of Γ such that Γ ⊂ U0 ⊂ U0 ⊂ U. For the C ∞ -domain Ω the normal satisﬁes n=

J

rj n =

j=1

since

∀j,

J j=1

rj

mj ∈ C ∞ (Γ; RN ) |mj |

mj ◦ hj ∈ C ∞ (B; RN ). |mj |

Given any ρ ∈ C ∞ (Γ) and t ≥ 0, consider the following perturbation Γt of Γ along the normal ﬁeld n: def

Γt = {x ∈ RN : x = X + tρ(X)n(X), ∀X ∈ Γ}.

(3.18)

We claim that for τ suﬃciently small and all t, 0 ≤ t ≤ τ , the set Γt is the boundary of a C ∞ -domain Ωt by constructing a transformation Tt of RN which maps Ω onto Ωt and Γ onto Γt . First construct an extension N ∈ D(RN , RN ) of the normal ﬁeld n on Γ. Deﬁne J def rj mj ∈ D(U, RN ). (3.19) m = j=1

By construction

m ∈ C ∞ (Γ; RN ),

(3.20)

m = 0 on Γ, and there exists a neighborhood U1 of Γ contained in U0 where m = 0 since m is at least C 1 . Now construct a function r0 in D(U1 ), 0 ≤ r0 (x) ≤ 1, and a neighborhood V of Γ such that r0 = 1 in V and Γ ⊂ V ⊂ V ⊂ U1 . Deﬁne the vector ﬁeld ∀x ∈ RN ,

def

N (x) = r0 (x)

m(x) . |m(x)|

(3.21)

Hence N belongs to D(RN , RN ) since supp N ⊂ V is compact. Moreover m N= in V ⇒ N (x) = n(x) on Γ = Γ ∩ V. |m| For each j, ρ ◦ hj ∈ C ∞ (B0 ) and the extensions ρhj (ζ) = ρhj (ζ , ζN ) = ρ(hj (ζ , 0)), def

∀ζ ∈ B,

def

ρ˜j = ρhj ◦ gj ,

3. Semiderivatives via Transformations Generated by Velocities

175

belong, respectively, to C ∞ (B) and C ∞ (Uj ). Then def

ρ˜ =

J

rj ρ˜j ∈ D(RN )

j=1

is an extension of ρ from Γ to RN with compact support since supp rj ⊂ Uj ⊂ U. Deﬁne the following transformation of RN : def

Tt (X) = X + t ρ˜(X)N (X),

t ≥ 0.

(3.22)

By construction, ρ˜ N is uniformly Lipschitzian in RN and, by Theorem 4.2 in section 4.2, there exists 0 < τ such that Tt is bijective and bicontinuous from RN onto itself. As a result from J. Dugundji [1] for 0 ≤ t ≤ τ Ωt = Tt (Ω) = [I + t˜ ρ N ](Ω), ∂Ωt = Tt (∂Ω) = [I + t˜ ρ N ](∂Ω) = [I + tρ n](∂Ω) = Γt . Since the domain Ωt is speciﬁed by its boundary Γt , it depends only on ρ and not on its extension ρ˜. The special transformation Tt introduced here is of class C ∞ , that is, Tt ∈ C ∞ (RN , RN ), and 1t [Tt − I] is proportional to the normal ﬁeld n on Γ, but it is not proportional to the normal nt on Γt for t > 0. In other words, at t = 0 the deformation is along n, but at t > 0 the deformation is generally not along nt . If J(Ω) is a real-valued shape functional deﬁned on C ∞ -domains in D, the semiderivative (if it exists) is deﬁned as follows: for all ρ ∈ C ∞ (Γ) J (I + t˜ ρN )Ω − J(Ω) def dn J(Ω; ρ˜) = lim . (3.23) t0 t It turns out that this limit depends only on ρ and not on its extension ρ˜. 3.3.2

C k -Domains

When Ω is a domain of class C k with boundary Γ, the normal ﬁeld n belongs to C k−1 (Γ, RN ). Therefore, choosing deformations along the normal would yield transformations {Tt } mapping C k -domains Ω onto C k−1 -domains Ωt = Tt (Ω). The obvious way to deal with C k -domains is to relax the constraint that the perturbation ρ˜N be carried by the normal. Choose vector ﬁelds θ in D k (RN , RN ) and consider the family of transformations Tt = I + tθ,

Ωt = Tt (Ω),

t ≥ 0.

(3.24)

This is a generalization of the family of transformations (3.22) in section 3.3.1 from ρ˜N to θ. For k ≥ 1, θ is again uniformly Lipschitzian in RN and, by Theorem 4.2 in section 4.2, there exists τ > 0 such that Tt is bijective and bicontinuous from RN onto itself. Thus for 0 ≤ t ≤ τ , Ωt = Tt (Ω) = [I + tθ](Ω)

and ∂Ωt = Tt (∂Ω) = [I + tθ](∂Ω).

176

Chapter 4. Transformations Generated by Velocities

A more restrictive approach to get around the lack of suﬃcient smoothness of the normal n to Γ would be to introduce a transverse ﬁeld p on Γ such that p ∈ C k (Γ; RN ),

∀x ∈ Γ,

p(x) · n(x) > 0.

Given p and ρ ∈ C k (Γ) deﬁne for t ≥ 0 def

Γt = x ∈ RN : x = X + tρ(X)p(X), ∀X ∈ Γ .

(3.25)

(3.26)

Choosing C k -extensions ρ˜ and p˜ of ρ and p we can go back to the case where θ = ρ˜p˜ ∈ Dk (RN , RN ).

(3.27)

For any θ ∈ Dk (RN , RN ) the semiderivative is deﬁned as J (I + tθ)(Ω) − J(Ω) def dk J(Ω; θ) = lim . t0 t 3.3.3

(3.28)

Cartesian Graphs

In many applications it is convenient to work with domains Ω which are the hypograph of some positive function γ in Cartesian coordinates. Such domains are typically of the form def (3.29) Ω = (x , xN ) ∈ RN : x ∈ U ⊂ RN −1 and 0 < xN < γ(x ) , where U is a connected open set in RN −1 and γ ∈ C 0 (U ; R+ ) is a positive function. Many free boundary and contact problems are formulated over such domains. Usually the domains Ω (and hence the functions γ) will be constrained. The function γ can be speciﬁed on U \U or not. In some examples the derivative of γ could also be speciﬁed, that is, ∂γ/∂ν = g on ∂U , where U is smooth and ν is the outward unit normal ﬁeld along ∂U . When γ (resp., ∂γ/∂ν) is speciﬁed along ∂U , the directions of deformation µ are chosen in C 0 (U ; R+ ) such that µ = 0 (resp., ∂µ/∂ν = 0) on ∂U.

(3.30)

For small t ≥ 0 and each such µ, deﬁne the perturbed domain

Ωt = (x , xN ) ∈ RN : 0 < xN < γ(x ) + tµ(x ) and the family of transformations (x , xN ) → Tt (x , xN ) =

x , xN +

(3.31)

xN tµ(x ) γ(x )

: Ω → Ωt ,

(3.32)

which can be extended to a neighborhood D of Ω containing the perturbed domains Ωt , 0 ≤ t ≤ t1 , for some small t1 > 0. In general, D will be such that D = U × [0, L] for some L > 0. This construction is also appropriate for domains that are Lipschitzian or of class C k , 1 ≤ k ≤ ∞, that is, when γ is a Lipschitzian or a C k -function. Again the

3. Semiderivatives via Transformations Generated by Velocities transformation Tt is equal to I + tθ, where µ(x ) def . θ(x , xN ) = 0, xN γ(x )

177

(3.33)

But of course this θ is not the only choice for that Ωt = Tt (Ω). For instance, let λ : R+ → R+ be any smooth increasing function such that λ(0) = 0 and λ(1) = 1. Then we could consider the transformations xN def ) . (3.34) µ(x Tt (x , xN ) = x , xN + tλ γ(x ) This example illustrates the general principle that the transformation Tt of D such that Ωt = Tt (Ω) is not unique. This means that, at least for smooth domains, only the trace Tt |Γt on Γt is important, while the displacement of the inner points does not contribute to the deﬁnition of Ωt . Nevertheless this statement is to be interpreted with caution. Here we implicitly assume that the objective function J(Ω) and the associated constraints are only a function of the shape of Ω. However, in some problems involving singularities at inner points of Ω (e.g., when the solution y(Ω) of the state equation has a singularity or when some constraints on the domain are active), the situation might require a ﬁner analysis. One such example is the internal displacement of the interior nodes of a triangularization τh when the solution y(Ω) of a partial diﬀerential equation is approximated by a piecewise polynomial solution over the triangularized domain Ωh in the ﬁnite element method. Such a displacement does not change the shape of Ωh , but it does change the solution yh of the problem. When the displacement of the interior nodes is a priori parametrized by the boundary nodes the solution yh will depend only on the position of the boundary nodes, but the interior nodes will contribute through the choice of the speciﬁed parametrization. 3.3.4

Polar Coordinates and Star-Shaped Domains

In some examples domains are star-shaped with respect to a point. Since a domain can always be translated, there is no loss of generality in assuming that this point is the origin. Then such domains Ω can be parametrized as follows: def

(3.35) Ω = x ∈ RN : x = ρζ, ζ ∈ SN −1 , 0 ≤ ρ < f (ζ) , where SN −1 is the unit sphere in RN , def

SN −1 = x ∈ RN : |x| = 1 ,

(3.36)

and f : SN −1 → R+ is a positive continuous mapping from SN −1 such that def

m = min {f (ζ) : ζ ∈ SN −1 } > 0.

(3.37)

Given any g ∈ C 0 (SN −1 ) and a suﬃciently small t ≥ 0 the perturbed domains are deﬁned as def

Ωt = x ∈ RN : x = ρζ, ζ ∈ SN −1 , 0 ≤ ρ < f (ζ) + tg(ζ) . (3.38)

178

Chapter 4. Transformations Generated by Velocities

For example, choose t, 0 ≤ t ≤ t1 , for some m t1 = >0 gC 0 (SN −1 )

(3.39)

and deﬁne the transformation Tt as Tt (X) = .0,

/

Tt (X) = ρ +

ρ t f (ζ) g(ζ)

if X = 0, ζ, if X = ρζ = 0.

(3.40)

As in the previous example, Tt is not unique and for any continuous increasing function λ : R+ → R+ such that λ(0) = 0 and λ(t) = 1 the transformation Tt (X) = .0,

Tt (X) = ρ + tλ

ρ f (ζ)

/

if X = 0,

g(ζ) ζ, if X = ρζ = 0,

(3.41)

yields the same domain Ωt . 3.3.5

Level Sets

In sections 3.3.1 to 3.3.4, the perturbed domain Ωt always appears in the form Ωt = Tt (Ω), where Tt is a bijective transformation of RN and Tt is of the form I +tθ. In some free boundary problems (e.g., plasma physics, propagation of fronts) the free boundary Γ is a level curve of a smooth function u deﬁned over an open domain D. Assume that D is bounded open with smooth boundary ∂D. Let u ∈ C 2 (D) be a positive function on D such that u ≥ 0 in D,

u = 0 on ∂D,

∃ a unique xu ∈ D such that ∀x ∈ D − {xu } , |∇u(x)| > 0.

If m = max u(x) : x ∈ D , then for each t in [0, m[ the level set Γt = u−1 (t) 2

(3.42)

(3.43)

N

is a C -submanifold of R in D, which is the boundary of the open set Ωt = {x ∈ D : u(x) > t} .

(3.44)

By deﬁnition Ω0 = D, for all t1 > t2 , Ωt1 ⊂ Ωt2 , and the domains Ωt converge in the Hausdorﬀ complementary topology5 to the point xu . The outward unit normal ﬁeld on Γt is given by x ∈ Γt ,

nt (x) = −|∇u(x)|−1 ∇u(x).

(3.45)

This suggests introducing the velocity ﬁeld ∀x ∈ D − {xu } ,

V (x) = |∇u(x)|−2 ∇u(x), def

(3.46)

which is continuous everywhere but at x = xu . If V were continuous everywhere, then for each X, the path x(t; X) generated by the diﬀerential equation dx (t) = V x(t) , x(0) = X (3.47) dt 5 Cf.

section 2 of Chapter 6.

3. Semiderivatives via Transformations Generated by Velocities would have the property that

u x(t) = u(X) + t

179

(3.48)

since formally

dx d u x(t) = ∇u x(t) · (t) = 1. (3.49) dt dt This means that the map X → Tt (X) = x(t; X) constructed from (3.47) would map the level set

(3.50) Γ0 = X ∈ D : u(X) = 0

onto the level set

Γt = x ∈ D : u(x) = t

(3.51)

and eventually Ω0 onto Ωt . Unfortunately, it is easy to see that this last property fails on the function u(x) = 1 − x2 deﬁned on the unit disk. To get around this diﬃculty, introduce for some arbitrarily small ε, 0 < ε < m/2, an inﬁnitely diﬀerentiable function ρε : RN → [0, 1] such that ρε (x) =

0, 1,

if |∇u(x)| < ε, if |∇u(x)| > 2ε,

(3.52)

x ∈ D.

(3.53)

and the velocity Vε (x) = ρε (x)V (x), As above, deﬁne the transformation X → Ttε (X) = x(t; X),

(3.54)

where x(t; X) is the solution of the diﬀerential equation dx (t) = Vε x(t) , t ≥ 0, x(0) = X ∈ D. (3.55) dt For 0 ≤ t < m − 2ε, Tt maps Γ0 onto Γt ; for 0 ≤ s < m − ε such that s + t < m − ε, Tt maps Γs onto Γt+s . However, for s > m − ε, Tt is the identity operator. As a result, for 0 ≤ t < m − 2ε, Tt (Ω0 ) = Ωt

and Tt (Γ0 ) = Γt .

(3.56)

Of course ε > 0 is arbitrary and we can make the construction for t’s arbitrary close to m. This is an example that can be handled by the velocity (speed) method and not by a perturbation of the identity. Here the domains Ωt are implicitly constrained to stay within the larger domain D. We shall see in section 5 how to introduce and characterize such a constraint. Another example of description by level sets is provided by the oriented distance function bΩ for some open domain Ω of class C 2 with compact boundary Γ (cf. Chapter 7). We shall see that there exists h > 0 and a neighborhood

Uh (Γ) = x ∈ RN : |bΩ (x)| < h such that bΩ ∈ C 2 (Uh (Γ)). Then for 0 ≤ t < h the ﬂow corresponding to the velocity ﬁeld V = ∇bΩ maps Ω and its boundary Γ onto

Tt (Ω) = Ωt = x ∈ RN : bΩ (x)| < t , Tt (Γ) = Γt = x ∈ RN : bΩ (x)| = t .

180

4

Chapter 4. Transformations Generated by Velocities

Unconstrained Families of Domains

´sio [12, Here we study equivalences between the velocity method (cf. J.-P. Zole 8]) and methods using a family of transformations. In section 4.1, we give some general conditions to construct a family of transformations of RN from a velocity ﬁeld. Conversely, we show how to construct a velocity ﬁeld from a family of transformations of RN . In section 4.2, this construction is applied to Lipschitzian perturbations of the identity. In section 4.3, the various equivalences of section 4.1 are specialized to velocities in C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0.

4.1

Equivalence between Velocities and Transformations

Let the real number τ > 0 and the map V : [0, τ ] × RN → RN be given. If t is interpreted as an artiﬁcial time, the map V can be viewed as the (time-dependent) velocity ﬁeld {V (t) : 0 ≤ t ≤ τ } deﬁned on RN : def

x → V (t)(x) = V (t, x) : RN → RN . Assume that there exists τ = τ (V ) > 0 such that ∀x ∈ RN , V (·, x) ∈ C [0, τ ]; RN , (V) ∃c > 0, ∀x, y ∈ RN , V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|,

(4.1)

(4.2)

where V (·, x) is the function t → V (t, x). Note that V is continuous on [0, τ ] × RN . Hence it is uniformly continuous on [0, τ ] × D for any bounded open subset D of RN and V (·) ∈ C([0, τ ]; C(D; RN )).

(4.3)

Associate with V the solution x(t; V ) of the vector ordinary diﬀerential equation dx (t) = V t, x(t) , dt

t ∈ [0, τ ],

x(0) = X ∈ RN

(4.4)

and deﬁne the transformations def

X → Tt (V )(X) = xV (t; X) : RN → RN

(4.5)

and the maps (whenever the inverse of Tt exists) def

(t, X) → TV (t, X) = Tt (V )(X) : [0, τ ] × RN → RN ,

(4.6)

(t, x) → TV−1 (t, x) = Tt−1 (V )(x) : [0, τ ] × RN → RN .

(4.7)

def

Notation 4.1. In what follows we shall drop the V in TV (t, X), TV−1 (t, x), and Tt (V ) whenever no confusion arises.

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181

Theorem 4.1. (i) Under assumptions (V) the map T speciﬁed by (4.4)–(4.6) has the following properties: ∀X ∈ RN , T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1) ∀X, Y ∈ RN , T (·, Y ) − T (·, X)C 1 ([0,τ ];RN ) ≤ c|Y − X|, (T2) (T3)

∀t ∈ [0, τ ], X → Tt (X) = T (t, X) : RN → RN is bijective, ∀x ∈ RN , T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0, ∀x, y ∈ RN ,

(4.8)

T −1 (·, y) − T −1 (·, x)C([0,τ ];RN ) ≤ c|y − x|.

(ii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1) to (T3), the map def

(t, x) → V (t, x) =

∂T t, Tt−1 (x) : [0, τ ] × RN → RN ∂t

(4.9)

satisﬁes conditions (V), where Tt−1 is the inverse of X → Tt (X) = T (t, X). If, in addition, T (0, ·) = I, then T (·, X) is the solution of (4.4) for that V . (iii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1) and (T2) and T (0, ·) = I, then there exists τ > 0 such that the conclusions of part (ii) hold on [0, τ ]. A more general version of this theorem for constrained domains (Theorem 5.1) will be given and proved in section 5.1. Proof. (i) Conditions (T1) follow by standard arguments. Condition (T2). Associate with X in RN the function y(s) = Tt−s (X), Then

dy (s) = −V (t − s, y(s)), ds

0 ≤ s ≤ t.

0 ≤ s ≤ t,

y(0) = Tt (X).

(4.10)

y(0) = x ∈ RN

(4.11)

For each x ∈ RN , the diﬀerential equation dy (s) = −V (t − s, y(s)), ds

0 ≤ s ≤ t,

has a unique solution in C 1 ([0, t]; RN ). The solutions of (4.11) deﬁne the map def

x → St (x) = y(t) : RN → RN such that ∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ RN ,

|St (y) − St (x)| ≤ c|y − x|.

(4.12)

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Chapter 4. Transformations Generated by Velocities

In view of (4.10) and (4.11) St (Tt (X)) = y(t) = Tt−t (X) = X

⇒ St ◦ Tt = I on RN .

To obtain the other identity, consider the function z(r) = y(t − r; x), where y(·, x) is the solution of (4.11) for some arbitrary x in RN . By deﬁnition dz (r) = V (r, z(r)), dr

z(0) = y(t, x),

and necessarily x = y(0; x) = z(t) = Tt (y(t; x)) = Tt (St (x)) ⇒ Tt ◦ St = I on RN

⇒ St = Tt−1 : RN → RN .

Conditions (T3). The uniform Lipschitz continuity in (T3) follows from (4.12), and we need only show that ∀x ∈ RN ,

T −1 (·, x) ∈ C([0, τ ]; RN ).

Given t in [0, τ ] pick an arbitrary sequence {tn }, tn → t. Then for each x ∈ RN there exists X ∈ RN such that Tt (X) = x and Ttn (X) → Tt (X) = x from the ﬁrst condition (T1). But (x) − Tt−1 (x) = Tt−1 (Tt (X)) − Tt−1 (Tt (X)) Tt−1 n n (Tt (X)) − Tt−1 (Ttn (X)) . = Tt−1 n n By the uniform Lipschitz continuity of Tt−1 |Tt−1 (x) − Tt−1 (x)| = |Tt−1 (Tt (X)) − Tt−1 (Ttn (X)) | ≤ c|Tt (X) − Ttn (X)|, n n n and the last term converges to zero as tn goes to t. (ii) The ﬁrst part of conditions (V) is satisﬁed since, for each x ∈ RN and t, s in [0, τ ], |V (t, x) − V (s, x)| ∂T ∂T ∂T ∂T (t, Tt−1 (x)) − (t, Ts−1 (x)) + (t, Ts−1 (x)) − (s, Ts−1 (x)) ≤ ∂t ∂t ∂t ∂t ∂T ∂T −1 −1 −1 −1 (t, Ts (x)) − (s, Ts (x)) . ≤ c|Tt (x) − Ts (x)| + ∂t ∂t

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183

Thus from (T3) and (T1) t → V (t, x) is continuous at s = t, and hence for all x in RN , V (., x) ∈ C([0, τ ]; RN ). The Lipschitzian property follows directly from the Lipschitzian properties (T1) and (T3): for all x and y in RN , ∂T ∂T −1 −1 |V (t, y) − V (t, x)| = (t, Tt (y)) − (t, Tt (x)) ∂t ∂t −1 −1 ≤ c Tt (y) − Tt (x) ≤ cc |y − x|. This proves that V satisﬁes condition (V). (iii) From (T1) and (T2) for f (t) = Tt − I and t ≥ s, t ∂T ∂T (r, Tr (y)) − (r, Tr (x)) dr, f (t)(y) − f (t)(x) = ∂t ∂t 0 t c |Tr (y) − Tr (x)| dr ≤ c2 t |y − x|. |f (t)(y) − f (t)(x)| ≤ 0

For τ = min{τ, 1/(2c )} and 0 ≤ t ≤ τ , c(f (t)) ≤ 1/2, 2

g(t) = Tt−1 − I = (I − Tt ) ◦ Tt−1 = −f (t) ◦ [I + g(t)], (1 − c(f (t))) c(g(t)) ≤ c(f (t))

⇒ c(g(t)) ≤ 1 and c(Tt−1 ) ≤ 2,

and the second condition (T3) is satisﬁed on [0, τ ]. The ﬁrst one follows by the same argument as in part (i). Therefore the conclusions of part (ii) are true on [0, τ ]. This equivalence theorem says that we can start either from a family of velocity ﬁelds {V (t)} on RN or a family of transformations {Tt } of RN provided that the map V , V (t, x) = V (t)(x), satisﬁes (V) or the map T , T (t, X) = Tt (X), satisﬁes (T1) to (T3). Starting from V , the family of homeomorphisms {Tt (V )} generates the family def

Ωt = Tt (V )(Ω) = {Tt (V )(X) : X ∈ Ω}

(4.13)

of perturbations of the initial domain Ω. Interior (resp., boundary) points of Ω are mapped onto interior (resp., boundary) points of Ωt . This is the basis of the velocity method which will be used to deﬁne shape derivatives.

4.2

Perturbations of the Identity

In examples it is usually possible to show that the transformation T satisﬁes assumptions (T1) to (T3) and construct the corresponding velocity ﬁeld V deﬁned in (4.9). For instance, consider perturbations of the identity to the ﬁrst (A = 0) or second order: for t ≥ 0 and X ∈ RN , def

Tt (X) = X + tU (X) +

t2 A(X), 2

(4.14)

where U and A are transformations of RN . It turns out that for Lipschitzian transformations U and A, assumptions (T1) to (T3) are satisﬁed in some interval [0, τ ].

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Chapter 4. Transformations Generated by Velocities

Theorem 4.2. Let U and A be two uniform Lipschitzian transformations of RN : there exists c > 0 such that for all X, Y ∈ RN , |U (Y ) − U (X)| ≤ c|Y − X|

and

|A(Y ) − A(X)| ≤ c|Y − X|.

There exists τ > 0 such that the map T given by (4.14) satisﬁes conditions (T1) to (T3) on [0, τ ]. The associated velocity V given by (t, x) → V (t, x) = U Tt−1 (x) + tA Tt−1 (x) : [0, τ ] × RN → RN (4.15) satisﬁes conditions (V) on [0, τ ]. Remark 4.1. Observe that from (4.14) and (4.15) V (0) = U,

∂V V˙ (0)(x) = (t, x) t=0 = A − [DU ]U, ∂t

(4.16)

where DU is the Jacobian matrix of U . The term V˙ (0) is an acceleration at t = 0 which will always be present even when A = 0, but it can be eliminated by choosing A = [DU ]U . Proof. (i) By the deﬁnition of T in (4.14), t → T (t, X) and t → ∂T (t, X) = U (X) + ∂t tA(X) are continuous on [0, ∞[. Moreover, for all X and Y , t2 |T (t, Y ) − T (t, X)| ≤ 1 + tc + c |Y − X| 2 and

∂T ∂T ≤ [c + tc] |Y − X|. (t, Y ) − (t, X) ∂t ∂t

Thus conditions (T1) are satisﬁed for any ﬁnite τ > 0. To check condition (T2), def

consider for any Y ∈ RN the mapping h(X) = Y −[Tt (X)−X]. For any X1 and X2 |h(X2 ) − h(X1 )| ≤ t|U (X2 ) − U (X1 )| +

t2 |A(X2 ) − A(X1 )| 2

t2 ≤ t c |X2 − X1 | + c |X2 − X1 | = t c [1 + t/2] |X2 − X1 |. 2

(4.17)

For τ = min{1, 1/(4c)}, and any t, 0 ≤ t ≤ τ , t c [1 + t/2] < 1/2 and h is a contraction. So for all 0 ≤ t ≤ τ and Y ∈ RN , there exists a unique X ∈ RN such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y and Tt is bijective. Therefore (T2) is satisﬁed in [0, τ ]. The last part of the proof is the uniform Lipschitzian property of Tt−1 . In view of (4.17), for all t, 0 ≤ t ≤ τ ,

4. Unconstrained Families of Domains

185

t c [1 + t/2] < 1/2 and |Tt (X2 ) − X2 − (Tt (X1 ) − X1 )| = |h(X2 ) − h(X1 )| ≤

1 |X2 − X1 | 2

1 |X2 − X1 | 2 ⇒ |X2 − X1 | ≤ 2 |Tt (X2 ) − Tt (X1 )|.

⇒ |X2 − X1 | − |Tt (X2 ) − Tt (X1 )| ≤

In view of condition (T2) for all x and y, |Tt−1 (y) − Tt−1 (x)| ≤ 2 |Tt (Tt−1 (y)) − Tt (Tt−1 (x))| = 2 |y − x|.

(4.18)

To complete our argument we prove the continuity with respect to t for each x. Let X = Tt−1 (x). For any s in [0, τ ] Ts−1 (x) − Tt−1 (x) = Ts−1 (Tt (X)) − Tt−1 (Tt (X)) = Ts−1 (Tt (X)) − Ts−1 (Ts (X)), and in view of (4.18) |Ts−1 (x) − Tt−1 (x)| ≤ 2|Tt (X) − Ts (X)|. The continuity of Ts−1 (x) at s = t now follows from the continuity of Ts (X) at s = t. Thus conditions (T3) are satisﬁed.

4.3

Equivalence for Special Families of Velocities

In this section we specialize Theorem 4.1 to velocities in C k,1 (RN , RN ), C0k+1 (RN , RN ), and C k+1 (RN , RN ), k ≥ 0. The following notation will be convenient: def

f (t) = Tt − I,

f (t) =

dTt , dt

g(t) = Tt−1 − I, def

whenever Tt−1 exists and the identities g(t) = −f (t) ◦ Tt−1 = −f (t) ◦ [I + g(t)], dTt ◦ Tt−1 = f (t) ◦ Tt−1 = f (t) ◦ [I + g(t)]. V (t) = dt Recall also for a function F : RN → RN the notation def

|F (y) − F (x)| and ∀k ≥ 1, |y − x| y=x

def

c(F ) = sup

ck (F ) =

c(∂ α F ).

|α|=k

Theorem 4.3. Let k ≥ 0 be an integer. (i) Given τ > 0 and a velocity ﬁeld V such that V ∈ C([0, τ ]; C k (RN , RN ))

and

ck (V (t)) ≤ c

(4.19)

186

Chapter 4. Transformations Generated by Velocities for some constant c > 0 independent of t, the map T given by (4.4)–(4.6) satisﬁes conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C k (RN , RN )) ∩ C([0, τ ]; C k,1 (RN , RN )),

ck (f (t)) ≤ c, (4.20)

for some constant c > 0 independent of t. Moreover, conditions (T3) are satisﬁed and there exists τ > 0 such that g ∈ C([0, τ ]; C k (RN , RN )),

ck (g(t)) ≤ ct,

(4.21)

for some constant c independent of t. (ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.20) and T (0, ·) = I, there exists τ > 0 such that the velocity ﬁeld V (t) = f (t) ◦ Tt−1 satisﬁes conditions (V) and (4.19) in [0, τ ]. Proof. We prove the theorem for k = 0. The general case is obtained by induction over k. (i) By assumption on V , the conditions (V) given by (4.2) are satisﬁed and by Theorem 4.1 the corresponding family T satisﬁes conditions (T1) to (T3). Conditions (4.20) on f . For any x and s ≤ t t Tt (x) − Ts (x) = V (r) ◦ Tr (x) dr, s t |Tt (x) − Ts (x)| ≤ c|Tr (x) − Ts (x)| + |V (r) ◦ Ts (x)| dr, s t c|f (r)(x) − f (s)(x)| + V (r)C dr. |f (t)(x) − f (s)(x)| ≤ s

By assumption on V and Gronwall’s inequality, ∀t, s ∈ [0, τ ],

f (t) − f (s)C ≤ c |t − s|

for another constant c independent of t. Moreover, |(f (t) − f (s))(y) − (f (t) − f (s))(x)| = |(Tt − Ts )(y) − (Tt − Ts )(x)| t ≤ |V (r) ◦ Tr (y) − V (r) ◦ Tr (x)| dr s t c|(Tr − Ts )(y) − (Tr − Ts )(x)| + c|Ts (y) − Ts (x)| dr ≤ s t ≤ c|(f (r) − f (s))(y) − (f (r) − f (s))(x)| + cc |y − x| dr s

for some other constant c by the second condition (T1). Again by Gronwall’s inequality there exists another constant c such that |(f (t) − f (s))(y) − (f (t) − f (s))(x)| ≤ c|t − s| |y − x| ⇒ c(f (t) − f (s)) ≤ c |t − s| ⇒ f ∈ C([0, τ ]; C 0,1 (RN , RN )) and f (t) − f (s)C0,1 ≤ c |t − s|.

(4.22)

4. Unconstrained Families of Domains

187

Moreover, f (t) = V (t) ◦ Tt and |f (t)(x) − f (s)(x)| ≤ |V (t)(Tt (x)) − V (s)(Tt (x))| + |V (s)(Tt (x)) − V (s)(Ts (x))| ≤ V (t) − V (s)C + c(V (s)) Tt − Ts C ≤ V (t) − V (s)C + c f (t) − f (s)C . Finally, |f (t)(y) − f (t)(x)| ≤ |V (t)(Tt (y)) − V (t)(Tt (x))| ≤ c(V (t)) |Tt (y) − Ts (x)| ≤ c c(Tt ) |y − x| and c(f (t)) ≤ c for some new constant c independent of t. Therefore, f ∈ C 1 ([0, τ ]; C(RN , RN )) and c(f (t)) ≤ c. Conditions (4.21) on g. Since conditions (T1) and (T2) are satisﬁed there exists τ > 0 such that conditions (T3) are satisﬁed by Theorem 4.1 (iii). Moreover, from conditions (4.20) |g(t)(y) − g(t)(x)| ≤ |f (t)(Tt−1 (y)) − f (t)(Tt−1 (x))| ≤ c(f (t))|Tt−1 (y) − Tt−1 (x)| ≤ c(f (t)) (|g(t)(y) − g(t)(x)| + |y − x|) ⇒ (1 − c(f (t))) |g(t)(y) − g(t)(x)| ≤ c(f (t)) |y − x| ≤ ct |y − x|. Choose a new τ = min{τ , 1/(2c)}. Then for 0 ≤ t ≤ τ , c(g(t)) ≤ 2ct. Now g(t) − g(s) = −f (t) ◦ [I + g(t)] + f (s) ◦ [I + g(s)], g(t) − g(s)C ≤f (t) ◦ [I + g(t)] − f (t) ◦ [I + g(s)]C + f (t) ◦ [I + g(s)] − f (s) ◦ [I + g(s)]C ≤c(f (t))g(t) − g(s)C + f (t) − f (s)C ≤ct g(t) − g(s)C + f (t) − f (s)C . For t in [0, τ ], ct ≤ 1/2, and g(t) − g(s)C ≤ 2f (t) − f (s)C ⇒ g ∈ C([0, τ ]; C(RN , RN )) and c(g(t)) ≤ 2ct. The conditions (4.20) on f are satisﬁed for k = 0. For k = 1 we start from the equation

t

DTt − DTs =

DV (r) ◦ Tr DTr dr s

188

Chapter 4. Transformations Generated by Velocities

and use the fact that DTt−1 = [DTt ]−1 ◦ Tt−1 in connection with the identity Dg(t) = −Df (t) ◦ Tt−1 DTt−1 = −(Df (t)[DTt ]−1 ) ◦ Tt−1 . (ii) From conditions (4.20) on f , the transformation T satisﬁes conditions (T1). To check condition (T2) we consider two cases: k ≥ 1 and k = 0. For k ≥ 1 the function t → Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t → det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ > 0 such that Tt is invertible for all t in [0, τ ] and (T2) is satisﬁed in [0, τ ]. In the case k = 0 consider for any Y the map h(X) = Y − f (t)(X). For any X1 and X2 , |h(X2 ) − h(X1 )| ≤ c(f (t)) |X2 − X1 |. But by assumption f ∈ C([0, τ ]; C 0,1 (RN )) and c(f (0)) = 0 since f (0) = 0. Hence there exists τ > 0 such that c(f (t)) ≤ 1/2 for all t in [0, τ ] and h is a contraction. So for all Y in RN there exists a unique X such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y, Tt is bijective, and condition (T2) is satisﬁed in [0, τ ]. By Theorem 4.1 (iii) from (T1) and (T2), there exists another τ > 0 for which conditions (T3) on g and (V) on V (t) = f (t) ◦ Tt−1 are also satisﬁed. Moreover, we have seen in the proof of part (i) that conditions (4.21) on g follow from (T2) and (4.20). Using conditions (4.20) and (4.21), |V (t)(y) − V (t)(x)| ≤ |f (t)(Tt−1 (y)) − f (t)(Tt−1 (x))| ≤ c(f (t)) |Tt−1 (y) − Tt−1 (x)| ≤ c(f (t)) [1 + c(g(t))] |y − x| ≤ c |y − x| and c(V (t)) ≤ c . Also |V (t)(x) − V (s)(x)| = |f (t)(Tt−1 (x)) − f (s)(Ts−1 (x))| ≤ |f (t)(Tt−1 (x)) − f (t)(Ts−1 (x))| + |f (t)(Ts−1 (x)) − f (s)(Ts−1 (x))| ≤ c(f (t)) |Tt−1 (x) − Ts−1 (x)| + f (t) − f (s)C ≤ c g(t) − g(s)C + f (t) − f (s)C . Therefore, since both g and f are continuous, V ∈ C([0, τ ]; C(RN , RN )) and c(V (t)) ≤ c for some constant c independent of t. This proves the result for k = 0. As in part (i), for k = 1 we use the identity DV (t) = Df (t) ◦ Tt−1 DTt−1 = (Df (t)[DTt ]−1 ) ◦ Tt−1 and proceed in the same way. The general case is obtained by induction over k.

4. Unconstrained Families of Domains

189

We now turn to the case of velocities in C0k (RN , RN ). As in Chapter 2, it will be convenient to use the notation C0k for the space C0k (RN , RN ), C k (RN ) for the space C k (RN , RN ), and C k,1 (RN ) for the space C k,1 (RN , RN ). Theorem 4.4. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity ﬁeld V such that V ∈ C([0, τ ]; C0k (RN , RN )),

(4.23)

the map T given by (4.4)–(4.6) satisﬁes conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C0k (RN , RN )).

(4.24)

Moreover, conditions (T3) are satisﬁed and there exists τ > 0 such that g ∈ C([0, τ ]; C0k (RN , RN )).

(4.25)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.24) and T (0, ·) = I, there exists τ > 0 such that the velocity ﬁeld V (t) = f (t) ◦ Tt−1 satisﬁes conditions (V) and (4.23) on [0, τ ]. Proof. As in the proof of Theorem 4.3, we prove only the theorem for k = 1. The general case is obtained by induction on k, the various identities on f , g, f , and V , and the techniques of Lemma 2.4, Theorems 2.11 and 2.12, and Lemmas 2.5 and 2.6 of section 2.5 in Chapter 3. (i) By the embedding C01 (RN , RN ) ⊂ C 1 (RN , RN ) ⊂ C 0,1 (RN , RN ), it follows from (4.23) that V ∈ C([0, τ ]; C 0,1 (RN , RN )) and condition (4.19) of Theorem 4.3 are satisﬁed. Therefore, conditions (4.20) and (4.21) of Theorem 4.3 are also satisﬁed in some interval [0, τ ], τ > 0. Conditions (4.24) on f . It remains to show that f (t) and f (t) belong to the subspace C0 (RN , RN ) of C(RN , RN ) and to prove the appropriate properties for Df (t) and Df (t). Recall from the proof of the previous theorems that there exists c > 0 such that t t |f (t)(x)| ≤ c |V (r)(x)| dr ≤ c |(V (r) − V (0))(x)| dr + c t |V (0)(x)|. 0

0

By assumption on V (0), for ε > 0 there exists a compact set K such that ∀x ∈ K,

|V (0)(x)| ≤ ε/(2c)

and there exists δ, 0 < δ < 1, such that ∀t, 0 ≤ t ≤ δ,

V (r) − V (0)C ≤ ε/(2c)

⇒ ∀t, 0 ≤ t ≤ δ, ∀x ∈ K,

|f (t)(x)| ≤ ε

⇒ f (t) ∈ C0 .

Proceeding in this fashion from the interval [0, δ] to the next interval [δ, 2δ] using the inequality t |(f (t) − f (s))(x)| ≤ c |(V (r) − V (δ))(x)| dr + c |t − δ| |V (δ)(x)|, s

190

Chapter 4. Transformations Generated by Velocities

the uniform continuity of V ∀t, s,

|t − s| < δ,

V (t) − V (s)C ≤ ε/(2c),

and the fact that V (δ) ∈ C0 , that is, that there exists a compact set K(δ) such that ∀x ∈ K(δ),

|V (δ)(x)| ≤ ε/(2c),

we get f (t) ∈ C0 , δ ≤ t ≤ 2δ, and hence f ∈ C([0, τ ]; C0 ). For f (t) we make use of the identity f (t) = V (t) ◦ Tt . Again by assumption for any ε > 0 there exists a compact set K(t) such that |V (t)(x)| ≤ ε on K(t). Thus by choosing the compact set Kt = Tt−1 (K(t)), |f (t)(x)| ≤ ε on Kt , and f ∈ C([0, τ ]; C0 ). In order to complete the proof, it remains to establish the same properties for Df (t) and Df (t). The matrix Df (t) is solution of the equations d Df (t) = DV (t) ◦ Tt DTt , dt

Df (0) = 0

⇒ Df (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt .

(4.26)

From the proofs of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, for each t the elements of the matrix def

A(t) = DV (t) ◦ Tt = DV (t) ◦ [I + f (t)] belong to C0 since DV (t) and f (t) do. By assumption, V ∈ C([0, τ ]; C0k ) and V and all its derivatives ∂ α V are uniformly continuous in [0, τ ] × RN . Therefore, for each ε > 0 there exists δ > 0 such that ∀ |t − s| < δ, ∀ |y − x | < δ,

|DV (t)(y ) − DV (s)(x )| < ε.

Pick 0 < δ < δ such that ∀ |t − s| < δ ,

Tt − Ts C = f (t) − f (s)C < δ ⇒ ∀x, ∀ |t − s| < δ , |Tt (x) − Ts (x)| < δ ⇒ |DV (t)(Tt (x)) − DV (s)(Ts (x))| < ε

⇒ A(t) − A(s)C < ε

⇒ A ∈ C([0, τ ]; (C0 )N ).

For each x, Df (t)(x) is the unique solution of the linear matrix equation (4.26). To show that Df (t) ∈ (C0 )N we ﬁrst show that Df (t)(x) is uniformly continuous for x in RN . For any x and y t |V (r, Tr (y)) − V (r, Tr (x))| dr |Df (t)(y) − Df (t)(x)| ≤ 0 t ≤ c|Tr (y) − Tr (x)| dr 0 t |f (r)(y) − f (r)(x)| + |y − x| dr. ≤c 0

4. Unconstrained Families of Domains

191

But f ∈ C([0, τ ]; C0 ) is uniformly continuous in (t, x): for each ε > 0 there exists δ, 0 < δ < ε/(2cτ ), such that ∀ |t − s| < δ, ∀ |y − x| < δ,

|f (t)(y) − f (s)(x)| < ε/(2cτ ).

Substituting in the previous inequality for each ε > 0, there exists δ > 0 such that ∀ t, ∀ |y − x| < δ,

|Df (t)(y) − Df (t)(x)| < ε.

Hence Df (t) is uniformly continuous in RN . Furthermore, from (4.26) we have the following inequality: t |Df (t)(x)| ≤ |DV (r)(Tr (x))| |Df (r)(x)| + |DV (r)(Tr (x))| dr t 0 (4.27) (|Df (r)(x)| + 1) dr ≤c 0

since V ∈ C([0, τ ]; C01 ). By Gronwall’s inequality |Df (t)(x)| ≤ c t for some other constant c independent of t. Hence Df (t) ∈ C(RN , RN )N . Finally to show that Df (t) vanishes at inﬁnity we start from the integral form of (4.26): t DV (r)(Tr (x)) DTr (x) dr, Df (t)(x) = t 0 |DV (r)(Tr (x)) − DV (r)(x)| + |DV (r)(x)| dr |Df (t)(x)| ≤ c t 0 ≤c |f (r)(x)| + |DV (r)(x)| dr. 0

By the same technique as for f (t), it follows that the elements of Df (t) belong to C0 since both f (s) and DV (r) do. Finally for the continuity with respect to t t A(r)Df (r) + A(r) dr, Df (t) − Df (s) = s t Df (t) − Df (s)C ≤ A(r)C Df (r) − Df (s)C s + A(r)C (1 + Df (r)C ) dr. Again, by Gronwall’s inequality, there exists another constant c such that Df (t) − Df (s)C ≤ c |t − s|. Therefore, Df ∈ C([0, τ ]; (C0 )N ) and f ∈ C([0, τ ]; C01 ). For Df we repeat the proof for f using the identity Df (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt to get Df ∈ C([0, τ ]; (C0 )N )

⇒ f ∈ C([0, τ ]; C01 ).

192

Chapter 4. Transformations Generated by Velocities

Conditions (4.25) on g. From the remark at the beginning of part (i) of the proof, the conclusions of Theorem 4.3 are true for g, and it remains to check the remaining properties for g and Dg using the identities g(t) = −f (t) ◦ [I + g(t)],

Dg(t) = −Df (t) ◦ [I + g(t)] (I + Dg(t)).

From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, g(t) ∈ C0 since Df (t) and g(t) belong to C0 . Therefore, g(t) ∈ C01 . The continuity follows by the same argument as for f and g ∈ C([0, τ ]; C01 ). (ii) By assumption from conditions (4.24), conditions (T1) are satisﬁed. For (T2) observe that for k ≥ 1 the function t → Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t → det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ > 0 such that Tt is invertible for all t in [0, τ ] and (T2) is satisﬁed in [0, τ ]. Furthermore, from the proof of part (i) conditions (T3) and (4.25) on g are also satisﬁed in some interval [0, τ ], τ > 0. Therefore, the velocity ﬁeld V (t) = f (t) ◦ Tt−1 = f (t) ◦ [I + g(t)] satisﬁes the conditions (V) speciﬁed by (4.2) in [0, τ ]. From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, V (t) ∈ C0k since f (t) and g(t) belong to C0k . By assumption, f ∈ C 1 ([0, τ ]; C0k ). Hence f and all its derivatives ∂ α f , |α| ≤ k, are uniformly continuous on [0, τ ] × RN ; that is, given ε > 0, there exists δ > 0 such that ∀t, s, |t − s| < δ, ∀y , x , |y − x | < δ,

|∂ α f (t)(y ) − ∂ α f (s)(x )| < ε.

Similarly g ∈ C([0, τ ]; C0k ) and there exists 0 < δ ≤ δ such that ∀t, s, |t − s| < δ , ∀y, x, |y − x| < δ ,

|∂ α g(t)(y) − ∂ α g(s)(x)| < δ.

Therefore, for |t − s| < δ Tt−1 − Ts−1 C = g(t) − g(s)C < δ, and since δ < δ ∀x, |f (t)(Tt−1 (x)) − f (t)(Ts−1 (x))| < ε ⇒ V (t) − V (s)C < ε ⇒ V ∈ C([0, τ ]; C0 ). We then proceed to the ﬁrst derivative of V , DV (t) = Df (t) ◦ Tt−1 DTt−1 = Df (t) ◦ [I + g(t)] [I + Dg(t)], and by uniform continuity of the right-hand side V ∈ C([0, τ ]; C01 ). By induction on k, we ﬁnally get V ∈ C([0, τ ]; C0k ). The proof of the last theorem is based on the fact that the vector functions involved are uniformly continuous. The fact that they vanish at inﬁnity is not an essential element of the proof. Therefore, the theorem is valid with C k (RN , RN ) in place of C0k (RN , RN ).

5. Constrained Families of Domains

193

Theorem 4.5. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity ﬁeld V such that V ∈ C([0, τ ]; C k (RN , RN )),

(4.28)

the map T given by (4.4)–(4.6) satisﬁes conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C k (RN , RN )).

(4.29)

Moreover, conditions (T3) are satisﬁed and there exists τ > 0 such that g ∈ C([0, τ ]; C k (RN , RN )).

(4.30)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.29) and T (0, ·) = I, there exists τ > 0 such that the velocity ﬁeld V (t) = f (t) ◦ Tt−1 satisﬁes conditions (V) and (4.28) on [0, τ ].

5

Constrained Families of Domains

We now turn to the case where the family of admissible domains Ω is constrained to lie in a ﬁxed larger subset D of RN or its closure. For instance, D can be an open set or a closed submanifold of RN .

5.1

Equivalence between Velocities and Transformations

Given a nonempty subset D of RN , consider a family of transformations T : [0, τ ] × D → RN

(5.1)

for some τ = τ (T ) > 0 with the following properties: ∀X ∈ D, T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1D ) ∀X, Y ∈ D, T (·, Y ) − T (·, X)C 1 ([0,τ ];RN ) ≤ c|Y − X|, (T2D ) (T3D )

X → Tt (X) = T (t, X) : D → D is bijective, ∀x ∈ D, T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0,

∀t ∈ [0, τ ], ∀x, y ∈ D,

(5.2)

T −1 (·, y) − T −1 (·, x)C([0,τ ];RN ) ≤ c|y − x|,

where under assumption (T2D ) T −1 is deﬁned from the inverse of Tt as (t, x) → T −1 (t, x) = Tt−1 (x) : [0, τ ] × D → RN . def

(5.3)

These three properties are the analogue for D of the same three properties obtained for RN . In fact, Theorem 4.1 extends from RN to D by adding one assumption to (V). Speciﬁcally, consider a velocity ﬁeld V : [0, τ ] × D → RN

(5.4)

194

Chapter 4. Transformations Generated by Velocities

for which there exists τ = τ (V ) > 0 such that (V1D ) (V2D )

∀x ∈ D, ∀x, y ∈ D,

V (·, x) ∈ C [0, τ ]; RN ,

∃c > 0,

V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|,

∀x ∈ D, ∀t ∈ [0, τ ],

(5.5)

±V (t, x) ∈ TD (x),

where TD (x) is Bouligand’s contingent cone6 to D at the point x in D % % * 1 def TD (X) = (D − X) + εB h ε>0 α>0

(5.6)

0 0 and a map T : [0, τ ] × D → D satisfying assumptions (T1D ) and (T2D ) and T (0, ·) = I, then there exists τ > 0 such that the conclusions of part (ii) hold on [0, τ ]. Remark 5.1. Assumption (V 2D ) is a double viability condition. M. Nagumo [1]’s usual viability condition (5.11) V (t, x) ∈ TD (x), ∀t ∈ [0, τ ], ∀x ∈ D is a necessary and suﬃcient condition for a viable solution to (5.9): ∀t ∈ [0, τ ], ∀X ∈ D, x(t; X) ∈ D or Tt (D) ⊂ D

(5.12)

(cf. J.-P. Aubin and A. Cellina [1, p. 174 and p. 180]). Condition (V 2D ), ∀t ∈ [0, τ ], ∀x ∈ D,

±V (t, x) ∈ TD (x),

(5.13)

is a strict viability condition which says that Tt maps D into D and ∀t ∈ [0, τ ],

Tt : D → D

is a homeomorphism.

(5.14)

In particular it maps interior points onto interior points and boundary points onto boundary points (cf. J. Dugundji [1, pp. 87–88]). Remark 5.2. Condition (V 2D ) is a generalization to an arbitrary set D of the following condition ´sio [12] in 1979: for all x in ∂D, used by J.-P. Zole V (t, x) · n(x) = 0, 0,

if the outward normal n(x) exists, otherwise.

Proof of Theorem 5.1. (i) Existence and uniqueness of viable solutions to (5.9). Apply M. Nagumo [1]’s theorem to the augmented system on [0, τ ] dx (t) = V t, x(t) , x(0) = X ∈ D, dt (5.15) dx0 (t) = 1, x0 (0) = 0, dt that is,

x dˆ (t) = Vˆ x ˆ(t) , dt x ˆ def ˆ(0) = (0, X) ∈ D = R+ ×D,

(5.16)

196

Chapter 4. Transformations Generated by Velocities

x) = 1, V˜ (ˆ x) , and where x ˆ(t) = x0 (t), x(t) ∈ RN +1 , Vˆ (ˆ V (x0 , x), 0 ≤ x0 ≤ τ ˜ V (x0 , x) = , x ∈ D. V (τ, x), τ < x0

(5.17)

It is easy that systems (5.15) and (5.16) are equivalent on [0, τ ] and that to check x ˆ(t) = t, x(t) . The new velocity ﬁeld on Vˆ ⊂ RN +1 is continuous at each point ˆ by the ﬁrst assumption (V 1D ) since x ˆ∈D Vˆ (ˆ y ) − Vˆ (ˆ x) = 0, V˜ (y0 , y) − V˜ (x0 , x) , and for 0 ≤ x0 , y0 ≤ τ |V (y0 , y) − V (x0 , x)| ≤ |V (y0 , y) − V (y0 , x)| + |V (y0 , x) − V (x0 , x)| ≤ n|y − x| + |V (y0 , x) − V (x0 , x)|. In addition, TDˆ (ˆ x) = TR+ (x0 ) × TD (x) ⇒ Vˆ (ˆ x) = 1, V (ˆ x) ∈ TR+ (x0 ) × TD (x). ˆ is bounded and RN +1 is ﬁnite-dimensional. By using the version Moreover Vˆ (D) of M. Nagumo[1]’s theorem given in J.-P. Aubin and A. Cellina [1, Thm. 3, part b), pp. 182–183], there exists a viable solution xˆ to (5.16) for all t ≥ 0. In particular, ˆ = R+ ×D, ∀t ∈ [0, τ ], x ˆ(t) ∈ D which is necessarily of the form x ˆ(t) = (t, x(t)). Hence there exists a viable solution x, x(t) ∈ D on [0, τ ], to (5.9). The uniqueness now follows from the Lipschitz condition (V 1D ). The Lipschitzian continuity (T 1D ) can be established by a standard argument. Condition (T2D ). Associate with X in D the function y(s) = Tt−s (X),

0 ≤ s ≤ t.

Then

dy (s) = −V t − s, y(s) , 0 ≤ s ≤ t, ds For each x ∈ D, the diﬀerential equation dy (s) = −V t − s, y(s) , ds

0 ≤ s ≤ t,

y(0) = Tt (X).

(5.18)

y(0) = x ∈ D

(5.19)

has a unique viable solution in C 1 ([0, t]; RN ): ∀s ∈ [0, t],

y(s) ∈ D,

since by assumption (V 2D ) ∀t ∈ [0, τ ], ∀x ∈ D,

−V (t, x) ∈ TD (x).

(5.20)

5. Constrained Families of Domains

197

The proof is the same as above. The solutions of (5.19) deﬁne a Lipschitzian mapping x → St (x) = y(t) : D → D such that ∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ D,

|St (y) − St (x)| ≤ c|y − x|.

Now in view of (5.18) and (5.19) St Tt (X) = y(t) = Tt−t (X) = X

⇒

(5.21)

St ◦ Tt = I on D.

To obtain the other identity, consider the function z(r) = y(t − r; x), where y(·, x) is the solution of (5.19). By deﬁnition, dz (r) = V r, z(r) , dr

z(0) = y(t, x)

and necessarily x = y(0; x) = z(t) = Tt y(t; x) = Tt St (x) ⇒ Tt ◦ St = I on D ⇒ St = Tt−1 : D → D. Condition (T3D ). The uniform Lipschitz continuity in (T3D ) follows from (5.21) and (T2D ), and we need only show that ∀x ∈ D, T −1 (·, x) ∈ C [0, τ ]; RN . Given t in [0, τ ] choose an arbitrary sequence {tn }, tn → t. Then for each x ∈ D there exists X ∈ D such that Tt (X) = x and Ttn (X) → Tt (X) = x from (T1D ). But Tt−1 Tt (X) − Tt−1 Tt (X) (x) − Tt−1 (x) = Tt−1 n n Tt (X) − Tt−1 Ttn (X) . = Tt−1 n n By the uniform Lipschitz continuity of Tt−1 Tt (X) − Tt−1 Ttn (X) | |Tt−1 (x) − Tt−1 (x)| = |Tt−1 n n n ≤ c|Tt (X) − Ttn (X)|, and the last term converges to zero as tn goes to t.

198

Chapter 4. Transformations Generated by Velocities ¯ and t, s in [0, τ ] (ii) The ﬁrst condition (V1D ) is satisﬁed since for each x ∈ D ∂T ∂T |V (t, x) − V (s, x)| ≤ t, Tt−1 (x) − t, Ts−1 (x) ∂t ∂t ∂T ∂T t, Ts−1 (x) − s, Ts−1 (x) + ∂t ∂t ≤ c|Tt−1 (x) − Ts−1 (x)| ∂T ∂T t, Ts−1 (x) − s, Ts−1 (x) . + ∂t ∂t

The second condition (V1D ) follows from (T1D ) and (T3D ) and the following inequality: for all x and y in D ∂T ∂T −1 −1 |V (t, y) − V (t, x)| = t, Tt (y) − t, Tt (x) ∂t ∂t ≤ c Tt−1 (y) − Tt−1 (x) ≤ cc |y − x|. To check condition (V2D ), recall deﬁnition (5.6) of the Bouligand contingent cone: % % * 1 TD (X) = (D − X) + εB , h ε>0 α>0 0 0, ∃h ∈ ]0, α[ , ∃u such that x (t) ∈ u + εB and x(t) + hu ∈ D. Choose δ, 0 < δ < α, such that ∀s,

|s − t| < δ

⇒

|x (s) − s (t)| < ε.

Then ﬁx t . For all t such that 0 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

dG ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Condition (6.3) on V coincides with condition (4.23) of Theorem 4.4, which implies conditions (4.24) and (4.25): f ∈ C 1 ([0, τ ]; C0k (RN , RN ))

and g ∈ C([0, τ ]; C0k (RN , RN ))

⇒ Tt − ICk (RN ) → 0 and Tt−1 − ICk (RN ) → 0 as t → 0. But by deﬁnition of the metric dG d([Tt ], [I]) ≤ Tt−1 − IC k + Tt − ICk → 0 as t → 0, and we get the convergence (6.2) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (6.3). (ii) Conversely, it is suﬃcient to prove that for any sequence {[Tn ]} such that dG ([Tn ], [I]) goes to zero, there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞. Indeed let = lim inf J(Tn (Ω)) n→∞

and L = lim sup J(Tn (Ω)). n→∞

By deﬁnition of the liminf, there is a subsequence, still indexed by n, such that = lim inf n→∞ J(Tn (Ω)). But since there exists a subsequence {Tnk } of {Tn } such that J(Tnk (Ω)) → J(Ω), then necessarily = J(Ω). The same reasoning applies to the limsup, and hence the whole sequence J(Tn (Ω)) converges to J(Ω) and we have the continuity of J at Ω. We prove that we can construct a velocity V associated with a subsequence of {Tn } verifying conditions (4.23) of Theorem 4.4 and hence conditions (6.3). By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequence {Tn } such that dG ([Tn ], [I]) → 0 a subsequence, still denoted by {Tn }, such that fn C k + gn Ck = Tn−1 − IC k + Tn − IC k ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Deﬁne the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ], def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

(Tn+1 (X) − Tn (X)),

def

T0 (X) = X,

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1).

204

Chapter 4. Transformations Generated by Velocities

Conditions on f . By deﬁnition, for all t, 0 ≤ t ≤ 1/2, f (t) = Tt − I ∈ C0k (RN ). Moreover, for 0 < t ≤ 1/2 ∂T ∂T (tn , X) = 0 = (tn+1 , X), ∂t ∂t ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t (t, X) = p , ∂t |tn − tn+1 | tn+1 − tn

Ttn (X) = Tn (X), Ttn+1 (X) = Tn+1 (X),

f (t) = ∂T /∂t(t, ·) ∈ C0k (RN ), and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By deﬁnition, f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and tn+1 − t (fn+1 − fn )C k f (t) − f (0)Ck = f (t)C k = fn + p tn+1 − tn ≤ 2fn Ck + fn+1 C k ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Deﬁne at t = 0, f (t) = 0. By the same technique, there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and f (t) − f (0)Ck = f (t)C k 3 3 3 ∂T 3 3 ≤ c Tn+1 − Tn Ck = c fn+1 − fn C k =3 (t, ·) 3 ∂t 3 k |tn+1 − tn | 2−(n+1) C

≤ c 2 2−2(n+2) /2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t ⇒ f (t)C k ≤ ct. So for each X the functions t → f (t)(X) and t → Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C0k (RN )) and the condition (4.24) of Theorem 4.4 is satisﬁed. Hence the corresponding velocity V satisﬁes conditions (4.23). Finally V satisﬁes conditions (6.3) and by (6.2) for all ε > 0 there exists δ > 0 such that ∀t, 0 ≤ t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ, and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε,

and this proves the d-continuity for the subsequence {Tn }. The case of the Courant metric associated with the space C k (RN ) = C k (RN , RN ) is a corollary to Theorem 6.1. Theorem 6.2. Let k ≥ 1 be an integer, B be a Banach space, and Ω be a nonempty open subset of RN . Consider a shape functional J : NΩ ([I]) → B deﬁned in a neighborhood NΩ ([I]) of [I] in F(C k (RN , RN ))/G(Ω).

6. Continuity of Shape Functions along Velocity Flows

205

Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(6.4)

t0

for all families of velocity ﬁelds {V (t) : 0 ≤ t ≤ τ } satisfying the condition V ∈ C([0, τ ]; C k (RN , RN )).

(6.5)

The proof of the theorem for the Courant metric topology associated with the space C k,1 (RN ) = C k,1 (RN , RN ) is similar to the proof of Theorem 6.1 with obvious changes. Theorem 6.3. Let k ≥ 0 be an integer, Ω be a nonempty open subset of RN , and B be a Banach space. Consider a shape functional J : NΩ ([I]) → B deﬁned in a neighborhood NΩ ([I]) of [I] in F(C k,1 (RN , RN ))/G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(6.6)

t0

for all families {V (t) : 0 ≤ t ≤ τ } of velocity ﬁelds in C k,1 (RN , RN ) satisfying the conditions V ∈ C([0, τ ]; C k (RN , RN ))

and

ck (V (t)) ≤ c

(6.7)

for some constant c independent of t. Proof. As in the proof of Theorem 6.1, it is suﬃcient to prove the theorem for a real-valued function J. (i) If J is dG -continuous at Ω, then for all ε > 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

dG ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Under condition (6.7) from Theorem 4.3 f = T − I ∈ C([0, τ ]; C k,1 (RN )) and Tt − IC k,1 → 0 as t → 0, g(t) = Tt−1 − I ∈ C k,1 (RN ) and Tt−1 − IC k,1 ≤ ct → 0 as t → 0. But by deﬁnition of the metric dG dG ([Tt ], [I]) ≤ Tt−1 − IC k,1 + Tt − IC k,1 → 0 as t → 0, and we get the convergence (6.6) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (6.7). (ii) Conversely, as in the proof of Theorem 6.1, it is suﬃcient to prove that given any sequence {[Tn ]} such that δ([Tn ], [I]) → 0 there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞.

206

Chapter 4. Transformations Generated by Velocities

By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequence {Tn } such that dG ([Tn ], [I]) → 0 a subsequence, still denoted by {Tn }, such that fn C k,1 + gn Ck,1 = Tn−1 − IC k,1 + Tn − ICk,1 ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Deﬁne the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ], def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

(Tn+1 (X) − Tn (X)),

def

T0 (X) = X,

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1). Conditions on f . By deﬁnition for all t, 0 ≤ t ≤ 1/2, f (t) = Tt −I ∈ C k,1 (RN ). Moreover, for 0 < t ≤ 1/2 ∂T ∂T Ttn (X) = Tn (X), Ttn+1 (X) = Tn+1 (X), (tn , X) = 0 = (tn+1 , X), ∂t ∂t ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t (t, X) = p , ∂t |tn − tn+1 | tn+1 − tn f (t) = ∂T /∂t ∈ C k,1 (RN ), and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By deﬁnition, f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and 3 3 3 3 tn+1 − t 3 (f f (t) − f (0)C k,1 = f (t)C k,1 = 3 f + p − f ) n+1 n 3 3 n tn+1 − tn C k,1 ≤ 2fn C k,1 + fn+1 C k,1 ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Deﬁne at t = 0, f (t) = 0. By the same technique there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and f (t) − f (0)C k,1 = f (t)C k,1 3 3 3 3 ∂T fn+1 − fn Ck,1 Tn+1 − Tn C k,1 3 (t, ·)3 =c =3 ≤c 3 ∂t |tn+1 − tn | 2−(n+1) k,1 C −2(n+2)

≤ c22

/2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t

⇒ f (t)C k,1 ≤ ct and ck (f (t)) ≤ ct and for each X the functions t → f (t)(X) and t → Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C k (RN )). Moreover, it can be shown that ck (f (t)) ≤ ct

⇒ ∀t, s ∈ [0, τ ],

ck (f (t) − f (s)) ≤ c |t − s|

for some c > 0. The result is straightforward for k = 0 and then the general case follows by induction on k. As a result f ∈ C([0, 1/2]; C k,1 (RN )) and the condition

6. Continuity of Shape Functions along Velocity Flows

207

(4.20) of Theorem 4.3 is satisﬁed. Hence the corresponding velocity V satisﬁes conditions (4.19). Finally the velocity ﬁeld V satisﬁes conditions (6.7), and by (6.6) for all ε > 0 there exists δ > 0 such that ∀t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε,

and we have the dG -continuity for the subsequence {Tn }. Remark 6.2. The conclusions of Theorems 6.1, 6.2, and 6.3 are generic. They also have their counterpart in the constrained case. The diﬃculty lies in the second part of the theorem, which requires a special construction to make sure that the family of transformations {Tt : 0 ≤ t ≤ τ } constructed from the sequence {Tn } are homeomorphisms of D.

Chapter 5

Metrics via Characteristic Functions 1

Introduction

The constructions of the metric topologies of Chapter 3 are limited to families of sets which are the image of a ﬁxed set by a family of homeomorphisms or diﬀeomorphisms. If it is connected, bounded, or C k , k ≥ 1, then the images will have the same properties under C k -transformations of RN . In this chapter we considerably enlarge the family of available sets by relaxing the smoothness assumption to the mere Lebesgue measurability and even just measurability to include Hausdorﬀ measures. This is done by identifying Ω ⊂ RN with its characteristic function & 1, if x ∈ Ω def def χΩ (x) = if Ω = ∅ and χ∅ = 0. (1.1) 0, if x ∈ /Ω We ﬁrst introduce an Abelian group structure1 on characteristic functions of measurable 2 subsets of a ﬁxed holdall D in section 2 and then construct complete metric spaces of equivalence classes of measurable characteristic functions via the Lp norms, 1 ≤ p < ∞, that turn out to be algebraically and topologically equivalent. Starting with section 3, we specialize to the Lebesgue measure in RN . The complete topology generated by the metric deﬁned as the Lp -norm of the diﬀerence of two characteristic functions is called the strong topology in section 3.1. In section 3.2 we consider the weak Lp -topology on the family of characteristic functions. Weak limits of sequences of characteristic functions are functions with values in [0, 1] which belong to the closed convex hull of the equivalence classes of measurable characteristic functions. This occurs in optimization problems where the function to be optimized depends on the solution of a partial diﬀerential equation on the variable domain, as was shown, for instance, by F. Murat [1] in 1971. It usually corresponds to the appearance of a microstructure or a composite material in mechanics. 1 Cf.

the group structure on subsets of a holdall D in section 2.2 of Chapter 2. in the sense of L. C. Evans and R. F. Gariepy [1]. It simultaneously covers Hausdorﬀ and Lebesgue measures. 2 Measurable

209

210

Chapter 5. Metrics via Characteristic Functions

One important example in that category was the analysis of the optimal thickness of a circular plate by K.-T. Cheng and N. Olhoff [2, 1] in 1981. The reader is referred to the work of F. Murat and L. Tartar [3, 1], initially published in 1985, for a comprehensive treatment of the calculus of variations and homogenization. Section 3.3 deals with the question of ﬁnding a nice representative in the equivalence class of sets. Can it be chosen open? We introduce the measure theoretic representative and characterize its interior, exterior, and boundary. It will be used later in section 6.4 to prove the compactness of the family of Lipschitzian domains verifying a uniform cone property. Section 3.4 shows that the family of convex subsets of a ﬁxed bounded holdall is closed in the strong topology. The use of the Lp -topologies is illustrated in section 4 by revisiting the optimal ´a and K. Malanowski [1] in 1970. By relaxing design problem studied by J. Ce the family of characteristic functions to functions with values in the interval [0, 1] a saddle point formulation is obtained. Functions with values in the interval [0, 1] can also be found in the modeling of a dam by H. W. Alt [1] and H. W. Alt and G. Gilardi [1], where the “liquid saturation” in the soil is a function with values in [0, 1]. Another problem amenable to that formulation is the buckling of columns as will be illustrated in section 5 using the work of S. J. Cox and M. L. Overton [1]. It is one of the very early optimal design problems, formulated by Lagrange in 1770. The Caccioppoli or ﬁnite perimeter sets of the celebrated Plateau problem are revisited in section 6. Their characteristic function is a function of bounded variation. They provide the ﬁrst example of compact families of characteristic functions in Lp (D)-strong, 1 ≤ p < ∞. This was developed mainly by R. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem of minimal surfaces. Section 6.3 exploits the embedding3 of BV(D) ∩ L∞ (D) into the Sobolev spaces W ε,p (D), 0 < ε < 1/p, p ≥ 1, to introduce a cascade of complete metrics between Lp (D) and BV(D) on characteristic functions. In section 6.4 we show that the family of Lipschitzian domains in a ﬁxed bounded holdall verifying the uniform cone property of section 6.4.1 in Chapter 2 is also compact. This condition naturally yields a uniform bound on the perimeter of the sets in the family and hence can be viewed as a special case of the ﬁrst compactness theorem. Section 7 gives an example of the use of the perimeter in the Bernoulli free boundary problem and in particular for the water wave. There the energy associated with the surface tension of the water is proportional to the perimeter, that is, the surface area of the free boundary.

2

Abelian Group Structure on Measurable Characteristic Functions

2.1

Group Structure on Xµ (RN )

From section 2.2 in Chapter 2, the Abelian group structure on the subsets of RN extends to the corresponding family of their characteristic functions. 3 BV(D)

is the space of functions of bounded variations in D.

2. Abelian Group Structure on Measurable Characteristic Functions

211

Theorem 2.1. Let ∅ = D ⊂ RN . The space of characteristic functions X(D) endowed with the multiplication and the neutral multiplicative element χ∅ = 0 def

χA χB = |χA − χB | = χ(A∩B)∪(B∩A) = χAB

(2.1)

is an Abelian group, that is, χ A χB = χ B χA ,

χA χ∅ = χA ,

χA χA = χ∅ ,

χ−1 A = χA .

(2.2)

Proof. Note that the underlying group structure on subsets of D is the one of section 2.2 of Chapter 2. The product is well-deﬁned in X(D) since it is the characteristic function of a subset of A ∪ B in D. The ﬁrst two identities (2.2) are obvious. As for the inverse χ A χB = χ ∅

⇔

|χA − χB | = 0

⇔

B = A,

and there is a unique multiplicative inverse χ−1 A = χA or χA χA = χ∅ . Remark 2.1. We chose the product terminology , but we could have chosen to call that algebraic operation an addition ⊕.

2.2

Measure Spaces

Deﬁnition 2.1. Let X be a set and P(X) be the collection of subsets of X. (i) A mapping µ : P(X) → [0, ∞] is called a measure 4 on X if (a) µ(∅) = 0, and ) ∞ (b) µ(A) ≤ ∞ k=1 µ(Ak ), whenever A ⊂ ∪k=1 Ak . (ii) A subset A ⊂ X is σ-ﬁnite with respect to µ if it can be written A = ∪∞ k=1 Bk , where Bk is µ-measurable and µ(Bk ) < ∞ for k = 1, 2, . . . . (iii) A measure µ on RN is a Radon measure if µ is Borel regular and µ(K) < ∞ for each compact set K ⊂ RN . The Lebesgue measure mN on RN is a Radon measure since RN is σ-ﬁnite with respect to mN . The s-dimensional Hausdorﬀ measure Hs , 0 ≤ s < N , on RN (cf. section 3.2.3 in Chapter 2) is Borel regular but not necessarily a Radon measure since bounded Hs -measurable subsets of RN are not necessarily σ-ﬁnite with respect to Hs . Yet, a Borel regular measure µ can be made σ-ﬁnite by restricting it to a µ-measurable set A ⊂ RN such that µ(A) < ∞ or, more generally, to a σ-ﬁnite µ-measurable set A ⊂ RN . For smooth submanifolds of dimension s, s an integer, Hs gives the same area as the integral of the canonical density in section 3.2.2 of Chapter 2. 4 A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outer measure in most texts.

212

Chapter 5. Metrics via Characteristic Functions

Given a µ-measurable subset D ⊂ RN , let Lp (D, µ) denote the Banach space of equivalence classes of µ-measurable functions such that |f |p dµ < ∞, 1 ≤ p < ∞, and let L∞ (D, µ) be the space of equivalence classes of µ-measurable functions such that ess supD |f | < ∞. Denote by [A]µ the equivalence class of µ-measurable subsets of D that are equal almost everywhere and def

Xµ (D) = {χA : A ⊂ D is µ-measurable} . Since A B is µ-measurable for A and B µ-measurable, {[A]µ : A ⊂ D is µ-measurable} is a subgroup of P(D) and Xµ (D) ∩ L1 (D, µ) = {χA : A ⊂ D is µ-measurable and µ(A) < ∞}

(2.3)

is a group in Lp (D, µ) with respect to . When µ = mN , we drop the subscript µ.

2.3

Complete Metric for Characteristic Functions in Lp -Topologies

Now that we have revealed the underlying group structure of Xµ (D) ∩ L1 (D, µ), we construct a metric on that group. We have a structure similar to the Courant metric of A. M. Micheletti [1] in Chapter 3 on groups of transformations of RN . We identify equivalence classes of µ-measurable subsets of D and characteristic functions via the bijection [A]µ ↔ χA . Theorem 2.2. Let 1 ≤ p < ∞, let µ be a measure on RN , and let ∅ = D ⊂ RN be µ-measurable. (i) Xµ (D) ∩ L1 (D, µ) is closed in Lp (D, µ). The function def

ρD ([A2 ]µ , [A1 ]µ ) = χA2 − χA1 Lp (D,µ) deﬁnes a complete metric structure on the Abelian group Xµ (D)∩L1 (D, µ) that makes it a topological group. If µ(D) < ∞, then Xµ (D) ∩ L1 (D, µ) = Xµ (D). (ii) If, in addition, D is σ-ﬁnite with respect to µ for a family {Dk } of µ-measurable subsets of D such that µ(Dk ) < ∞, for all k ≥ 1, then Xµ (D) is closed in Lploc (D, µ) and def

ρ([A2 ]µ , [A1 ]µ ) =

∞ χA2 − χA1 Lp (Dk ,µ) 1 2n 1 + χA2 − χA1 Lp (Dk ,µ) k=1

deﬁnes a complete metric structure on the Abelian group Xµ (D) that makes it a topological group. When µ is a Radon measure on RN , the assumption that D is σ-ﬁnite with respect to µ can be dropped. Remark 2.2. For the Lebesgue measure mN and D measurable (resp., measurable and bounded), X(D) is a topological Abelian group in Lploc (D, mN ) (resp., Lp (D, mN )). For Hausdorﬀ measures Hs , 0 ≤ s < N , the theorem does not say more.

2. Abelian Group Structure on Measurable Characteristic Functions

213

Proof. (i) Let {χAn } be a Cauchy sequence of µ-measurable characteristic functions in Xµ (D) ∩ L1 (D, µ) ⊂ Lp (D, µ) converging to some f ∈ Lp (D, µ). There exists a subsequence, still denoted by {χAn }, such that χAn (x) → f (x) in D except for a subset Z of zero µ-measure. Hence 0 = χAn (x) (1 − χAn (x)) → f (x) (1 − f (x)). Deﬁne the set def

A = {x ∈ D\Z : f (x) = 1}. Clearly A is µ-measurable and χA = f on D\Z since f (x) (1 − f (x)) = 0 on D\Z. Hence f = χA almost everywhere on D, χAn → χA in Lp (D, µ), and χA ∈ Xµ (D) ∩ L1 (D, µ). Finally, the metric is compatible with the group structure −1 ρD ([Ω2 ]µ , [Ω1 ]µ ) = χΩ2 − χΩ1 Lp = χΩ2 χ−1 Ω1 Lp = χΩ1 χΩ2 Lp def

as in Chapter 3 and Xµ (D) ∩ L1 (D, µ) is a topological group. (ii) From part (i) using the Fr´echet topology associated with the family of seminorms qk (f ) = f Lp (Dk ) , k ≥ 1. For p = 1 and µ-measurable subsets A1 and A2 in D, the metric ρD ([A2 ]µ , [A1 ]µ ) is the µ-measure of the symmetric diﬀerence A1 ∆A2 = (A1 ∩ A2 ) ∪ (A2 ∩ A1 ) and for 1 ≤ p < ∞ all the topologies are equivalent on Xµ (D) ∩ L1 (D, µ). Theorem 2.3. Let 1 ≤ p < ∞, let µ be a measure on RN , and let ∅ = D ⊂ RN be µ-measurable. (i) The topologies induced by Lp (D, µ) on Xµ (D) ∩ L1 (D, µ) are all equivalent for 1 ≤ p < ∞. (ii) If, in addition, D is σ-ﬁnite with respect to µ for a family {Dk } of µ-measurable subsets of D such that µ(Dk ) < ∞, for all k ≥ 1, then the topologies induced by Lploc (D, µ) on Xµ (D) are all equivalent for 1 ≤ p < ∞. Proof. For D µ-measurable and 1 ≤ p < ∞, Xµ (D) ∩ Lp (D, µ) = Xµ (D) ∩ L1 (D, µ), and for any χ and χ in Xµ (D) ∩ L1 (D, µ) χ − χpLp (D,µ) = |χ − χ(x)|p dµ = |χ − χ(x)| dµ = χ − χL1 (D,µ) , D

D

since χ(x) and χ(x) are either 0 or 1 almost everywhere in D. Therefore the linear (identity) mapping between Xµ (D)∩Lp (D, µ) and Xµ (D)∩L1 (D, µ) is bicontinuous and the topologies are equivalent. Remark 2.3. At this stage, it is not clear what the tangent space to this topological Abelian group is. If µ is a Radon measure, then for all ϕ ∈ D(RN )

1 1 χA χBε (x) χ−1 ϕ dµ = χB (x) ϕ dµ A µ(Bε (x)) RN µ(Bε (x)) RN ε

214

Chapter 5. Metrics via Characteristic Functions

and by the Lebesgue–Besicovitch diﬀerentiation theorem

1 N χA χBε (x) χ−1 A ϕ dµ → ϕ(x), µ a.e. in R . µ(Bε (x)) RN

3

Lebesgue Measurable Characteristic Functions

In this section we specialize to the Lebesgue measure and the corresponding family of characteristic functions def

X(RN ) = χΩ : ∀Ω Lebesgue measurable in RN . (3.1) Clearly X(RN ) ⊂ L∞ (RN ), and for all p ≥ 1, X(RN ) ⊂ Lploc (RN ). Also associate with ∅ = D ⊂ RN the set def

X(D) = {χΩ : ∀Ω Lebesgue measurable in D} .

(3.2)

The above deﬁnitions are special cases of the deﬁnitions of section 2.3, and Theorems 2.2 and 2.3 apply to mN as a Radon measure.

3.1

Strong Topologies and C ∞ -Approximations

In view of the equivalence Theorem 2.3, we introduce the notion of strong convergence. Deﬁnition 3.1. (i) If D is a bounded measurable subset of RN , a sequence {χn } in X(D) is said to be strongly convergent in D if it converges in Lp (D)-strong for some p, 1 ≤ p < ∞. (ii) A sequence {χn } in X(RN ) is said to be locally strongly convergent if it converges in Lploc (RN )-strong for some p, 1 ≤ p < ∞. The following approximation theorem will also be useful Theorem 3.1. Let Ω be an arbitrary Lebesgue measurable subset of RN . There exists a sequence {Ωn } of open C ∞ -domains in RN such that χΩn → χΩ

in L1loc (RN ).

Proof. The construction of the family of C ∞ -domains {Ωn } can be found in many places (cf., for instance, E. Giusti [1, sect. 1.14, p. 10 and Lem. 1.25, p. 23]). Associate with χ = χΩ the sequence of convolutions fn = χ ∗ ρn for a sequence {ρn } of symmetric molliﬁers. By construction, 0 ≤ fn ≤ 1. For t, 0 < t < 1, deﬁne def

Fnt = x ∈ RN : fn (x) > t . By deﬁnition, fn − χ > t in Fnt \Ω, χ − fn > 1 − t in Ω\Fnt , and 1 |χ − fn | a.e. in RN . |χ − χFnt | ≤ min{t, 1 − t}

3. Lebesgue Measurable Characteristic Functions

215

By Sard’s Theorem 4.3 of Chapter 2, the set Fnt is a C ∞ -domain for almost all t in (0, 1). Fix α, 0 < α < 1/2, and choose a sequence {tn } such that, for all n, def

α ≤ tn ≤ 1 − α, and deﬁne Ωn = Fntn . Therefore |χΩ − χΩn | ≤

1 1 |χΩ − fn | ≤ |χΩ − fn | a.e. in RN min{tn , 1 − tn } α

and for any bounded measurable subset D of RN χΩ − χΩn L1 (D) ≤ α−1 χΩ − fn L1 (D) , which goes to zero as n goes to inﬁnity.

3.2

Weak Topologies and Microstructures

Some shape optimization problems lead to apparent paradoxes. Their solution is no longer a geometric domain associated with a characteristic function, but a fuzzy domain associated with the relaxation of a characteristic function to a function with values ranging in [0, 1]. The intuitive notion of a geometric domain is relaxed to the notion of a probability distribution of the presence of points of the set. When the underlying problem involves two diﬀerent materials characterized by two constants k1 = k2 , the occurrence of such a solution can be interpreted as the mixing or homogenization of the two materials at the microscale. This is also referred to as a composite material or a microstructure. Somehow this is related to the fact that the strong convergence needs to be relaxed to the weak Lp -convergence and the space X(D) needs to be suitably enlarged. Even if X(D) is strongly closed and bounded in Lp (D), it is not strongly compact. However, for 1 < p < ∞ its closed convex hull co X(D) is weakly compact in the reﬂexive Banach space Lp (D). In fact, co X(D) = {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D} .

(3.3)

Indeed, by deﬁnition co X(D) ⊂ {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D}. Conversely any χ that belongs to the right-hand side of (3.3) can be approximated by a sequence of convex combinations of elements of X(D). Choose χn =

n 1 χBnm , n m=1

m Bnm = x : χ(x) ≥ , n

for which |χn (x) − χ(x)| < 1/n. The elements of co X(D) are not necessarily characteristic functions of a domain; that is, the identity χ(x) (1 − χ(x)) = 0 a.e. in D is not necessarily satisﬁed. We ﬁrst give a few basic results and then consider a classical example from the theory of homogenization of diﬀerential equations.

216

Chapter 5. Metrics via Characteristic Functions

Lemma 3.1. Let D be a bounded open subset of RN , let K be a bounded subset of R, and let def

K = {k : D → R : k is measurable and k(x) ∈ K a.e. in D} . (i) For any p, 1 ≤ p < ∞, and any sequence {kn } ⊂ K the following statements are equivalent: (a) {kn } converges in L∞ (D)-weak; (b) {kn } converges in Lp (D)-weak; (c) {kn } converges in D(D) , where D(D) is the space of scalar distributions on D. (ii) If K is bounded, closed, and convex, then K is convex and compact in L∞ (D)weak, Lp (D)-weak, and D(D) . The above results remain true in the vectorial case when K is the set of mappings k : D → K for some bounded subset K ⊂ Rp and a ﬁnite integer p ≥ 1. Proof. (i) It is clear that (a) ⇒ (b) ⇒ (c). To prove that (c) ⇒ (a) recall that since K is bounded, there exists a constant c > 0 such that K ⊂ cB, where B denotes the unit ball in RN . By density of D(D) in L1 (D), any ϕ in L1 (D) can be approximated by a sequence {ϕm } ⊂ D(D) such that ϕm → ϕ in L1 (D). So for each ε > 0 there exists M > 0 such that ∀m ≥ M,

ϕm − ϕL1 (D) ≤

ε . 4c

Moreover, there exists N > 0 such that ε ∀n ≥ N, ∀ ≥ N, ϕM (kn − k ) dx ≤ . 2 D Hence for each ε > 0, there exists N such that for all n ≥ N and ≥ N ϕ(kn − k ) dx ≤ ϕM (kn − k ) dx + (ϕ − ϕM )(kn − k ) dx D D D ε ≤ + 2cϕ − ϕM L1 (D) ≤ ε. 2 (ii) When K is bounded, closed, and convex, K is also bounded, closed, and convex. Since L2 (D) is a Hilbert space, K is weakly compact. In view of the equivalences of part (i), K is also sequentially compact in all the other weak topologies.5 5 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E , any closed ball is a compact metrizable space for the weak topology. Since K is a bounded subset of the normed reﬂexive separable Banach space Lp (D), 1 ≤ p < ∞, the weak compactness of K coincides ´ [1, Vol. 2, Chap. XII, sect. 12.15.9, p. 75]). with its weak sequential compactness (cf. J. Dieudonne

3. Lebesgue Measurable Characteristic Functions

217

In view of this equivalence we adopt the following terminology. Deﬁnition 3.2. Let ∅ = D ⊂ RN be bounded open. A sequence {χn } in X(D) is said to be weakly convergent if it converges for some topology between L∞ (D)-weak and D(D) . It is interesting to observe that working with the weak convergence makes sense only when the limit element is not a characteristic function. Theorem 3.2. Let the assumptions of the theorem be satisﬁed. Let {χn } and χ be elements of X(RN ) (resp., X(D)) such that χn χ weakly in L2 (D) (resp., L2loc (RN )). Then for all p, 1 ≤ p < ∞, χn → χ strongly in Lp (D) resp., Lploc (RN ) . Proof. It is suﬃcient to prove the result for D. The strong L2 (D)-convergence follows from the property |χn |2 dx = χn 1 dx → χ1 dx = |χ|2 dx D D D D 2 2 2 |χn − χ| dx = |χn | − 2 χχn + |χ| dx → |χ|2 − 2 χχ+ |χ|2 dx = 0. ⇒ D

D

D

The convergence for any p ≥ 1 now follows from Theorem 2.2. Working with the weak convergence creates new phenomena and diﬃculties. For instance when a characteristic function is present in the coeﬃcient of the higherorder term of a diﬀerential equation the weak convergence of a sequence of characteristic functions {χn } to some element χ of co X(D), χn χ in L2 (D)-weak

⇒ χn χ in L∞ (D)-weak,

does not imply the weak convergence in H 1 (D) of the sequence {y(χn )} of solutions to the solution of the diﬀerential equation corresponding to y(χ), y(χn ) y(χ) in H 1 (D)-weak. By compactness6 of the injection of H 1 (D) into L2 (D) this would have implied convergence in L2 (D)-strong: y(χn ) → y(χ) in L2 (D)-strong. This fact was pointed out in 1971 by F. Murat [1] in the following example, which will be rewritten to emphasize the role of the characteristic function. 6 This is true since D is a bounded Lipschitzian domain. Examples of domains between two spirals can be constructed where the injection is not compact.

218

Chapter 5. Metrics via Characteristic Functions

Example 3.1. Consider the following boundary value problem: d dy − k + ky = 0 in D = ]0, 1[ , dx dx y(0) = 1 and y(1) = 2,

(3.4)

where K = {k = k1 (x)χ + k2 (x)(1 − χ) : χ ∈ X(D)} with

4 k1 (x) = 1 −

1 x2 − , 2 6

4 k2 (x) = 1 +

1 x2 − . 2 6

(3.5)

(3.6)

This is equivalent to K = {k ∈ L∞ (0, 1) : k(x) ∈ {k1 (x), k2 (x)} Associate with the integer p ≥ 1 the function 4 m 2m + 1 1 x2 1 − 2 − 6 , p < x ≤ 2p , k p (x) = 4 1 x2 2m + 1 m+1 1 + − , <x≤ , 2 6 2p p and the characteristic function m 2m + 1 <x≤ , 1, p 2p χp (x) = m+1 2m + 1 0, <x≤ , 2p p

a.e. in [0, 1]} .

0 ≤ m ≤ p − 1,

0 ≤ m ≤ p − 1.

(3.7)

(3.8)

(3.9)

It is readily seen that kp = k1 χp + k2 (1 − χp ). He shows that for each p, k p ∈ K (resp., χp ∈ X(D)) and that 1 k p k∞ = 1 in L∞ (0, 1)-weak, resp., χp (3.10) 2 1 1 1 1 1 = in L∞ (0, 1)-weak . + (3.11) p k 2 k1 k2 1/2 + x2 /6 Moreover, yp y in H 1 (0, 1)-weak,

(3.12)

where yp denotes the solution of (3.4) corresponding to k = k p and y the solution of the boundary value problem 1 x2 dy − d + + y = 0 in ]0, 1[ , dx 2 6 dx (3.13) y(0) = 1, y(1) = 2.

3. Lebesgue Measurable Characteristic Functions

219

Deﬁne the function kH (x) = which corresponds to

1 x2 + , 2 6

4 1 1 x2 χH (x) = 1+ − ∈ co X(D). 2 2 6

(3.14)

(3.15)

Notice that kH appears in the second-order term and k∞ in the zeroth-order term in (3.13): dy d (3.16) kH + k∞ y = 0 in ]0, 1[ . − dx dx It is easy to check that y(x) = 1 + x2 in [0, 1],

(3.17)

which is not equal to the solution y∞ of (3.4) for the weak limit k∞ = 1: y∞ (x) =

2(ex − e−x ) + e1−x − e−(1−x) . e − e−1

(3.18)

To our knowledge this was the beginning of the theory of homogenization in France. This is only one part of F. Murat [1]’s example. He also constructs an objective function for which the lower bound is not achieved by an element of K. This is a nonexistence result. The above example uses space varying coeﬃcients k1 (x) and k2 (x). However, it is still valid for two positive constants k1 > 0 and k2 > 0. It is easy to show that 1 k1 + k2 1 1 1 1 k2 , χH = k∞ = , χ∞ = , and = + , 2 2 kH 2 k1 k2 k1 + k2 and the solution y of the boundary value problem (3.16) is given by k1 + k2 2 sinh cx + sinh c(1 − x) ≥ 1, , where c = √ sinh c 2 k1 k2 and the solution y∞ by 2 sinh x + sinh(1 − x) . y∞ (x) = sinh 1 Thus for k1 = k2 , or equivalently c > 1, y = y∞ . In fact, using the same sequence of χp ’s, the sequence yp = y(χp ) of solutions of (4.5) weakly converges to yH = y(χH ), which is diﬀerent from the solution y∞ = y(χ∞ ) for k1 = k2 . This is readily seen by noticing that since χ∞ and χH are constant y(x) =

−∆(kH yH ) = χ∞ f = −∆(k∞ y∞ ) ⇒ kH yH = k∞ y∞

⇒ yH =

(k1 + k2 )2 y∞ = y∞ , for k1 = k2 . 4k1 k2

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Chapter 5. Metrics via Characteristic Functions

Despite this example, we shall see in the next section that in some cases we obtain the existence (and uniqueness) of a maximizer in co X(D) which belongs to X(D).

3.3

Nice or Measure Theoretic Representative

Since the strong and weak Lp -topologies are deﬁned on equivalence classes [Ω] of (Lebesgue) measurable subsets Ω of RN , it is natural to ask if there is a nice representative that is generic of the class [Ω]. For instance we have seen in Chapter 2 that within the equivalence class of a set of class C k there is a unique open and a unique closed representative and that all elements of the class have the same interior, boundary, and exterior. The same question will again arise for ﬁnite perimeter sets. As an illustration of what is meant by a nice representative, consider the smiling and the expressionless suns in Figure 5.1. The expressionless sun is obtained by adding missing points and lines “inside” Ω and removing the rays “outside” Ω. This “restoring/cleaning” operation can be formalized as follows.

Ω

Ω

Figure 5.1. Smiling sun Ω and expressionless sun Ω. Deﬁnition 3.3. Associate with a Lebesgue measurable set Ω in RN the sets def

Ω0 = x ∈ RN : ∃ρ > 0 such that m(Ω ∩ B(x, ρ) = 0 , def

Ω1 = x ∈ RN : ∃ρ > 0 such that m Ω ∩ B(x, ρ) = m B(x, ρ) , def

Ω• = x ∈ RN : ∀ρ > 0 such that 0 < m Ω ∩ B(x, ρ) < m B(x, ρ) and the measure theoretic exterior O, interior I, and boundary ∂∗ Ω m(B(x, r) ∩ Ω) def O = x ∈ RN : lim =0 , r0 m(B(x, r)) m(B(x, r) ∩ Ω) def I = x ∈ RN : lim =1 , r0 m(B(x, r)) def

∂∗ Ω =

x∈R

N

& m(B(x, r) ∩ Ω) m(B(x, r) ∩ Ω) : 0 ≤ lim inf < lim sup ≤1 . r0 m(B(x, r)) m(B(x, r)) r0

We shall say that I is the nice or measure theoretic representative of Ω.

3. Lebesgue Measurable Characteristic Functions

221

The six sets Ω0 , Ω1 , Ω• , O, I, and ∂∗ Ω are invariant for all sets in the equivalence class [Ω] of Ω. They are two diﬀerent partitions of RN Ω0 ∪ Ω 1 ∪ Ω • = R N

and

O ∪ I ∪ ∂∗ Ω = RN

(3.19)

like int Ω ∪ int Ω ∪ ∂Ω = RN . The next theorem links the six invariant sets and describes some of the interesting properties of the measure theoretic representative. Theorem 3.3. Let Ω be a Lebesgue measurable set in RN and let Ω0 , Ω1 , Ω• , O, I, and ∂∗ Ω be the sets constructed from Ω in Deﬁnition 3.3. (i) The sets O, I, and ∂∗ Ω are invariants of the equivalence class [Ω]. Moreover, they are Borel measurable and χI = χΩ ,

χO = χΩ ,

and

χ∂∗ Ω = 0

a.e. in RN .

(ii) The sets Ω0 , Ω1 , and Ω• are invariants of the equivalence class [Ω]. The sets Ω0 and Ω1 are open, Ω• is closed, and int Ω ⊂ Ω1 = int I, int Ω ⊂ Ω0 = int O, ∂Ω ⊃ I ∩ O = Ω• ⊃ ∂∗ Ω, (3.20) ∂∗ Ω ∪ ∂O ∪ ∂I ⊂ Ω• ⊂ ∂Ω. (3.21) (iii) The condition m(∂Ω) = 0 implies m(∂I) = 0. When m(∂I) = 0, then int I and I can be chosen as the respective open and closed representatives of Ω. In particular, this is true for Lipschitzian sets and hence for convex sets. Proof. (i) All the sets involved depend only on χΩ almost everywhere. They are invariants of the equivalence class [Ω]. From L. C. Evans and R. F. Gariepy [1, Lem. 2, sect. 5.11, p. 222, Cor. 3, sect. 1.7.1, p. 45], I, O, and ∂∗ Ω are Borel measurable and m(I\Ω ∪ Ω\I) = 0, and lim

r0

m(B(x, r) ∩ Ω) = 1 a.e. in Ω, m(B(x, r))

lim

r0

m(B(x, r) ∩ Ω) = 0 a.e. in Ω. m(B(x, r))

Therefore, m(∂∗ Ω) = 0, χI = χΩ , and χO = χΩ almost everywhere in RN . (ii) We use the proof given by E. Giusti [1, Prop. 4.1, pp. 42–43] for Ω0 and Ω1 . To show that Ω0 is open, pick any point x in Ω0 . By construction, there exists ρ > 0 such that m Ω ∩ B(x, ρ) = 0. For any y ∈ B(x, ρ) (that is, |y − x| < ρ), ρ0 = ρ − |x − y| > 0. Then B(y, ρ0 ) ⊂ B(x, ρ) and choose m B(y, ρ0 ) ∩ Ω ≤ m B(x, ρ) ∩ Ω = 0. So B(x, ρ) ⊂ Ω0 and Ω0 is open. Similarly, by repeating the argument for Ω, we obtain that Ω1 is open. By complementarity, Ω• is closed. By deﬁnition, int Ω ⊂ Ω1 ⊂ I and, since Ω1 is open, Ω1 ⊂ int I. For all x ∈ int I, there exists r > 0 such that B(x, r) ⊂ I and m B(x, r) = m B(x, r) ∩ I = m B(x, r) ∩ Ω . Hence int I ⊂ Ω1 and necessarily int I = Ω1 . Similarly, int Ω ⊂ Ω0 = int O. By taking the union term by term of the two chains int Ω ⊂ int I ⊂ Ω1 = int I ⊂ I

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Chapter 5. Metrics via Characteristic Functions

and int Ω ⊂ Ω0 = int O ⊂ O and then the complement, we get ∂Ω ⊃ I ∩ O = Ω• ⊃ ∂∗ Ω. From the ﬁrst two relations (3.20), Ω1 = int I implies Ω1 = I ⊂ O and Ω0 = int O implies Ω0 = O ⊂ I. Therefore, Ω• = Ω1 ∩ Ω0 ⊃ I ∩ I = ∂I and similarly Ω• = Ω1 ∩ Ω0 ⊃ O ∩ O = ∂O. (iii) From part (ii). The inclusions (3.20) indicate that the measure theoretic interior and exterior are enlarged and that the measure theoretic boundary is reduced. In general those operations do not commute with the set theoretic operations. For instance, they do not commute with the closure, as can be seen from the following simple example. Example 3.2. Consider the set Ω of all rational numbers in [0, 1]. Then O = Ω0 = R I = Ω1 = ∅ ⇒ I(Ω) = ∅, ∂∗ Ω = Ω• = ∅ O(Ω) = (Ω)0 = R \[0, 1] I(Ω) = (Ω)1 = ]0, 1[ ⇒ I(Ω) = ]0, 1[ . ∂ (Ω) = (Ω) = {0, 1} ∗

•

However, the complement operation commutes with the other set of operations and determines the same sets (Ω)0 = Ω1 , (Ω)1 = Ω0 , I(Ω) = O(Ω), I(Ω) = O(Ω),

(Ω)• = Ω• , ∂∗ (Ω) = ∂∗ (Ω).

(3.22) (3.23)

The next theorem is a companion to Lemma 5.1 in Chapter 2. Theorem 3.4. The following conditions are equivalent: (i) int Ω = Ω (ii) int Ω = Ω1 ,

and

int Ω = Ω.

int Ω = Ω0 ,

∂Ω = ∂Ω0 = ∂Ω1 (= Ω• ).

If either of the above two conditions is satisﬁed, then int I = int Ω,

int O = int Ω,

I ∩ O = ∂Ω = ∂Ω = ∂Ω.

Proof. (i) ⇒ (ii). First note that Ω1 = (Ω0 ∪ Ω• ) = Ω0 ∩ Ω• ⊂ Ω0 . Similarly, Ω0 ⊂ Ω1 . From the ﬁrst two relations (3.20), int Ω ⊂ Ω1 and int Ω ⊂ Ω0 . Combining them with the two identities in (i), Ω = int Ω ⊂ Ω1 ⊂ Ω0 and Ω = int Ω ⊂ Ω0 ⊂ Ω1 . By taking the intersection of the two chains, ∂Ω ⊂ Ω0 ∩ Ω1 = (Ω0 ∪ Ω1 ) = Ω• . But int Ω ∪ int Ω ∪ ∂Ω and Ω1 ∪ Ω0 ∪ Ω• are two disjoint partitions of RN such that int Ω ⊂ Ω1 , int Ω ⊂ Ω0 , and ∂Ω ⊂ Ω• . We conclude that int Ω = Ω1 , int Ω = Ω0 , ∂Ω = Ω• , and ∂Ω = Ω• = ∂Ω0 = ∂Ω1 since,

3. Lebesgue Measurable Characteristic Functions

223

by Lemma 5.1 in Chapter 2, int Ω = Ω and int Ω = Ω imply ∂ int Ω = ∂Ω = ∂ int Ω. (ii) ⇒ (i). By assumption, Ω = int Ω ∪ ∂Ω = Ω1 ∪ ∂Ω1 = Ω1 = int Ω, Ω = int Ω ∪ ∂Ω = Ω0 ∪ ∂Ω0 = Ω0 = int Ω ⇒ ∂Ω = Ω ∩ Ω = int Ω ∩ Ω = Ω ∩ Ω = ∂ Ω ⇒ ∂Ω = Ω ∩ Ω = Ω ∩ int Ω = Ω ∩ Ω = ∂ Ω.

3.4

The Family of Convex Sets

Convex sets will play a special role here and in the subsequent chapters. We shall say that an equivalence class [Ω] of Lebesgue measurable subsets of D is convex if there exists a convex Lebesgue measurable subset Ω∗ of D such that [Ω] = [Ω∗ ]. We also introduce the notation def

C(D) = {χΩ : Ω convex subset of D} .

(3.24)

Theorem 3.5. Let Ω = ∅ be a convex subset of RN . (i) Ω and int Ω are convex.7 (ii) (int Ω) = Ω = Ω = int Ω. If int Ω = ∅, then int Ω = Ω. If int Ω = ∅ and Ω = RN , then ∂Ω = ∅ and int Ω = ∅. (iii) Given a measurable (resp., bounded measurable) subset D in RN ∀1 ≤ p < ∞,

C(D) is closed in Lploc (D) (resp., Lp (D)).

Proof. (i) For all x and y in Ω, there exist {xn } and {yn } in Ω such that xn → x and yn → y. Then for all λ ∈ [0, 1] Ω λ xn + (1 − λ) yn → λ x + (1 − λ) y ∈ Ω. If int Ω = ∅, then for all x and y in int Ω, there exist rx > 0 and ry > 0 such that B(x, rx ) ⊂ Ω and B(y, ry ) ⊂ Ω. So for all λ ∈ [0, 1] def

xλ = λx + (1 − λ)y ∈B(xλ , λrx + (1 − λ)ry ) ⊂ λB(x, rx ) + (1 − λ)B(y, ry ) ⊂ Ω and xλ ∈ int Ω. (ii) By deﬁnition, int Ω = Ω and int Ω = Ω. If Ω = RN , Ω ⊂ Ω = ∅. This implies int Ω = Ω = Ω = ∅ and int Ω = Ω. If Ω = RN and int Ω = ∅, 7 By

convention ∅ is convex.

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Chapter 5. Metrics via Characteristic Functions

from Theorem 5.6 in Chapter 2, Ω is locally Lipschitzian, and from Theorem 5.4 in Chapter 2, ∂Ω = ∅, int Ω = ∅, int Ω = Ω, and Ω = Ω. Finally, if int Ω = ∅, then Ω = RN and Ω belongs to a linear subspace L of RN of dimension strictly less than N . Therefore Ω ⊃ Ω ⊃ L, RN = Ω ⊃ Ω ⊃ L = RN , and Ω = Ω. (iii) It is suﬃcient to prove the result for D bounded. For any Cauchy sequence {χn } in C(D), there exist a sequence of convex sets {Ωn } and χΩ in X(D) such that χΩn → χΩ in Lp (D). In particular, there exists a subsequence χk = χΩnk such that χk (x) → χΩ (x) almost everywhere in D. Deﬁne Ω∗ = {x ∈ D : χk (x) → 1 as k → ∞}. def

To show that Ω∗ is convex consider x and y in Ω∗ . There exists K ≥ 1 such that ∀k ≥ K,

|χk (x) − 1| < 1/2,

|χk (y) − 1| < 1/2,

and, since χk is either 0 or 1, for all k ≥ K, χk (x) = 1 = χk (y). By convexity for all λ ∈ [0, 1], xλ = λx + (1 − λ)y ∈ Ωnk and ∀k ≥ K,

χk (λx + (1 − λ)y) = 1 → 1

⇒ x λ ∈ Ω∗ .

But χk (x) → χΩ (x) almost everywhere and lim χk (x) ∈ {0, 1} a.e.

k→∞

So for x ∈ D\Ω∗ , χk (x) → 0 = χΩ∗ (x) almost everywhere in D. By the Lebesguedominated convergence theorem, χk → χΩ∗ in Lp (D) and necessarily χΩ = χΩ∗ . This means that [Ω] is a convex equivalence class. We shall show in section 6.1 (Corollary 1) that for bounded domains D, C(D) is also compact for the Lp (D) topology, 1 ≤ p < ∞.

3.5

Sobolev Spaces for Measurable Domains

The lack of a priori smoothness on Ω may introduce technical diﬃculties in the formulation of some boundary value problems. However, it is possible to relax such boundary value problems from smooth bounded open connected domains Ω to ´sio [7]). For instance consider the homogeneous measurable domains (cf. J.-P. Zole Dirichlet boundary value problem −y = f in Ω,

y = 0 on Γ

(3.25)

over a bounded open connected domain Ω with a boundary Γ of class C 1 and associate with its solution y = y(Ω) the objective function and volume constraint 1 2 J(Ω) = |y − g| dx, dx = π. (3.26) 2 Ω Ω There is a priori no reason to assume that an optimal (minimizing) domain Ω∗ is of class C 1 or is connected. So the problem must be suitably relaxed to a large

3. Lebesgue Measurable Characteristic Functions

225

enough class of domains, which preserves the meaning of the underlying function spaces, the well-posedness of the original problem, and the volume. To extend problem (3.25)–(3.26) to Lebesgue measurable sets, we ﬁrst have to make sense of the Sobolev space for measurable subsets Ω of D. Theorem 3.6. Let D be an open domain in RN . For any Lebesgue measurable subset Ω of RN , the spaces def

H•1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )∇ϕ = 0 a.e. in D , (3.27)

def H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D (3.28) are closed subspaces of H01 (D) and hence Hilbert spaces.8 Similarly, for any χ ∈ co X(D), def

H•1 (χ; D) = ϕ ∈ H01 (D) : (1 − χ)∇ϕ = 0 a.e. in D , (3.29)

def H1 (χ; D) = ϕ ∈ H01 (D) : (1 − χ)ϕ = 0 a.e. in D (3.30) are also closed subspaces of H01 (D) and hence Hilbert spaces. Furthermore9 H1 (Ω; D) ⊂ H•1 (Ω; D),

H1 (χ; D) ⊂ H•1 (χ; D).

Proof. We give only the proof for H•1 (Ω; D). Let {ϕn } in H•1 (Ω; D) be a Cauchy sequence. It converges to an element ϕ in the H01 (D)-topology. Hence {∇ϕn } converges to ∇ϕ in L2 (D). But for all n (1 − χΩ )∇ϕn = 0 in L2 (D). By Schwarz’s inequality the map ϕ → (1 − χΩ )∇ϕ is continuous and (1 − χΩ )∇ϕ = 0 in L2 (D). So ﬁnally ϕ ∈ H•1 (Ω; D) and H•1 (Ω; D) is a closed subspace of H01 (D). Assuming that D is bounded, the variational problems 1 ﬁnd y = y(Ω) ∈ H• (Ω; D) such that ∀ϕ ∈ H•1 (Ω; D), ∇y · ∇ϕ dx = χΩ f ϕ dx, D D 1 ﬁnd y = y(Ω) ∈ H (Ω; D) such that ∀ϕ ∈ H1 (Ω; D), ∇y · ∇ϕ dx = χΩ f ϕ dx D

(3.31)

(3.32)

D

8 Observe that for two open domains in the same equivalence class, the spaces deﬁned by (3.27)– (3.28) coincide. Therefore their functions do not see cracks in the underlying domain. The case of cracks will be handled in Chapter 8 by capacity methods. 9 From L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 130], ∇ϕ = 0 a.e. on {ϕ = 0}.

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Chapter 5. Metrics via Characteristic Functions

now make sense and have unique solutions for measurable subsets Ω of D (or even χ ∈ co X(D)), and the associated objective function J(Ω) = h χΩ , y(Ω) ,

def

h(χ, ϕ) =

1 2

χ|ϕ − g|2 dx

(3.33)

D

can be minimized over all measurable subsets Ω of D (or all χ ∈ co X(D)) with ﬁxed measure m(Ω) = π. The above problems are now well-posed, and their restriction to smooth bounded open connected domains coincides with the initial problem (3.25)–(3.26). Indeed, if Ω is a connected open domain with a boundary Γ of class C 1 , then Γ has a zero Lebesgue measure and the deﬁnition (3.28) of H•1 (Ω; D) coincides with H1 (Ω; D). Therefore, the problem speciﬁed by (3.31)–(3.33) is a well-deﬁned extension of problem (3.25)–(3.26). In general, when Ω contains holes, the elements of H•1 (Ω; D) are not necessarily equal to zero on the boundary of each hole and can be equal to diﬀerent constants from hole to hole, as in physical problems involving a potential. Example 3.3. Let D = ]−2, 2[ , Ω = ]−2, −1[ ∪ ]1, 2[ , and f = 1 on D. The solution of (3.31) is given by d2 y − 2 =1 dx y(−2) = 0,

on ]−2, −1[ ,

d2 y − 2 = 1 on ]1, 2[ , dx y(2) = 0, dy (1) = 0, dx

dy (−1) = 0, dx 1 y= on [−1, 1], 2 x −(x + 2) 2 in ]−2, −1[ , ⇒ y(x) = 12 in [−1, 1], −(x − 2) x2 in ]1, 2[ .

Example 3.4. Let D = B(0, 2), the open ball of radius 2 centered in 0 in R2 , H = B(0, 1), Ω = D H, and f = 1. In polar coordinates the solution of (3.31) is given by 1 d dy r = 1 in ]1, 2[ , − 1 1 3 r dr dr andy(r) = ln + 2 2 4 y(2) = 0, dy (1) = 0, dr or explicitly y(r) =

1 2 1 2

ln 2r + 1 − ln 12 + 34

r2 4

in ]1, 2[ , in [0, 1].

in [0, 1],

3. Lebesgue Measurable Characteristic Functions

227

The last example is a special case that retains the characteristics of the onedimensional example. In higher dimensions the normal derivative is not necessarily zero on the “internal boundary” of Ω. Example 3.5. Let D = B(0, 1) in R2 , let H be a bounded open connected hole in D such that H ⊂ D, and let Ω = D H. Then it can be checked that the solution of (3.31) is of the form −∆y = f in Ω,

y = 0 on ∂D,

where the constant c on H is determined by the condition ∂y ∀ϕ ∈ H•1 (Ω; D), ϕ dγ = 0, ∂Ω ∂n or equivalently

∂H

∂y dγ = 0. ∂n

Example 3.6. Let D = ]−2, 2[ × ]−2, 2[ and let Ω1 and Ω2 be two open squares in G = ]−1, 1[ × ]−1, 1[ such that Ω1 ⊂ G, Ω2 ⊂ G, and Ω1 ∩ Ω2 = ∅. Deﬁne the domain as Ω = Ω0 ∪ Ω1 ∪ Ω2 ,

Ω0 = D\G

(cf. Figure 5.2).

Then the solution of (3.31) is characterized by −∆y = f in Ω,

y = 0 on ∂D,

y = c on G\ Ω1 ∪ Ω2 ,

where the constant c is determined by the condition ∂y ∂y ∂y dγ + dγ + dγ = 0 int ∂n ∂n ∂n ∂Ω0 ∂Ω1 ∂Ω2 and ∂Ωint 0 = ∂G, the interior boundary of Ω0 .

Ω2

Ω0 = D\G

Ω1

Figure 5.2. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 .

228

4

Chapter 5. Metrics via Characteristic Functions

Some Compliance Problems with Two Materials

As a ﬁrst example consider the optimal compliance problem where the optimization variable is the distribution of two materials with diﬀerent physical characteristics within a ﬁxed domain D. It cannot a priori be assumed that the two regions are separated by a smooth boundary and that each region is connected. The optimal solution may lead to a nonsmooth interface and even to the mixing of the two materials. This type of solution occurs in control or optimization problems over a bounded nonconvex subset of a function space. In general, their relaxed solution lies in the closed convex hull of that subset. In control theory, the phenomenon is known as chattering control. To illustrate this approach it is best to consider a generic example. In section 4.1, we consider a variation of the optimal design problem ´a and K. Malanowski [1] in 1970 and then discuss its original studied by J. Ce version in section 4.2. The variation of the problem is constructed in such a way that the form of the associated variational equation is similar to the one of Example 3.1, which provided a counterexample to the weak continuity of the solution. Yet the maximization of the minimum energy yields as solution a characteristic function in both cases. However, this is no longer true for the minimization of the same minimum energy as was shown by F. Murat and L. Tartar [1, 3].

4.1

Transmission Problem and Compliance

Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assume that the domain D is partitioned into two subdomains Ω1 and Ω2 separated by a smooth boundary ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 5.3. Domain Ω1 (resp., Ω2 ) is made up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y be the solution of the transmission problem − k1 y = f in Ω1 , −k2 y = 0 in Ω2 , (4.1) ∂y ∂y y = 0 on ∂D, k1 + k2 = 0 on ∂Ω1 ∩ ∂Ω2 , ∂n1 ∂n2 where n1 (resp., n2 ) is the unit outward normal to Ω1 (resp., Ω2 ) and f is a given function in L2 (D). Our objective is to maximize the equivalent of the compliance J(Ω1 ) = − f y dx (4.2) Ω1

Ω1

Ω2

Figure 5.3. Fixed domain D and its partition into Ω1 and Ω2 .

4. Some Compliance Problems with Two Materials

229

over all domains Ω1 in D. In mechanics the compliance is associated with the total work of the body forces f . Denote by Ω the domain Ω1 . By complementarity Ω2 = D Ω and let χ = χΩ . Problem (4.1) can be rewritten in the following variational form: ﬁnd y = y(χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D), [k1 χ + k2 (1 − χ)] ∇y · ∇ϕ dx = χf ϕ dx. D

(4.3)

D

For k1 > 0 and k2 > 0 and all χ in X(D) def

k(x) = k1 χ(x) + k2 (1 − χ(x)), 0 < min{k1 , k2 } ≤ k(x) ≤ max{k1 , k2 } a.e. in D.

(4.4)

By the Lax–Milgram theorem, the variational equation (4.3) still makes sense and has a unique solution y = y(χ) in H01 (D) that coincides with the solution of the boundary value problem −div (k ∇y) = χf in D,

y = 0 on ∂D.

(4.5)

Notice that the high-order term has the same form as the term in Example 3.1. As for the objective function, it can be rewritten as J(χ) = − χf y(χ) dx. (4.6) D

Thus the initial boundary value problem (4.1) has been transformed into the variational problem (4.3), and the initial objective function (4.2) into (4.6). Both make sense for χ in X(D) and even in co X(D). The family of characteristic functions X(D) and its closed convex hull co X(D) have been deﬁned and characterized in (3.2) and (3.3). The objective function (4.6) can be further rewritten as a minimum J(χ) = for the energy function def

min

ϕ∈H01 (D)

E(χ, ϕ)

(4.7)

(k1 χ + k2 (1 − χ)) |∇ϕ|2 − 2 χf ϕ dx

E(χ, ϕ) =

(4.8)

D

associated with the variational problem (4.3). The initial optimal design problem becomes a max-min problem max J(χ) = max χ∈X(D)

min

χ∈X(D) ϕ∈H01 (D)

E(χ, ϕ),

where X(D) can be considered as a subset of Lp (D) for some p, 1 ≤ p < ∞.

(4.9)

230

Chapter 5. Metrics via Characteristic Functions This problem can also be relaxed to functions χ with value in [0, 1] max

J(χ) =

χ∈co X(D)

max

min

χ∈co X(D) ϕ∈H01 (D)

E(χ, ϕ).

(4.10)

We shall refer to this problem as the relaxed problem. ´a and K. Malanowski These formulations have been introduced by J. Ce [1], who used the variable k(x) and the following equality constraint on its integral: γ − k2 m(D) k(x) dx = γ or χ(x) dx = k1 − k 2 D D for some appropriate γ > 0. Moreover the force f in (4.1)–(4.3) was exerted everywhere in D and not only in Ω. So the objective function (4.2)–(4.6) was the integral of f y over all of D. We shall see in section 4.2 that the fact that the support of f is Ω or all of D does not aﬀect the nature of the results. In both maximization problems (4.1)–(4.2) and (4.3)–(4.6) it is necessary to introduce a volume constraint in order to avoid a trivial solution. Notice that by using (4.3) with ϕ = y(χ) the objective function (4.6) becomes (k1 χ + k2 (1 − χ)) |∇y(χ)|2 dx ≤ 0. J(χ) = − D

So maximizing J(χ) is equivalent to minimizing the integral of k(x) |∇y|2 . Therefore, χ = 0 (only material k2 ) is a maximizer since the corresponding solution of (4.3) is y = 0. In order to eliminate this situation we introduce the following constraint on the volume of material k1 : χ dx ≥ α > 0 (4.11) D

for some α, 0 < α ≤ m(D). The case α = m(D) yields the unique solution χ = 1 (only material k1 ). So we can further assume that α < m(D). For 0 < α < m(D), the optimal design problem becomes max J(χ) = max χ∈X(D) χ∈X(D) D

χ dx≥α

D

min

E(χ, ϕ)

(4.12)

min

E(χ, ϕ).

(4.13)

ϕ∈H01 (D)

χ dx≥α

and its relaxed version max

χ∈co X(D) χ dx≥α D

J(χ) =

max

χ∈co X(D) ϕ∈H01 (D) χ dx≥α D

We shall now show that problem (4.13) has a unique solution χ∗ in co X(D) and that χ∗ is a characteristic function, χ∗ ∈ X(D), for which the inequality constraint is saturated: ∗ χ∗ dx = α. (4.14) χ ∈ X(D) and D

4. Some Compliance Problems with Two Materials

231

At this juncture it is advantageous to incorporate the volume inequality constraint into the problem formulation by introducing a Lagrange multiplier λ ≥ 0. The formulation of the relaxed problem becomes max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ),

def

χ dx − α

G(χ, ϕ, λ) = E(χ, ϕ) + λ

(4.15)

D

and E(χ, ϕ) is the energy function given in (4.7). We ﬁrst establish the existence of saddle point solutions to problem (4.15). We use a general result in I. Ekeland and R. Temam [1, Prop. 2.4, p. 164]. The set co X(D) is a nonempty bounded closed convex subset of L2 (D) and the set H01 (D)×[R+ ∪{0}] is trivially closed and convex. The function G is concave-convex with the following properties: ∀χ ∈ co X(D), (ϕ, λ) → G(χ, ϕ, λ) is convex, continuous, and (4.16) G(χ0 , ϕ, λ) = +∞; ∃χ0 ∈ co X(D) such that ϕ 1lim +|λ|→∞ H

∀ϕ ∈

H01 (D),

∀λ ≥ 0,

0

the map χ → G(χ, ϕ, λ) is aﬃne and

2

continuous for L (D)-strong.

(4.17)

For the ﬁrst condition recall that 0 < α < m(D) and pick χ0 = 1 on D. To check the second condition pick any sequence {χn } in co X(D) which converges to some χ in L2 (D)-strong. Then the sequence also converges in L2 (D)-weak and by Lemma 3.1 in L∞ (D)-weak∗ . Hence G(χn , ϕ, λ) converges to G(χ, ϕ, λ). ˆ is of the form X × Y ⊂ co X(D) × {H 1 (D) × The set of saddle points (χ, ˆ y, λ) 0 + [R ∪ {0}]} and is completely characterized by the following variational equation and inequalities (cf. I. Ekeland and R. Temam [1, Prop. 1.6, p. 157]): 1 [k2 + (k1 − k2 )χ] ˆ ∇y · ∇ϕ − χf ˆ ϕ dx = 0, (4.18) ∀ϕ ∈ H0 (D), D. / ˆ (χ − χ) (k1 − k2 )|∇y|2 − 2f y + λ ˆ dx ≤ 0, (4.19) ∀χ ∈ co X(D), D ˆ = 0, ˆ ≥ 0. χ ˆ dx − α λ χ ˆ dx − α ≥ 0, λ (4.20) D

D

But for each χ ∈ co X(D) there exists a unique y(χ) solution of (4.18) and then

∀χ ˆ ∈ X, {χ} ˆ × Y ⊂ {χ} ˆ × {y(χ)} ˆ × [R+ ∪ {0}] . Therefore, y(χ) ˆ is independent of χ ˆ ∈ X, that is, Y = {y} × Λ and ˆ ∈ Λ, ∀χ ˆ ∈ X, λ ˆ inequality (4.19) is For each λ, terization of the maximizer χ ˆ ∈ X: if 1, χ(x) ˆ = ∈ [0, 1], if 0, if

y(χ) ˆ = y.

completely equivalent to the following characˆ > 0, (k1 − k2 )|∇y|2 − 2f y + λ 2 ˆ = 0, (k1 − k2 )|∇y| − 2f y + λ 2 ˆ < 0. (k1 − k2 )|∇y| − 2f y + λ

(4.21)

232

Chapter 5. Metrics via Characteristic Functions

Associate with an arbitrary λ ≥ 0 the sets D+ (λ) = {x ∈ D : (k1 − k2 )|∇y|2 − 2f y + λ > 0}, D0 (λ) = {x ∈ D : (k1 − k2 )|∇y|2 − 2f y + λ = 0}.

(4.22)

In order to complete the characterization of the optimal triplets, we need the following general result. Lemma 4.1. Consider a function G:A×B →R

(4.23)

for some sets A and B. Deﬁne def

g = inf sup G(x, y), x∈A y∈B

def

A0 =

x ∈ A : sup G(x, y) = g ,

def

h = sup inf G(x, y), y∈B x∈A

def

B0 =

(4.24)

y∈B

y ∈ B : inf G(x, y) = h . x∈A

(4.25)

When g = h the set of saddle points (possibly empty) will be denoted by def

S = {(x, y) ∈ A × B : g = G(x, y) = h} .

(4.26)

Then the following hold. (i) In general h ≤ g and ∀(x0 , y0 ) ∈ A0 × B0 ,

h ≤ G(x0 , y0 ) ≤ g.

(4.27)

(ii) If h = g, then S = A0 × B0 . Proof. (i) If A0 × B0 = ∅, there is nothing to prove. If there exist x0 ∈ A0 and y0 ∈ B0 , then by deﬁnition h = inf G(x, y0 ) ≤ G(x0 , y0 ) ≤ sup G(x0 , y) = g. x∈A

(4.28)

y∈B

(ii) If h = g, then in view of (4.26)–(4.27), A0 × B0 ⊂ S. Conversely if there exists (x0 , y0 ) ∈ S, then h = G(x0 , y0 ) = g and by the deﬁnitions of A0 and B0 , (x0 , y0 ) ∈ A0 × B0 . ˆ the characteristic function Associate with an arbitrary solution (χ, ˆ y, λ), χλˆ =

χD+ (λ) ˆ , 0,

ˆ = ∅, if D+ (λ) ˆ = ∅. if D+ (λ)

(4.29)

Then from (4.21) ˆ = χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ ˆ a.e. in D χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ λ

4. Some Compliance Problems with Two Materials and

233

ˆ [χk ˆ 1 + (1 − χ)k ˆ 2 ] |∇y| − 2χf ˆ y dx + λ

ˆ = G(χ, ˆ y, λ) D

.

χ ˆ dx − α

2

/

D

ˆ dx − λα ˆ k2 |∇y|2 + χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ

= D = D

. / ˆ dx − λα ˆ k2 |∇y|2 + χλˆ (k1 − k2 )|∇y|2 − 2f y + λ

ˆ = G(χλˆ , y, λ). ˆ is also a saddle point. So there exists a maximizer From Lemma 4.1, (χλˆ , y, λ) χλˆ ∈ X(D) that is a characteristic function and necessarily max

min

χ∈X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ) =

max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ).

ˆ ∈ Λ such that λ ˆ > 0, then by But we can show more than that. If there exists λ construction χ ˆ ≥ χλˆ and by (4.20) α= χ ˆ dx ≥ χλˆ dx = α. D

D

ˆ then χ ˆ = χλˆ . But, always by Since χ ˆ = χλˆ = 1 almost everywhere in D+ (λ), construction, χλˆ is independent of χ. ˆ Thus the maximizer is unique, it is a characteristic function, and its integral is equal to α. The case Λ = {0} is a degenerate one. Set ϕ = y in (4.18) and regroup the terms as follows: (k2 + (k1 − k2 )χ) ˆ |∇y|2 − f y χ ˆ dx 0= D k1 − k2 k1 − k2 2 2 χ ˆ |∇y| dx + χ ˆ |∇y| − f y χ ˆ dx. = k2 + 2 2 D D The integrand of the ﬁrst integral is positive. From the characterization of χ, ˆ the integrand of the second one is also positive. Hence they are both zero almost everywhere in D. As a result m(D+ (0)) = 0

and ∇y = 0.

ˆ = Therefore y = 0 in D. Hence the saddle points are of the general form (χ, ˆ y, λ) (χ, ˆ 0, 0), χ ˆ ∈ X. In particular, D+ (0) = ∅ and from our previous considerations χ∅ ; that is, χ ˆ = 0 is a solution. But this is impossible since D χ ˆ dx ≥ α > 0. ˆ = 0 cannot occur. Therefore λ In conclusion, there exists a (unique for α > 0) maximizer χ∗ in co X(D) that is in fact a characteristic function, and necessarily max χ∈X(D) D

χ dx≥α

min

ϕ∈H01 (D)

E(χ, ϕ) =

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α D

E(χ, ϕ).

(4.30)

234

Chapter 5. Metrics via Characteristic Functions

Moreover, for 0 < α ≤ m(D),

χ∗ dx = α

(4.31)

D

and χ∗ is the unique solution of the problem with an equality constraint: max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ) =

χ dx=α

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α

E(χ, ϕ).

(4.32)

D

As a numerical illustration of the theory consider (4.1) over the diamond-shaped domain D = {(x, y) : |x| + |y| < 1}

(4.33)

with the function f of Figure 5.4. This function has a sharp peak in (0, 0) which has been scaled down in the picture. The variational form (4.3) of the boundary value problem was approximated by continuous piecewise linear ﬁnite elements on each triangle, and the function χ by a piecewise constant function on each triangle. The constant on each triangle was constrained to lie between 0 and 1 together with the global constraint on its integral over the whole domain D. Figure 5.5 shows the optimal partition (the domain has been rotated by 45 degrees to save space). The ˆ m ), where χ grey triangles correspond to the region D0 (λ ˆ ∈ [0, 1]. The presence of this grey zone in the approximated problem is due to the fact that equality for the total area where χ ˆ = 1 could not be exactly achieved with the chosen triangulation of the domain. Thus the problem had to adjust the value of χ ˆ between 0 and 1 in

60 50 40 30 20

1 0.8

10

0.6 0.4

0 1

0.2 0.8

0 0.6

0.4

−0.2 0.2

−0.4 0

−0.2

−0.6 −0.4

−0.6

−0.8 −0.8

−1

−1

Figure 5.4. The function f (x, y) = 56 (1 − |x| − |y|)6 .

4. Some Compliance Problems with Two Materials

235

Figure 5.5. Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1 (white) for the problem of section 4.1. a few triangles in order to achieve equality for the integral of χ. ˆ For this example the Lagrange multiplier associated with the problem is strictly positive.

4.2

The Original Problem of C´ ea and Malanowski

We now put the force f everywhere in the ﬁxed domain D. The same technique and the same conclusion can be drawn: there exists at least one maximizer of the compliance, which is a characteristic function. For completeness we give the main elements below. Fix the bounded open Lipschitzian domain D in RN and let Ω be a smooth subset of D. Let y be the solution of the transmission problem − k1 y = f in Ω, −k2 y = f in D\Ω, y = 0 on ∂D, (4.34) ∂y ∂y k1 + k2 = 0 on ∂Ω ∩ D, ∂n1 ∂n2 where n1 (resp., n2 ) is the unit outward normal to Ω (resp., D Ω) and f is a given function in L2 (D). Again problem (4.34) can be reformulated in terms of the characteristic function χ = χΩ : ﬁnd y = y(χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D), [k1 χ + k2 (1 − χ)] ∇y · ∇ϕ dx = f ϕ dx, D

(4.35)

D

with the objective function

J(χ) = −

f y(χ) dx D

(4.36)

236

Chapter 5. Metrics via Characteristic Functions

to be maximized over all χ ∈ X(D). As in the previous case, the function J(χ) can be rewritten as the minimum of the energy function def [k1 χ + k2 (1 − χ)] |∇ϕ|2 − 2f ϕ dx (4.37) E(χ, ϕ) = D

over H01 (D), J(χ) =

min

ϕ∈H01 (D)

E(χ, ϕ),

(4.38)

and we have the relaxed max-min problem max

min

χ∈co X(D) ϕ∈H01 (D)

E(χ, ϕ).

(4.39)

Without constraint on the integral of χ, the problem is trivial and χ = 1 (resp., χ = 0) if k1 > k2 (resp., k2 > k1 ). In other words it is optimal to use only the strong material. In order to make the problem nontrivial assume that the strong material is k1 , that is, k1 > k2 , and put an upper bound on the volume of material k1 which occupies the part Ω of D: χ dx ≤ α, 0 < α < m(D). (4.40) D

The case α = 0 trivially yields χ = 0. Under assumption (4.40) the case χ = 1 with only the strong material k1 is no longer admissible. Thus we consider for 0 < α < m(D) the problem max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≤α

E(χ, ϕ).

(4.41)

D

We shall now show that problem (4.41) has a unique solution χ∗ in co X(D) and that in fact χ∗ is a characteristic function for which the inequality constraint is saturated: χ∗ dx = α. (4.42) D

This solves the original problem of C´ea and Malanowski with the equality constraint: max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ).

(4.43)

χ dx=α

As in section 4.1 it is convenient to reformulate the problem with a Lagrange multiplier λ ≥ 0 for the constraint inequality (4.40): max min G(χ, ϕ, λ), G(χ, ϕ, λ) = E(χ, ϕ) − λ χ dx − α . (4.44) 1 χ∈X(D) ϕ∈H0 (D) λ≥0

D

4. Some Compliance Problems with Two Materials

237

We then relax the problem to co X(D): max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ).

(4.45)

By the same arguments as the ones used in section 4.1 we have the existence of ˆ which are completely characterized by saddle points (χ, ˆ y, λ) ∀ϕ ∈ H01 (D), [k2 + (k1 − k2 )χ]∇y ˆ · ∇ϕ − f ϕ dx = 0, (4.46) D . / ˆ (χ − χ) (k1 − k2 )|∇y|2 − λ ˆ dx ≤ 0, (4.47) ∀χ ∈ co X(D), D ˆ = 0, ˆ ≥ 0. χ ˆ dx − α λ χ ˆ dx − α ≤ 0, λ (4.48) D

D

As before y is unique and the set of saddle points of G is the closed convex set

X × {y} × Λ ⊂ X(D) × {y} × {λ : λ ≥ 0} . ˆ m ≥ 0. The closed convex set Λ has a minimal element λ ˆ each maximizer χ For each λ, ˆ ∈ X is necessarily of the form ˆ > 0, if (k1 − k2 )|∇y|2 − λ 1, 2 ˆ = 0, χ(x) ˆ = ∈ [0, 1], if (k1 − k2 )|∇y| − λ 2 ˆ < 0. 0, if (k1 − k2 )|∇y| − λ Associate with each λ > 0 the sets

D+ (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ > 0 ,

D0 (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ = 0 ,

D− (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ < 0 .

(4.49)

(4.50) (4.51) (4.52)

Deﬁne the characteristic function χm = χD+ (λm ) . By construction all χ ˆ ∈ X are of the form ˆ χ ˆ dx ≤ α. (4.53) χm ≤ χ, D

ˆ m ) is also a saddle point of G. Therefore, we Again it is easy to show that (χm , y, λ have a maximizer χm ∈ X(D) over co X(D) which is a characteristic function and χm dx ≤ χ ˆ dx ≤ α. D

D

ˆ > 0 in Λ, then for all χ If there exists λ ˆ∈X χm dx = α = χ ˆ dx. D

D

238

Chapter 5. Metrics via Characteristic Functions

As a result the maximizer χm is unique, it is a characteristic function, and its integral is equal to α. The case Λ = {0} cannot occur since the triplet (1, y1 , 0) would be a saddle point, where y1 is the solution of the variational equation (4.34) for χ = 1. To see this, ﬁrst observe that for k1 > k2 ∀χ, ϕ,

G(χ, ϕ, 0) = E((χ, ϕ) ≤ E((1, ϕ) = G(1, ϕ, 0).

As a result inf G(χ, ϕ, λ) ≤ inf G(χ, ϕ, 0) ≤ inf G(1, ϕ, 0), ϕ

ϕ,λ

ϕ

inf G(1, ϕ, 0) = G(1, y1 , 0) ≤ sup inf G(χ, ϕ, 0) ϕ

χ

ϕ

⇒ sup inf G(χ, ϕ, λ) ≤ sup inf G(χ, ϕ, 0) = G(1, y1 , 0). χ ϕ,λ

χ

ϕ

But we know that there exists a unique y ∈ H01 (D) such that sup inf G(χ, ϕ, 0) ≤ sup G(χ, y, 0) = inf sup G(χ, ϕ, λ). χ

ϕ

ϕ,λ χ

χ

So from the above two inequalities, sup inf G(χ, ϕ, λ) = G(1, y1 , 0) = inf sup G(χ, ϕ, λ), χ ϕ,λ

ϕ,λ χ

(1, y1 , 0) is a saddle point of G, and α≥ χ dx = m(D). D

This contradicts the fact that α < m(D). In conclusion in all cases there exists a (unique when 0 ≤ α < m(D)) maximizer χ∗ in co X(D), which is in fact a characteristic function, and max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ) =

χ dx≥α

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α

E(χ, ϕ).

D

Moreover, for 0 ≤ α < m(D)

χ∗ dx = α. D

This is precisely the solution of the original problem of C´ea and Malanowski and max χ∈X(D) D

χ dx=α

min

ϕ∈H01 (D)

E(χ, ϕ) =

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α

E(χ, ϕ).

D

Again, as an illustration of the theoretical results, consider (4.34) over the diamondshaped domain D deﬁned in (4.33) with the function f of Figure 5.4 in section 4.1.

4. Some Compliance Problems with Two Materials

239

The variational form (4.35) of the boundary value problem was approximated in the same way as the variational form (4.3) of (4.1) in section 4.1. Figure 5.6 shows the optimal partition of the domain (rotated 45 degrees). The grey triangles ˆ m ), where χ ˆ ∈ [0, 1]. The black region corresponds correspond to the region D0 (λ ˆ m /(k1 − k2 ) and the white region to the ones where to the points where |∇y|2 > λ ˆ m /(k1 − k2 ), as can be readily seen in Figure 5.6. For this example the |∇y|2 < λ Lagrange multiplier associated with the problem is strictly positive. It is interesting to compare this computation with the one of Figure 5.5 in section 4.1, where the support of the force was restricted to Ω.

Figure 5.6. Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1 (white) for the problem of C´ea and Malanowski. Remark 4.1. The formulation and the results remain true if the generic variational equation is replaced by a variational inequality (unilateral problem) max χ∈X(D) D

4.3

min

ϕ∈{ψ∈H01 (D):ψ≥0 in D}

E(χ, ϕ).

χ dx=α

Relaxation and Homogenization

In sections 4.1 and 4.2 the possible homogenization phenomenon predicted in Example 3.1 of section 3.2 did not take place, and in both examples the solution was a characteristic function. Yet, it occurs when the maximization is changed to a ´a and K. Malanowski [1] of section 4.2: minimization in the problem of J. Ce inf χ∈X(D) D

χ dx=α

min

ϕ∈H01 (D)

E(χ, ϕ).

240

Chapter 5. Metrics via Characteristic Functions

In 1985 F. Murat and L. Tartar [1] gave a general framework to study this class of problems by relaxation. It is based on the use of L. C. Young [2]’s generalized functions (measures). They present a fairly complete analysis of the homogenization theory of second-order elliptic problems of the form ∂ ∂u aij (x) = f. − ∂xi ∂xj ij

They give as examples the maximization and minimization versions of the problem of section 4.2. This material, which would have deserved a whole chapter in this book, is fortunately available in English (cf. F. Murat and L. Tartar [3]). More results on composite materials can be found in the book edited by A. Cherkaev and R. Kohn [1], which gathers a selection of translations of key papers originally written in French and Russian.

5

Buckling of Columns

One of the very early optimal design problem was formulated by Lagrange in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by T. Clausen in 1849. It consists in ﬁnding the best proﬁle of a vertical column to prevent buckling. This problem and other problems related to columns have been revisited in a series of papers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, but a complete theoretical and numerical solution for the buckling of a column was given only in 1992 by S. J. Cox and M. L. Overton [1]. Consider a normalized column of unit height and unit volume. Denote by the magnitude of the normalized axial load and by u the resulting transverse displacement. Assume that the potential energy is the sum of the bending and elongation energies 1 1 2 EI |u | dx − |u |2 dx, 0

0

where I is the second moment of area of the column’s cross section and E is its Young’s modulus. For suﬃciently small load the minimum of this potential energy with respect to all admissible u is zero. Euler’s buckling load λ of the column is the largest for which this minimum is zero. This is equivalent to ﬁnding the following minimum: 1 EI |u |2 dx def 0 λ = inf , (5.1) 1 0=u∈V |u |2 dx 0 where V = H02 (0, 1) corresponds to the clamped case, but other types of boundary conditions can be contemplated. This is an eigenvalue problem with a special Rayleigh quotient. Assume that E is constant and that the second moment of area I(x) of the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

5. Buckling of Columns

241

cross-sectional area A(x),

I(x) = c A(x)

1

⇒

A(x) dx = 1. 0

Normalizing λ by cE and taking into account the engineering constraints ∃0 < A0 < A1 , ∀x ∈ [0, 1],

0 < A0 ≤ A(x) ≤ A1 ,

we ﬁnally get

1

A |u |2 dx , 1 0=u∈V A∈A |u |2 dx 0 1 def 2 A(x) dx = 1 . A = A ∈ L (0, 1) : A0 ≤ A ≤ A1 and sup λ(A),

def

λ(A) =

inf

0

(5.2) (5.3)

0

This problem can also be reformulated by rewriting 1 def 1 − A0 A(x) = A0 + χ(x) (A1 − A0 ), χ(x) dx = α = A1 − A0 0 for some χ ∈ co X([0, 1]). Clearly the problem makes sense only for 0 < A0 ≤ 1. Then 1 [A0 + (A1 − A0 ) χ] |u |2 dx def 0 ˜ ˜ λ(χ), λ(χ) = inf . (5.4) sup 1 0=u∈V |u |2 dx χ∈co X([0,1]) 0 1

0

χ(x) dx=α

Rayleigh’s quotient is not a nice convex concave function with respect to (v, A), and its analysis necessitates tools diﬀerent from the ones of section 4. One of the original elements of the paper of S. J. Cox and M. L. Overton [1] was to replace Rayleigh’s quotient by G. Auchmuty [1]’s dual variational principle for the eigenvalue problem (5.1). We ﬁrst recall the existence of solution to the minimization of Rayleigh’s quotient. In what follows we shall use the norm u L2 for the space H01 (0, 1) and u L2 for the space H02 (0, 1). Theorem 5.1. There exists at least one nonzero solution u ∈ V to the minimization problem 1 A |u |2 dx def 0 λ(A) = inf . (5.5) 1 0=u∈V |u |2 dx 0 Then λ(A0 ) > 0 and for all A ∈ A, λ(A) ≥ λ(A0 ), and 1 1 ∀v ∈ V, |v |2 dx ≤ λ(A0 )−1 A |v |2 dx. 0

(5.6)

0

The solutions are completely characterized by the variational equation: 1 1 A u v dx = λ(A) u v dx. ∃u ∈ V, ∀v ∈ V, 0

0

(5.7)

242

Chapter 5. Metrics via Characteristic Functions

Proof. The inﬁmum is bounded below by 0 and is necessarily ﬁnite. Let {un } be a minimizing sequence such that un L2 = 1. Then the sequence un L2 (D) is bounded. Hence {un } is a bounded sequence in H 2 (0, 1) and there exist u ∈ H 2 (0, 1) and a subsequence, still indexed by n, such that un u in H02 (0, 1)-weak. Therefore the subsequence strongly converges in H01 (0, 1) and 1 1 1 1 2 2 2 |un | dx → |u | dx and A |u | dx ≤ lim inf A |un |2 dx 1= 0

1 ⇒

0

0

A |u |2 dx

1 0

n→∞

0

|u |2

dx

1

A |un |2 dx

≤ lim inf 0 1 n→∞

0

|un |2 dx

0

= λ(A),

and, by the deﬁnition of λ(A), u ∈ H02 (0, 1) is a minimizing element. Notice that & 1 2 A |u | dx 0 0 ∀A ∈ A, λ(A0 ) = inf : ∀u ∈ H02 (0, 1), u = 0 1 |2 dx |u 0 & 1 A |u |2 dx 2 0 : ∀u ∈ H0 (0, 1), u = 0 = λ(A) ≤ inf 1 |u |2 dx 0 and similarly λ(A) ≤ λ(A1 ). If λ(A0 ) = 0, we repeat the above construction and end up with an element u ∈ H02 (0, 1) such that 1 1 1 |u |2 dx = 1 and A0 |u |2 dx = A0 |u |2 dx = 0, 0

0

0

which is impossible in H02 (0, 1) since A0 = 0 and u = 0 implies u = 0. If u = 0, then Rayleigh’s quotient is diﬀerentiable and its directional semiderivative in the direction v is given by 1 1 1 A u v dx u v dx 2 0 −2 A |u | dx 0 4 . 2 u 2L2 u L2 0 A nonzero solution u of the minimization problem is necessarily a stationary point, and for all v ∈ H02 (0, 1) 1 1 A0 |u |2 dx 1 A u v dx = 0 1 u v dx = λ(A) u v dx. |2 dx |u 0 0 D 0 Conversely any nonzero solution of (5.7) is necessarily a minimizer of Rayleigh’s quotient. The dual variational principle of G. Auchmuty [1] for the eigenvalue problem (5.5) can be chosen as

def µ(A) = inf L(A, v) : v ∈ H02 (0, 1) , (5.8) 1 1/2 1 def 1 L(A, v) = A |v |2 dx − |v |2 dx . (5.9) 2 0 0

5. Buckling of Columns

243

Theorem 5.2. For each A ∈ A, there exists at least one minimizer of L(A, v), µ(A) = −

1 , 2λ(A)

(5.10)

and the set of minimizers of (5.8) is given by u is solution of (5.7) and def 1/2 E(A) = u ∈ H02 (0, 1) : 1 . |u |2 dx = 1/λ(A)

(5.11)

0

Proof. The existence of the solution follows from the fact that v → L(A, v) is weakly lower semicontinuous and coercive. From Theorem 5.1 the set E(A) is not empty, and for any u ∈ E(A) µ(A) ≤ L(A, u) = −

1 < 0. 2λ(A)

(5.12)

Therefore, the minimizers of (5.7) are diﬀerent from the zero functions. For u = 0, the function L(A, u) is diﬀerentiable and its directional semiderivative is given by

1

dL(A, u; v) = 0

1 A u v dx − u L2

1

u v dx,

(5.13)

0

and any minimizer u of L(A, v) is a stationary point of dL(A, u; v), that is,

1

∀v ∈ H02 (0, 1),

A u v dx −

0

1 u L2

1

u v dx = 0.

(5.14)

0

Therefore, u is a solution of the eigenvalue problem with λ =

1 u

L2

⇒ −

1 1 = µ(A) ≤ − 2λ 2λ(A)

from inequality (5.12). By the minimality of λ(A), we necessarily have λ = λ(A), and this concludes the proof of the theorem. Theorem 5.3. (i) The set A is compact in the L2 (0, 1)-weak topology. (ii) The function A → µ(A) is concave and upper semicontinuous with respect to the L2 (0, 1)-weak topology. (iii) There exists A in A which maximizes µ(A) over A and, a fortiori, which maximizes λ(A) over A. Proof. (i) A is convex, bounded, and closed in L2 (0, 1). Hence it is compact in L2 (0, 1) weak.

244

Chapter 5. Metrics via Characteristic Functions (ii) Concavity. For all λ in [0, 1] and A and A in A, L(λA + (1 − λ)A , v) = λL(A, v) + (1 − λ)L(A , v) ≥ λ inf L(A, v) + (1 − λ) inf L(A , v) v∈V

v∈V

⇒ µ(λA + (1 − λ)A ) = inf L(λA + (1 − λ)A , v) ≥ λµ(A) + (1−λ)µ(A ). v∈V

Upper semicontinuity. Let uA ∈ A be a minimizer of L(A, v) and {An } be a sequence converging to A in the L2 (0, 1)-weak topology. By Lemma 3.1, {An } converges to A in the L∞ (0, 1)-weak topology. Then 1 1 µ(An ) − µ(A) ≤ L(An , uA ) − L(A, uA ) = (An − A) |uA |2 dx 1 2 0 1 (An − A) |uA |2 dx = 0. ⇒ lim sup µ(An ) − µ(A) ≤ lim n→∞ 2 0 n→∞ (iii) The existence of solution follows from the concavity and upper semicontinuity of µ(A). Hence it is weakly upper semicontinuous over the weakly compact subset A of L2 (0, 1). The existence of a minimizer of λ(A) follows from identity 5.10.

6

Caccioppoli or Finite Perimeter Sets

The notion of a ﬁnite perimeter set has been introduced and developed mainly by R. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem, named after the Belgian physicist and professor (1801–1883), who did experimental observations on the geometry of soap ﬁlms. A modern treatment of this subject can be found in the book of E. Giusti [1]. One of the diﬃculties in studying the minimal surface problem is the description of such surfaces in the usual language of diﬀerential geometry. For instance, the set of possible singularities is not known. Finite perimeter sets provide a geometrically signiﬁcant solution to Plateau’s problem without having to know ahead of time what all the possible singularities of the solution can be. The characterization of all the singularities of the solution is a diﬃcult problem which can be considered separately. This was a very fundamental contribution to the theory of variational problems, where the optimization variable is the geometry of a domain. This point of view has been expanded, and a variational calculus was developed by F. J. Almgren, Jr. [1]. This is the theory of varifolds. Perhaps the one most important virtue of varifolds is that it is possible to obtain a geometrically signiﬁcant solution to a number of variational problems, including Plateau’s problem, without having to know ahead of time what all the possible singularities of the solution can be. (cf. F. J. Almgren, Jr. [1, p. viii]). The treatment of this topic is unfortunately much beyond the scope of this book, and the interested reader is referred to the above reference for more details.

6. Caccioppoli or Finite Perimeter Sets

6.1

245

Finite Perimeter Sets

Given an open subset D of RN , consider L1 -functions f on D with distributional gradient ∇f in the space M 1 (D)N of (vectorial) bounded measures; that is, def f div ϕ

dx : D1 (D; RN ) → R ϕ

→ ∇f, ϕ

D = − D

is continuous with respect to the topology of uniform convergence in D: def

∇f M 1 (D)N = def

def

sup 1

N

∇f, ϕ

D < ∞, ϕC = sup | ϕ(x)|RN ,

ϕ ∈D (D;R ) ϕC ≤1 0

(6.1)

x∈D

where M 1 (D) = D (D) is the topological dual of D0 (D), and N ∇f ∈ L D0 (D; RN ), R ≡ L D0 (D), R = M 1 (D)N . Such functions are known as functions of bounded variation. The space def

BV(D) = f ∈ L1 (D) : ∇f ∈ M 1 (D)N

(6.2)

endowed with the norm f BV(D) = f L1 (D) + ∇f M 1 (D)N

(6.3)

is a Banach space. Deﬁnition 6.1 (L. C. Evans and R. F. Gariepy [1]). A function f ∈ L1loc (U ) in an open subset U of RN has locally bounded variation if for each bounded open subset V of U such that V ⊂ U , f ∈ BV(V ). The set of all such functions will be denoted by BVloc (U ). Theorem 6.1. Given an open subset U of RN , f belongs to BVloc (U ) if and only if for each x ∈ U there exists ρ > 0 such that B(x, ρ) ⊂ U and f ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x. Proof. If f ∈ BVloc (U ), then the result is true by specializing to balls in each point of U . Conversely, by assumption, the open set U is covered by a family F(U ) of balls B(x, ρx ) such that B(x, ρx ) ⊂ U and f ∈ BV(B(x, ρx )). Given a bounded open subset V of U such that V ⊂ U , the compact subset V can be covered by a ﬁnite number of balls {B(xi , ρi )}, 1 ≤ i ≤ n, in F(U ) for points xi ∈ V , 1 ≤ i ≤ n, V ⊂

n *

Bi ,

def

Bi = B(xi , ρxi ).

i=1

Denote by {ψi ∈ D(Bi )}ni=1 a partition of unity for the family {Bi } such that ∀i, 0 ≤ ψi ≤ 1 in Bi , and

n i=1

ψi = 1 on V .

246

Chapter 5. Metrics via Characteristic Functions

Then for each ϕ

∈ V 1 (V )N ϕ

=

n

ϕ

i on V,

def

ϕ

i = ψi ϕ

,

i=1

∈ V 1 (V )N such that and by construction ϕ

i ∈ V (Bi ∩ V )N . Therefore, for all ϕ ϕC(V ) ≤ 1 n n − f div ϕ

dx = − f div ϕ

i dx = − f div ϕ

i dx 1

V

i=1

V

i=1

n f div ϕ

dx ≤ ⇒ V

i=1

Bi ∩V

V ∩Bi

f div ϕ

i dx

n n ⇒ f div ϕ

dx ≤ ∇f M 1 (Bi ∩V ) ≤ ∇f M 1 (Bi ) < ∞, V

i=1

since V (Bi ∩ V ) ⊂ V (Bi ) and f ∈ BVloc (RN ). 1

N

1

N

i=1

and ϕi C(Bi ) ≤ 1. Therefore, ∇f M 1 (V ) is ﬁnite

For more details and properties see also F. Morgan [1, p. 117], H. Federer [5, sect. 4.5.9], L. C. Evans and R. F. Gariepy [1], W. P. Ziemer [1], and R. Temam [1]. As in the previous section, consider measurable subsets Ω of a ﬁxed bounded open subset D of RN . Their characteristic functions χΩ ∈ X(D) are L1 (D)-functions χΩ dx = m(Ω) ≤ m(D) < ∞ (6.4) χΩ L1 (D) = D

with distributional gradient

∀ ϕ ∈ D(D)

N

,

def

∇χΩ , ϕ

D = −

χΩ div ϕ

dx.

(6.5)

D

When Ω is an open domain with boundary Γ of class C 1 , then by the Stokes divergence theorem div ϕ

dx = − ϕ

· n dΓ = − ϕ

· n dΓ, − Ω∩D

∂(Ω∩D)

Γ∩D

where n is the outward normal ﬁeld along ∂(Ω ∩ D). Since Γ is of class C 1 the normal ﬁeld n along Γ belongs to C 0 (Γ), and from the last identity the maximum is def 2 |n| dΓ = dΓ = HN −1 (Γ ∩ D), PD (Ω) = ∇χΩ M 1 (D) = Γ∩D

Γ∩D

the (N − 1)-dimensional Hausdorﬀ measure of Γ ∩ D. As a result HN −1 (Γ) = PD (Ω) + HN −1 (Γ ∩ ∂D).

(6.6)

Thus the norm of the gradient provides a natural relaxation of the notion of perimeter to the following larger class of domains.

6. Caccioppoli or Finite Perimeter Sets

247

Deﬁnition 6.2. Let Ω be a Lebesgue measurable subset of RN . (i) The perimeter of Ω with respect to an open subset D of RN is deﬁned as def

PD (Ω) = ∇χΩ M 1 (D)N .

(6.7)

Ω is said to have ﬁnite perimeter with respect to D if PD (Ω) is ﬁnite. The family of measurable characteristic functions with ﬁnite measure and ﬁnite relative perimeter in D will be denoted as def

BX(D) = {χΩ ∈ X(D) : χΩ ∈ BV(D)} . (ii) Ω is said to have locally ﬁnite perimeter if for all bounded open subsets D of RN , χΩ ∈ BV(D), that is, χΩ ∈ BVloc (RN ). (iii) Ω is said to have ﬁnite perimeter if χΩ ∈ BV(RN ). Theorem 6.2. Let Ω be a Lebesgue measurable subset of RN . Ω has locally ﬁnite perimeter, that is, χΩ ∈ BVloc (RN ), if and only if for each x ∈ ∂Ω there exists ρ > 0 such that χΩ ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x. Proof. We use Theorem 6.1. In one direction the result is obvious. Conversely, if x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ

∈ D 1 (B(x, ρ)) − χΩ div ϕ

dx = − div ϕ

dx = − ϕ

· n dHN −1 = 0 B(x,ρ)

B(x,ρ)

∂B(x,ρ)

since ϕ

= 0 on the boundary of B(x, ρ). Thus χΩ ∈ BV(B(x, ρ)). If x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ

∈ D1 (B(x, ρ)) − χΩ div ϕ

dx = 0. B(x,ρ)

So again χΩ ∈ BV(B(x, ρ)). Finally, if x ∈ ∂Ω, then by assumption there exists ρ > 0 such that χΩ ∈ BV(B(x, ρ)). Therefore, by Theorem 6.1, χΩ ∈ BVloc (RN ). The interest behind this construction is twofold. First, the notion of perimeter of a set is extended to measurable sets; second, this framework provides a ﬁrst compactness theorem which will be useful in obtaining existence of optimal domains. Theorem 6.3. Assume that D is a bounded open domain in RN with a Lipschitzian boundary ∂D. Let {Ωn } be a sequence of measurable domains in D for which there exists a constant c > 0 such that ∀n,

PD (Ωn ) ≤ c.

(6.8)

Then there exist a measurable set Ω in D and a subsequence {Ωnk } such that χΩnk → χΩ in L1 (D) as k → ∞

and

PD (Ω) ≤ lim inf PD (Ωnk ) ≤ c. k→∞

(6.9)

248

Chapter 5. Metrics via Characteristic Functions

in Moreover ∇χΩnk “converges in measure” to ∇χΩ in M 1 (D)N ; that is, for all ϕ D0 (D, RN ), lim ∇χΩnk , ϕ

M 1 (D)N → ∇χΩ , ϕ

M 1 (D)N .

k→∞

(6.10)

Proof. This follows from the fact that the injection of the space BV(D) endowed with the norm (6.3) into L1 (D) is continuous and compact (cf. E. Giusti [1, Thm. 1.19, p. 17], V. G. Maz’ja [1, Thm. 6.1.4, p. 300, Lem. 1.4.6, p. 62], C. B. Morrey, Jr. [1, Thm. 4.4.4, p. 75], and L. C. Evans and R. F. Gariepy [1]). Example 6.1 (The staircase). In (6.9) the inequality can be strict, as can be seen from the following example (cf. Figure 5.7). For each n ≥ 1, deﬁne the set n * j j+1 j Ωn = , × 0, 1 − . n n n j=1 Its limit is the set Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x}. It is easy to check that the sets Ωn are contained in the holdall D = ]−1, 2[ × ]−1, 2[ and that √ ∀n ≥ 1, PD (Ωn ) = 4, PD (Ω) = 2 + 2, χΩn → χΩ in Lp (D), 1 ≤ p < ∞. Each Ωn is uniformly Lipschitzian, but the Lipschitz constant and the two neighborhoods of Deﬁnition 3.2 in Chapter 2 cannot be chosen independently of n.

1/n

Ωn

Ω Figure 5.7. The staircase.

Corollary 1. Let D be open bounded and Lipschitzian. There exists a constant c > 0 such that for all convex domains Ω in D PD (Ω) ≤ c,

m(Ω) ≤ c

and the set C(D) of convex subsets of D is compact in Lp (D) for all p, 1 ≤ p < ∞.

6. Caccioppoli or Finite Perimeter Sets

249

Proof. If Ω is a convex set with zero volume, m(Ω) = 0, then its perimeter PD (Ω) = 0 (cf. E. Giusti [1, Rem. 1.7 (iii), p. 6]). If m(Ω) > 0, then int Ω = ∅. For convex sets with a nonempty interior the perimeter and the volume enjoy a very nice monotonicity property. If C • denotes the set of nonempty closed convex subsets of RN , then the map Ω → m(Ω) : C • → R is strictly increasing: ∀A, B ∈ C • ,

A B =⇒ m(A) < m(B)

(cf. M. Berger [1, Vol. 3, Prop. 12.9.4.3, p. 141]). Similarly if P (Ω) denotes the perimeter of Ω in RN , the map Ω → P (Ω) : C • → R is strictly increasing (cf. M. Berger [1, Vol. 3, Prop. 12.10.2, p. 144]). As a result ∀n,

m(Ωn ) ≤ m(D) < ∞ and PD (Ωn ) ≤ P (Ωn ) ≤ P (co D) < ∞,

since the perimeter of a bounded convex set is ﬁnite. Then the conditions of the theorem are satisﬁed and the conclusions of the theorem follow. But we have seen in Theorem 3.5 (iii) that the set C(D) is closed in L1 (D). Therefore, Ω can be chosen convex in the equivalence class. This completes the proof. This theorem and the lower (resp., upper) semicontinuity of the shape function Ω → J(Ω) will provide existence results for domains in the class of ﬁnite perimeter sets in D. One example is the transmission problem (4.1) to (4.3) of section 4.1 with the objective function 1 J(Ω) = |y(Ω) − g|2 dx + αPD (Ω), α > 0, (6.11) 2 Ω for some g ∈ L2 (D). We shall come back later to the homogeneous Dirichlet boundary value problem (3.25)–(3.26). It is important to recall that even if a set Ω in D has a ﬁnite perimeter PD (Ω), its relative boundary Γ ∩ D can have a nonzero N -dimensional Lebesgue measure. To illustrate this point consider the following adaptation of Example 1.10 in E. Giusti [1, p. 7]. Example 6.2. Let D = B(0, 1) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let {xi } be an ordered sequence of all points in D with rational coordinates. Associate with each i the open ball Bi = {x ∈ D : |x − xi | < ρi },

0 < ρi ≤ min{2−i , 1 − |xi |}.

Deﬁne the new sequence of open subsets of D, Ωn = and notice that for all n ≥ 1, m(∂Ωn ) = 0,

n *

Bi ,

i=1

PD (Ωn ) ≤ 2π,

250

Chapter 5. Metrics via Characteristic Functions

where m = m2 is the Lebesgue measure in R2 and ∂Ωn is the boundary of Ωn . Moreover, since the sequence of sets {Ωn } is increasing, χΩn → χΩ in L1 (D),

Ω=

∞ *

Bi ,

PD (Ω) ≤ lim inf PD (Ωn ) ≤ 2π. n→∞

i=1

Observe that Ω = D and that ∂Ω = Ω ∩ Ω ⊃ D ∩ Ω. Thus 2π m(∂Ω) = m(D ∩ Ω) ≥ m(D) − m(Ω) ≥ 3 since ∞ π π2−2i = . m(D) = π and m(Ω) ≤ 3 1=1 Recall that m(Ωn ) ≤

n i=1

m(Bi ) ≤

n

π2−2i

⇒ m(D) ≤

i=1

∞ i=1

m(Bi ) ≤

∞

π2−2i .

i=1

For p, 1 ≤ p < ∞, the sequence of characteristic functions {χΩn } converges to χΩ in Lp (D)-strong. However, for all n, m(∂Ωn ) = 0, but m(∂Ω) > 0. In fact we can associate with the perimeter PD (Ω) the reduced boundary, ∂ ∗ Ω, which is the set of all x ∈ ∂Ω for which the normal n(x) exists. We quote the following interesting theorems from W. H. Fleming [1, p. 455] (cf. also E. De Giorgi [2], L. C. Evans and R. F. Gariepy [1, Thm. 1, sect. 5.1, p. 167, Notation, pp. 168–169, Lem. 1, p. 208], and H. Federer [2]). Theorem 6.4. Let χΩ ∈ BVloc (RN ). There exist a Radon measure ∂Ω on RN and a ∂Ω-measurable function νΩ : RN → RN such that (i) |νΩ (x)| = 1, ∂Ω almost everywhere and (ii) RN χΩ div ϕ dx = − RN ϕ · νΩ d∂Ω, for all ϕ ∈ Cc1 (RN ; RN ). Deﬁnition 6.3. Let χΩ ∈ BVloc (RN ). The point x ∈ RN belongs to the reduced boundary ∂ ∗ Ω if (i) ∂Ω(Br (x)) > 0, for all r > 0, (ii) limr0 m(B1r (x)) Br (x) νΩ d∂Ω = νΩ (x), and (iii) |νΩ (x)| = 1. Theorem 6.5. Let Ω have ﬁnite perimeter P (Ω). Let ∂ ∗ Ω denote the reduced boundary of Ω. Then (i) ∂ ∗ Ω ⊂ ∂∗ Ω ⊂ Ω• ⊂ ∂Ω and ∂ ∗ Ω = ∂Ω, (ii) P (Ω) = HN −1 (∂ ∗ Ω) (cf. E. Giusti [1, Chap. 4]),

6. Caccioppoli or Finite Perimeter Sets

251

(iii) the Gauss–Green theorem holds with ∂ ∗ Ω, and (iv) m(∂ ∗ Ω) = m(∂∗ Ω) = 0,

HN −1 (∂∗ Ω\∂ ∗ Ω) = 0.

We quote the following density theorem (cf. E. Giusti [1, Thm. 1.24 and Lem. 1.25, p. 23]), which complements Theorem 3.1. Theorem 6.6. Let Ω be a bounded measurable domain in RN with ﬁnite perimeter. Then there exists a sequence {Ωj } of C ∞ -domains such that as j goes to ∞ |χΩj − χΩ | dx → 0 and P (Ωj ) → P (Ω). RN

6.2

Decomposition of the Integral along Level Sets

We complete this section by giving some useful theorems on the decomposition of the integral along the level sets of a function. The ﬁrst one was used by J.´sio [6] (Grad’s model in plasma physics) in 1979 and [9, sect. 4.4, p. 95] in P. Zole 1981, by R. Temam [1] (monotone rearrangements) in 1979, by J. M. Rakotoson and R. Temam [1] in 1987, and more recently in 1995 by M. C. Delfour and ´sio [19, 20, 21, 25] (intrinsic formulation of models of shells). We quote J.-P. Zole the version given in L. C. Evans and R. F. Gariepy [1, Prop. 3, p. 118]. The original theorem can be found in H. Federer [3] and L. C. Young [1]. Theorem 6.7. Let f : RN → R be Lipschitz continuous with |∇f | > 0

a.e.

For any Lebesgue summable function g : RN → R, we have the following decomposition of the integral along the level sets of f :

∞

0

g dx = {f >t}

t

{f =s}

g dHN −1 |∇f |

1 ds.

The second theorem (W. H. Fleming and R. Rishel [1]) uses a BV function instead of a Lipschitz function. Theorem 6.8. Let D be an open subset of RN . For any f ∈ BV(D) and real t, let def

Et = {x : f (x) < t}. Then ∇f M 1 (D)N =

∞ −∞

P (Et ) dt

(cf., for instance, H. Whitney [1, Chap. 11]). This is known as the co-area formula (see also E. Giusti [1, Thm. 1.23, p. 20] and E. De Giorgi [4]).

252

6.3

Chapter 5. Metrics via Characteristic Functions

Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces

There is a general property enjoyed by functions in BV(D). Theorem 6.9. Let D be a bounded open Lipschitzian domain in RN . (i) BV(D) ⊂ W ε,1 (D),

0 ≤ ε < 1.

(ii) BV(D) ∩ L∞ (D) ⊂ W ε,p (D),

0 ≤ ε < 1p , 1 ≤ p < ∞.

This theorem says that for a Caccioppoli set Ω in D, ∀p ≥ 1, 0 ≤ ε

0 small we break the above integral into two parts: |f (y) − f (x)| |f (y) − f (x)| dx dy , I = dx dy . I1 = 2 N +ε |y − x| |y − x|N +ε D D∩B(x,α) D D B(x,α) The second integral I2 is bounded since |y − x| ≥ α. For the ﬁrst one we change the variable y to t = y − x, |(χD f )(x + t) − (χD f )(x)| dx dt , I1 = |t|N +ε D B(0,α) and after a change in the order of integration, 1 I1 = dt N +ε dx |(χD f )(x + t) − (χD f )(x)|. |t| B(0,α) D But the integral over D is bounded by |t| ∇f M 1 (D) . The result is true for smooth functions (cf. V. G. Maz’ja [1, Lem. 1.4.6, p. 62]). Pick a sequence of smooth functions such that fn → f in L1 (D),

∇fn → ∇f in M 1 (D)-weak;

then ∇f M 1 (D) ≤ lim sup ∇fn M 1 (D) = lim sup ∇fn L1 (D) n→∞

n→∞

6. Caccioppoli or Finite Perimeter Sets

253

is bounded, and by going to the limit on both sides of the inequality, dx |(χD fn )(x + t) − (χD fn )(x)| ≤ |t|∇fn L1 (D) , D

we obtain the desired result. Now coming back to the estimate of I1 , α1−ε ∇f M 1 (D) , 0 < ε < 1. I1 ≤ |t|1−N −ε dt ∇f M 1 (D) ≤ c(α) 1−ε B(x,α) (ii) The function f belongs to W ε,p (D) if the integral |f (y) − f (x)|p dx dy < +∞. I= |y − x|N +εp D D For each f ∈ BV(D) ∩ L∞ (D), there exists M > 0 such that |f (x)| ≤ M

a.e. in D.

Then

I = (2 M )p

dx D

and since

dy D

f (y)−f (x) p 2M |y − x|N +εp

,

f (y) − f (x) ≤ 1, 2M

then for each p, 1 ≤ p < ∞, f (y) − f (x) p f (y) − f (x) ≤ . 2M 2M As a result

I ≤ (2 M )

p−1

dx D

dy D

|f (y) − f (x)| . |y − x|N +εp

But we have seen in part (i) that for f ∈ BV(D), the double integral is ﬁnite if 0 ≤ εp < 1 ⇒ 0 ≤ ε

0, we have 0 ≤ ε < 1/p for all p ≥ 1 and |χΩ (x) − χΩ (y)| dx dy + χΩ pLp (D) . (6.12) χΩ pW ε,p (D) = |x − y|N +pε D D

254

Chapter 5. Metrics via Characteristic Functions

The W ε,p (D)-norm is equivalent to the W ε ,p (D) for all pairs (p , ε ), p ≥ 1, 0 ≥ ε < 1/p , such that p ε = p ε. At this stage, it is not clear whether that norm is related to some Hausdorﬀ measure of the relative boundary Γ ∩ D of the set. For p = 2 and a characteristic function, χΩ , of a measurable set Ω, we get 2 χΩ H ε (D) = 2 |x − y|−(N +2ε) dx dy + m(Ω ∩ D). (6.13) Ω

D Ω

ε,2

The space X(D)∩W (D) is a closed subspace of the Hilbert space W ε,2 (D) and the square of its norm is diﬀerentiable. A direct consequence for optimization problems is that a penalization term of the form χΩ 2W ε,2 (D) is now diﬀerentiable and can be used in various minimization problems or to regularize the objective function to obtain existence of approximate solutions. For instance, to obtain existence results when minimizing a function J(Ω) (deﬁned for all measurable sets Ω in D), we can consider a regularized problem in the following form: Jα (Ω) = J(Ω) + αχΩ 2H ε (D) ,

α > 0.

(6.14)

Section 6.1 provided the important compactness Theorem 6.3 for Caccioppoli or ﬁnite perimeter sets. For a bounded open holdall D, this also provides a new complete metric topology on X(D) ∩ BV(D) when endowed with the metric def

ρBV (χ1 , χ2 ) = χ2 − χ1 BV(D) = χ2 − χ1 L1 (D) + ∇χ2 − ∇χ1 M 1 (D) . Similarly, the intermediate spaces of Theorem 6.9 between BV(D) and Lp (D) provide little rougher metrics for p ≥ 1 and ε < 1/p on X(D) ∩ W ε,p (D) when endowed with the metric def

ρW ε,p (χ1 , χ2 ) = χ2 − χ1 W ε,p (D) . Theorem 6.10. Given a bounded open holdall D in RN , X(D)∩BV(D) and X(D)∩ W ε,p (D), p ≥ 1, ε < 1/p, are complete metric spaces and X(D) ∩ BV(D) ⊂ X(D) ∩ W ε,p (D) ⊂ X(D).

6.4

Compactness and Uniform Cone Property

In Theorem 6.3 of section 6.1 we have seen a ﬁrst compactness theorem for a family of sets with a uniformly bounded perimeter. In this section we present a second compactness theorem for measurable domains satisfying a uniform cone property due to D. Chenais [1, 4, 6]. We provide a proof of this result that emphasizes the fact that the perimeter of sets in that family is uniformly bounded. It then becomes a special case of Theorem 6.3. This result will also be obtained as Corollary 2 to Theorem 13.1 in section 13 of Chapter 7. It will use completely diﬀerent arguments and apply to families of sets that do not even have a ﬁnite boundary measure. Theorem 6.11. Let D be a bounded open holdall in RN with uniformly Lipschitzian boundary ∂D. For r > 0, ω > 0, and λ > 0 consider the family def Ω is Lebesgue measurable and satisﬁes L(D, r, ω, λ) = Ω ⊂ D : the uniform cone property for (r, ω, λ) . (6.15)

6. Caccioppoli or Finite Perimeter Sets

255

For p, 1 ≤ p < ∞, the set def

X(D, r, ω, λ) = {χΩ : ∀Ω ∈ L(D, r, ω, λ)}

(6.16)

is compact in Lp (D) and there exists a constant p(D, r, ω, λ) > 0 such that ∀Ω ∈ L(D, r, ω, λ),

HN −1 (∂Ω) ≤ p(D, r, ω, λ).

(6.17)

We ﬁrst show that the perimeter of the sets of the family L(D, r, ω, λ) is uniformly bounded and use Theorem 6.8 to show that any sequence of X(D, r, ω, λ) has a subsequence converging to the characteristic function of some ﬁnite perimeter set Ω. We use the full strength of the compactness of the injection of BV(D) into L1 (D) rather than checking directly the conditions under which a subset of Lp (D) is relatively compact. The proof will be completed by showing that the nice representative (Deﬁnition 3.3) I of Ω satisﬁes the same uniform cone property. The proof uses some elements of D. Chenais [4, 6]’s original proof. Proof of Theorem 6.11. Each Ω ∈ L(D, r, ω, λ) satisﬁes the same uniform cone property (cf. Deﬁnition 6.3 (ii)) of Chapter 2 with parameters r, λ, and rλ . By Theorem 6.8 of Chapter 2 it is uniformly Lipschitzian in the sense of Deﬁnition 5.2 (iii) of Chapter 2. As a result ρ, the largest radius such that BH (0, ρ) ⊂ {PH (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω}, and the neighborhoods V = BH (0, ρ) and U = B(0, rλ ) ∩ {y : PH y ∈ BH (0, ρ)} can be chosen as speciﬁed in Theorem 6.6 (i) of Chapter 2. They are the same for all Ω ∈ L(D, r, ω, λ): for all Ω ∈ L(D, r, ω, λ) and all x ∈ ∂Ω, Vx = V , and there exists Ax ∈ O(N) such that U(x) = x + Ax U . By construction B(x, ρ) ⊂ B(x, rλ ) ∩ {y : PH A−1 x (y − x) ∈ BH (0, ρ)} = x + Ax U = U(x).

The family B (z, ρ/2) : z ∈ D is an open cover of D. Since D is compact, there exists a ﬁnite subcover {Bi }m i=1 , Bi = B (zi , ρ/2), of D. Now for any Ω ∈ L(D, r, ω, λ), m ∂Ω ⊂ D and there is a subcover {Bik }K k=1 of {Bi }i=1 such that ∂Ω ⊂

K *

Bik and ∀k,

1 ≤ k ≤ K,

∂Ω ∩ Bik = ∅.

k=1

Pick a sequence {xk }K k=1 such that xk ∈ ∂Ω ∩ Bik , 1 ≤ k ≤ K, and notice that ρ Bik = B zik , ⊂ B(xk , ρ) 2 or ∃K ≤ m,

∃{xk }K k=1 ⊂ ∂Ω,

∂Ω ⊂

K * k=1

B(xk , ρ).

256

Chapter 5. Metrics via Characteristic Functions

Thus from the estimate (5.37) in Theorem 5.7 of Chapter 2 with cx = tan ω, HN −1 (∂Ω) ≤

K

K HN −1 ∂Ω ∩ B(xk , ρ) ≤ HN −1 ∂Ω ∩ U(xk )

k=1

≤

K

k=1

ρN −1 αN −1

'

1 + (tan ω)2 ≤ m ρN −1 αN −1

' 1 + (tan ω)2 ,

k=1

where αN −1 is the volume of the unit (N − 1)-dimensional ball. So the right-hand side of the above inequality is a constant that is equal to p = p(D, r, ω, λ) > 0 and is independent of Ω in L(D, r, ω, λ): ∀Ω ∈ L(D, r, ω, λ),

PD (Ω) = HN −1 (∂Ω) ≤ p.

Now from Theorem 6.3 for any sequence {Ωn } ⊂ L(D, r, ω, λ), there exists Ω such that χΩ ∈ BV(D) and a subsequence, still denoted by {Ωn }, such that χΩn → χΩ in L1 (D) and PD (Ω) ≤ p. (ii) To complete the proof we consider the representative I of Ω (cf. Deﬁnition 3.3) and show that I ∈ L(D, r, ω, λ). We need the following lemma. Lemma 6.1. Let χΩn → χΩ in Lp (D), 1 ≤ p < ∞, for Ω ⊂ D, Ωn ⊂ D, and let I be the measure theoretic representative of Ω. Then ∀x ∈ I, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), m B(x, R) ∩ Ωn > 0. Moreover, ∀x ∈ ∂I, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), m B(x, R) ∩ Ωn > 0 and m B(x, R) ∩ Ωn > 0 and B(x, R) ∩ ∂Ωn = ∅. Proof. We proceed by contradiction. Assume that ∃x ∈ I,

∃R > 0,

∀N > 0,

∃n ≥ N,

m B(x, R) ∩ Ωn = 0.

So there exists a subsequence {Ωnk }, nk → ∞, such that m B(x, R) ∩ Ω = lim m B(x, R) ∩ Ωnk = 0 k→∞ ⇒ ∃R > 0, m B(x, R) ∩ Ω = 0 =⇒ x ∈ Ω0 = int I = I. But this is in contradiction with the fact that x ∈ I. For x ∈ ∂I = I ∩ I, simultaneously apply the ﬁrst statement to I and χΩn → χΩ and to I and χΩn → χΩ since χΩ = χI almost everywhere by Theorem 3.3 (i) and choose the larger of the two integers. As for the last property it follows from the fact that the open ball B(x, R) cannot be partitioned into two nonempty disjoint open subsets.

6. Caccioppoli or Finite Perimeter Sets

257

We wish to prove that ∀x ∈ ∂I,

∃Ax ∈ O(N),

∀y ∈ I ∩ B(x, r),

y + Ax C(λ, ω) ⊂ int I.

Since Ωn is Lipschitzian, Ωn is also Lipschitzian and χΩn = χΩn

and χΩ = χΩn n

almost everywhere. So we apply the second part of the lemma to x ∈ ∂I, χΩn → χΩ and χΩ → χΩ . So for each x ∈ ∂I, for all k ≥ 1, there exists nk ≥ k such that n

r B x, k ∩ ∂Ωnk = ∅. 2

Denote by xk an element of that intersection: r ∀k ≥ 1, xk ∈ B x, k ∩ ∂Ωnk . 2 By construction, xk → x. Next consider y ∈ B(x, r) ∩ I. By Lemma 6.1, there exists a subsequence of {Ωnk }, still denoted by {Ωnk }, such that r ∀k ≥ 1, B y, k ∩ Ωnk = ∅. 2 For each k ≥ 1 denote by yk a point of that intersection. By construction, yk ∈ Ωnk → y ∈ I ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this note that y ∈ B(x, r) and that ρ ∃ρ > 0, B(y, ρ) ⊂ B(x, r) and |y − x| + < r. 2 Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | . r r ρ ρ/ r 2 + . 2 log 2

So we have constructed a subsequence {Ωnk } such that for k ≥ K, xk ∈ ∂Ωnk → x ∈ ∂I

and yk ∈ Ωnk ∩ B(xk , r) → y ∈ I ∩ B(x, r).

For each k, there exists Ak ∈ O(N) such that yk + Ak C(λ, ω) ⊂ int Ωnk .

258

Chapter 5. Metrics via Characteristic Functions

Pick another subsequence of {Ωnk }, still denoted by {Ωnk }, such that ∃A ∈ O(N),

Ak → A,

since O(N) is compact. Now consider z ∈ y + AC(λ, ω). Since z is an interior point ∃ρ > 0,

B(z, ρ) ⊂ y + AC(λ, ω).

So there exists K ≥ K such that ρ ∀k ≥ K , B z, ⊂ yk + Ak C(λ, ω) ⊂ int Ωnk . 2 Therefore, for k ≥ K ρ ρ m B z, = m B z, ∩ Ωnk , 2 2 and, as k → ∞, ρ ρ m B z, = m B z, ∩Ω , 2 2 and by Deﬁnition 3.3, z ∈ Ω1 and y + AC(λ, ω) ⊂ Ω1 = int I. This proves that I ⊂ D satisﬁes the uniform cone property and I ∈ L(D, λ, ω, r).

7 7.1

Existence for the Bernoulli Free Boundary Problem An Example: Elementary Modeling of the Water Wave

Consider a ﬂuid in a domain Ω in R3 and assume that the velocity of the ﬂow u satisﬁes the Navier–Stokes equations ut + Du u − ν ∆u + ∇p = −ρ g in Ω,

div u = 0 in Ω,

(7.1)

where ν > 0 and ρ > 0 are the respective viscosity and density of the ﬂuid, p is the pressure, and g is the gravity constant. The second equation characterizes the incompressibility of the ﬂuid. A standard example considered by physicists is the water wave in a channel. The boundary conditions on ∂Ω are the sliding conditions at the bottom S and on the free boundary Γ, that is, u · n = 0 on S ∪ Γ.

(7.2)

Assume that a stationary regime has been reached so that the velocity of the ﬂuid is no longer a function of the time. Furthermore assume that the motion of the ﬂuid is irrotational. By the classical Hodge decomposition, the velocity can be written in the form u = ∇ϕ + curl ξ.

(7.3)

7. Existence for the Bernoulli Free Boundary Problem

259

As curl u = curl curl ξ = 0 we conclude that ξ = 0, since curl curl is a good isomorphism. Then u = ∇ϕ and the incompressibility condition becomes ∆ϕ = 0 in Ω.

(7.4)

Therefore the Navier–Stokes equation reduces to D(∇ϕ) ∇ϕ + ν ∇(∆ϕ) + ∇p = −∇(ρgz),

(7.5)

but D(∇ϕ) ∇ϕ = so that

∇

1 ∇(|∇ϕ|2 ), 2

1 |∇ϕ|2 + p + ρgz 2

(7.6)

= 0 in Ω.

(7.7)

Then if Ω is connected, there exists a constant c such that 1 |∇ϕ|2 + p + ρgz = c in Ω. 2

(7.8)

In the domain Ω, that Bernoulli condition yields explicitly the pressure p as a function of the velocity |∇ϕ| and the height z in the ﬂuid. The ﬂow is now assumed to be independent of the transverse variable y so that the initial three-dimensional problem in a perfect channel reduces to a two-dimensional one. Since the problem has been reduced to a two-dimensional one, introduce the harmonic conjugate ψ, the so-called stream function, so that the boundary condition ∂ϕ/∂n = 0 takes the form ψ = constant on each connected component of the boundary ∂Ω. On the free boundary at the top of the wave, the pressure is related to the existing atmospheric pressure pa through the surface tension σ > 0 and the mean curvature H of the free boundary. Actually p − pa = −σ H

(7.9)

on the free boundary of the stationary wave, where H is the mean curvature associated with the ﬂuid domain. Also, from the Cauchy conditions, we have |∇ψ| = |∇ϕ| so that the boundary condition (7.8) on the free boundary of the wave becomes 1 |∇ψ|2 + ρgz + σH = pa . 2

(7.10)

In order to simplify the presentation we replace the equation ∆ψ = 0 by ∆ψ = f in Ω to avoid a forcing term on a part of the boundary, and we show that the resulting free boundary problem has the following shape variational formulation. Let D be a ﬁxed, suﬃciently large, smooth, bounded, and open domain in R2 . Let a be a real number such that 0 < a < m(D). To ﬁnd Ω ⊂ D, m(Ω) = a and ψ ∈ H01 (D) such that −∆ψ = f in Ω

(7.11)

260

Chapter 5. Metrics via Characteristic Functions

and ψ = constant and satisﬁes the boundary condition (7.10) on the free part ∂Ω ∩ D of the boundary. For a ﬁxed Ω, the solution of this problem is a minimizing element of the following variational problem: 1 def 2 |∇ϕ| J(Ω) = inf − f ϕ + ρgz dx + σ PD (Ω), (7.12) 2 ϕ∈H1 (Ω;D) Ω where def

(7.13) H1 (Ω; D) = u ∈ H01 (D) : u(x) = 0 a.e. x in D \ Ω is the relaxation of the deﬁnition of the Sobolev space H01 (Ω)10 for any measurable subset Ω of D. Its properties were studied in Theorem 3.6. With that formulation there exists a measurable Ω∗ ⊂ D, |Ω∗ | = a, such that ∀Ω ⊂ D, |Ω| = a,

J(Ω∗ ) ≤ J(Ω).

(7.14)

By using the methods of Chapter 10 it follows that if ∂Ω∗ is suﬃciently smooth, the shape Euler condition dJ(Ω∗ ; V ) = 0 yields the original free boundary problem and the free boundary condition (7.10). The existence of a solution will now follow from Theorem 6.3 in section 6.1. The case without surface tension is physically important. It occurs in phenomena with “evaporation ” (cf. M. Souli and J.´sio [2]). In that case the previous Bernoulli condition becomes P. Zole 2 ∂ξ = g 2 (g ≥ 0) ∂n on the free boundary, so that if we assume (in the channel setting) that there is no cavitation or recirculation in the ﬂuid, then ∂ξ/∂n > 0 on the free boundary and we get the Neumann-like condition ∂ξ/∂n = g together with the Dirichlet condition. In section 7.4 we shall consider the case with surface tension.

7.2

Existence for a Class of Free Boundary Problems

´sio [25, 26, Consider the following free boundary problem, studied in J.-P. Zole 29]: ﬁnd Ω in a ﬁxed holdall D and a function y on Ω such that −∆y = f in Ω,

y = 0 and

∂y = Q2 on ∂Ω, ∂n

(7.15)

where f and Q are appropriate functions deﬁned in D. To study this type of problem H. W. Alt and L. A. Caffarelli [1] have introduced the following function: 1 def Q2 χϕ>0 dx, (7.16) J(ϕ) = |∇ϕ|2 − f ϕ dx + 2 D D 10 In Chapter 8 the space H 1 (Ω; D) of extensions by zero to D of elements of H 1 (Ω) is deﬁned 0 0 for Ω open. It is then characterized in Lemma 6.1 of Chapter 8 by a capacity condition on the complement D\Ω. This characterization extends to quasi-open sets Ω as deﬁned in section 6 of Chapter 8.

7. Existence for the Bernoulli Free Boundary Problem

261

which is minimized over def

K = {u ∈ H01 (D) : u(x) ≥ 0 a.e. in D},

(7.17)

where χϕ>0 (resp., χϕ=0 ) is the characteristic function of the set {x ∈ D : ϕ(x) > 0}11 (resp., {x ∈ D : ϕ(x) = 0}). The existence of a solution is based on the following lemma. Lemma 7.1. Let {un } and {χn } be two converging sequences such that un → u in L2 (D)-strong, and let the χn ’s be characteristic functions, χn (1 − χn ) = 0, which converge to some function λ in L2 (D)-weak. Then ∀n,

(1 − χn )un = 0

⇒

λ ≥ χu=0 .

(7.18)

Proof. We have (1 − χn )un = 0, and in the limit (1 − λ)u = 0. Thus on the set {x : u(x) = 0} we have λ = 1; elsewhere λ lies between 0 and 1 as the weak limit of a sequence of characteristic functions. Proposition 7.1. Let f and Q be two elements of L2 (D) such that f ≥ 0 almost everywhere. There exists u in K which minimizes the function J over the positive cone K of H01 (D). Proof. Let {un } ⊂ K be a minimizing sequence for the function J over the convex set K. Denote by χn the characteristic function of the set {x ∈ D : un (x) > 0}, which is in fact equal to the subset {x ∈ D : un (x) = 0}. It is easy to verify that the sequence {un } remains bounded in H01 (D). Still denote by {un } a subsequence that weakly converges in H01 (D) to an element u of K. That convergence holds in L2 (D)-strong so that Lemma 7.1 applies and we get λ ≥ χu=0 for any weak limiting element λ of the sequence {χn } (which is bounded in L2 (D)). Denote by j the minimum of J over K. Then J(un ) weakly converges to j. We get 1 1 2 (7.19) |∇u| − f u dx ≤ lim inf |∇un |2 − f un dx, n→∞ D 2 D 2 χu=0 Q2 dx ≤ λQ2 dx = lim χn Q2 dx. (7.20) D

D

n→∞

D

Finally, by adding these two estimates we get J(u) ≤ j. Obviously in the upper bound in (7.20), Q2 is a positive function. Moreover (i) the set Ω is given by {x ∈ D : u(x) > 0}, and (ii) the restriction u|Ω of u to Ω is a weak solution of the free boundary problem −∆u = f in Ω,

u = 0 and

∂u = Q2 on ∂Ω. ∂n

(7.21)

11 This set deﬁned up to a set of zero measure is a quasi-open set in the sense of section 6 in Chapter 8.

262

Chapter 5. Metrics via Characteristic Functions

Eﬀectively, the minimization problem (7.16)–(7.17) can be written as a shape optimization problem. First introduce for any measurable Ω ⊂ D the positive cone def

H01 (Ω)+ = u ∈ H1 (Ω; D) : u(x) ≥ 0 a.e. x ∈ D in the Hilbert space H1 (Ω; D). Then consider the following shape optimization problem: inf {E(Ω) : Ω is measurable subset in D} , where the energy function E is given by 1 def 2 E(Ω) = |∇ϕ| min − f ϕ dx + Q2 dx. ϕ∈H01 (Ω)+ Ω 2 Ω

(7.22)

(7.23)

The necessary condition associated with the minimum could be obtained by the techniques introduced in section 2.1 of Chapter 10. The important diﬀerence with the previous formulation (7.16)–(7.17) is that the independent variable in the objective function is no longer a function but a domain. The shape formulation (7.22)– (7.23) now makes it possible to handle constraints on the volume or the perimeter of the domain, which were diﬃcult to incorporate in the ﬁrst formulation. As we have seen in Theorem 3.6, H1 (Ω; D) endowed with the norm of H01 (D) is a Hilbert space, and H01 (Ω)+ a closed convex cone in H1 (Ω; D), so that for any measurable subset Ω in D, problem (7.23) has a unique solution y in H01 (Ω)+ . Thus we have the following equivalence between problems (7.22)–(7.23) and the minimization (7.16)–(7.17) of J over K. Proposition 7.2. Let u be a minimizing element of J over K. Then def

Ω = {x ∈ D : u(x) > 0} is a solution of problem (7.22) and y = u|Ω is a solution of (7.23). Conversely, if Ω is a measurable subset of D and y is a solution of (7.22)–(7.23) in H01 (Ω)+ , then the element u deﬁned by def

u(x) =

y(x), 0,

if x ∈ Ω, if x ∈ D \ Ω,

belongs to K and minimizes J over K.

7.3 7.3.1

Weak Solutions of Some Generic Free Boundary Problems Problem without Constraint

Problem (7.22)–(7.23) can be relaxed as follows: given any f in L2 (D), G in L1 (D), (P0 )

inf {E(Ω) : Ω ⊂ D measurable},

(7.24)

7. Existence for the Bernoulli Free Boundary Problem

263

where for any measurable subset Ω of D the energy function is now deﬁned by 1 def def min E (ϕ) + G dx, E (ϕ) = (7.25) E(Ω) = |∇ϕ|2 − f ϕ dx, D D 1 ϕ∈H (Ω;D) Ω D 2 where H1 (Ω; D) is deﬁned in (7.13). By Theorem 3.6 H1 (Ω; D) is contained in H•1 (Ω; D), and for any element u in H1 (Ω; D) we have ∇u(x) = 0 for almost all x in D \ Ω so that 1 1 (7.26) ∀ϕ ∈ H (Ω; D), ED (ϕ) = |∇ϕ|2 − f ϕ dx Ω 2 1 G dx . (7.27) ⇒ E(Ω) = min |∇ϕ|2 − f ϕ dx + 1 ϕ∈H (Ω;D) Ω 2 Ω We have the following existence result for problem (P0 ). Theorem 7.1. For any f in L2 (D), G = Q2 in L1 (D), there exists at least one solution to problem (P0 ). Proof. Let {Ωn } be a minimizing sequence for problem (P0 ), and for each n let un be the solution to problem (7.25) with Ω = Ωn . If χn is the characteristic function of the measurable set Ωn , we have un in H1 (Ωn ; D), which implies that (1 − χn )un = 0. On the other hand, the sequence {un } remains uniformly bounded in H01 (D). Taking ϕ = 0 in (7.25) we get 1 |∇un |2 − f un dx ≤ 0, D 2 and the conclusion follows from the equivalence of norms in H01 (D). We can assume that the sequence {χn } weakly converges in L2 (D) to an element λ and that the sequence {un } weakly converges in H01 (D) to an element u. From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Deﬁne def

Ω(u) = {x ∈ D : u(x) = 0}. Then u belongs to H01 Ω(u) and we have m(Ω(u)) = m({x ∈ D : λ(x) = 1}) ≤ α, since we have α = m({x : λ(x) = 1}) + m({x : 0 ≤ λ(x) < 1}). In the limit with Ω = Ω(u) we get 1 1 1 2 2 |∇u| − f u dx = |∇u| − f u dx ≤ lim inf |∇un |2 − f un dx, (7.28) 2 2 Ω D D 2 G dx ≤ λG dx = lim G dx. (7.29) Ω

D

n→∞

Ωn

By adding (7.28) and (7.29) we get that Ω minimizes E and u minimizes E(Ω).

264 7.3.2

Chapter 5. Metrics via Characteristic Functions Constraint on the Measure of the Domain Ω

Consider an important variation of problem (7.22)–(7.23). Given any f in L2 (D), G in L1 (D), and a real number α, 0 < α < m(D), (P0α )

inf {E(Ω) : Ω ⊂ D measurable and m(Ω) = α} .

(7.30)

We have the following existence result for problem (P0α ). Theorem 7.2. For any f in L2 (D), G = 0, and any real number α, 0 < α < m(D), there exists at least one solution to problem (P0α ). Proof. Let {Ωn } be a minimizing sequence for problem (P0 ), and for each n let un be the unique solution to problem (7.25). If χn is the characteristic function of the measurable set Ωn , we have un in H1 (Ωn ; D), which implies that (1 − χn )un = 0. On the other hand, by picking ϕ = 0 in (7.25), {un } remains bounded in H01 (D), 1 |∇un |2 − f un dx ≤ 0, D 2 and the conclusion follows from the equivalence of norms in H01 (D). We can assume that {χn } weakly converges in L2 (D) to an element λ and that {un } weakly converges in H01 (D) to an element u ∈ H01 (D). In the limit we get λ(x) dx = lim χn (x) dx = α. n→∞

D

D

From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Deﬁne def

Ω(u) = {x ∈ D : u(x) = 0}. Then u belongs to H1 (Ω(u); D) and we have m(Ω(u)) = m({x ∈ D : λ(x) = 1}) ≤ α (as we have α = m({x : λ(x) = 1}) + m({x : 0 ≤ λ(x) < 1})). In the limit we get for Ω = Ω(u) 1 1 1 |∇u|2 − f u dx = |∇u|2 − f u dx ≤ lim inf |∇un |2 − f un dx, (7.31) n→∞ 2 2 2 Ω D D λG dx = lim G dx, (7.32) n→∞

D

so that

Ω

1 |∇u|2 − f u dx + 2

Ωn

λG dx ≤ D

inf

Ω ⊂D m(Ω )=α

If G ≥ 0 almost everywhere in D, we get E(Ω) ≤

inf

Ω ⊂D m(Ω )=α

{E(Ω )} ,

E(Ω ).

7. Existence for the Bernoulli Free Boundary Problem

265

but Ω does not necessarily satisfy the constraint m(Ω) = α. Note that for any measurable set Ω such that Ω ⊂ Ω ⊂ D, we have 1 1 2 |∇u| − f u dx = |∇u|2 − f u dx 2 Ω Ω 2 so that in expression (7.31) Ω can be enlarged to any such Ω . The inclusion of Ω in Ω implies the inclusion of H1 (Ω; D) in H1 (Ω ; D). So if G ≤ 0 almost everywhere in D, then it is readily seen that from (7.25) E(Ω ) ≤ E(Ω). In view of the previous assumption on G, now G = 0 almost everywhere in D. To conclude the proof we just have to select Ω with m(Ω ) = α. That measurable set Ω is admissible and minimizes the objective function in (7.30), and we have E(Ω ) = E(Ω) =

inf

{E(Ω )}.

Ω ⊂D m(Ω )=α

Corollary 1. Assume that G = 0 and f in L2 (D) and f = Q2 in L1 (D). Then the following problem has an optimal solution: (P0α− )

inf

Ω⊂D m(Ω)≤α

E(Ω).

(7.33)

Proof. The proof is similar to the proof of the theorem where the minimizing sequence is chosen such that m(Ωn ) ≤ α, so that in the weak limit we get λ(x) dx ≤ α. m(Ω) ≤ D

7.4

Weak Existence with Surface Tension

Problems (P0 ), (P0α ), and (P0α− ) have optimal solutions, but as they are associated with the homogeneous Dirichlet boundary condition, u in H1 (Ω; D), the optimal domain Ω is in general not allowed to have holes, that is to say, roughly speaking, that the topology of Ω is a priori speciﬁed. In many examples it turns out that the solution u physically corresponds to a potential and the classical homogeneous Dirichlet boundary condition is not the appropriate one. The physical condition is that the potential u should be constant on each connected component of the boundary ∂Ω in D. When Ω is a simply connected domain in R2 , then ∂Ω has a single connected component so the constant can be taken as zero. In general this constant can be ﬁxed only in one connected component; in the others the constant is an unknown of the problem. The minimization problems (P0 ), (P0α ), and (P0α− ) associated with the Hilbert space H•1 (Ω; D) fail (in the sense that the previous techniques for existence of an optimal Ω fail). The main reason is that Lemma 7.1 is no longer true when un is replaced by ∇un weakly converging in L2 (D)N . The key idea is to recover the equivalent of Lemma 7.1 by imposing the strong L2 (D)-convergence of the sequence {un }. In practice {un } corresponds to the sequence of characteristic functions

266

Chapter 5. Metrics via Characteristic Functions

χΩn of a minimizing sequence {Ωn }. To obtain the strong L2 (D)-convergence of a subsequence we add a constraint on the perimeters. Consider the family of ﬁnite perimeter sets in D of Deﬁnition 6.2 (ii) in section 6.1, def

BPS(D) = {Ω ⊂ D : χΩ ∈ BV(D)}, where BV(D) is deﬁned in (6.2). Introduce the following problem indexed by σ > 0: (Pσα )

inf

Ω⊂D m(Ω)=α def

E(Ω) =

Eσ (Ω),

min

ϕ∈H•1 (Ω;D)

Ω

def

Eσ (Ω) = E(Ω) + σPD (Ω), 1 |∇ϕ|2 − f ϕ dx + 2

(7.34)

G dx .

(7.35)

Ω

Theorem 7.3. Let f in L2 (D), G in L1 (D), σ > 0, and 0 ≤ α < m(D). Then problem (Pσα ) has at least one optimal solution Ω in BPS(D). Proof. Let {Ωn } be a minimizing sequence for (Pσα ) and let {χn } be the corresponding sequence of characteristic functions associated with {Ωn }. By picking ϕ = 0 in (7.35), −1 2 |G| dx + σPD (D) . cf L2 (D) + PD (Ωn ) ≤ σ D

By Theorem 6.3 there exist a subsequence of {Ωn }, still indexed by n, and Ω ⊂ D with ﬁnite perimeter such that χn → χ = χΩ in L1 (D)-strong and α = m(Ωn ) = m(Ω). For each n, let un in H•1 (Ωn ; D) be the unique minimizer of (7.35). That sequence remains bounded in H01 (D). Pick a subsequence, still indexed by n, such that {un } weakly converges to an element u in H01 (D). From the identity (1 − χn )∇un = 0 almost everywhere in D, we get in the limit (1 − χΩ )∇u = 0 almost everywhere in D, so that the limiting element u belongs to H•1 (Ω; D). Hence we have 1 1 f u dx |∇u|2 − f u dx = |∇u|2 dx − Ω 2 D 2 Ω (7.36) 1 2 ≤ lim inf |∇un | dx − f un dx, n→∞ D 2 Ωn f u dx = lim f un dx, G dx = lim G dx, (7.37) Ω

so that

Ω

n→∞

Ωn

1 |∇u|2 − f u dx + 2

Ω

n→∞

Ωn

G dx + σPD (Ω) ≤ Ω

inf

Ω⊂D m(Ω)=α

Eσ (Ω).

(7.38)

Chapter 6

Metrics via Distance Functions 1

Introduction

In Chapter 5 the characteristic function was used to embed the equivalence classes of measurable subsets of D into Lp (D) or Lploc (D), 1 ≤ p < ∞, and induce a complete metric on the equivalence classes of measurable sets. This construction is generic and extends to other set-dependent functions embedded in an appropriate function space. The Hausdorﬀ metric is the result of such a construction, where the distance function plays the role of the characteristic function. The distance function embeds equivalence classes of subsets A of a closed holdall D with the same closure A into the space C(D) of continuous functions. When D is bounded the C 0 -norm of the diﬀerence of the distance functions of two subsets of D is the Hausdorﬀ metric. The Hausdorﬀ topology has many much desired properties. In particular, for D bounded, the set of equivalence classes of nonempty subsets A of D is compact. Yet, the volume and perimeter are not continuous with respect to the Hausdorﬀ topology. This can be ﬁxed by changing the space C(D) to the space W 1,p (D) since distance functions also belong to that space. With that metric the volume is now continuous. The price to pay is the loss of compactness when D is bounded. But other sequentially compact families can easily be constructed. By analogy with Caccioppoli sets we introduce the sets for which the elements of the Hessian matrix of second-order derivatives of the distance function are bounded measures. They are called sets of bounded curvature. Their closure is a Caccioppoli set, and they enjoy other interesting properties. Convex sets belong to that family. General compactness theorems are obtained for such families under global or local conditions. This chapter also deals with the family of open sets characterized by the distance function to their complement. They are discussed in parallel along with the distance functions to a set. The properties of distance functions and Hausdorﬀ and Hausdorﬀ complementary metric topologies are studied in section 2. The projections, skeletons, cracks, and diﬀerentiability properties of distance functions are discussed in section 3. W 1,p topologies are introduced in section 4 and related to characteristic functions. The 267

268

Chapter 6. Metrics via Distance Functions

compact families of sets of bounded and locally bounded curvature are characterized in section 5. The notion of reach and Federer’s sets of positive reach are studied in section 6. The smoothness of smooth closed submanifolds is related to the smoothness of the square of the distance function in a neighborhood. Approximation of distance functions by dilated sets/tubular neighborhoods and their critical points are presented in section 7. Convex sets are characterized in section 8 along with the special family of convex sets. Both sets of positive reach and convex sets will be further investigated in Chapter 7. Finally section 9 gives several compactness theorems under global and local conditions on the Hessian matrix of the distance function.

2

Uniform Metric Topologies

2.1

Family of Distance Functions Cd (D)

In this section we review some properties of distance functions and present the general approach that will be followed in this chapter. Deﬁnition 2.1. Given A ⊂ RN the distance function from a point x to A is deﬁned as def

dA (x) =

inf |y − x|,

A = ∅,

+∞,

A = ∅,

y∈A

(2.1)

and the family of all distance functions of nonempty subsets of D as def

Cd (D) = dA : ∀A, ∅ = A ⊂ D .

(2.2)

When D = RN the family Cd (RN ) is denoted by Cd . By deﬁnition dA is ﬁnite in RN if and only if A = ∅. Recall the following properties of distance functions. Theorem 2.1. Assume that A and B are nonempty subsets of RN . (i) The map x → dA (x) is uniformly Lipschitz continuous in RN : ∀x, y ∈ RN ,

|dA (y) − dA (x)| ≤ |y − x|

(2.3)

0,1 (RN ).1 and dA ∈ Cloc

(ii) There exists p ∈ A such that dA (x) = |p − x| and dA = dA in RN . (iii) A = {x ∈ RN : dA (x) = 0}. 1 A function f belongs to C 0,1 (RN ) if for all bounded open subsets D of RN its restriction to loc D belongs to C 0,1 (D).

2. Uniform Metric Topologies

269

(iv) dA = 0 in RN ⇐⇒ A = RN . (v) A ⊂ B ⇐⇒ dA ≥ dB . (vi) dA∪B = min{dA , dB }. (vii) dA is (Fr´echet) diﬀerentiable almost everywhere and |∇dA (x)| ≤ 1

a.e. in RN .

(2.4)

Proof. (i) For all z ∈ A and x, y ∈ RN |z − y| ≤ |z − x| + |y − x|, dA (y) = inf |z − y| ≤ inf |z − x| + |y − x| = dA (x) + |y − x| y∈A

y∈A

and hence the Lipschitz continuity. (ii) As the function y → |y − x| is continuous (hence upper semicontinuous), dA (x) = inf |y − x| = inf |y − x| = dA (x). y∈A

y∈A

Since the function |y − x| ≥ 0 is bounded below, its inﬁmum over y ∈ A is ﬁnite. Let {yn } ⊂ A be a minimizing sequence dA (x) = inf |y − x| = lim |yn − x|. y∈A

n→∞

Then {yn − x} and hence {yn } are bounded sequences. Hence there exists a subsequence converging to some y ∈ A and by continuity dA (x) = |y − x|. (iii) By the continuity of dA , d−1 A {0} is closed and A ⊂ d−1 A {0}

⇒ A ⊂ d−1 A {0}.

Conversely, if dA (x) = 0, there exists y in A such that 0 = dA (x) = |y − x| and necessarily x = y ∈ A. (iv) follows from (iii). (v) From (ii) for x ∈ A, dA (x) = 0 and dB (x) ≤ dA (x) = 0 necessarily implies dB (x) = 0 and x ∈ B. Conversely, if A ⊂ B, then dB (x) = dB (x) = inf |y − x| ≤ inf |y − x| = dA (x) = dA (x). y∈B

y∈A

(vi) is obvious. (vii) follows from Rademacher’s theorem.

2.2

Pomp´ eiu–Hausdorﬀ Metric on Cd (D)

Let D be a nonempty subset of RN and associate with each nonempty subset A of D the equivalence class def

[A]d = B : ∀B, B ⊂ D and B = A ,

270

Chapter 6. Metrics via Distance Functions

since from Theorem 2.1 (v) dA = dB if and only if A = B. So A is the invariant closed representative of the class [A]d . Consider the set def

Fd (D) = [A]d : ∀A, ∅ = A ⊂ D that can be identiﬁed with the set

A : ∀A, ∅ = A closed ⊂ D . In general, there is no open representative in the class [A]d since dA = dA ≤ dint A , where int A = A denotes the interior of A. By the deﬁnition of [A]d the map [A]d → dA : Fd (D) → Cd (D) ⊂ C(D) is injective. So Fd (D) can be identiﬁed with the subset of distance functions Cd (D) in C(D). The distance function plays the same role as the set X(D) in Lp (D) of equivalence classes of characteristic functions of measurable sets. When D is bounded, C(D) is a Banach space when endowed with the norm f C(D) = sup |f (x)|.

(2.5)

x∈D

As for the characteristic functions of Chapter 5, this induces a complete metric def

ρ([A]d , [B]d ) = dA − dB C(D) = sup |dA (x) − dB (x)|

(2.6)

x∈D

on Fd (D), which turns out to be equal to the classical Pomp´eiu–Hausdorﬀ metric2

def

ρH ([A]d , [B]d ) = max sup dA (x), sup dB (y) x∈B

y∈A

(cf. J. Dugundji [1, Chap. IX, Prob. 4.8, p. 205] for the deﬁnition of ρH ). Indeed for x ∈ D, xA ∈ A, and xB ∈ B |x − xA | ≤ |x − xB | + |xB − xA | ⇒ dA (x) ≤ |x − xB | + dA (xB ) ≤ |x − xB | + sup dA (y) y∈B

⇒

dA (x) ≤ dB (x) + sup dA (y) y∈B

⇒

dA (x) − dB (x) ≤ sup dA (y). y∈B

2 The “´ ´iu [1] in his thesis presented ecart mutuel” between two sets was introduced by D. Pompe in Paris in March 1905. This is the ﬁrst example of a metric between two sets in the literature. It was studied in more detail by F. Hausdorff [2, “Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914.

2. Uniform Metric Topologies

271

By interchanging the roles of A and B ∀x ∈ D,

|dA (x) − dB (x)| ≤ max sup dA (z), sup dB (y) . z∈B

y∈A

The converse inequality follows from the fact that A ⊂ D and B ⊂ D. 0 When D is open but not necessarily bounded, we use Cloc (D) (cf. (2.19) in section 2.6.1 of Chapter 2), the space C(D) of continuous functions on D endowed with the Fr´echet topology of uniform convergence on compact subsets K of D, which is deﬁned by the family of seminorms ∀K compact ⊂ D,

def

qK (f ) = max |f (x)|. x∈K

(2.7)

The Fr´echet topology on C(D) is metrizable since the topology induced by the family of seminorms {qK } is equivalent to the one generated by the subfamily {qKk }k≥1 , where the compact sets {Kk }k≥1 are chosen as follows: 1 def Kk = x ∈ D : dD (x) ≥ and |x| ≤ k , k ≥ 1. (2.8) k It is equivalent to the topology deﬁned by the metric def

δ(f, g) =

∞ 1 qKk (f − g) . 2k 1 + qKk (f − g)

(2.9)

k=1

It will be shown below that Cd (D) is a closed subset of Cloc (D) and that this will induce the following complete metric on Fd (D): def

ρδ ([A]d , [B]d ) = δ(dA , dB ) =

∞ 1 qKk (dA − dB ) , 2k 1 + qKk (dA − dB )

(2.10)

k=1

which is a natural extension of the Hausdorﬀ metric to an unbounded domain D. Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN . (i) The set Cd (D) is closed in Cloc (D) (resp., C(D)) and ρδ (resp., ρ) deﬁnes a complete metric topology on Fd (D). (ii) When D is a bounded open subset of RN , the set Cd (D) is compact in C(D) and the metrics ρ and ρH are equal. Proof. (i) We give the proof of C. Dellacherie [1, Thm. 2, p. 42 and Rem. 1, p. 43]. The basic constructions will again be used in Chapter 7. Consider a sequence {An } of nonempty subsets of D such that dAn converges to some element f of Cloc (D). We wish to prove that f = dA for def

A = {x ∈ D : f (x) = 0}

272

Chapter 6. Metrics via Distance Functions

and that the closed subset A of D is nonempty. Fix x in D; then ∀n, ∃yn ∈ An ,

|yn − x| = inf |z − x| = dAn (x) z∈An

⇒ lim |yn − x| = lim dAn (x) = f (x). n→∞

n→∞

Hence {yn − x} and {yn } ⊂ D are bounded and there exists a subsequence, still indexed by n, which converges to some y ∈ D yn → y, and |y − x| = f (x). In particular, f (y) = 0, since in the inequality f (y) ≤ f (y) − dAn (y) + dAn (y) − dAn (yn ) + dAn (yn ), the last term is zero and dAn is Lipschitz continuous with constant equal to 1: |f (y)| ≤ |f (y) − dAn (y)| + |y − yn |, and both terms go to zero. By the deﬁnition of A, y ∈ A and A is not empty. Therefore for each x ∈ D, there exists y ∈ A such that f (x) = |y − x| ≥ inf |z − x| = dA (x). z∈A

Next we prove the inequality in the other direction. By construction for any An ⊂ D ∀x, y ∈ D,

|dAn (x) − dAn (y)| ≤ |x − y|

and |f (x) − f (y)| ≤ |f (x) − dAn (x)| + |dAn (x) − dAn (y)| + |dAn (y) − f (y)|. By uniform convergence the ﬁrst and last terms converge to zero, and by Lipschitz continuity of dAn , ∀x, y ∈ D, |f (x) − f (y)| ≤ |x − y|. Hence for all x ∈ D and y ∈ A, f (y) = 0 and f (x) ≤ f (y) + |x − y| = |x − y| ⇒ ∀x ∈ D,

f (x) ≤ inf |x − y| = dA (x). y∈A

This proves the reverse equality. (ii) Observe that on a compact set D and for any A ⊂ D def

dA (x) = inf |y − x| ≤ sup |y − x| ≤ c = sup |y − x| < ∞, y∈A

∀x, y ∈ D,

y∈A

y,x∈D

|dA (y) − dA (x)| ≤ |y − x|.

2. Uniform Metric Topologies

273

The compactness of Cd (D) now follows by Ascoli–Arzel`a Theorem 2.4 of Chapter 2 and part (i). By construction, ρ and ρδ are metrics on Fd (D). By deﬁnition, for A and B in the compact D, ρ(A, B) = max |dA (x) − dB (x)| x∈D ≥ max max |dA (x) − dB (x)|, max |dA (x) − dB (x)| x∈B x∈A ≥ max max dA (x), max dB (x) = ρH (A, B). x∈B

x∈A

Conversely, for any x ∈ D and y in A, dA (x) − dB (x) ≤ |y − x| − dB (x) and there exists xB ∈ B such that dB (x) = |x − xB |. Therefore, ∀y ∈ A, dA (x) − dB (x) ≤ |y − x| − |x − xB | ≤ |y − xB | ⇒ dA (x) − dB (x) ≤ inf |y − xB | = dA (xB ) ≤ max dA (x). y∈A

x∈B

Similarly dB (x) − dA (x) ≤ max dB (x), x∈A

and for all x in D

|dB (x) − dA (x)| ≤ max max dA (x), max dB (x) ⇒ ρ(A, B) ≤ ρH (A, B). x∈B

x∈A

When D is bounded the family Fd (D) enjoys more interesting properties. Theorem 2.3. Let D be a nonempty open (resp., bounded open) subset of RN . Deﬁne for a subset S of RN the sets def

H(S) = dA ∈ Cd (D) : S ⊂ A , def

I(S) = dA ∈ Cd (D) : A ⊂ S , def

J(S) = dA ∈ Cd (D) : A ∩ S = ∅ . (i) Let S be a subset of RN . Then H(S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then I(S) is closed in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then J(S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then J(S) is open in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then I(S) is open in Cloc (D) (resp., C(D)). (iv) For a nonempty bounded holdall D, the map def

[A]d → #(A) : F# (D) = [A]d : ∀A, ∅ = A ⊂ D and #(A) < ∞ → R

274

Chapter 6. Metrics via Distance Functions is lower semicontinuous, where #(A) is the number of connected components as deﬁned in Deﬁnition 2.2 of Chapter 2. For a ﬁxed number c > 0 the subset {dA ∈ Cd (D) : #(A) ≤ c} of Cd (D) is compact in C(D). In particular, the subset {dA ∈ Cd (D) : A is connected} of Cd (D) is compact in C(D).

Proof. (i) If S ⊂ D, then H(S) = ∅ and there is nothing to prove. Assume that S ⊂ D. From Theorem 2.1 (v), S ⊂ A ⇒ dS ≥ dA . So for any sequence {dAn } in H(S) converging to dA in Cloc (D), ∀n,

dS ≥ dAn

⇒ dS ≥ dA

⇒ S⊂S⊂A

⇒ dA ∈ H(S).

(ii) We shall use the same technique for I(S) as for H(S), but here we need S = S to conclude. For D ∩ S = ∅, J(S) = ∅ and there is nothing to prove. Assume that D ∩ S = ∅ and consider a sequence {dAn } in J(S) converging to dA in Cloc (D). Assume that A ∩ S = ∅. Then A ⊂ D implies A ∩ [S ∩ D] = A ∩ S = ∅. By assumption, S ∩ D is compact and ∃δ > 0, ∀x ∈ S ∩ D, dA (x) ≥ δ, ∃N ≥ 1, ∀n ≥ N, dAn − dA C(D) ≤ δ/2. So for all x ∈ D ∩ S dAn (x) ≥ dA (x) − |dAn (x) − dA (x)| ≥ δ − δ/2 > 0 ⇒ An ∩ S = An ∩ [D ∩ S] = ∅, and this contradicts the fact that An ∈ J(S). (iii) By deﬁnition, for any S, J(S) = I(S),

J(S) = dA ∈ Cd (D) : A ∩ S = ∅ = dA ∈ Cd (D) : A ⊂ S = I(S). Since S is closed, J(S) is closed from part (ii) and J(S) is open. Similarly, replacing S by S in the previous identity, I(S) = J(S). So from part (ii) if D ∩ S is compact, then I(S) is closed and I(S) is open. From part (ii) if D ∩ S is compact, then I(S) is closed and I(S) is open. (iv) Let {An } and A be nonempty subsets of D such that dAn converges to dA in C(D). Assume that #(A) = k is ﬁnite. Then there exists a family of disjoint open sets G1 ,. . . , Gk such that A ⊂ G = ∪ki=1 Gi

and ∀i, A ∩ Gi = ∅.

In view of the deﬁnitions of I(S) and J(S) in part (i) A ∈ U = ∩ki=1 J(Gi ) ∩ I(G).

2. Uniform Metric Topologies

275

But U is not empty and open as the ﬁnite intersection of k + 1 open sets. As a result there exist ε > 0 and an open neighborhood of [A]d : Nε ([A]d ) = {[B]d : dB − dA < ε} ⊂ U. Hence since dAn converges to dA , there exists n > 0 such that, for all n ≥ n, [An ]d ∈ U, and necessarily An ⊂ G,

∀n ≥ n,

An ∩ Gi = ∅, ∀i

⇒ #(An ) ≥ #(A).

Therefore, [A]d → #(A) is lower semicontinuous. This theorem has many interesting corollaries. For instance part (iii) says something about the function that gives the number of connected components of A (cf. T. J. Richardson [1] for an application to image segmentation).

2.3

Uniform Complementary Metric Topology and Cdc (D)

In the previous section we dealt with a theory of closed sets since the equivalence class of the subsets of D was completely determined by their unique closure. In partial diﬀerential equations the underlying domain is usually open. To accommodate this point of view, consider the set of open subsets Ω of a ﬁxed nonempty open holdall D in RN , endowed with the Hausdorﬀ topology generated by the distance functions dΩ to the complement of Ω. This approach has been used in ´sio [6, sect. 1.3, p. 405] in the conseveral contexts, for instance, in J.-P. Zole ˇ ´ k [2] uses the family Cdc (D) for text of free boundary problems. Also, V. Sver a open sets Ω such that #([Ω]) ≤ c for some ﬁxed c > 0. His main result is that in dimension 2 the convergence of a sequence {Ωn } to Ω of such sets implies the convergence of the corresponding projection operators {PΩn : H01 (D) → H01 (Ωn )} to PΩ : H01 (D) → H01 (Ω), where the projection operators are directly related to homogeneous Dirichlet linear boundary value problems on the corresponding domains {Ωn } and Ω. In dimension 1 the constraint on the number of components can be dropped. This result will be discussed in Theorem 8.2 of section 8 in Chapter 8. By analogy with the constructions of the previous section, consider for a nonempty subset D of RN the family

dA : ∅ = A and A ⊂ D . By deﬁnition, A ⊃ D

⇒ dA = 0 in D,

and, by Lipschitz continuity, dA = 0 in D and dA = 0 on ∂D = D ∩ D. If int D = ∅, then dA = 0 in RN and for all sets A in the family int A = ∅. If int D = ∅, associate with each A the open set def

Ω = int A = A

⇒ Ω = A and dΩ = dA .

276

Chapter 6. Metrics via Distance Functions

By deﬁnition and the previous considerations, A ⊃ D

⇒ Ω = A ⊂ D = int D.

So ﬁnally, for int D = ∅,

dA : ∅ = A and A ⊂ D = dΩ : Ω = RN and Ω open ⊂ int D . The invariant set int A is the open representative in the equivalence class def [A]cd = B : B = A = ∅ , but, in general, there is no closed representative in that class since dint A = dA = dA ≤ dA , as in the case of dA , where dA = dA ≤ dint A . From this analysis it will be suﬃcient to consider the family of open subsets of an open holdall D. Deﬁnition 2.2. Let D be a nonempty open subset of RN . Deﬁne the family of functions def

Cdc (D) = dΩ : ∀Ω open subset of D and Ω = RN corresponding to the family of open sets def

G(D) = Ω ⊂ D : ∀Ω open and Ω = RN .

(2.11)

(2.12)

For D bounded deﬁne the Hausdorﬀ complementary metric ρcH on G(D): def

ρcH (Ω2 , Ω1 ) = dΩ2 − dΩ1 C(D) .

(2.13)

Note that for Ω1 ⊂ D and Ω2 ⊂ D, dΩ1 = dΩ2 = 0 in D. Therefore, dΩ1 = dΩ2 in D ⇐⇒ Ω1 = Ω2 ⇐⇒ Ω1 = Ω2 . Theorem 2.4. Let D be a nonempty open subset of RN . (i) The set Cdc (D) is closed in Cloc (D). (ii) If, in addition, D is bounded, then Cdc (D) is compact in C0 (D) and (G(D), ρcH ) is a compact metric space. (iii) (Compactivorous property) Let {Ωn } and Ω be sets in G(D) such as dΩn → dΩ

in Cloc (D).

Then for any compact subset K ⊂ Ω, there exists an integer N (K) > 0 such that ∀n ≥ N (K), K ⊂ Ωn .

2. Uniform Metric Topologies

277

Proof. (i) Let {Ωn } be a sequence of open subsets in D such that {dΩn } is a Cauchy sequence in Cloc (D). For all n, dΩn = 0 in D and {dΩn } is also a Cauchy sequence in Cloc (RN ). By Theorem 2.2 (i) the sequence {dΩn } converges in Cloc (RN ) to some distance function dA . By construction Ωn ⊂ D =⇒ Ωn ⊃ D =⇒ dΩn ≤ dD =⇒ dA ≤ dD =⇒ A ⊃ D =⇒ A ⊂ D. By choosing the open set Ω = A in D we get dΩn → dΩ

in Cloc (RN ) and C0,loc (D).

(ii) To prove the compactness we use the compactness of Cd (D) from Theorem 2.2 (ii) and the fact that Cdc (D) is closed in C0 (D). Observe that since Ωn ⊃ D, dΩn = 0 in D, dΩn ∈ C0 (D), and Ωn = [Ωn ∩ D] ∪ [Ωn ∩ D] ⊂ [Ωn ∩ D] ∪ D ⊂ Ωn ⇒ dΩn = d[Ωn ∩D]∪D = min{dΩn ∩D , dD }. There exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that dΩn ∩D → dA

in C0 (D)

⇒

dΩn = min{dΩn ∩D , dD } → min{dA , dD } = dA∪D

⇒

dΩn → dA∪D

in C(D)

in C(RN )

since dΩn = 0 = dA∪D in D. The result now follows by choosing the open set def

Ω = [A ∪ D] = A ∩ D ⊂ D. (iii) Deﬁne

def

m = inf inf |x − y| = inf dΩ (x). x∈K y∈Ω

x∈K

Since K ⊂ Ω ⊂ D, there exist x ˆ ∈ K, yˆ ∈ ∂Ω such that x − yˆ|. m = inf inf |x − y| = |ˆ x∈K y∈∂Ω

Necessarily, m > 0 since Ω is open, x ˆ ∈ K ⊂ Ω, and yˆ ∈ ∂Ω. Now ∃N > 0, ∀n ≥ N, dΩn − dΩ C(K) < m/2 ⇒ ∀x ∈ K, dΩn (x) ≥ dΩ (x) − |dΩn (x) − dΩ (x)| ≥ m − m/2 > 0 ⇒ x = Ωn

⇒ x ∈ Ωn = int Ωn = Ωn

Notation 2.1. It will be convenient to write

⇒ K ⊂ Ωn .

Hc

Ωn → Ω for the Hausdorﬀ complementary convergence of open sets of G(D) dΩn → dΩ

in Cloc (D).

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Chapter 6. Metrics via Distance Functions

We have the analogue of Theorem 2.3 for the distance function to the complement of an open subset of D. Theorem 2.5. Let D be a nonempty open (resp., bounded open) subset of RN . Deﬁne for a subset S of RN and open subsets Ω of D the following families: def

H c (S) = dΩ ∈ Cdc (D) : S ⊂ Ω , def

I c (S) = dΩ ∈ Cdc (D) : Ω ⊂ S , def

J c (S) = dΩ ∈ Cdc (D) : Ω ∩ S = ∅ . (i) Let S be a subset of RN . Then H c (S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then I c (S) is closed in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then J c (S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then J c (S) is open in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then I c (S) is open in Cloc (D) (resp., C(D)). (iv) For a nonempty bounded open holdall D, the map def

Ω → #(Ω) : G# (D) = Ω : Ω open ⊂ D and # Ω < ∞ → R is lower semicontinuous, where #(Ω) is the number of connected components of Ω as deﬁned in Deﬁnition 2.2 of Chapter 2. For a ﬁxed number c ≥ 0, the subset {dΩ ∈ Cdc (D) : # Ω ≤ c} is compact in C(D). In particular, the subset {dΩ ∈ Cdc (D) : Ω is hole-free} is compact in C(D), where hole-free is in the sense of Deﬁnition 2.2 of Chapter 2.

2.4

c Families Cdc (E; D) and Cd,loc (E; D)

Now and then, it is desirable to avoid having the empty set as a solution of a shape optimization problem. This can occur for the C(D)-topology on Cdc (D). One way to get around this is to assume that each set of the family contains a ﬁxed nonempty set or, more generally, that each set of the family contains a translated and rotated image of a ﬁxed nonempty set. Theorem 2.6. Let D, ∅ = D ⊂ RN , be bounded open, let E, ∅ = E ⊂ D, and let def

Cdc (E; D) = {dΩ : E ⊂ Ω open ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open ⊂ D N

dΩ :

(2.14)

& .

(2.15)

3. Projection, Skeleton, Crack, and Diﬀerentiability

279

c (E; D) are compact in C(D). (i) If E is open, Cdc (E; D) and Cd,loc c (ii) If E is closed, Cdc (E; D) and Cd,loc (E; D) are open in C(D).

Proof. (a) We ﬁrst deal with Cdc (E; D) using the fact that

Cdc (E; D) = dΩ : Ω ⊂ E and Ω open ⊂ D = I c (E) and Theorem 2.5. If E is open, E is closed and I c (E) is closed and hence compact since D is bounded. If E is closed, E is open, E ∩ D is compact, and I c (E) is open. c (D) (b) Assume that E is open. Let {dΩn } be a Cauchy sequence in CD converging to some dΩ . Denote by xn ∈ RN and An ∈ O(N) the translation and rotation of E such that xn + An E ⊂ Ωn and dxn +An E ≤ dΩ . Since, for all n, xn + An E ⊂ D and D is bounded, there exist x ∈ RN and A ∈ O(N) and subsequences, still indexed by n, such that xn → x and An → A. But dxn +An E (z) = dE (An (z − xn )) since A−1 = A in O(N) and dE (An (z − xn )) → dE (A(z − x)). Finally, dx+AE = dE (A(z − x)) ≤ dΩ and, since x + AE and Ω are both closed, x + AE ⊂ Ω, c c which implies x + AE ⊃ Ω and dΩ ∈ Cd,loc (E; D). Therefore Cd,loc (E; D) as c a closed subset of the compact set Cd (D) is compact. We leave the proof that c (E; D) is open when E is closed to the reader. Cd,loc

3

Projection, Skeleton, Crack, and Diﬀerentiability

In this section we study the connection between the gradient of dA , the set of projections onto A, and the characteristic function of A. We partition the set of singularities of the gradient of dA into the skeleton 3 and set of cracks of A. Deﬁnition 3.1. Let A ⊂ RN , ∅ = A, and denote by def

Sing (∇dA ) = x ∈ RN : ∇dA (x)

(3.1)

the set of singularities of the gradient of dA . (i) Given x ∈ RN , a point p ∈ A such that |p − x| = dA (x) is called a projection onto A. The set of all projections onto A will be denoted by def

ΠA (x) = p ∈ A : |p − x| = dA (x) . (3.2) When ΠA (x) is a singleton, its element will be denoted by pA (x). 3 Our

deﬁnition of a skeleton does not exactly coincide with the one used in morphological mathematics, where it is deﬁned as the closure Sk (A) of our skeleton Sk (A) (cf., for instance, `re [1], and J. Serra [1] for the pioneering applications in mining G. Matheron [1] or A. Rivie engineering in 1968 and later in image processing).

280

Chapter 6. Metrics via Distance Functions

(ii) The skeleton of A is the set of all points of RN whose projection onto A is not unique. It will be denoted by def

Sk (A) = x ∈ RN : ΠA (x) is not a singleton . (3.3) (iii) The set of cracks is deﬁned as def

Ck (A) = Sing ∇dA \Sk (A).

(3.4)

Note that Sk (A) = Sk (A), Ck (A) = Ck (A), and Sing (A) = Sing (A) since dA = dA by Theorem 2.1 (ii). Even for a set A with smooth boundary ∂A, the gradient ∇dA (x) may not exist far from the boundary as shown in Figure 6.1, where ∇dA (x) exists everywhere outside A except on Sk (A), a semi-inﬁnite line. Yet, ∇dA (x) exists close to the boundary. An example of the set of cracks is given by the eyes, nose, mouth, and rays of the smiling sun in Figure 5.1 of Chapter 5 that belong to the boundary of the set as opposed to the skeleton Sk (A) that lies outside of A. Sk (A)

Sk (A) A

Figure 6.1. Skeletons Sk (A), Sk (A), and Sk (∂A) = Sk (A) ∪ Sk (A). The level sets of dA generate dilations/tubular neighborhoods of A. Deﬁnition 3.2. Let ∅ = A ⊂ RN and let h > 0. (i) The open h-tubular neighborhood , the closed h-tubular neighborhood , and the h-boundary of A are respectively deﬁned as def

def

Uh (A) = x ∈ RN : dA (x)| < h , Ah = x ∈ RN : dA (x)| ≤ h , and d−1 A {h}. We shall also use the terminology open or closed dilated sets. (ii) For 0 < s < h < ∞ def

def

Us,h (A) = x ∈ RN : s < dA (x)| < h and As,h = x ∈ RN : s ≤ dA (x)| ≤ h . This terminology is justiﬁed by the fact that a dilated set is a relatively nice set.

3. Projection, Skeleton, Crack, and Diﬀerentiability

281

Theorem 3.1. Let ∅ = A ⊂ RN . For r > 0, Ar = Ur (A), ∂Ar = ∂Ur (A) = d−1 A {r}, Ar = Ur (A), int Ar = Ur (A).

(3.5)

In particular, for 0 < s < r, Us,r (A) = As,r . Proof. By deﬁnition A ⊂ Ur (A) ⊂ Ar and Ur (A) ⊂ Ar . For any x ∈ Ar , dA (x) ≤ r. / A and there If dA (x) < r, then x ∈ Ur (A) ⊂ Ur (A). If dA (x) = r > 0, then x ∈ exists p ∈ ∂A such that dA (x) = |x − p| = r > 0. Consider the points x−p x−p 1 def r →p+r = x. xn = p + 1 − n |x − p| |x − p| Then p ∈ ΠA (xn ), and dA (xn ) = r (1 − 1/n) < r, dA (xn ) → dA (x) = r, and x ∈ Ur (A). Therefore, since Ur (A) is open, Ar = Ur (A) = Ur (A) ∪ ∂Ur (A) implies that ∂Ur (A) = d−1 A {r}. Moreover, since Ur (A) is closed, Ur (A) = int Ur (A) ∪ ∂Ur (A) = Ar ∪ ∂Ur (A) implies that Ar = Ur (A). The proof is the same for Us,r (A) = As,r . We now give several important technical results that will be used later for sets of positive reach and in Chapter 7. We ﬁrst give a few additional deﬁnitions. Deﬁnition 3.3. (i) A function f : U → R deﬁned in a convex subset U of RN is convex if for all x and y in U and all λ ∈ [0, 1], f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y). It is concave if the function −f is convex. A function f : U → R deﬁned in a convex subset U of RN is semiconvex (resp., semiconcave) if ∃c ≥ 0,

fc (x) = c |x|2 + f (x)

(resp., fc (x) = c |x|2 − f (x))

is convex in U . (ii) A function f : U → R deﬁned in a subset U of RN is locally convex (resp., locally concave) if it is convex (resp., concave) in every convex subset of U . A function f : U → R deﬁned in a subset U of RN is locally semiconvex (resp., locally semiconcave) if it is semiconvex (resp., semiconcave) in every convex subset of U . When U is convex the local deﬁnitions coincide with the global ones. The next theorem is a slight extension of a result in J. H. G. Fu [1, Prop. 1.2]. Theorem 3.2. Let A ⊂ RN be such that A = ∅. (i) Given h > 0, the function def

kA,h (x) =

|x|2 − 2 h dA (x), |x|2 − d2A (x) − h2 ,

if dA (x) ≥ h, if dA (x) < h,

282

Chapter 6. Metrics via Distance Functions is convex (and continuous) in RN and the function x →

kA,h (x) |x|2 = − dA (x) : RN \Uh (A) → R 2h 2h

is locally convex (and continuous) in RN \Uh (A). (ii) The function def

x → fA (x) =

1 2 |x| − d2A (x) : RN → R 2

is convex and continuous, d2A (x) is the diﬀerence of two convex functions d2A (x) = |x|2 − |x|2 − d2A (x) , (3.6) N ∇fA and ∇d2A belong to BVloc (RN )N , and ∇dA ∈ BVloc RN \∂A . Proof. (i) For all p ∈ A deﬁne the convex function p (x) =

|x − p| − h, 0,

|x − p| ≥ h, |x − p| < h.

Since p is nonnegative, 2p is convex and |x|2 − 2 h |x − p| + |p|2 + h2 − 2 x · p, 0,

2p (x) =

|x − p| ≥ h, |x − p| < h.

By subtracting the constant term |p|2 + h2 and the linear term −2 x · p from 2p , we get the new convex function mp (x) =

|x|2 − 2 h |x − p|, |x|2 − |p − x|2 − h2 ,

|x − p| ≥ h, |x − p| < h.

Then the function def

k(x) = sup mp (x) = p∈A

|x|2 − 2 h dA (x),

& dA (x) ≥ h,

|x|2 − d2A (x) − h2 ,

dA (x) < h

is convex in RN . Moreover, it is ﬁnite for each x ∈ RN and hence continuous in x. Indeed, if x is such that dA (x) ≥ h, by Theorem 2.1 (ii) there exists p¯ ∈ A such that |x − p¯| = dA (x) ≥ h and |x − p¯| = dA (x) =

inf p∈A |x−p|≥h

|x − p|

3. Projection, Skeleton, Crack, and Diﬀerentiability

283

and k(x) = |x|2 − 2 h dA (x). If dA (x) < h, then there exists p¯ ∈ A such that |x − p¯| = dA (x) < h and |x − p¯| = dA (x) =

inf

|x − p|

p∈A |x−p| 0, |x|2 − d2A (x) is locally convex in Uh (A) it is locally convex and continuous in RN = ∪h>0 Uh (A) and hence convex in RN . Therefore, from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240], ∇fA ∈ BVloc (RN )N and ∇d2A ∈ BVloc (RN )N . Finally, RN \∂A is the union of the two disjoint open sets RN \A and int A. On int A, ∇dA = 0. For any point x in RN \A, there exists ρ, 0 < 3 ρ < dA (x), such that the open ball B(x, 2 ρ) is contained in RN \Uρ (A), where kA,ρ is locally convex by part (i). Hence kA,ρ is convex in B(x, 2 ρ) and ∇kA,ρ , and ∇dA belong to BVloc (B(x, 2ρ))N and, by Deﬁnition 6.1 in Chapter 5, ∇dA belong to BV(B(x, ρ))N . Finally, by Theorem 6.1 in Chapter 5, ∇dA belongs to BVloc (RN \A)N . We now give some basic results. Theorem 3.3. Let A, ∅ = A ⊂ RN , and let x ∈ RN . (i) The set ΠA (x) is nonempty, compact, and ∀x ∈ /A

ΠA (x) ⊂ ∂A

and

∀x ∈ A

ΠA (x) = {x}.

(ii) For all x and v in RN , the Hadamard semiderivative4 of d2A always exists, d2A (x + tv) − d2A (x) = min 2(x − p) · v t0 t p∈ΠA (x)

dH d2A (x; v) = lim

= 2(x · v − σΠA (x) (v)), fA (x + tv) − fA (x) = σΠA (x) (v) = σco ΠA (x) (v), dH fA (x; v) = lim t0 t where σB is the support function of the set B, σB (v) = sup z · v, z∈B

and co B is the convex hull of B. In particular,

Sk (A) = x ∈ RN : ∇d2A (x) ⊂ RN \A. 4A

function f : RN → R has a Hadamard semiderivative in x in the direction v if def

dH f (x; v) = lim

t0 w→v

(cf. Chapter 9, Deﬁnition 2.1 (ii)).

f (x + tw) − f (x) exists t

(3.7)

284

Chapter 6. Metrics via Distance Functions Given v ∈ RN , for all x ∈ RN \∂A the Hadamard semiderivative of dA exists, dH dA (x; v) =

min 2 p∈ΠA (x)

x−p · v, |x − p|

(3.8)

and for x ∈ ∂A, dH dA (x; v) exists if and only if lim

t0

dA (x + tv) t

exists.

(3.9)

(iii) The following statements are equivalent: (a) d2A (x) is (Fr´echet) diﬀerentiable at x. (b) d2A (x) is Gateaux diﬀerentiable at x. (c) ΠA (x) is a singleton. (iv) ∇fA (x) exists if and only if ΠA (x) = {pA (x)} is a singleton. In that case 1 pA (x) = ∇fA (x) = x − ∇d2A (x). 2 For all x and y in RN , ∀p(x) ∈ ΠA (x),

1 2 1 2 |y| − d2A (y) ≥ |x| − d2A (x) + p(x) · (y − x), (3.10) 2 2

or equivalently, ∀p(x) ∈ ΠA (x),

d2A (y) − d2A (x) − 2 (x − p(x)) · (y − x) ≤ |x − y|2 . (3.11)

For all x and y in RN ∀p(x) ∈ ΠA (x), ∀p(y) ∈ ΠA (y),

(p(y) − p(x)) · (y − x) ≥ 0.

(3.12)

(v) The functions pA : RN \Sk (A) → RN

and

∇d2A : RN \Sk (A) → RN

are continuous. For all x ∈ RN \Sk (A) 1 pA (x) = x − ∇d2A (x) = ∇fA (x). 2 In particular, for all x ∈ A, ΠA (x) = {x} and ∇d2A (x) = 0,

Sk (A) = x ∈ RN : ∇d2A (x) and ∇dA (x) ⊂ RN \A,

Ck (A) = x ∈ RN : ∇d2A (x) ∃ and ∇dA (x) ⊂ ∂A, and Sing (∇dA ) = Sk (A) ∪ Ck (A).

(3.13)

(3.14)

3. Projection, Skeleton, Crack, and Diﬀerentiability

285

(vi) The function ∇dA : RN \ Sk (A) ∪ ∂A → RN is continuous. For all x ∈ int A, ∇dA (x) = 0 and for all x ∈ RN \ Sk (A) ∪ A ∇dA (x) =

∇d2A (x) x − pA (x) = 2 dA (x) |x − pA (x)|

|∇dA (x)| = 1.

and

(3.15)

If ∇dA (x) exists for x ∈ A, then ∇dA (x) = 0. In particular, ∇dA = 0 almost everywhere in A. (vii) The functions d2A and dA are diﬀerentiable almost everywhere and m(Sk (A)) = m(Ck (A)) = m(Sing (∇dA )) = 0. For ∂A = ∅ Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A).

(3.16) (3.17)

If ∂A = ∅, then either A = RN or A = ∅. (viii) Given A ⊂ RN , ∅ = A (resp., ∅ = A), χA (x) = 1 − |∇dA (x)|,

χint A (x) = |∇dA (x)| in RN \Sing (∇dA )

(resp., χA (x) = 1 − |∇dA (x)|, χint A (x) = |∇dA (x)| in RN \Sing (∇dA )) and the above identities hold for almost all x in RN . def

(ix) Given x ∈ RN , α ∈ [0, 1], p ∈ ΠA (x), and xα = p + α (x − p), then dA (xα ) = |xα − p| = α |x − p| = α dA (x)

and

∀α ∈ [0, 1], ΠA (xα ) ⊂ ΠA (x).

In particular, if ΠA (x) is a singleton, then ΠA (xα ) is a singleton and ∇d2A (xα ) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = A, then for all 0 < α ≤ 1 ∇dA (xα ) exists and ∇dA (xα ) = ∇dA (x). Remark 3.1. In general, for each v ∈ RN ∃ˆ p = p(v) ∈ ΠA (x),

dH d2A (x; v) = 2(x − pˆ) =

inf p∈ΠA (x)

2(x − p) · v,

since ΠA (x) is compact. However, when ΠA (x) is not a singleton, pˆ is not necessarily unique since 2(x − p) depends on the direction v. For points x outside of A and Sk (A) the norm of ∇dA (x) is equal to 1. When A is suﬃciently smooth, ∇dA (x) coincides with the outward unit normal to A at the point pA (x). When A is not smooth, this normal is not always unique as shown in Figure 6.2.

286

Chapter 6. Metrics via Distance Functions x pA (x) = pA (y) ∇dA (x) A

∇dA (y)

y Figure 6.2. Nonuniqueness of the exterior normal.

It is useful to consider examples to illustrate the subtleties of the theorem. The set of cracks Ck (A) of a set A has zero measure, but its boundary ∂A can have a nonempty interior and a nonzero measure. Example 3.1. Let B1 (0) be the open ball of radius 1 at the origin in RN and let A = {x ∈ B1 (0) : x has rational coordinates}. Then A = B1 (0), A = RN , ∂A = B1 (0), ∂A = ∂B1 (0), ∇dA (x) = ∇d2A (x) = 0 in B1 (0)

x = pA (x) in B1 (0), |x| Ck (A) = ∂B1 (0).

and ∇dA (x) =

Sk (A) = ∅ and

In general ∂A ⊂ ∂A and this inclusion can be strict as illustrated by this example. Also note that the boundary ∂A has a nonempty interior int ∂A = B1 (0) and a nonzero Lebesgue measure (nonzero volume in dimension N = 3). Thus the boundary of two members of the equivalence class [A] can be diﬀerent. The boundary ∂A of a closed or an open set A has no interior; that is, ∂A is nowhere dense. Yet, its boundary ∂A still does not necessarily have a zero measure. If A is closed, then A = int A ∪ ∂A. If int A = ∅, there exist x ∈ ∂A and r > 0 such that Br (x) ⊂ ∂A ⊂ A and x ∈ int A, a contradiction. If A is open, A is closed, A = int A ∪ ∂A, and ∂A has no interior. Example 3.2 (Modiﬁed version of Example 1.10 in E. Giusti [1, p. 7]). Let B1 (0) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let {xi : i ∈ N} be the sequence of all points in B1 (0) with rational coordinates for some choice of ordering. Associate with each i the open ball Bi = {x ∈ R2 : |x − xi | < ρi },

0 < ρi ≤ min{2−i , 1 − |xi |}

⇒ Bi ⊂ B1 (0).

3. Projection, Skeleton, Crack, and Diﬀerentiability

287

Consider the increasing sequence of open subsets of B1 (0) def

Ωn =

n *

def

Bi Ω =

i=1

∞ *

Bi

i=1

def

and the closed set A = Ω. We get A = A = Ω,

A = Ω,

A = Ω = B1 (0),

∂A = B1 (0) ∩ Ω.

As a result, we have the following decreasing sequence of closed subsets of B1 (0): def

Dn = B1 (0) ∩ Ωn ∂A = B1 (0) ∩ Ω 2 n n 1 m(Bi ) = π − π ⇒ m(Dn ) ≥ m(B1 (0)) − 2i i=1 i=1 2 ∞ 2 1 ⇒ m(∂A) ≥ π − π = π. i 2 3 i=1 Proof of Theorem 3.3. (i) Existence. The function z → |z − x|2 : RN → R

(3.18)

is continuous and, a fortiori, upper semicontinuous, and hence inf |a − x|2 = inf |a − x|2 .

a∈A

a∈A

If A is bounded, there exists p ∈ A such that |p − x|2 = inf a∈A |a − x|2 . If A is unbounded, the function (3.18) is coercive with respect to A, |a − x|2 = +∞, |a| a∈A, |a|→∞ lim

and we also have the existence of p ∈ A such that |p − x|2 = inf a∈A |a − x|2 . By deﬁnition, ΠA (x) ⊂ A. For x ∈ RN \A, dA (x) > 0. Assume that there exists p ∈ ΠA (x) such that p ∈ int A. Therefore, there exists r, 0 < r < dA (x)/2, such that Br (p) ⊂ A. Choose the point r x−p 2 |x − p|

⇒ |y − p| = r/2 ⇒ y ∈ Br (p) ⊂ A r r x−p = 1− (p − x) ⇒ y−x=p−x+ 2 |x − p| 2dA (x) r ⇒ ∃y ∈ A such that |y − x| = 1 − |p − x| < dA (x); 2dA (x) y =p+

this contradicts the minimality of dA (x). Hence x ∈ ∂A.

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(ii) Semidiﬀerentiability of fA . The semiderivative in a direction v can be readily computed by using a theorem on the diﬀerentiability of the min with respect to a parameter (cf. Chap. 10, sect. 2.3, Thm. 2.1). It is suﬃcient to prove the semidiﬀerentiability of d2A in any direction v, that is, for each x and v the existence of the limit d2 (x + tv) − d2A (x) def . dd2A (x; v) = lim A t0 t Recall that for t ≥ 0,

ΠA (x + t v) = pt ∈ A¯ : |x + tv − pt |2 = d2A (x + tv) and consider for t > 0 the quotient qt =

d2A (x + tv) − d2A (x) . t

For all p ∈ ΠA (x) and pt ∈ ΠA (x + t v) |x + tv − pt |2 − |x − p|2 t |x + tv − p|2 − |x − p|2 = t |v|2 + 2 v · (x − p), ≤ t

qt =

and for all p ∈ ΠA (x) lim sup qt ≤ 2 v · (x − p) t0

⇒ q¯ = lim sup qt ≤ 2 t0

inf p∈ΠA (x)

v · (x − p).

In the other direction choose a sequence tk > 0 such that tk → 0 and qtk → q = lim inf t0 . The corresponding sequence ptk in A is uniformly bounded since |ptk | ≤ |x + tk v − ptk | + |x + tk v| ≤ dA (x + tk v) − dA (x) + dA (x) + |x + tk v| ≤ tk |v| + dA (x) + |x + tk v|, and we can ﬁnd p0 ∈ A¯ and a subsequence of {ptk }, still denoted by {ptk }, such that ptk → p0 . But by continuity dA (x + tk v) → dA (x) and |x + tk v − ptk | → |x − p0 | ⇒ ∃p0 ∈ A¯ such that dA (x) = |x − p0 | ⇒ p0 ∈ ΠA (x). Therefore, |x + tk v − ptk |2 − |x − p0 |2 tk |x + tk v − ptk |2 − |x − ptk |2 ≥ = tk |v|2 + 2 v · (x − ptk ) tk ⇒ q ≥ 2 v · (x − p0 ) ≥ 2 inf v · (x − p).

qtk =

p∈ΠA (x)

3. Projection, Skeleton, Crack, and Diﬀerentiability

289

Hence dd2A (x; v) = 2

inf p∈ΠA (x)

v · (x − p).

Finally, by Theorem 2.4 (ii), dH d2A (x; v) = dd2A (x; v) since d2A is locally Lipschitzian. The same remark applies to ddA (x; v) in RN \∂A. (iii) From part (ii), d2A (x) is Gateaux diﬀerentiable at x if and only if ΠA (x) is a singleton. The (Fr´echet) diﬀerentiability is a standard part of Rademacher’s theorem, but we reproduce it here for completeness as a lemma. Lemma 3.1. Let f : RN → R be a locally Lipschitzian function. Then f is (Fr´echet) diﬀerentiable at x, that is, lim

y→x

f (y) − f (x) − ∇f (x) · (y − x) =0 |y − x|

if and only if it is Gateaux diﬀerentiable at x, that is, ∀v ∈ RN ,

f (x + tv) − f (x) exists t0 t

df (x; v) = lim

and the map v → df (x; v) is linear and continuous. Proof. It is suﬃcient to prove that the Gateaux diﬀerentiability implies that f (y) − f (x) − ∇f (x) · (y − x) → 0. y→x |y − x| lim

The function f is locally Lipschitzian and there exist δ1 > 0 and c1 > 0 such that f is uniformly Lipschitzian in the open ball B(x, δ1 ) with Lipschitz constant c1 . Therefore, |∇f (x)| ≤ c1 . The unit sphere S(0, 1) = {v ∈ RN : |v| = 1} is compact and can be covered by the family of open balls {B(v, ε/(4c1 )) : v ∈ S(0, 1)} for an arbitrary ε > 0. Hence there exists a ﬁnite subcover {B(vn , ε/(4c1 )) : 1 ≤ n ≤ N } of S(0, 1). As a result, given any v ∈ S(0, 1), there exists n, 1 ≤ n ≤ N , such that |v − vn | ≤ ε/(4c1 ). Deﬁne the function g(x, v, t) =

f (x + tv) − f (x) − ∇f (x) · v t

for x ∈ RN , v ∈ S(0, 1), and t > 0. Since f (x) is Gateaux diﬀerentiable ∃δ, 0 < δ < δ1 , ∀n, 1 ≤ n ≤ N, ∀t < δ,

|g(x, vn , t)| < ε/2.

Hence for |y − x| < δ g x, y − x , |y − x| |y − x| y−x ≤ |g(x, vn , |y − x|)| + g x, , |y − x| − g(x, vn , |y − x|) |y − x| y−x ε ε − vn ≤ + 2 c1 ≤ |g(x, vn , |y − x|)| + 2 c1 = ε, |y − x| 2 (4c1 ) and we conclude that f (x) is diﬀerentiable at x.

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(iv) From part (ii), ∇fA (x) exists if and only if ΠA (x) is a singleton. Inequality (3.10) follows directly from the inequality ∀x and y ∈ RN , ∀ p(x) ∈ ΠA (x),

d2A (y) ≤ |p(x) − y|2

since d2A (y) − |y|2 ≤ |p(x) − x + x − y|2 − |y|2 ≤ |p(x) − x|2 + |x − y|2 + 2 (p(x) − x) · (x − y) − |y|2 ≤ |p(x) − x|2 − |x|2 + 2 (p(x) − x) · (x − y) + |x − y|2 + |x|2 − |y|2 and − 2fA (y) ≤ −2fA (x) + 2 p(x) · (x − y) − 2 x · (x − y) + |x − y|2 + |x|2 − |y|2 ≤ −2fA (x) − 2 p(x) · (y − x) ⇒ fA (y) ≥ fA (x) + 2 p(x) · (y − x). Inequality (3.11) is (3.10) rewritten. Inequality (3.12) follows by adding inequality (3.10) to the same inequality with x and y permuted. (v) From the equivalences in (ii), when d2A is diﬀerentiable at x, then ΠA (x) = {pA (x)} is a singleton, and from part (i) ∇d2A (x) = 2(x − pA (x)), which yields the expression for pA (x). When x ∈ A, ΠA (x) = {x}, and by substitution ∇d2A (x) = 0. Finally, it is suﬃcient to prove the continuity of pA . Given x ∈ RN \Sk (A), consider a sequence {xn } ⊂ RN \Sk (A) such that xn → x and the associated sequence {pA (xn )} ⊂ A. For all n |pA (xn ) − x| ≤ |pA (xn ) − xn | + |xn − x| ≤ |pA (x) − xn | + |xn − x| ≤ |pA (x) − x| + 2|xn − x| < dA (x) + 2r,

def

r = sup |xn − x| < ∞. n

So the sequence {pA (xn )} ⊂ A is bounded and there exists p ∈ A and a subsequence, still denoted by {pA (xn )}, such that pA (xn ) → p. Therefore dA (x) ← dA (xn ) = |xn − pA (xn )| → |x − p| ⇒ ∃p ∈ A,

dA (x) = |x − p|

⇒ pA (x) = p

and pA is continuous at x since p = pA (x) is independent of the choice of the converging subsequence. To complete the proof of identities (3.14), recall that the projection of each point of A onto A being a singleton, Sk (A) ⊂ RN \A. Therefore, for any x ∈ Sk (A), we have dA (x) > 0 and if ∇d2A (x) does not exist, then ∇dA (x) cannot exist. This sharpens the characterization (3.7) of Sk (A). From this we readily get the characterization of Ck (A) = Sing (∇dA )\Sk (A) and Ck (A) ⊂ A. This last

3. Projection, Skeleton, Crack, and Diﬀerentiability

291

property can be improved to Ck (A) ⊂ ∂A by noticing that for x ∈ int A, dA (x) = 0 and ∇d2A (x) = ∇dA (x) = 0. (vi) For all x ∈ int A, dA (x) = 0 and hence ∇dA (x) = 0 is constant and, a fortiori, continuous in int A. For x ∈ RN \(Sk (A) ∪ A), dA (x) > 0 and for all v ∈ RN and t > 0 suﬃciently small dA (x + tv)2 − dA (x)2 1 dA (x + tv) − dA (x) = t t dA (x + tv) + dA (x) dd2 (x; v) 1 1 ⇒ ddA (x; v) = A = ∇d2 (x) · v ⇒ ∇dA (x) = ∇d2 (x). 2dA (x) 2dA (x) A 2dA (x) A By continuity of ∇d2A (x) from part (iv) and continuity of dA , ∇dA (x) is continuous since dA (x) > 0. Moreover, from part (iv), ∇dA (x) = 2(x − pA (x))/2dA (x) and |∇dA (x)| = 1. If ∇dA (x) exists in some x ∈ A, then dA (x) = 0 and for all v ∈ RN , the diﬀerential quotient converges: dA (x + tv) dA (x + tv) − dA (x) = lim = ∇dA (x) · v t0 t t ⇒ ∀v, 0 ≤ ±∇dA (x) · v ⇒ ∇dA (x) = 0.

0 ≤ lim

t0

Since ∇dA exists almost everywhere, ∇dA = 0 almost everywhere in A. (vii) The function dA is Lipschitzian and the function d2A is locally Lipschitzian. By Rademacher’s theorem they are both diﬀerentiable almost everywhere. The other properties follow from parts (iii), (iv), and (v). By deﬁnition Sk (∂A) = {x ∈ RN : ∇d2∂A (x)} ⊂ RN \∂A = RN \A∪RN \A. If x ∈ Sk (∂A) ∩ RN \A, then d∂A (x) = dA (x), ∇d2A (x), and x ∈ Sk (A); if x ∈ Sk (∂A) ∩ RN \A, then d∂A (x) = dA (x), ∇d2A (x), and x ∈ Sk (A). Conversely, if x ∈ Sk (A) ⊂ RN \A, then d∂A (x) = dA (x), ∇d2∂A (x), and x ∈ Sk (∂A); if x ∈ Sk (A) ⊂ RN \A, then d∂A (x) = dA (x), ∇d2∂A (x), and x ∈ Sk (∂A). Finally, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ ∂A ∪ ∂A = ∂(∂A). The property that Ck (∂A) = Ck (A) ∪ Ck (A) is a consequence of the following property for x ∈ ∂A: ∇d∂A (x) exists if and only if ∇dA (x) and ∇dA (x) exist. Indeed, for x ∈ ∂A, d∂A (x) = dA (x) = dA (x) = 0. From part (v), if ∇d∂A (x) exists, it is 0. But ∂A ⊂ A implies d∂A ≥ dA and for all v ∈ RN and t > 0 0≤

dA (x + tv) − dA (x) d∂A (x + tv) − d∂A (x) ≤ →0 t t

⇒ ∇dA (x) = 0

and similarly ∇dA (x) = 0. Conversely, from part (v), if ∇dA (x) and ∇dA (x) exist, they are 0. Moreover, d∂A (x) = max{dA (x), dA (x)}, where dA (x + tv) − dA (x) dA (x + tv) − dA (x) d∂A (x + tv) − d∂A (x) 0≤ = max , . t t t Since both ∇dA (x) = 0 and ∇dA (x) = 0, then ∇d∂A (x) = 0. Finally, it is easy to check that ∂A ∪ ∂A = ∂(∂A). The other properties for ∂A = ∅ are obvious.

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Chapter 6. Metrics via Distance Functions

(viii) From part (i), if ∇dA (x) exists for x ∈ A, it is equal to 0. Hence χA (x) = 1 = 1 − |∇dA (x)| on A\Ck (A). From the deﬁnition of Sk (A), ∇dA (x) exists and |∇dA (x)| = 1 for all points in int A\Sk (A). Hence from part (i), χA (x) = 0 = 1 − |∇dA (x)| = 0 on A\Sk (A). From this, we get the ﬁrst line of identities in RN \Sing (∇dA ). Since m(Sing (∇dA )) = 0, the above identities are satisﬁed almost everywhere in RN . We get the same results with A in place of A. (ix) If x ∈ A, ΠA (x) = {x} and there is nothing to prove. If x ∈ / A, then dA (x) = |x − p| > 0 and p = x for all p ∈ ΠA (x). First dA (xα ) ≤ |xα − p| = α |x − p|. If the inequality is strict, then there exists pα ∈ A such that |xα − pα | = dA (xα ) and |x − pα | ≤ |x − xα | + |xα − pα | = (1 − α)|x − p| + dA (xα ) < |x − p| = dA (x). This contradicts the minimality of dA (x) with respect to A. Therefore, dA (xα ) = α |x − p| = |xα − p|

⇒ p ∈ ΠA (xα ).

Now for any pα ∈ ΠA (xα ), pα ∈ A and |pα − x| ≤ |pα − xα | + |xα − x| = α |x − p| + (1 − α) |p − x| = |p − x| = dA (x) and pα ∈ ΠA (x). Hence ΠA (xα ) ⊂ ΠA (x).

4 4.1

W 1,p -Metric Topology and Characteristic Functions Motivations and Main Properties

0,1 Since distance functions are locally Lipschitzian, they belong to Cloc (RN ) and hence 1,p N to Wloc (R ) for all p ≥ 1. Thus the constructions of section 2.1 can be repeated 1,p with Wloc (D) in place of Cloc (D) to generate new W 1,p -metric topologies on the family Cd (D). One big advantage is that the W 1,p -convergence of sequences will imply the Lp -convergence of the corresponding characteristic functions of the closure of the sets and hence the convergence of volumes (cf. Theorem 3.3 (viii)). In the uniform Hausdorﬀ topology, the convergence of volumes and perimeters is usually lost. In general, the measure of a closed set is only upper semicontinuous with respect to that topology. This includes Lebesgue and Hausdorﬀ measures. The next example shows that the Hausdorﬀ metric convergence is not suﬃcient to get the Lp -convergence of the characteristic functions of the closure of the corresponding sets in the sequence. The volume function is only upper semicontinuous with respect to the Hausdorﬀ topology. The perimeters do not converge either as illustrated by the example of the staircase of Example 6.1 and Figure 5.7 in Chapter 5, where the volumes converge but not the perimeters.

4. W 1,p -Metric Topology and Characteristic Functions

293

Example 4.1. Denote by D = ] − 1, 2[ × ] − 1, 2[ the open unit square in R2 , p ≥ 1 a real number, and for each n ≥ 1 deﬁne the sequence of closed sets 2k 2k + 1 def ≤ x1 ≤ , 0 ≤ k < n, 0 ≤ x2 ≤ 1 . An = (x1 , x2 ) ∈ D : 2n 2n

set A4

set A8

set S

Figure 6.3. Vertical stripes of Example 4.1. This deﬁnes n vertical stripes of equal width 1/2n each distant of 1/2n (cf. Figure 6.3). Let S = [0, 1] × [0, 1] be the closed unit square. Clearly, for all n ≥ 1, m(An ) = ∀x ∈ S,

1 , 2

dAn (x) ≤

PD (An ) = 2n + 1, 1 , 4n

∇dAn Lp (D) ≥ 2−1/p ,

where m(An ) is the surface and PD (An ) the perimeter of An . Hence dAn → dS

in C(D),

m(S) = 1,

PD (S) = 4.

But m(An ) = m(An ) =

1 1 = m(S) 2

PD (An ) PD (S),

χAn

⇒ χAn χS in Lp (D), 1 χS in Lp (D)-weak. 2

Since the characteristic functions do not converge, the sequence {∇dAn } does not converge in Lp (D)N and {dAn } does not converge in W 1,p (D). Theorem 4.1. Let µ be a measure5 and let D ⊂ RN be bounded open such that µ(D) < ∞. (i) The map dA → µ(A) : Cd (D) → R

(4.1)

is upper semicontinuous with respect to the topology of uniform convergence (Hausdorﬀ topology on Cd (D)). 5 A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outer measure in most texts.

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Chapter 6. Metrics via Distance Functions

(ii) The map dA → µ(int A) : Cdc (D) → R

(4.2)

is lower semicontinuous with respect to the topology of uniform convergence (Hausdorﬀ complementary topology on Cdc (D)). Note that µ can be the Lebesgue measure but also any of the Hausdorﬀ measures. Proof. We prove (ii). The proof of (i) is similar and simpler. It is suﬃcient to work with open sets Ω = int A = A. Let {Ωn } be a sequence of open subsets of D such that dΩn → dΩ in C(D) for some open Ω ⊂ D. Given ε > 0, there exists N such that for all n > N dΩn − dΩ }C(D) ≤ ε

⇒ Ωn ⊂ (Ω)ε .

By deﬁnition of a measurable set µ(Ωn ) = µ(Ωn ∩ D) = µ(D) − µ(D ∩ Ωn ) ≥ µ(D) − µ(D ∩ (Ω)ε ) ⇒ lim inf µ(Ωn ) ≥ µ(D) − µ(D ∩ (Ω)ε ). n→∞

But D ∩ (Ω)ε is monotonically decreasing to ∩ε>0 D ∩ (Ω)ε = D ∩ Ω and hence lim inf µ(Ωn ) ≥ µ(D) − lim µ(D ∩ (Ω)ε ) = µ(D) − µ(D ∩ Ω) = µ(Ω) n→∞

ε→0

since D ∩ Ω = Ω. Theorem 4.2. Let D be an open (resp., bounded open) subset of RN . 1,p (D) (resp., W 1,p (D)) on Cd (D) and Cdc (D) (i) The topologies induced by Wloc are all equivalent for p, 1 ≤ p < ∞. 1,p (D) (resp., W 1,p (D)) for p, 1 ≤ p < ∞, and (ii) Cd (D) is closed in Wloc def

ρD ([A2 ], [A1 ]) =

∞ dA2 − dA1 W 1.p (B(0,n)) 1 n 2 1 + dA2 − dA1 W 1.p (B(0,n)) n=1 def

(resp., ρD ([A2 ], [A1 ]) = dA2 − dA1 W 1,p (D) ) deﬁnes a complete metric structure on F(D). For p, 1 ≤ p < ∞, the map 1,p dA → χA = 1 − |∇dA | : Cd (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and nonempty subsets A1 and A2 of D χA2 − χA1 Lp (K) ≤ ∇dA2 − ∇dA1 Lp (K) ≤ dA2 − dA1 W 1,p (K) .

4. W 1,p -Metric Topology and Characteristic Functions

295

1,p (D) (resp., W01,p (D)) for p, 1 ≤ p < ∞, and (iii) Cdc (D) is closed in Wloc def

ρD (Ω2 , Ω1 ) =

∞ dΩ2 − dΩ1 W 1.p (B(0,n)) 1 n 2 1 + dΩ2 − dΩ1 W 1.p (B(0,n)) n=1 def

(resp., ρD (Ω2 , Ω1 ) = dΩ2 − dΩ1 W 1,p (D) ) deﬁnes a complete metric structure on the family G(D) of open subsets of D. For p, 1 ≤ p < ∞, the map 1,p dΩ → χΩ = |∇dΩ | : Cdc (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and open subsets Ω1 = RN and Ω2 = RN of D χΩ2 − χΩ1 Lp (K) ≤ ∇dΩ2 − ∇dΩ1 Lp (K) ≤ dΩ2 − dΩ1 W 1,p (K) . Proof. (i) The proof is similar to the proof of Theorem 2.3 in Chapter 5. It is suﬃcient to prove it for D bounded open. Since D is bounded it is contained in a suﬃciently large ball of radius c. Therefore, dA (x) = inf |x − y| ≤ |x| + |y| ≤ 2c y∈A

since y ∈ A ⊂ D, and by Theorem 2.1 (vii) |∇dA (x)| ≤ 1 a.e. in D. For all p > 1 the injection of W 1,p (D) into W 1,1 (D) is continuous since dA W 1,1 (D) ≤ dA W 1,p (D) m(D)1/q with 1/p + 1/q = 1. Conversely, we have the continuity in the other direction. For p > 1 and any dA and dB in Cd (D), |dA − dB |p + |∇dA − ∇dB |p dx D |dA − dB | |dA − dB |p−1 + |∇dA − ∇dB | |∇dA − ∇dB |p−1 dx = D p−1 p−1 |dA − dB | + |∇dA − ∇dB | dx ≤ max{(2c) , 2 } D

⇒ dA − dB pW 1,p (D) ≤ (2 max{c, 1})p−1 dA − dB W 1,1 (D) . Therefore, for any ε > 0, pick δ = εp /(2 max{c, 1})p−1 and dA − dB W 1,1 (D) ≤ δ

⇒ dA − dB W 1,p (D) ≤ ε.

(ii) It is suﬃcient to prove it for D bounded open. Let {dAn } be a Cauchy sequence in Cd (D) which converges to some f in W 1,p (D)-strong. By Theorem 2.2 (ii),

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Cd (D) is compact in C(D) and there exist a subsequence, still denoted by {dAn }, and ∅ = A ⊂ D such that dAn → dA in C(D) and, a fortiori, in Lp (D)-strong since D is bounded. By uniqueness of the limit in Lp (D)-strong, f = dA and dAn converges to dA in W 1,p (D)-strong. Therefore Cd (D) is closed in W 1,p (D). For the Lipschitz continuity, recall that the distance function dA is diﬀerentiable almost everywhere in RN for A = ∅. In view of Theorem 3.3 (viii) χA = 1 − |∇dA (x)| almost everywhere in RN . Given two nonempty subsets A1 and A2 of D

⇒ χ A1 ≤ χ A2

|∇dA2 | ≤ |∇dA1 | + |∇dA2 − ∇dA1 | + |∇dA2 − ∇dA1 | ⇒ |χA1 − χA2 |p dx ≤ dA2 − dA1 pW 1,p (D) D

for 1 ≤ p < ∞ and with the ess-sup norm for p = ∞. (iii) Again it is suﬃcient to prove the result for D bounded. In that case Ω = ∅ for all open subsets Ω of D. Let {Ωn } be a sequence of open subsets of D such that {dΩn } is Cauchy in W 1,p (D). By assumption Ωn ⊂ D, Ωn ⊃ D, ∀n ≥ 1,

dΩn = 0 in D

⇒ dΩn ∈ W01,p (D),

and the Cauchy sequence converges to some f ∈ W01,p (D). By Theorem 2.4 (ii), Cdc (D) is compact in C(D) and there exist a subsequence, still denoted by {dΩn }, and an open set Ω ⊂ D such that dΩn → dΩ in C(D) and hence in Lp (D)-strong since D is bounded. By uniqueness of the limit in Lp (D), f = dΩ and the Cauchy sequence dΩn converges to dΩ in W01,p (D). The other part of the proof is similar to that of part (ii).

4.2

Weak W 1,p -Topology

We have the following general result. Theorem 4.3. Let D be a bounded open domain in RN . (i) If {dAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. (ii) If {dAn } converges in C(D), then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. Conversely if {dAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, it converges in C(D). (iii) Cd (D) is compact in W 1,p (D)-weak for all p, 1 ≤ p < ∞.6 (iv) Parts (i) to (iii) also apply to Cdc (D). 6 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E , any closed ball is a compact metrizable space for the weak topology. Since Cd (D) is a bounded subset of the normed reﬂexive separable Banach space W 1,p (D), 1 ≤ p < ∞, the weak compactness of ´ [1, Vol. II, Chap. XII, Cd (D) coincides with the weak sequential compactness (cf. J. Dieudonne sect. 12.15.9, p. 75]).

4. W 1,p -Metric Topology and Characteristic Functions

297

Proof. (i) For D bounded there exists a constant c > 0 such that for all dA ∈ Cd (D) dA (x) ≤ c and |∇dA (x)| ≤ 1 a.e. in D. If {dAn } weakly converges in W 1,p (D), then {dAn } weakly converges in Lp (D),

{∇dAn } weakly converges in Lp (D)N .

By Lemma 3.1 (iii) in Chapter 5 both sequences weakly converge for all p ≥ 1, and hence {dAn } weakly converges in W 1,p (D) for all p ≥ 1. (ii) If {dAn } converges in C(D), then by Theorem 2.2 (i) there exists dA ∈ Cb (D) such that dAn → dA in C(D) and hence in Lp (D). So for all ϕ ∈ D(D)N , ∇dAn · ϕ dx = − dAn div ϕ dx → − dA div ϕ dx = ∇dA · ϕ dx. D

D

D

D

By density of D(D) in L2 (D), ∇dAn → ∇dA in L2 (D)N -weak and hence dAn → dA in W 1,2 (D)-weak. From part (i) it converges in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Conversely, the weakly convergent sequence converges to some f in W 1,p (D). By compactness of Cb (D) there exist a subsequence, still indexed by n, and dA such that dAn → dA in C(D) and hence in W 1,p (D)-weak. By uniqueness of the limit dA = f . Therefore, all convergent subsequences in C(D) converge to the same limit, so the whole sequence converges in C(D). (iii) Consider an arbitrary sequence {dAn } in Cd (D). From Theorem 2.2 (ii) Cd (D) is compact and there exist a subsequence {dAnk } and dA ∈ Cd (D) such that dAnk → dA in C(D). From part (ii) the subsequence weakly converges in W 1,p (D) and hence Cd (D) is compact in W 1,p (D)-weak. Theorem 4.4. Let D be a closed subset of RN and {An }, ∅ = An ⊂ D, be a sequence such that dAn → dA in Cloc (D) for some A, ∅ = A ⊂ D. Then ∀x ∈ RN \A, lim χAn (x) = χA (x) = 0 and ∀x ∈ RN , lim sup χAn (x) ≤ χA (x), n→∞

n→∞

and for any compact subset K of D χAn dx ≤ χA dx. lim sup n→∞

K

K

Corollary 1. Let D = ∅ be a bounded open subset of RN and {Ωn }, ∅ = Ωn ⊂ D, be a sequence of open subsets of D converging to an open subset Ω, ∅ = Ω ⊂ D, of ¯ Then D in the Hausdorﬀ complementary topology: dΩn → dΩ in C(D). ∀x ∈ Ω, lim χΩn (x) = χΩ (x) = 1, n→∞

and

and

n→∞

χΩn dx ≥

lim inf n→∞

∀x ∈ RN , lim inf χΩn (x) ≥ χΩ (x),

D

χΩ dx. D

298

Chapter 6. Metrics via Distance Functions

Proof of Theorem 4.4. For all x ∈ / A, dA (x) > 0 and ∃Nx > 0, ∀n ≥ Nx , dA − dAn ≤ dA (x)/2 ⇒ ∃Nx > 0, ∀n ≥ Nx , dAn (x) ≥ dA (x)/2 > 0 ⇒ ∃Nx > 0, ∀n ≥ Nx ,

x∈ / An and χAn (x) = 0.

So for all x ∈ /A lim χAn (x) = χA (x) = 0.

n→∞

Finally, for all x ∈ A and all n ≥ 1 χAn (x) ≤ 1 = χA (x) and the result follows trivially by taking the limsup of each term. We conclude that lim sup χAn ≤ χA , n→∞

and by using the analogue of Fatou’s lemma for the limsup we get for all compact subsets K of D lim sup χAn dx ≤ χA dx. lim sup χAn dx ≤ n→∞

K n→∞

K

K

In order to get the Lp -convergence of the characteristic functions of the closure of the sets in the sequence, we need the Lp -convergence of the gradients of the distance functions which are related to the characteristic functions of the closure of the sets (cf. Theorem 3.3 (viii)). We have seen in Example 4.1 that the weak convergence of the characteristic functions is not suﬃcient to obtain the strong convergence of the sequence {dAn } to dA in W 1,2 (D). However, if we assume that {χAn } is strongly convergent, it converges to the characteristic function χB of some measurable subset B of D. Is this suﬃcient to conclude that χB = χA ? The answer is negative. The counterexample is provided by Example 6.2 in Chapter 5, where dAn dD in W 1,2 (D)-weak

and

χAn → χB in L2 (D)-strong

for some B ⊂ D such that m(D) = π >

π ≥ m(B) 3

⇒ χD = χB .

Remark 4.1. In view of part (ii) of Theorem 4.2 an optimization problem with respect to the characteristic functions χΩ of open sets Ω in D for which we have the continuity with respect to χΩ can be transformed into an optimization problem with respect to dΩ in W 1,1 (D) since χΩ = |∇dΩ |. This would apply to the transmission problem (4.3)–(4.6) in section 4.1 of Chapter 5.

5. Sets of Bounded and Locally Bounded Curvature

5

299

Sets of Bounded and Locally Bounded Curvature

From Theorem 6.1 in Chapter 5 and Theorem 3.2, it is readily seen that N ∇dA ∈ BVloc RN

⇐⇒

∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N .

The global properties of ∇dA depend only on its local properties around ∂A. This local property is fairly general and is veriﬁed for sets with corners, that is, sets with discontinuities in the orientation of the normal along the boundary. This suggests introducing a new family of sets and motivates the study of their general properties, We shall later see that convex sets have this property and later in Chapter 7 that the larger family of sets with positive reach (cf. section 6) also does. One remarkable property is that such sets are Caccioppoli sets (cf. section 6 in Chapter 5). We give the deﬁnitions for an arbitrary set A, but they can be specialized to A and ∂A (associated with the distance functions dA and d∂A ) that would require a ﬁner terminology such as exterior, interior, or boundary curvature to distinguish them. Deﬁnition 5.1. (i) Given a bounded open holdall D ⊂ RN and A, ∅ = A ⊂ D, the set A is said to be of bounded curvature with respect to D if ∇dA ∈ BV(D)N .

(5.1)

The family of sets with bounded curvature will be denoted by def

BCd (D) = dA ∈ Cd (D) : ∇dA ∈ BV(D)N . (ii) A set A, A = ∅, in RN is said to be of locally bounded curvature if ∇dA ∈ BVloc (RN )N . The family of sets with locally bounded curvature will be denoted by def

BCd = dA ∈ Cd (RN ) : ∇dA ∈ BVloc (RN )N . As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, it is suﬃcient to satisfy the BV property in a neighborhood of each point of the boundary ∂A. Theorem 5.1. A, ∅ = A ⊂ RN , is of locally bounded curvature if and only if ∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N ,

(5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x. Proof. From Theorem 6.1 and Deﬁnition 6.1 in Chapter 5, a function belongs to BVloc (RN ) if and only if for each x ∈ RN it belongs to BV(B(x, ρ)) for some ρ > 0. The equivalence now follows from Theorem 3.2.

300

Chapter 6. Metrics via Distance Functions

Recall that the relaxation of the perimeter of a set was obtained from the norm of the gradient of its characteristic function for Caccioppoli sets. For distance functions χA = 1 − |∇dA | almost everywhere and the gradient of dA also has a jump discontinuity along the boundary ∂A of magnitude at most 1. So it is not too surprising that a set A with bounded curvature is a Caccioppoli set. Theorem 5.2. (i) Let D ⊂ RN be a bounded open holdall and let A, ∅ = A ⊂ D. If ∇dA ∈ BV(D)N , then ∇χA M 1 (D) ≤ 2 D2 dA M 1 (D) ,

χA ∈ BV(D),

and A has ﬁnite perimeter with respect to D. (ii) For any subset A of RN , A = ∅, ∇dA ∈ BVloc (RN )N

⇒

χA ∈ BVloc (RN )

and A has locally ﬁnite perimeter. Proof. Given ∇dA in BV(D)N , there exists a sequence {uk } in C ∞ (D)N such that uk → ∇dA in L1 (D)N

and Duk M 1 (D) → D 2 dA M 1 (D)

as k goes to inﬁnity, and since |∇dA (x)| ≤ 1, this sequence can be chosen in such a way that |uk (x)| ≤ 1 for all k ≥ 1. This follows from the use of molliﬁers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N − χA div V dx = (|∇dA |2 − 1)div V dx = |∇dA |2 div V dx. D

D

D

For each uk |uk |2 div V dx = −2 D

∗

D

[Duk ]uk · V dx = −2

uk · [Duk ]V dx, D

where ∗ Duk is the transpose of the Jacobian matrix Duk and |uk |2 div V dx ≤ 2 |uk | |Duk | |V | dx D

D

≤ 2 Duk L1 V C(D) ≤ 2 Duk M 1 V C(D) since for W 1,1 (D)-functions ∇f L1 (D)N = ∇f M 1 (D)N . Therefore, χ div V dx = |∇dA |2 div V dx ≤ 2D2 dA M 1 V A C(D) D

D

as k goes to inﬁnity, where D2 dA is the Hessian matrix of second-order partial derivatives of dA . Finally, ∇χA ∈ M 1 (D)N .

5. Sets of Bounded and Locally Bounded Curvature

5.1

301

Examples

It is useful to consider the following three simple illustrative examples (cf. Figure 6.4). By convexity, they are all of locally bounded curvature. Example 5.1 (Half-plane in R2 ). Consider the domain A = {(x1 , x2 ) : x1 ≤ 0},

∂A = {(x1 , x2 ) : x1 = 0}.

It is readily seen that (0, 0), x1 < 0, dA (x1 , x2 ) = max{x1 , 0}, ∇dA (x1 , x2 ) = (1, 0), x1 > 0, ϕ dH1 , ∂12 dA = ∂21 dA = ∂22 dA = 0, ∂11 dA , ϕ = ∂A ∆dA , ϕ = ϕ dH1 ⇒ ∆dA = H1 . ∂A

Thus ∆dA is the one-dimensional Hausdorﬀ measure of ∂A.

Figure 6.4. ∇dA for Examples 5.1, 5.2, and 5.3. Example 5.2 (Ball of radius R > 0 in R2 ). Consider the domain A = {x ∈ R2 : |x| ≤ R},

∂A = {x ∈ R2 : |x| = R}.

Clearly dA (x) = max{0, |x| − R},

∇dA (x) =

2π

∂11 dA , ϕ =

2

R cos (θ)ϕ dθ + 0

2π

A

x/|x|, (0, 0),

|x| > R, |x| < R,

x22 ϕ dx, (x21 + x22 )3/2

x21 ϕ dx, 2 2 3/2 0 A (x1 + x2 ) 2π x2 x1 R cos(θ) sin(θ)ϕ dθ + ϕ dx, ∂12 dA , ϕ = ∂21 dA , ϕ = 2 + x2 )3/2 (x 0 A 1 2 1 ϕ dH1 + ϕ dx, ∆dA , ϕ = 2 2 1/2 ∂A A (x1 + x2 ) ∂22 dA , ϕ =

R sin2 (θ)ϕ dθ +

302

Chapter 6. Metrics via Distance Functions

where H1 is the one-dimensional Hausdorﬀ measure. ∆dA contains the onedimensional Hausdorﬀ measure of the boundary ∂A plus a term that corresponds to the volume integral of the mean curvature over the level sets of dA in A. Example 5.3 (Unit square in R2 ). Consider the domain A = {x = (x1 , x2 ) : |x1 | ≤ 1, |x2 | ≤ 1}. Since A is symmetrical with respect to both axes, it is suﬃcient to specify dA in the ﬁrst quadrant. We use the notation Q1 , Q2 , Q3 , and Q4 for the four quadrants in the counterclockwise order and c1 , c2 , c3 , and c4 for the four corners of the square in the same order. We also divide the plane into three regions: D1 = {(x1 , x2 ) : |x2 | ≤ min{1, |x1 |}} , D2 = {(x1 , x2 ) : |x1 | ≤ min{1, |x2 |}} , D3 = {(x1 , x2 ) : |x1 | ≥ 1 and |x2 | ≥ 1} . Hence

min{x2 − 1, 0}, x ∈ D2 ∩ Q1 , dA (x) = |x − c1 |, x ∈ D3 ∩ Q1 , min{x1 − 1, 0}, x ∈ D1 ∩ Q1 ,

∇dA (x) =

(0, 1), x−c1

x ∈ D2 ∩ Q1 and x2 > 1, |x−c1 | , x ∈ D3 ∩ Q1 ,

(1, 0), (0, 0),

x ∈ D1 ∩ Q1 and x1 > 1, x ∈ Q1 , x1 < 1 and x2 < 1,

(x2 − ci,2 )2 ϕ dx + ϕ dH1 , |x − ci |2 ∂A∩Qi ∩D1 i=1 D3 ∩Qi 4 (x1 − ci,1 )2 ∂22 dA , ϕ ϕ dx + ϕ dH1 , |x − ci |2 ∂A∩Qi ∩D2 i=1 D3 ∩Qi 4 (x2 − ci,2 )(x1 − ci,1 ) ∂12 dA , ϕ = ∂21 dA , ϕ = ϕ dx, |x − ci |2 i=1 D3 ∩Qi 4 1 ϕ dx + ∆dA , ϕ = ϕ dH1 . |x − ci | ∂A i=1 D3 ∩Qi ∂11 dA , ϕ =

4

Notice that the structure of the Laplacian is similar to that observed in the previous examples. The norm D2 dA M 1 (Uh (A)) is decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary ∂A.

6. Reach and Federer’s Sets of Positive Reach

303

Example 5.4. Let N = 2. For the ﬁnite square and the ball of ﬁnite radius, lim ∆dA M 1 (Uh (∂A)) = H1 (∂A),

h0

where H1 is the one-dimensional Hausdorﬀ measure.

6

Reach and Federer’s Sets of Positive Reach

The material in this section is mostly taken from the pioneering 1959 work of H. Federer [3, sect. 4], where he introduced the notion of reach and the sets of positive reach.

6.1

Deﬁnitions and Main Properties

By deﬁnition, A∩Sk (A) = ∅. For a ∈ int A, there exists r > 0 such that Br (a) ⊂ A. Thus Sk (A) ∩ Br (a) = ∅ and sup {r > 0 : Sk (A) ∩ Br (a) = ∅} > 0.

(6.1)

/ Sk (A) or a ∈ Sk (A). If a ∈ / Sk (A), there exists r > 0 If a ∈ ∂A, then either a ∈ such that Br (a) ∩ Sk (A) = ∅. Thus Br (a) ∩ Sk (A) = ∅ and (6.1) is veriﬁed. If a ∈ Sk (A), then for all r > 0, Br (a) ∩ Sk (A) = ∅. This leads to the following deﬁnition. Deﬁnition 6.1. Given A, ∅ = A ⊂ RN , and a point a ∈ A def

reach (A, a) =

0, if a ∈ ∂A ∩ Sk (A), sup {r > 0 : Sk (A) ∩ Br (a) = ∅} , otherwise, def

reach (A) = inf {reach (A, a) : a ∈ A} . The set A is said to have positive reach if reach (A) > 0. Remark 6.1. (1) Since Sk (A) = Sk (A), reach (A) = reach (A). (2) For any set A, Sk (A) ⊂ Sk (∂A) and ∀a ∈ ∂A,

0 ≤ reach (∂A, a) ≤ reach (A, a) ≤ +∞.

(3) For all a ∈ / ∂A ∩ Sk (A), reach (A, a) > 0. Lemma 6.1. Let ∅ = A ⊂ RN . The following statements are equivalent: (i) There exists a ∈ A such that reach (A, a) = +∞. (ii) Sk (A) = ∅.

304

Chapter 6. Metrics via Distance Functions

(iii) For all a ∈ A, reach (A, a) = +∞. (iv) reach (A) = +∞. Proof. (i) ⇒ (ii) If there exists a ∈ A such that reach (A, a) = +∞, then RN = ∪r>0 Br (a) ⊂ RN \Sk (A) and Sk (A) = ∅. (ii) ⇒ (iii) If Sk (A) = ∅, for all a ∈ A reach (A, a) = +∞. (iii) ⇒ (iv) If for all a ∈ A reach (A, a) = +∞, then reach (A) = inf a∈A reach (A, a) = +∞. (iv) ⇒ (i) is obvious. Remark 6.2. We shall see later that all nonempty convex sets A have positive reach and that reach (A) = +∞. Bounded domains and submanifolds of class C 2 also have positive reach. Theorem 6.1. Let A, ∅ = A ⊂ RN , be such that Sk (A) = ∅. (i) The function a → reach (A, a) : A → R is continuous. (ii) If reach (A) > 0, then for all h, 0 < h < reach (A), Uh (A) ∩ Sk (A) = ∅. Conversely, if there exists h > 0 such that Uh (A) ∩Sk (A) = ∅, reach (A) ≥ h, and A has positive reach. (iii) If reach (A) > 0, then A ∩ Sk (A) = ∅. If A ∩ Sk (A) = ∅ and A is bounded, then reach (A) > 0. Remark 6.3. When A is unbounded and A ∩ Sk (A) = ∅, we can possibly have reach (A) = 0 even if for all a ∈ A, reach (A, a) > 0 as readily seen from the following example: def

A = {(x1 , x2 ) : |x2 | ≥ ex1 } , where Sk (A) = {(x1 , 0) : x1 ∈ R} . Proof. (i) We prove that the function is both lower and upper semicontinuous. For h, reach (A, a) > h, there exists r, h < r < reach (A, a), such that Br (a) ∩ Sk (A) = ∅. Let a ∈ Bδ (a), δ = (r − h)/2 > 0. Then r − δ = (r + h)/2 > h and for all a ∈ Bδ (a)∩A, Br−δ (a ) ⊂ Br (a). Thus Br−δ (a )∩Sk (A) ⊂ Br (a)∩Sk (A) = ∅, reach (A, a ) ≥ r − δ > h, and reach (A, a) is lower semicontinuous. Similarly, for k, reach (A, a) < k, there exists r, reach (A, a) > r > k, such that Br (a) ∩ Sk (A) = ∅. Let a ∈ Bδ (a), δ = (k − r)/2 > 0. Then r + δ = (r + k)/2 < k and for all a ∈ Bδ (a) ∩ A, Br (a) ⊂ Br+δ (a ). Thus ∅ = Br (a) ∩ Sk (A) ⊂ Br+δ (a ) ∩ Sk (A), reach (A, a ) ≤ r + δ < k, and reach (A, a) is upper semicontinuous. (ii) By deﬁnition of reach (A) as an inﬁmum. Conversely, for all 0 < r < h and all x ∈ A, Br (x) ∩ Sk (x) = ∅, and reach (A, x) ≥ r. Since this is true for all 0 < r < h, reach (A, x) ≥ h and reach (A) = inf x∈A reach (A, x) ≥ h > 0. (iii) By separation of a closed set and a compact set. Theorem 6.2. Let ∅ = A ⊂ RN . (i) Associate with a ∈ A the sets def

P (a) = v ∈ RN : ΠA (a + v) = {a} ,

def

Q(a) = v ∈ RN : dA (a + v) = |v| .

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta A)∗ .

6. Reach and Federer’s Sets of Positive Reach

305

(ii) Given a ∈ A and v ∈ RN , assume that def

0 < r(a, v) = sup {t > 0 : ΠA (a + tv) = {a}} . Then for all t, 0 ≤ t < r(a, v), ΠA (a + tv) = {a} and dA (a + tv) = t |v|. Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (A). (iii) If there exists h such that 0 < h ≤ reach (A), then for all x ∈ Uh (A)\A and 0 < t < h, ΠA (x(t)) = {pA (x)} and dA (x(t)) = t, where def

x(t) = pA (x) + t ∇dA (x) = pA (x) + t

x − pA (x) . dA (x)

(6.2)

(iv) Given b ∈ A, x ∈ RN \Sk (A), a = pA (x), such that reach (A, a) > 0, then (x − a) · (a − b) ≥ −

|a − b|2 |x − a| . 2 reach (A, a)

(6.3)

(v) Given 0 < r < q < ∞ and x ∈ RN \Sk (A) such that dA (x) ≤ r and reach (A, pA (x)) ≥ q, y ∈ RN \Sk (A) such that dA (y) ≤ r and reach (A, pA (y)) ≥ q, then |pA (y) − pA (x)| ≤

q |y − x|. q−r

(6.4)

(vi) If 0 < reach (A) < +∞, then for all h, 0 < h < reach (A), reach (A) |y − x|, ∀x, y ∈ Ah = {x ∈ RN : dA (x) ≤ h}, reach (A) − h 2 ∇dA (y) − ∇d2A (x) ≤ 2 1 + reach (A) |y − x|, ∀x, y ∈ Ah ; (6.5) reach (A) − h

for all 0 < s < h < reach (A) and all x, y ∈ As,h = x ∈ RN : s ≤ dA (x) ≤ h , reach (A) 1 2+ |y − x| . |∇dA (y) − ∇dA (x)| ≤ s reach (A) − h

|pA (y) − pA (x)| ≤

If reach (A) = +∞, ∀x, y ∈ RN , |pA (y) − pA (x)| ≤ |y − x|, 2 ∇d (y) − ∇d2 (x) ≤ 4 |y − x|; A A for all 0 < s and all x, y ∈ {x ∈ RN : s ≤ dA (x)}, |∇dA (y) − ∇dA (x)| ≤

3 |y − x| . s

(6.6)

306

Chapter 6. Metrics via Distance Functions

(vii) If reach (A) > 0, then, for all 0 < r < reach (A), Ur (A) is a set of class C 1,1 , Ur (A) = Ar , ∂Ur (A) = {x ∈ RN : dA (x) = r}, and Ar = Ur (A) = {x ∈ RN : dA (x) ≥ r}. (viii) For all a ∈ A,

dA (a + th) =0 . h ∈ R : lim inf t0 t

Ta A =

N

In particular, if a ∈ int A, Ta A = RN . Remark 6.4. For part (viii), more properties of Ta A and the normal cone (−Ta A)∗ (cf. Deﬁnitions 2.3 and 2.4 in Chapter 2) can be found in H. Federer [3, sect. 4]. Proof. (i) By deﬁnition for all a ∈ A, a ∈ P (A) a ∈ Q(A)

⇐⇒ ⇐⇒

∀b ∈ A\{a}, ∀b ∈ A\{a},

|a + v − b| > |v|, |a + v − b| ≥ |v|

and P (a) ⊂ Q(a). To prove the convexity, we need the identity |a + v − b|2 − |v|2 = (a − b) · (2v + a − b). The convexity of P (a) and Q(a) now follow from the following identities: for all λ ∈ [0, 1] and v, w ∈ RN , |a + (λv + (1 − λ)w) − b|2 − |(λv + (1 − λ)w|2 = (a − b) · (2((λv + (1 − λ)w) + a − b) = λ (a − b) · (2v + a − b) + (1 − λ) (a − b) · (2w + a − b) = λ |a + v − b|2 − |v|2 + (1 − λ) |a + w − b|2 − |v|2 . Finally, for h ∈ Ta A, there exist sequences {bn } ⊂ A and {εn > 0}, εn 0, such that (bn − a)/εn → h and bn → a. For all v ∈ Q(a), 2

bn − a |bn − a|2 ·v ≤ |bn − a| ≤ 0 εn εn

⇒ h·v ≤0

⇒ h ∈ [−Ta A]∗ .

def

(ii) Let S = {t > 0 : ΠA (a + tv) = {a}}. Since, by assumption, r(a, v) = sup S > 0, then S = ∅ and v = 0. For each t ∈ S, ΠA (a + tv) = {a} and dA (a + tv) = t |v|. In particular for all 0 < s < t, dA (a + sv) ≤ s |v|. For any p ∈ ΠA (a + sv) ⊂ A, dA (a + sv) = |a + sv − p| ≤ s |v| and dA (a + tv) ≤ |a + tv − p| = |a + sv − p + (t − s)v| ≤ |a + sv − p| + (t − s)|v| ≤ s |v| + (t − s)v| = t |v| = dA (a + tv). As a result p ∈ ΠA (a + tv) = {a}, p = a, and necessarily ΠA (a + sv) = {a} and dA (a + sv) = s |v|. This proves that for all t ∈ S, {s : 0 < s ≤ t} ⊂ S and hence for all 0 ≤ t < r(a, v).

6. Reach and Federer’s Sets of Positive Reach

307

Assume that when r = r(a, v) < ∞, a + rv ∈ / Sk (A). Then there exists δ > 0 such that Bδ (a+rv)∩(Sk (A)∪A) = ∅. By continuity of the projection, ΠA (a+rv) = = r |v|> 0 so that a+rv ∈ / A. By Theorem 3.3 (vi), the function {a} and dA (a+rv) ∇dA : RN \ Sk (A) ∪ ∂A → R is continuous and for all x ∈ RN \ Sk (A) ∪ A , ∇dA (x) = (x − pA (x))/|x − pA (x)|. For 0 < t < r, pA (a + tv) = a, dA (a + tv) = t|v|, and ∇dA (a + tv) = v/|v|. By continuity, ∇dA (a + rv) = v/|v|. Since ∇dA is continuous in Bδ (a+rv), the Caratheodory conditions are veriﬁed and there exists ε, 0 < ε < δ, such that the diﬀerential equation x (t) = ∇dA (x(t)),

x(0) = a + rv = a + r |v|

v |v|

has a solution in (−ε, ε). This yields d dA (x(t)) = ∇dA (x(t)) · x (t) = ∇dA (x(t)) · ∇dA (x(t)) = 1 dt ⇒ dA (x(t)) = r|v| + t in (−ε, ε). For −ε < p < q < ε, the length of the curve x(t) is given by q q d dA (x(t)) dt = dA (x(q)) − dA (x(p)) ≤ |x(q) − x(p)| |x (t)| dt = p p dt so that the length of the curve x(t) between the points x(p) and x(q) is less than or equal to the distance |x(q) − x(p)| between x(p) and x(q). Therefore x(t) is a line between x(p) and x(q): x(q) − x(p) , q−p

x(q) − x(p) q−p v x(q) − x(p) ⇒ = x (0) = ∇dA (x(0)) = ∇dA (a + rv) = q−p |v| v v = a + rv + t ⇒ ∀t, 0 < t < ε, x(t) = x(0) + (t − 0) |v| |v| ⇒ |x(t) − a| = r|v| + t = dA (x(t)) ⇒ pA (x(t)) = a and ΠA (a + r|v| + t) = {a} x(t) = x(p) + (t − p)

x (t) =

and we get the contradiction sup S ≥ r + t/|v| > r = sup S. Therefore, a + rv ∈ / Sk (A). (iii) If there exists 0 < h ≤ reach (A), then for all a ∈ A and all 0 < s < h, Bs (a) ∩ Sk (A) = ∅. For x ∈ Uh (A)\A, 0 < dA (x) < h and for all p ∈ ΠA (x) (dA (x) + h)/2 < h ≤ reach (A)

⇒ B(dA (x)+h)/2 (p) ∩ Sk (A) = ∅. def

Since x ∈ B(dA (x)+h)/2 (p), ΠA (x) is a singleton and p = pA (x). From part (ii), S = def

{t > 0 : ΠA (pA (x) + t ∇dA (x)) = {pA (x)}} = ∅ and rA = r(pA (x), ∇dA (x)) ≥ dA (x) > 0. If rA = +∞, then for all t ≥ 0 we have ΠA (x(t)) = {pA (x)} and dA (x(t)) = t. If rA < +∞, then x(rA ) ∈ Sk (A) and for all ρ > 0, BrA +ρ (pA (x)) ∩ Sk (A) = ∅. This implies that, for all ρ > 0, reach (pA (x), A) ≤ rA + ρ and

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reach (A) ≤ reach (pA (x), A) ≤ rA . Thus, for all 0 < t < h, t < h ≤ reach (A) ≤ rA and we have ΠA (x(t)) = {pA (x)} and dA (x(t)) = t. (iv) If b = a, the inequality is veriﬁed. For b = a, let def

v =

x−a , |x − a|

def

S = {t > 0 : ΠA (a + tv) = {a}} .

By assumption pA (x) = a and x = a + |x − a| v

⇒ |x − a| ∈ S,

sup S ≥ |x − a| > 0,

and, from part (ii), a + sup S v ∈ Sk (A). If reach (A, a) > sup S, then for ρ, reach (A, a) > ρ > sup S, Bρ (a) ∩ Sk (A) = ∅

⇒ Bsup S (a) ∩ Sk (A) = ∅

and this contradicts the fact that a + sup S v ∈ Sk (A). Therefore, reach (A, a) ≤ sup S. Given t, 0 < t < reach (A, a), deﬁne xt = a + tv. Then t ∈ S and t < sup S. Thus ΠA (xt ) = {a}, pA (xt ) = a, and dA (xt ) = |xt − a| = t. For b ∈ A, |xt − b| ≥ |xt − a| = t > 0 and 2 a + t x − a − b ≥ t2 |x − a|

⇒ (a − b) · (x − a) ≥ −|a − b|2

|x − a| . 2t

To conclude, choose a sequence {tn }, 0 < tn < reach (A, a), such that tn → reach (A, a) in the above inequality. (v) Apply the inequality of part (iii) twice to the triplets (a, x, b) and (b, y, a): |x − a| r ≥ −|a − b|2 , 2 reach (A, a) 2q |y − b| r ≥ −|a − b|2 . (b − a) · (y − b) ≥ −|a − b|2 2 reach (A, b) 2q

(a − b) · (x − a) ≥ −|a − b|2

By adding the two inequalities r (x − a + b − y) · (a − b) ≥ − |a − b|2 q

⇒ (x − y) · (a − b) ≥

⇒ |pA (x) − pA (y)| = |a − b| ≤

r 1− q

|a − b|2

q |x − y|. q−r

(vi) Given 0 < h < r < reach (A), for all a ∈ A, 0 < h < r < reach (A) ≤ reach (A, a)

and Br (a) ∩ Sk (A) = ∅.

Therefore Ah ⊂ Ur (A) = ∪a∈A Br (a) ⊂ RN \Sk (A) and from (iv) ∀x, y ∈ Ah , ∀r, h < r < reach (A),

|pA (x) − pA (y)| ≤

r |x − y|. r−h

6. Reach and Federer’s Sets of Positive Reach

309

The function r → r/(r − h) : (h, ∞) → R is strictly decreasing to 1. If reach (A) = +∞, then for all a ∈ A, reach (A, a) = +∞ and by letting r go to inﬁnity ∀x, y ∈ Ah ,

|pA (x) − pA (y)| ≤ |x − y|.

If reach (A) is ﬁnite, then for all a ∈ A, reach (A, a) ≥ reach (A) > r and by letting r go to reach (A) ∀x, y ∈ Ah ,

|pA (x) − pA (y)| ≤

reach (A) |x − y|. reach (A) − h

The inequalities for ∇d2A follow from identity (3.13) in Theorem 3.3 (v). Finally, the Lipschitz continuity of ∇dA follows from identity (3.15) in Theorem 3.3 (vi). For all pairs 0 < s < h < reach (A) and all x, y ∈ {x ∈ RN : s ≤ dA (x) ≤ h} y − pA (y) x − pA (x) |∇dA (y) − ∇dA (x)| = − dA (y) dA (x) |(y − pA (y)) dA (x) − (x − pA (x)) dA (y)| = dA (x) dA (y) |(y − pA (y)) (dA (x) − dA (y))| |[(y − pA (y)) − (x − pA (x))] dA (y)| + ≤ dA (x) dA (y) dA (x) dA (y) 1 ≤ (|dA (x) − dA (y)| + |y − x| + |pA (y) − pA (x)|) dA (x) 1 reach (A) ≤ 2+ |y − x| . s reach (A) − h When reach (A) = +∞, for all s > 0 and all x, y ∈ {x ∈ RN : s ≤ dA (x)} |∇dA (y) − ∇dA (x)| ≤

3 |y − x| . s

(vii) For 0 < r < reach (A), there exists h such that 0 < r < h < reach (A). N The open set Ur (A) is deﬁned via the r-level set d−1 A {r} = {x ∈ R : dA (x) = r} of −1 the function dA . From part (vi), for each x ∈ dA {r}, there exists a neighborhood V (x) = Bρ (x), ρ = min{r, h − r}/2, of x in Uh (A)\A such that dA ∈ C 1,1 (V (x)). Therefore, ∇dA exists and is not zero on the level set ∂Ar = d−1 A {r} since |∇dA | = 1. From Theorem 4.2 in Chapter 2, the set Ur (A) is of class C 1,1 , Ur (A) = Ar , and ∂Ur (A) = d−1 A {r}, and Ar = Ur (A). (viii) Given h ∈ Ta A (cf. Deﬁnition 2.3 in Chapter 2), there exist sequences {an } ⊂ A and {εn }, εn 0, such that (an − a)/εn → h. For each n, dA (a + th) dA (a + εn h) |a + εn h − an | an − a → 0 ⇒ lim inf = 0. 0≤ ≤ = h − t0 εn εn εn t

310

Chapter 6. Metrics via Distance Functions

Conversely, there exists {tn > 0}, tn 0, such that dA (a + tn h)/tn → 0. This means that there exists a sequence of projections {an } ⊂ A such that h − an − a = |a + tn h − an | = dA (a + tn h) → 0 tn tn tn and h ∈ Ta A. The next theorem summarizes several characterizations of sets of positive reach. Note that condition (ii) is a global condition on the smoothness of d2A in the tubular neighborhood Uh (A). Theorem 6.3. Given ∅ = A ⊂ RN , the following conditions are equivalent: (i) A has positive reach, that is, reach (A) > 0. (ii) There exists h > 0 such that d2A ∈ C 1,1 (Uh (A)). (iii) There exists h > 0 such that for all s, 0 < s < h, dA ∈ C 1,1 (Us,h (A)). (iv) There exists h > 0 such that dA ∈ C 1 (Uh (A)\A). (v) There exists h > 0 such that for all x ∈ Uh (A), ΠA (x) is a singleton. Proof. (i) ⇒ (ii) For all h, 0 < h < reach (A), and all a ∈ A, Bh (a) ∩ Sk (A) = ∅. Hence for all x ∈ Uh (A) ΠA (x) is a singleton and ∇d2A (x) exists. The functions d2A and ∇d2A are bounded and uniformly continuous in Uh (A): ∀x ∈ Uh (A),

d2A (x) ≤ h2 ,

∇d2A (x) = |2(x − pA (x)| = 2 dA (x) ≤ 2h,

|d2A (y) − d2A (x)| = (|dA (y) − dA (x)) |dA (y) − dA (x)| ≤ 2h |y − x|, 2 reach (A) 2 ∀x, y ∈ Uh (A), ∇dA (y) − ∇dA (x) ≤ 2 1 + |y − x| reach (A) − h

∀x, y ∈ Uh (A),

from inequality (6.5) of Theorem 6.2 (vi). By deﬁnition, d2A ∈ C 1,1 (Uh (A)). (ii) ⇒ (iii) From the proof of Theorem 6.2 (vi). (iii) ⇒ (iv) By assumption, for all 0 < s < h, dA ∈ C 1 (Us,h (A)). Since Uh (A)\A = ∪0<s 0 and ∇d2A (x) = 2 dA (x)∇dA (x) exists; from Theorem 3.3 (v), for all x ∈ A, ∇d2A (x) = 0. This means that ΠA (x) is a singleton in Uh (A) = Uh (A)\A ∪ A. (v) ⇒ (i) By assumption, for all a ∈ A and all x ∈ Bh (a), Bh (a) ∩ Sk (A) = ∅. This implies that for all a ∈ A, reach (A, a) ≥ h, which ﬁnally implies that reach (A) ≥ h > 0.

6.2

C k -Submanifolds

For arbitrary closed submanifolds M of RN of codimension greater than or equal to one, ∇dM has a discontinuity along M and generally does not exist. In that case the smoothness of M is related to the existence and the smoothness of ∇d2M in a neighborhood of M that is, a fortiori, locally of positive reach. The following analysis of the smoothness of M was given by J.-B. Poly and G. Raby [1] in 1984.

6. Reach and Federer’s Sets of Positive Reach

311

Theorem 6.4. Let A be a closed nonempty subset of RN and k ≥ 2 be an integer (k = ∞ and ω included7 ). Then sing k A = A ∩ sing k d2A , where sing k d2A = RN \reg k d2A , sing k A = RN \reg k A, and def

reg k d2A = x ∈ RN : d2A is C k in a neighborhood of x , def

reg k A = x ∈ A : A is a C k -submanifold of RN in a neighborhood of x . It is more interesting to rephrase their theorem in terms of regularities than singularities. Theorem 6.5. Let ∅ = A ⊂ RN and let k ≥ 2 be an integer (k = ∞ and ω included), and x ∈ A.8 (i) d2A is C k in a neighborhood of a point x ∈ A if and only if A is a C k submanifold in a neighborhood of x. (ii) Under conditions (i), the dimension of A in x is equal to the rank of DpA (x) (dimension of the subspace DpA (RN )) and DpA (x) is the orthogonal projector onto the tangent space to A in x. (iii) If ∂A = ∅, then A = RN . If ∂A = ∅ and d2A is C 2 in a neighborhood of x ∈ A, then there exists an open neighborhood V (x) of x such that (a) if x ∈ int A, then dA = dRN = 0 in V (x); (b) if x ∈ ∂A, then dA = d∂A in V (x), int A ∩ V (x) = ∅, dA = 0 in V (x), A is a C 2 -submanifold of dimension less than or equal to N − 1 in V (x), and m(∂A ∩ V (x)) = 0. Proof. (i) This follows from Theorem 6.4. (ii) For y ∈ V (x) 1 ∇fA (y) = y − ∇d2A (y) = pA (y), def 1 2 2 2 |y| − dA (y) fA (y) = ⇒ 1 2 2 D fA (y) = I − D2 d2A (y) = DpA (y). 2 Since pA is a projection, pA (pA (y)) = pA (y) implies that DpA ◦ pA DpA = DpA and (DpA )2 = DpA on A. Therefore, if R(x) denotes the image of DpA (x), DpA (x) is an orthogonal projector onto R(x) and I − DpA (x) is an orthogonal projector onto R(x)⊥ and ∀v ∈ RN , 7ω

v = DpA (x)v + [I − DpA (x)]v ∈ R(x) ⊕ R(x)⊥ .

indicates the analytical case. that A can have several connected components with the same smoothness k but diﬀerent dimension. When A = RN , dA is identically zero. Another set of results for H¨ olderian sets was obtained by G. M. Lieberman [1] (see also V. I. Burenkov [1]) by introducing a regularized distance function. 8 Note

312

Chapter 6. Metrics via Distance Functions

Consider the C 1 -mapping Tx (y) = (DpA (x)(pA (y) − x), [I − DpA (x)](y − pA (y))) : W (x) → R(x) ⊕ R(x)⊥ . def

Tx is a C 1 -diﬀeomorphism from a neighborhood U (x) ⊂ V (x) of x onto a neighborhood W (0) of 0 since DTx is continuous in V (x) and def

DTx (y) = (DpA (x)DpA (y), [I − DpA (x)](I − DpA (y))), def

DTx (x) = (DpA (x)DpA (x), [I − DpA (x)](I − DpA (x))) = (DpA (x)DpA (x), [I − DpA (x)](I − DpA (x)) = (DpA (x), I − DpA (x)) = I. By construction Tx (U (x) ∩ A) = V (0) ∩ R(x), ⊥

V (0) ∩ R(x) = {(0, [I − DpA (x)](y − pA (y))) : ∀y ∈ U (x)} , and U (x) ∩ A is a C 1 -submanifold in RN of codimension r, where N − r is the rank of DpA (x). (iii) If ∂A = ∅, A ⊂ A and A is both open and closed. Since A = ∅, then A = RN . Let ∂A = ∅. If x ∈ ∂A, then A = ∅. We claim that int A = ∅. If this is not true, there exists a neighborhood V (x) of x where d2A is C 2 and int A ∩ V (x) = ∅ where dA = 0 and D2 d2A = 0. This implies that DpA = I in int A ∩ V (x). In A ∩ V (x), D2 d2A is continuous and dA > 0. Therefore, ∇dA and D2 dA exist, |∇dA | = 1, and 1 2 2 D dA = ∇dA ∗∇dA + dA D 2 dA 2 ⇒ DpA = I − ∇dA ∗∇dA − dA D 2 dA and DpA ∇dA = 0 since |∇dA |2 = 1 implies that D2 dA ∇dA = 0. As a result the rank of DpA is less than or equal to N − 1 in A ∩ V (x) and equal to N in int A ∩ V (x). This contradicts the continuity of D2 d2A and DpA in x. Therefore, A∩V (x) = ∂A∩V (x), and d∂A = dA in V (x) and, by the previous argument in A ∩ V (x), A is a C 2 submanifold of dimension less than or equal to N − 1 in a neighborhood of x. At this juncture, several interesting remarks can be made. (a) The equivalence in (i) fails in the direction (⇒) for d2A ∈ C 1,1 as seen from Example 6.1 of J.-B. Poly and G. Raby [1]. (b) The equivalence in (i) also fails in the direction (⇐) for a C 1, submanifold, 0 ≤ < 1, as seen from Example 6.2 of S. G. Krantz and H. R. Parks [1]. (c) The square d2A is really pertinent for C k -submanifolds of codimension greater than or equal to one and not for sets of class C k . Indeed, when ∂A = ∅ and d2A is C k , k ≥ 2, in a neighborhood of each point of A, then A is necessarily a C k -submanifold of codimension greater than or equal to one. In particular, A = RN , A = ∂A, and m(A) = 0.

6. Reach and Federer’s Sets of Positive Reach

313

Example 6.1. Consider the half-space in RN def

A = x ∈ RN : x · eN ≤ 0 , dA (x) =

0, x · eN ,

x · eN ≤ 0, x · eN > 0,

∇d2A (x) =

x · eN ≤ 0, x · eN > 0,

0, 2 (x · eN ) eN ,

and d2A is locally C 1,1 . For any point x ∈ ∂A = {x ∈ RN : x · eN = 0}, A ∩ V (x) is not even a manifold in any neighborhood V (x) of x. The equivalence in (i) also fails in the direction (⇐) for C 1, , 0 ≤ < 1, as seen from Example 6.2, that shows that for a set of class C 1, , 0 ≤ < 1, we do not get bA ∈ C 1,λ in a neighborhood of the boundary ∂A. Example 6.2. Consider the two-dimensional domain Ω deﬁned as the epigraph of the function f : def

Ω = {(x, z) : f (x) > z, ∀x ∈ R} ,

def

1

f (x) = |x|2− n

for some arbitrary integer n ≥ 1. Ω is a set of class C 1,1−1/n and ∂Ω is a C 1,1−1/n submanifold of dimension 1. In view of the presence of the absolute value in (0, 0), it is the point where the smoothness of ∂Ω is minimum. We claim that Sk (∂Ω) = Sk (Ω) = {(0, y) : y > 0}

and ∂Ω ∩ Sk (∂Ω) = (0, 0) = ∅.

Because the skeleton is a line touching ∂Ω in (0, 0), d2∂Ω and d2Ω cannot be C 1 in any neighborhood of (0, 0). Indeed, for each (0, y)

d2∂Ω (0, y) = inf x2 + |f (x) − y|2 . x∈R

The point (0, y), y > 0, belongs to Sk (∂Ω) since there exist two diﬀerent points x ˆ that minimize the function def

F (x, y) = x2 + +|f (x) − y|2 .

(6.7)

Since f is symmetric with respect to the y-axis it is suﬃcient to show that there exists a strictly positive minimizer x ˆ > 0. A locally minimizing point x ˆ ≥ 0 must satisfy the conditions 1 1 1 2− n x ˆ1− n = 0, x, y) = 2ˆ x + 2 |ˆ x| −y 2− (6.8) Fx (ˆ n 2 1 2 1 1 1 1 F (ˆ 3− x ˆ2− n − 2 − 1− yx ˆ− n ≥ 0. (6.9) x, y) = 1 + 2 − 2 x n n n n Equation (6.8) can be rewritten as . 1 / 1 1 2ˆ x1− n x ˆn + x ˆ2− n − y (2 − 1/n) = 0,

314

Chapter 6. Metrics via Distance Functions

and x ˆ = 0 is a solution. The second factor can be written as an equation in the def 1/n new variable X = x ˆ and def

g(X) =

n X + X 2n−1 − y = 0. 2n − 1

ˆ since It has exactly one solution X ∀X,

n dg (X) = + (2n − 1)X 2n−2 > 0, dX 2n − 1

ˆ > 0 for g(0) = −y, and g(X) goes to inﬁnity as X goes to inﬁnity. In particular, X n ˆ ˆ y > 0 and X = 0 for y = 0. For y > 0 and x ˆ=X 1 x, y) = 0 F (ˆ 2 x and

1

⇒ y=x ˆ2− n +

1 n x ˆn 2n − 1

(6.10)

2 1 2 1 F (ˆ 3− x ˆ2− n x, y) =1 + 2 − 2 x n n 2 1 1 n 2− n 1− x ˆ − 2− + n n 2n − 1 2 2 2 2n − 1 2n − 1 1− 1 n−1 1 + x ˆ n > 0. =1 − x ˆ2− n = + n n n n

Therefore, x ˆ is a local minimum. To complete the proof for y > 0 we must compare ˆ = 0 with F (0, y) = y 2 for the solution corresponding to x 1

F (ˆ x, y) = x ˆ2 + |ˆ x2− n − y|2 for the solution x ˆ > 0. Again using identity (6.10) 2 2 n F (ˆ x, y) = x ˆ + x ˆn, 2n − 1 2 2 n 2 2 2− 1 1 1 n n x ˆ n = x x ˆ2 ˆ2− n + x ˆn + 2 F (0, y) = x ˆ n+ 2n − 1 2n − 1 2n − 1 2 2 1 1 2n 1 −1 x ˆ2 + x x ˆ2 + x ⇒ F (0, y) − F (ˆ x, y) = ˆ2− n = ˆ2− n > 0. 2n − 1 2n − 1

2

This proves that for y > 0, x ˆ > 0 is the minimizing positive solution. Therefore, for all n ≥ 1, f (x) = |x|2−1/n yields a domain Ω for which the strictly positive part of the y-axis is the skeleton. It is interesting to note that, for each point x, the points (x, f (x)) ∈ ∂Ω are the projections of the point 1 1 n (0, y) = 0, |x|2− n + |x| n 2n − 1

6. Reach and Federer’s Sets of Positive Reach

315

and that the square of the distance function is equal to 2 n 2 xn , b2Ω (0, y) = x2 + 2n − 1 1 ∇bΩ (x, f (x)) = ' (f (x), −1) 1 + f (x)2 1 1 −n = 2n−1 2 2− 2 ((2 − 1/n) x |x| , −1). 1+ |x| n n Finally for all n ≥ 1, f ∈ / C 1,1 . For n = 1, f ∈ C 0,1 , and for n > 1, f ∈ C 1,1−1/n . This gives an example in dimension 2 of a domain Ω of class C 1,λ , 0 ≤ λ < 1, with skeleton converging to the boundary.

6.3

A Compact Family of Sets with Uniform Positive Reach

The following family of sets of positive reach and the associated compactness theorem were ﬁrst given in H. Federer [3, Thm. 4.13, Rem. 4.14]. Theorem 6.6. Given an open (resp., bounded open) holdall D in RN and h > 0, the family def

Cd,h (D) = dA : ∅ = A ⊂ D, reach (A) ≥ h

(6.11)

is closed in Cloc (D) (resp., compact in C(D)). Proof. It is suﬃcient to prove the theorem for a bounded open D. Given any C(D)Cauchy sequence {dAn } ⊂ Cd,h (D), there exists dA ∈ Cd (D) such that dAn → dA in C(D). Choose W open such that W ⊂ Uh (A). Then supx∈W dA (x) < h. Choose r, supx∈W dA (x) < r < h. Then W ⊂ Ur (A) ⊂ Ar ⊂ Uh (A). Choose N such that ∀n > N,

sup dAn (x) < r < h.

x∈W

Then for all n > N , W ⊂ Ur (An ). From Theorem 6.2 (v), for all n > N and ∀x, y ∈ (An )r ,

2 ∇dA (y) − ∇d2A (x) ≤ 2 1 + n n

h h−r

|y − x|.

(6.12)

But W ⊂ Ur (A) ⊂ Ar ⊂ Uh (A), d2An → d2A in C(D), and ∀n > N,

d2An ∈ C 1,1 (W ) ⊂ C 1 (W ).

From Theorem 2.5 in Chapter 2, the embedding C 1,1 (W ) ⊂ C 1 (W ) is compact. Therefore, for all open subsets such that W ⊂ Uh (A), d2An → d2A ∈ C 1 (W ), d2A ∈ C 1 (Ur (A)), and reach (A) ≥ r. Since this is true for all r < h, we ﬁnally get reach (A) ≥ h and dA ∈ Cd,h (D).

316

Chapter 6. Metrics via Distance Functions The compactness of Theorem 6.7 can also be expressed as follows.

Theorem 6.7. Let D be a ﬁxed bounded open subset of RN . Let {An }, An = ∅, be a sequence of subsets of D. Assume that there exists h > 0 such that ∀n,

d2An ∈ C 1,1 (Uh (An )).

(6.13)

Then there exist a subsequence {Ank } and A ⊂ D, A = ∅, such that reach (A) ≥ h and for all r, 0 < r < h, d2A ∈ C 1,1 (Ur (A)), d2An → d2A in C 1 (Ur (A)) k

and

dAnk → dA in C(D).

(6.14)

This condition in an h-tubular neighborhood is generic of other such conditions that will be expressed in diﬀerent function spaces.

7

Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points

The W 1,p -topology turns out to be an appropriate setting for the approximation of the closure of a set by its dilated sets as deﬁned in Deﬁnition 3.2. Theorem 7.1. Let ∅ = A ⊂ RN . For r > 0, Ar = Ur (A), ∂Ar = ∂Ur (A) = d−1 A {r}, Ar = Ur (A), int Ar = Ur (A), dUr (A) = dAr (x) = ∀x ∈ RN ,

dA (x) − r, 0,

if dA (x) ≥ r, if dA (x) < r,

0 ≤ dA (x) − dAr (x) ≤ r,

dUr (A) = dAr (x) → dA (x) uniformly in R

N

dUr (A) = dAr → dA in

1,p Wloc (RN )

and

(7.1) (7.2) (7.3)

as r → 0,

χAr → χA in Lploc (RN )

(7.4)

for all p, 1 ≤ p < ∞. Proof. The ﬁrst line of properties follows from Theorem 3.1. For r > 0, A ⊂ Ur (A) ⊂ Ar and dA (x) ≥ dAr (x) for all x ∈ RN . For x ∈ RN such that dA (x) < r, dAr (x) = 0 and 0 ≤ dA (x) − dAr (x) < r. For x ∈ RN such that dA (x) ≥ r and all a ∈ A and ar ∈ Ar , |x − a| ≤ |x − ar | + |x − ar | ⇒ dA (x) ≤ |x − ar | + dA (ar ) ≤ |x − ar | + r ⇒ dA (x) ≤ dAr (x) + r ⇒ 0 ≤ dA (x) − dAr (x) ≤ r. We show that dA (x) − dAr (x) = r. Assume that dA (x) − dAr (x) < r. There exists a unique p ∈ A such that dA (x) = |x − p| ≥ r > 0. Consider the point def

xr = p + r

x−p |x − p|

⇒ |x − xr | = dA (x) − r

and |xr − p| = r.

7. Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points

317

By deﬁnition, dA (xr ) ≤ |xr − p| = r and xr ∈ Ar . As a result, dA (x) < dAr (x) + r ≤ |x − xr | + r = dA (x) − r + r = dA (x) yields a contradiction. This establishes identity (7.2) and for all x ∈ RN , |dAr (x) − dA (x)| ≤ r → 0 as r goes to zero. When ∇dAr (x) exists at a point ∈ Ar , ∇dAr (x) = 0 (cf. Theorem 3.3 (vi)). This means that ∇dAr = 0 almost everywhere in Ar and that the gradient of dAr is almost everywhere equal to ∇dAr (x) =

∇dA (x), 0,

dA (x) > r, dA (x) ≤ r.

For any compact K ⊂ RN χAr − χA dx. |∇dAr − ∇dA | dx = |∇dA | χAr dx = (1 − χA ) χAr dx = K

K

K

K

Since {K ∩ Ar } is a decreasing family of closed sets as r → 0 and ∩r>0 Ar = A, m(K ∩ Ar ) → m(K ∩ A) (cf. W. Rudin [1, Thm. 1.1.9 (e), p. 16]). Finally, 1,1 ∇dAr − ∇dA L1 → 0 and dAr → dA in Wloc (RN ). S. Ferry [1] proved in 1975 that if N = 2 or 3, then d−1 A {r} is an (N − 1)manifold for almost all r, and that if A is a ﬁnite polyhedron in RN , then d−1 A {r} is an (N − 1)-manifold for all suﬃciently small r. On the other hand, there is a Cantor set K in R4 such that d−1 K {r} is not a 3-manifold for any r between 0 and 1. The set K generalizes to higher dimensions. Generalizing those results, J. H. G. Fu [1] investigated the regularity properties of ∂Ar and proved the following theorem involving the notion of critical point of dA and of Lipschitz manifold. In order to make things simple we use a speciﬁc characterization of the critical points of dA as a deﬁnition (cf. J. H. G. Fu [1, 2] and Figure 6.5). Deﬁnition 7.1. Given ∅ = A ⊂ RN , x ∈ RN \A is said to be a critical point of dA if x ∈ co ΠA (x). The set of all critical points of dA will be denoted by crit (dA ). Deﬁnition 7.2. An m-dimensional Lipschitz manifold is a paracompact metric space M such that there is a system of open sets {Uα } covering M and, for each α, a bi-Lipschitzian homeomorphism ϕα of Uα onto an open subset of Rm . Theorem 7.2. Let A ⊂ RN be compact. The subset C(A) = dA (crit (dA )) ⊂ R is compact: 1/2 N C(A) ⊂ 0, diam (A) and H(N −1)/2 (C(A)) = 0 2N + 2 (the assertion about the measure of C(A) is nontrivial only for N ≤ 3). For all r∈ / C(A), Ar has positive reach and ∂Ar is a Lipschitz manifold.

318

Chapter 6. Metrics via Distance Functions

critical points of A

Figure 6.5. Set of critical points of A.

8

Characterization of Convex Sets

8.1

Convex Sets and Properties of dA

In the convex case the squared distance function is diﬀerentiable everywhere and this property can be used to characterize the convexity of a set. Theorem 8.1. Let ∅ = A ⊂ RN . The following statements are equivalent: (i) There exists a ∈ A such that reach (A, a) = +∞. (ii) Sk (A) = ∅. (iii) For all a ∈ A, reach (A, a) = +∞. (iv) reach (A) = +∞. (v) For all x ∈ RN , ΠA (x) is a singleton.9 (vi) A is convex. (vii) dA is convex. In particular, ∀x, ∀y ∈ RN ,

|pA (y) − pA (x)| ≤ |y − x|

(8.1)

1,1 2,∞ (RN ) (and, a fortiori, to Wloc (RN )). Moreover, if A is and d2A belongs to Cloc convex, then A = A and dA = dA .

Remark 8.1. The convexity of dA implies the convexity of A but not of A. The set A of all points of the unit open ball with rational coordinates has a convex closure but is not convex. 9 This part of the theorem is related to deeper results on the convexity of Chebyshev sets in metric spaces. A subset A of a metric space X is called a Chebyshev set provided that every point x of X has a unique projection pA (x) in A. The reader is referred to V. L. Klee [1] for details and background material.

8. Characterization of Convex Sets

319

Proof. The equivalence of properties (i) to (v) follows from Lemma 6.1. (v) ⇒ (vi) Given x, y ∈ A and λ ∈ [0, 1], consider the point xλ = λx + (1 − λ)y and the continuous function λ → dA (xλ ) : [0, 1] → R is continuous. There exists µ ∈ [0, 1] such that dA (xµ ) = supλ∈[0,1] dA (xλ ). If supλ∈[0,1] dA (xλ ) = 0, then xλ ∈ A. If supλ∈[0,1] dA (xλ ) > 0, then x = y, dA (xµ ) > 0, µ ∈ (0, 1), and xµ ∈ / A. Since Sk (A) = ∅, ∇dA (xµ ) exists and the maximum is characterized by d = 0 ⇒ ∇dA (xµ ) · (x − y) = 0. dA (xλ ) dλ λ=µ By assumption there exists a unique projection pµ = pA (xµ ) ∈ A such that dA (xµ ) = |xµ − pµ | and by Theorem 3.3 (vi) ∇dA (xµ ) =

xµ − pµ . |xµ − pµ |

From inequality (6.6) in Theorem 6.2 (vi) |pµ − x| = |pA (xµ ) − pA (x)| ≤ |xµ − x| = (1 − µ) |x − y|, |pµ − y| = |pA (xµ ) − pA (y)| ≤ |xµ − y| = µ |x − y| ⇒ |x − y| ≤ |pµ − x| + |pµ − y| ≤ |x − y|. Therefore the point pµ lies on the segment between x and y: there exists µ ∈ [0, 1], µ = µ, such that pµ = xµ and xµ − pµ = xµ − xµ = (µ − µ ) (x − y). Finally, µ − µ xµ − xµ xµ − pµ |x − y| = · (x − y) = · (x − y) = ∇dA (xµ ) · (x − y) = 0 |µ − µ | |xµ − xµ | |xµ − pµ | and we get the contradiction x = y. (vi) ⇒ (v) By deﬁnition for each x ∈ RN d2A (x) = inf |z − x|2 = inf |z − x|2 , z∈A

z∈A

and since A is closed and convex and z → |z − x|2 is strictly convex and coercive, there exists a unique minimizing point pA (x) in A and ΠA (x) is a singleton. (vi) ⇒ (vii) Given x and y in RN , there exist x and y in A such that dA (x) = |x−x| and dA (y) = |y−y|. By convexity of A, for all λ, 0 ≤ λ ≤ 1, λ x+(1−λ) y ∈ A and dA (λ x + (1 − λ) y) ≤|λ x + (1 − λ) y − (λ x + (1 − λ) y)| ≤λ |x − x| + (1 − λ) |y − y| = λ dA (x) + (1 − λ) dA (y) and dA is convex in RN . (vii) ⇒ (vi) If dA is convex, then ∀λ ∈ [0, 1], ∀x, y ∈ A,

dA (λx + (1 − λ)y) ≤ λdA (x) + (1 − λ)dA (y).

320

Chapter 6. Metrics via Distance Functions

But x and y in A imply that dA (x) = dA (y) = 0 and hence ∀λ ∈ [0, 1], dA (λx + (1 − λ)y) = 0. Thus λx + (1 − λ)y ∈ A and A is convex. Properties (8.1) follow from Theorem 6.2 and the identity on the complements follows from Theorem 3.5 (ii) in Chapter 5. The next theorem is a consequence of the fact that the gradient of a convex function is locally of bounded variations. Theorem 8.2. For all subsets A of RN such that ∂A = ∅ and A is convex, (i) ∇dA belongs to BVloc (RN )N , (ii) the Hessian matrix D2 dA of second-order derivatives is a matrix of signed Radon measures that are nonnegative on the diagonal, and (iii) dA has a second-order derivative almost everywhere and for almost all x and y in RN , dA (y) − dA (x) − ∇dA (x) · (y − x) − 1 (y − x) · D2 dA (x) (y − x) = o(|y − x|2 ) 2 as y → x. Proof. From the equivalence of (vi) and (vii) in Theorem 8.1 dA is convex and continuous and the result follows from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240, Thm. 2, p. 239, and Aleksandrov’s Thm., p. 242].

8.2

Semiconvexity and BV Character of dA

Since the BV properties in Theorem 8.2 arise solely from the convexity of the distance function dA , the BV property will extend to sets A such that dA is semiconvex. Theorem 8.3. Let A be a nonempty subset of RN such that ∀x ∈ ∂A, ∃ρ > 0 such that dA is semiconvex in B(x, ρ). Then ∇dA ∈ BVloc (RN )N . Proof. For all x ∈ int A, dA = 0 and there exists ρ > 0 such that B(x, ρ) ⊂ int A and the result is trivial. For each x ∈ RN \A, dA (x) > 0. Pick h = dA (x)/2. Then for all y ∈ B(x, h) and all a ∈ A, |y − a| ≥ |x − a| − |y − x| ≥ dA (x) − |y − x| > 2h − h = h and B(x, h) ⊂ RN \Ah . By Theorem 3.2 (i) the function |x|2 /(2h) − dA (x) is locally convex in RN \Ah and, a fortiori, convex in B(x, h). Finally, by assumption for all x ∈ ∂A, there exists ρ > 0 such that dA is semiconvex. Hence for each x ∈ RN , ∇dA belongs to BV(B(x, ρ/2)) for some ρ > 0. Therefore, ∇dA belongs to BVloc (RN )N .

8. Characterization of Convex Sets

321

However, it is never semiconcave in RN for nontrivial sets. Theorem 8.4. (i) Given a subset A of RN , A = ∅, ∃c ≥ 0,

fc (x) = c |x|2 − dA (x) is convex in RN

if and only if ∃h > 0, ∃c ≥ 0,

fc (x) = c |x|2 − dA (x)

is locally convex in Uh (A). (ii) Given a subset A of RN , ∅ = A = RN , c ≥ 0,

fc (x) = c |x|2 − dA (x) is convex in RN .

Proof. (i) If fc is convex in RN , it is locally convex in Uh (A). Conversely, assume that there exist h > 0 and c ≥ 0 such that fc is locally convex in Uh (A). From Theorem 3.2 (i), 1 2 |x| − dA (x) h is locally convex in RN \Ah/2 . Therefore, for c = max{c, 1/h}, the function fc is locally convex in RN and hence convex on RN . (ii) Assume the existence of a c ≥ 0 for which fc is convex in RN . For each y ∈ A, the function x → Fc (x, y) = c |x − y|2 − dA (x) is also convex since it diﬀers from fc (x) by the linear term x → c (|y|2 − 2 x · y). Since ∅ = A = RN , there exist x ∈ A and p ∈ A such that 1 . 2c For any t > 0, deﬁne xt = p − t (x − p) and λ = t/(1 + t) ∈ ]0, 1[ and observe that 0 < dA (x) = |x − p| ≤

def

xλ = λ x + (1 − λ) xt = p and Fc (xλ , p) = 0. But λ Fc (x, p) + (1 − λ) Fc (xt , p) 1 t = [c |x − p|2 − dA (x)] + [c |xt − p|2 − dA (xt )] 1+t 1+t t ≤ [c dA (x)2 − dA (x) + c tdA (x)2 ] 1+t t ≤ dA (x) [(1 + t) c dA (x) − 1] 1+t dA (x) t t − 1 t 1 dA (x) (1 + t) − 1 = , ≤ 1+t 2 1+t 2

322

Chapter 6. Metrics via Distance Functions

since by construction dA (x) > 0 and c dA (x) < 1/2. Therefore for t, 0 < t < 1, the above quantity is strictly negative and we have constructed two points x and xt and a λ, 0 < λ < 1, such that Fc (λ x + (1 − λ) xt , p) = 0 > λ Fc (x, p) + (1 − λ) Fc (xt , p). This contradicts the convexity of the function x → Fc (x, p) and, a fortiori, of fc .

8.3

Closed Convex Hull of A and Fenchel Transform of dA

Another interesting property is that the closed convex hull of A is completely characterized by the convex envelope of dA that can be obtained from the double Fenchel transform of dA . Theorem 8.5. Let A be a nonempty subset of RN : (i) The Fenchel transform d∗A (x∗ ) = sup x∗ · x − dA (x) x∈RN

of dA is given by

d∗A (x∗ ) = σA (x∗ ) + IB(0,1) (x∗ ),

(8.2)

where σA (x∗ ) is the support function of A and IB(0,1) (x∗ ) =

x∈ / B(0, 1), x ∈ B(0, 1),

+∞, 0,

is the indicator function of the closed unit ball B(0, 1) at the origin. (ii) The Fenchel transform of d∗A is given by d∗∗ A = dco A = dco A .

(8.3)

In particular, dco A is the convex envelope of dA . The function dA and the indicator function IA are both zero on A and the Fenchel transform of IA , ∗ = σA = σA , IA

coincides with d∗A on B(0, 1). Proof. (i) From the deﬁnition, d∗A (x∗ ) = sup x∗ · x − dA (x) = sup x∈RN

x∈RN

.

x∗ · x − inf |x − p|

/

p∈A

∗

= sup sup x · x − |x − p| = sup sup x∗ · x − |x − p| x∈RN p∈A

p∈A x∈RN

∗

∗

= sup sup x · p + x · (x − p) − |x − p| p∈A x∈RN ∗

= sup x · p + sup [ x∗ · (x − p) − |x − p| ] x∈RN

p∈A ∗

= sup x · p + sup [ x∗ · x − |x| ] = σA (x∗ ) + IB(0,1) (x∗ ). p∈A

x∈RN

8. Characterization of Convex Sets

323

(ii) We now use the property that σA = σco A : ∗∗ ∗∗ · x∗ − d∗A (x∗ ) = sup x∗∗ · x∗ − σA (x∗ ) − IB(0,1) (x∗ ) d∗∗ A (x ) = sup x x∗ ∈RN x∗ ∈RN . / ∗∗ ∗ ∗ = sup x · x − σA (x ) = sup x∗∗ · x∗ − sup x∗ · x |x∗ |≤1

= sup

|x∗ |≤1

inf (x

∗∗

|x∗ |≤1 x∈co A

x∈co A

∗

− x) · x

≤ sup (x∗∗ − p(x∗∗ )) · x∗ ≤ |x∗∗ − p(x∗∗ )| = dco A (x∗∗ ). |x∗ |≤1

But in fact we have a saddle point. If x∗∗ ∈ co A, pick x∗ = 0 and ∗∗ d∗∗ A (x ) = sup

inf (x∗∗ − x) · x∗ ≥ 0 = dco A (x∗∗ ).

|x∗ |≤1 x∈co A

For x∗∗ ∈ / co A rewrite the function as (x∗∗ − x) · x∗ = (x∗∗ − p(x∗∗ )) · x∗ + (p(x∗∗ ) − x) · x∗ . Since p(x∗∗ ) is the minimizing point of the function |x−x∗∗ |2 over the closed convex set co A it is completely characterized by the variational inequality ∀x ∈ co A,

(p(x∗∗ ) − x∗∗ ) · (x − p(x∗∗ )) ≥ 0.

(8.4)

By choosing the following special x∗ , x∗ =

x∗∗ − p(x∗∗ ) , |x∗∗ − p(x∗∗ )|

we get ∗∗ d∗∗ A (x ) = sup

inf (x∗∗ − x) · x∗

|x∗ |≤1 x∈co A

≥ |x∗∗ − p(x∗∗ )| + (p(x∗∗ ) − x) ·

x∗∗ − p(x∗∗ ) |x∗∗ − p(x∗∗ )|

≥ |x∗∗ − p(x∗∗ )| = dco A (x∗∗ ) in view of (8.4). Therefore sup

inf (x∗∗ − x) · x∗ = dco A (x∗∗ ) = inf

|x∗ |≤1 x∈co A

sup (x∗∗ − x) · x∗ .

x∈co A |x∗ |≤1

Finally from the previous theorem dco A is a convex function that coincides with the convex envelope since it is the bidual of dA .

8.4

Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and c (E; D) Cd,loc

Theorem 8.6. Let D be a nonempty open subset of RN . The subfamily def

Cd (D) = {dA ∈ Cd (D) : A = ∅ convex}

(8.5)

324

Chapter 6. Metrics via Distance Functions

of Cd (D) is closed in Cloc (D). It is compact in C(D) when D is bounded. The subfamily def

Cdc (D) = {dΩ ∈ Cdc (D) : Ω open and convex}

(8.6)

and for E open, ∅ = E ⊂ D, the subfamilies def

Cdc (E; D) = {dΩ : E ⊂ Ω open and convex ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open and convex ⊂ D N

dΩ :

&

(8.7) (8.8)

of Cdc (D) are closed in Cloc (D). If D is bounded, they are compact in C(D). Proof. The set of all convex functions in Cloc (D) is a closed convex cone with vertex at 0. Hence its intersection with Cd (D) is closed. Any Cauchy sequence in Cd (D) converges to some convex dA . From part (i) A is convex. But dA = dA and the nonempty convex set A can be chosen as the limit set. For the complementary distance function, it is suﬃcient to prove the compactness for D bounded. Given any sequence {dΩn }, there exist an open subset Ω of D and a subsequence, still indexed by n, such that dΩn → dΩ in C(D). If Ω is empty, there is nothing to prove. If Ω is not empty, consider two points x and y in Ω and λ ∈ [0, 1]. There exists r > 0 such that B(x, r) ⊂ Ω and B(y, r) ⊂ Ω

⇒ dΩ (x) ≥ r and dΩ (y) ≥ r.

There exists N > 0 such that for all n > N , dΩn − dΩ C(D) < r/2 and dΩn (x) ≥ dΩ (x) − r/2 = r/2 ⇒ B(x, r/2) ⊂ Ωn , dΩn (y) ≥ dΩ (y) − r/2 = r/2 ⇒ B(y, r/2) ⊂ Ωn . By convexity of Ωn , xλ = λx + (1 − λ)y ∈ Ωn , B(xλ , r/2) ⊂ λ B(x, r/2) + (1 − λ)B(y, r/2) ⊂ Ωn ⇒ ∀n > N, dΩn (xλ ) ≥ r/2 ⇒ dΩ (xλ ) ≥ r/2 ⇒ B(xλ , r/2) ⊂ Ω. Therefore xλ ∈ Ω, Ω is convex, and Cdc (D) is compact. Finally, Cdc (E; D) = c c (E; D) = Cd,loc (E; D) ∩ Cdc (D) are closed since Cdc (D) is Cdc (E; D) ∩ Cdc (D) and Cd,loc c c closed and Cd (E; D) and Cd,loc (E; D) are closed by Theorem 2.6.

9

Compactness Theorems for Sets of Bounded Curvature

We have seen two compactness conditions in Theorems 6.6 and 6.7 of section 6.3 for sets of positive reach from a condition in a tubular neighborhood. In this section we give compactness theorems for families of sets of global and local bounded curvature.

9. Compactness Theorems for Sets of Bounded Curvature

325

For the family of sets with bounded curvature, the key result is the compactness of the embeddings

(9.1) BCd (D) = dA ∈ Cd (D) : ∇dA ∈ BV(D)N → W 1,p (D),

BCcd (D) = dΩ ∈ Cdc (D) : ∇dΩ ∈ BV(D)N → W 1,p (D) (9.2) for bounded open Lipschitzian subsets D of RN and p, 1 ≤ p < ∞. It is the analogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets BX(D) = {χ ∈ X(D) : χ ∈ BV(D)} → Lp (D),

(9.3)

which is a consequence of the compactness of the embedding BV(D) → L1 (D)

(9.4)

for bounded open Lipschitzian subsets D of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 5, we give a ﬁrst version involving global conditions on a ﬁxed bounded open Lipschitzian holdall D. In the second version, the sets are contained in a bounded open holdall D with local conditions in their tubular neighborhood or the tubular neighborhood of their boundary.

9.1

Global Conditions in D

Theorem 9.1. Let D be a nonempty bounded open Lipschitzian holdall in RN . The embedding (9.1) is compact. Thus for any sequence {An }, ∅ = An , of subsets of D such that (9.5) ∃c > 0, ∀n ≥ 1, D2 dAn M 1 (D) ≤ c, there exist a subsequence {Ank } and A = ∅, such that ∇dA ∈ BV(D)N and dAnk → dA in W 1,p (D)-strong for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D), lim ∂ij dAnk , ϕ = ∂ij dA , ϕ, 1 ≤ i, j ≤ N,

n→∞

and

D2 dA M 1 (D) ≤ c.

(9.6)

Proof. Given c > 0 consider the set def

Sc = dA ∈ Cd (D) : D 2 dA M 1 (D) ≤ c . By compactness of the embedding (9.4), given any sequence {dAn }, there exist a subsequence, still denoted by {dAn }, and f ∈ BV(D)N such that ∇dAn → f in L1 (D)N . But by Theorem 2.2 (ii), Cd (D) is compact in C(D) for bounded D and there exist another subsequence {dAnk } and dA ∈ Cd (D) such that dAnk → dA in C(D) and, a fortiori, in L1 (D). Therefore, dAnk converges in W 1,1 (D) and also in

326

Chapter 6. Metrics via Distance Functions

L1 (D). By uniqueness of the limit, f = ∇dA and dAnk converges in W 1,1 (D) to dA . For Φ ∈ D1 (D)N ×N as k goes to inﬁnity −→ −→ ∇dAnk · div Φ dx → ∇dA · div Φ dx D D −→ −→ ∇dA · div Φ dx = lim ∇dAnk · div Φ dx ≤ cΦC(D) , ⇒ k→∞

D

D

D dA M 1 (D) ≤ c, and ∇dA ∈ BV(D) . This proves the compactness of the embedding for p = 1 and properties (9.6). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cd (D) in Theorem 4.2 (i). 2

N

When D is bounded open, Cdc (D) is compact in C(D) and closed in W01,p (D), 1 ≤ p < ∞, and we have the analogue of the previous compactness theorem. Theorem 9.2. Let D, ∅ = D ⊂ RN , be bounded open Lipschitzian. The embedding (9.2) is compact. Thus for any sequence {Ωn } of open subsets of D such that ∃h > 0, ∃c > 0, ∀n,

D2 dΩn M 1 (D) ≤ c,

(9.7)

there exist a subsequence {Ωnk } and an open subset Ω of D such that ∇dΩ ∈ BV(D)N and dΩn → dΩ in W01,p (D)

(9.8)

k

for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D)N ×N , D2 dΩn , ϕ → D2 dΩ , ϕ,

D 2 dΩ M 1 (D) ≤ c,

and

χΩ ∈ BV(D). (9.9)

Proof. Given c > 0 consider the set def

Scc = dΩ ∈ Cdc (D) : D 2 dΩ M 1 (D) ≤ c . By compactness of the embedding (9.4), given any sequence {dΩn } there exist a subsequence, still denoted by {dΩn }, and f ∈ BV(D)N such that ∇dΩn → f in L1 (D)N . But by Theorem 2.4 (ii), Cdc (D) is compact in C0 (D) for bounded D and there exist another subsequence {dΩn } and dΩ ∈ Cdc (D) such that dΩn → dΩ k

k

in C0 (D) and, a fortiori, in L1 (D). Therefore, dΩn converges in W01,1 (D) and also k

in L1 (D). By uniqueness of the limit, f = ∇dΩ and dΩn converges in W01,1 (D) k to dΩ . For all Φ ∈ D1 (D)N ×N as k goes to inﬁnity −→ −→ ∇dΩn · divΦ dx → ∇dΩ · divΦ dx k D D −→ −→ ∇dΩ · divΦ dx = lim ∇dΩn · divΦ dx ≤ cΦC(D) , ⇒ D

k→∞

D

k

D 2 dΩ M 1 (D) ≤ c, ∇dΩ ∈ BV(D)N , and χΩ ∈ BV(D). This proves the compactness of the embedding for p = 1 and properties (9.9). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cdc (D) in Theorem 4.2 (i).

9. Compactness Theorems for Sets of Bounded Curvature

9.2

327

Local Conditions in Tubular Neighborhoods

The global conditions (9.5) and (9.7) can be weakened to local ones in a neighborhood of each set of the sequence, and the Lipschitzian condition on D can be removed since only the uniform boundedness of the sets of the sequence is required. Theorem 9.3. Let D, ∅ = D ⊂ RN , be a bounded open holdall and {An }, ∅ = An , be a sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that ∀n,

D2 dAn M 1 (Uh (∂An )) ≤ c.

(9.10)

Then there exist a subsequence {Ank } and a subset A, ∅ = A, of D such that ∇dA ∈ BVloc (RN )N , and for all p, 1 ≤ p < ∞, dAnk → dA in W 1,p (Uh (D))-strong.

(9.11)

Moreover, for all ϕ ∈ D0 (Uh (A)) lim ∂ij dAnk , ϕ = ∂ij dA , ϕ, 1 ≤ i, j ≤ N,

k→∞

D2 dA M 1 (Uh (A)) ≤ c,

(9.12)

and χA belongs to BVloc (RN ). Proof. First notice that, since An is bounded and nonempty, ∂An = ∅. Further Uh (∂An ) can be replaced by Uh (An ) in condition (9.10) since dAn = 0 in An . So it is suﬃcient to prove the theorem for that case. Lemma 9.1. For any A, ∂A = ∅ and h > 0, D2 dA M 1 (Uh (A)) = D2 dA M 1 (Uh (∂A)) . The proof of the lemma will be given after the proof of the theorem. (i) The assumption An ⊂ D implies Uh (An ) ⊂ Uh (D). Since Uh (D) is bounded, there exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that dAn → dA in C(Uh (D)) and another subsequence, still denoted by {dAn }, such that dAn → dA

in H 1 (Uh (D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and x in Uh (D) dAn (x) ≤ dA (x) + ε,

dA (x) ≤ dAn (x) + ε.

Therefore, An ⊂ Uh−2 ε (An ) ⊂ Uh−ε (A) ⊂ Uh (An ),

(9.13)

Uh−ε (A) ⊂ Uh−2 ε (An ) ⊂ An .

(9.14)

From (9.10) and (9.13) ∀n ≥ N,

D2 dAn M 1 (Uh−ε (A)) ≤ c.

328

Chapter 6. Metrics via Distance Functions

In order to use the compactness of the embedding (9.4) as in the proof of Theorem 9.1, we would need Uh−ε (A) to be Lipschitzian. To get around this, we construct a bounded Lipschitzian set between Uh−2ε (A) and Uh−ε (A). Indeed, by deﬁnition, Uh−ε (A) = ∪x∈A B(x, h − ε)

and Uh−2ε (A) ⊂ Uh−ε (A),

and by compactness, there exists a ﬁnite sequence of points {xi }ni=1 in A such that |xi − xj | < (h − ε)/2 for all i = j so that no two balls are tangent, and def

Uh−2ε (A) ⊂ UB = ∪ni=1 B(xi , h − ε) ⊂ Uh (D). Since UB is Lipschitzian as the union of a ﬁnite number of nontangent balls, it now follows by compactness of the embedding (9.4) for UB that there exist a subsequence, still denoted by {dAn }, and f ∈ BV(UB )N such that ∇dAn → f in L1 (UB )N . Since Uh (D) is bounded, Cd (Uh (D)) is compact in C(Uh (D)) and there exist another subsequence, still denoted by {dAn }, and ∅ = A ⊂ D such that dAn → dA in C(Uh (D)) and, a fortiori, in L1 (Uh (D)). Therefore, dAn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇dA on UB and dAn converges to dA in W 1,1 (UB ). By Deﬁnition 3.3 and Theorem 5.1, ∇dA and ∇dAn all belong to BVloc (RN )N since they are BV in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), χA ∈ BVloc (RN ). The above conclusions also hold for the subset Uh−2ε (A) of UB . (ii) Convergence in W 1,p (Uh (D)). Consider the integral |∇dAn − ∇dA |2 dx Uh (D) 2 = |∇dAn − ∇dA | dx + |∇dAn − ∇dA |2 dx. Uh−2ε (A)

Uh (D)\Uh−2ε (A)

From part (i) the ﬁrst integral on the right-hand side converges to zero as n goes to inﬁnity. The second integral is on a subset of Uh−2ε (A). From (9.14) for all n ≥ N, |∇dAn (x)| = 1 a.e. in An ⊃ Uh−3ε (An ) ⊃ Uh−2ε (A), |∇dA (x)| = 1 a.e. in A ⊃ Uh−2ε (A). The second integral reduces to |∇dAn − ∇dA |2 dx = Uh (D)\Uh−2ε (A)

2 (1 − ∇dAn · ∇dA ) dx,

Uh (D)\Uh−2ε (A)

which converges to zero since ∇dAn ∇dA in L2 (Uh (D))N -weak in part (i) and the fact that |∇dA | = 1 almost everywhere in Uh (D)\Uh−2ε (A). Therefore, since dAn → dA in C(Uh (D)), dAn → dA in H 1 (Uh (D))-strong, and by Theorem 4.2 (i) the convergence is true in W 1,p (Uh (D)) for all p ≥ 1.

9. Compactness Theorems for Sets of Bounded Curvature

329

(iii) Properties (9.12). Consider the initial subsequence {dAn } which converges to dA in H 1 (Uh (D))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of dAn to dA in H 1 (Uh (D))-weak for each Φ ∈ D1 (Uh (A))N ×N , −→ −→ ∇dAn · div Φ dx = ∇dA · div Φ dx. lim n→∞

Uh (A)

Uh (A)

Each such Φ has compact support in Uh (A), and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (A). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (An ) ⊂ Uh−ε (A) ⊂ Uh (An ).

For n ≥ N (ε) consider the integral −→ ∇dAn · div Φ dx = Uh (A)

⇒

Uh (A)

−→

Uh−2ε (A)

∇dAn

∇dAn · div Φ dx =

−→

∇dAn · div Φ dx U (A )

h n · div Φ dx ≤ D2 dAn M 1 (Uh (An )) ΦC(Uh (An )) ≤ c ΦC(Uh−2ε (A)) = c ΦC(Uh (A)) .

−→

By convergence of ∇dAn to ∇dA in L2 (D∪Uh (A))-weak, for all Φ ∈ D 1 (Uh (A))N ×N −→ ∇dA · div Φ dx ≤ c ΦC(Uh (A)) ⇒ D2 dA M 1 (Uh (A)) ≤ c. Uh (A)

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof. Proof of Lemma 9.1. First check that Uh (A) = Uh (∂A) ∪x∈A−h B(x, h),

def

A−h = x ∈ RN : B(x, h) ⊂ A .

It is suﬃcient to show that all points x of int A are contained in the right-hand side of the above expression. If d∂A (x) < h, then x ∈ Uh (∂A); if d∂A (x) ≥ h, then B(x, h) ⊂ A and x ∈ A−h . For all Φ ∈ D1 (Uh (A))N ×N , Φ has compact support and there exists ε > 0, 0 < 2ε < h, such that supp Φ ⊂ Uh−2ε (A). But Uh−ε (A) ⊂ Uh (A) and there exists {xj ∈ A−h : 1 ≤ j ≤ m} such that Uh−ε (A) ⊂ Uh (∂A)∪m j=1 B(xj , h). Let ψ0 ∈ D(Uh (∂A)) and let ψj ∈ D(B(xj , h)) be a partition of unity such that m ψj = 1 on Uh−2ε (A). 0 ≤ ψj ≤ 1, j=0

330

Chapter 6. Metrics via Distance Functions

Consider the integral −→ ∇dA · div Φ dx = Uh (A)

∇dA · div Φ dx = −→

Uh−2ε (A)

=

−→ ∇dA · div ψ0 Φ dx + Uh−2ε (A)

m ∇dA · div ψj Φ dx −→

Uh−2ε (A)

m

j=1

j=0

−→ ∇dA · div ψj Φ dx. Uh−2ε (A)

By construction for ΦC(Uh (A)) ≤ 1 ψ0 Φ ∈ D1 (Uh−2ε (A) ∩ Uh (∂A))N ×N , ψj Φ ∈ D1 (Uh−2ε (A) ∩ B(xj , h))N ×N ,

ψ0 ΦC(Uh (∂A)) ≤ 1,

ψj ΦC(B(xj ,h)) ≤ 1,

1 ≤ j ≤ m.

But for j, 1 ≤ j ≤ m, B(xj , h) ⊂ int A, where both dA and ∇dA are identically zero. As a result the above integral reduces to −→ −→ −→ ∇dA · div Φ dx = ∇dA · div (ψ0 Φ) dx = ∇dA· div (ψ0 Φ) dx Uh (A)

⇒

Uh−2ε (A)

U (∂A)

h ∇dA · div Φ dx ≤ D2 dA M 1 (Uh (∂A)) ψ0 ΦC(Uh (∂A))

−→

Uh (A)

⇒ D2 dA M 1 (Uh (A)) ≤ D 2 dA M 1 (Uh (∂A)) , and this completes the proof. Theorem 9.4. Let D, ∅ = D ⊂ RN , be a bounded open holdall. Let {Ωn }, ∅ = Ωn ⊂ D, be a sequence of open subsets and assume that ∃h > 0 and ∃c > 0 such that

∀n,

D2 dΩn M 1 (Uh (∂Ωn )) ≤ c.

(9.15)

Then there exist a subsequence {Ωnk } and an open subset Ω of D such that ∇dΩ ∈ BVloc (RN )N and for all p, 1 ≤ p < ∞, dΩn → dΩ in W01,p (D)-strong. k

(9.16)

Moreover for all ϕ ∈ D0 (Uh (Ω)) lim ∂ij dΩn , ϕ = ∂ij dΩ , ϕ, 1 ≤ i, j ≤ N,

k→∞

k

D2 dΩ M 1 (Uh (Ω)) ≤ c, (9.17)

and χΩ belongs to BVloc (RN ). Proof. First note that ∂Ωn = ∅ since Ωn is bounded and nonempty. By Lemma 9.1, Uh (∂Ωn ) can be replaced by Uh (Ωn ) in condition (9.15). Moreover since Ωn can be unbounded, we shall work with the bounded neighborhoods Uh (D Ωn ) of D Ωn rather than with Uh (Ωn ). Lemma 9.2. Let h > 0 and let ∅ = Ω ⊂ D ⊂ RN be bounded open. Then D2 dΩ M 1 (Uh (

D

Ω))

= D2 dΩ M 1 (Uh (Ω)) .

If ∂Ω = ∅, D 2 dΩ M 1 (Uh (∂Ω)) = D2 dΩ M 1 (Uh (Ω)) .

9. Compactness Theorems for Sets of Bounded Curvature

331

The proof of this lemma will be given after the proof of the theorem. (i) By Theorem 2.4 (ii) since D is bounded, there exist a subsequence, still denoted by {dΩn }, and an open subset Ω of D such that dΩn → dΩ

in C0 (D) and Cc (D ∪ Uh (Ω))

and another subsequence, still denoted by {dΩn }, such that dΩn → dΩ

in H01 (D)-weak and H01 (D ∪ Uh (Ω))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N dΩn (x) ≤ dΩ (x) + ε,

dΩ (x) ≤ dΩn (x) + ε.

Clearly, since Ω ⊂ D and Ωn ⊂ D and D is open, D Ωn = ∅ and D Ω = ∅. Furthermore, D Ωn ⊂ Uh−2 ε (D Ωn ) ⊂ Uh−ε (D Ω) ⊂ Uh (D Ωn ), D Uh−ε (D Ω) ⊂ D Uh−2 ε (D Ωn ) ⊂ Ωn .

(9.18) (9.19)

From (9.15) and (9.18) ∀n ≥ N,

D 2 dΩn M 1 (Uh−ε (

D

Ω))

≤ c.

(9.20)

As in part (i) of the proof of Theorem 9.3, we can construct a bounded open Lipschitzian set UB between Uh−2ε (D Ω) and Uh−ε (D Ω) such that Uh−2ε (D Ω) ⊂ UB ⊂ Uh−ε (D Ω). Since UB is bounded and Lipschitzian, it now follows by compactness of the embedding (9.4) for UB that there exist a subsequence, still denoted by {dΩn }, and f ∈ BV(UB )N such that ∇dΩn → f in L1 (UB )N . Since D ∪ UB is bounded, Cdc (D ∪ UB ) is compact in C0 (D ∪ UB ), and there exist another subsequence, still denoted by {dΩn }, and an open subset Ω of D such that dΩn → dΩ in C0 (D ∪ UB ) and, a fortiori, in L1 (D ∪ UB ). Therefore dΩn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇dΩ on BD and dΩn converges to dΩ in W 1,1 (UB ). By Deﬁnition 3.3 and Theorem 5.1, ∇dΩ and ∇dΩn all belong to BVloc (RN )N since they are BV in tubular neighborhoods Uh (D Ω) and Uh (D Ωn ) of their respective boundaries ∂Ω and ∂Ωn . Moreover, by Theorem 5.2 (ii), χΩ ∈ BVloc (RN ) and, a fortiori, χΩ since Ω is open. The above conclusions also hold for the subset Uh−2ε (D Ω) of UB . (ii) Convergence in W01,p (D). Consider the integral |∇dΩn − ∇dΩ |2 dx D |∇dΩn − ∇dΩ |2 dx + |∇dΩn − ∇dΩ |2 dx. = D∩Uh−2ε (D Ω)

D\Uh−2ε (D Ω)

332

Chapter 6. Metrics via Distance Functions

From part (i) the ﬁrst integral on the right-hand side converges to zero as n goes to inﬁnity. The second integral is on a subset of D Uh−2ε (D Ω). From the relations (9.19) for all n ≥ N , |∇dΩn (x)| = 1 a.e. in Ωn ⊃ D Uh−3ε (D Ωn ) ⊃ D Uh−2ε (D Ω), |∇dΩ (x)| = 1 a.e. in Ω ⊃ D Uh−2ε (D Ω). The second integral reduces to |∇dΩn − ∇dΩ |2 dx = D\Uh−2ε (D Ω)

2 (1 − ∇dΩn · ∇dΩ ) dx,

D\Uh−2ε (D Ω)

which converges to zero by weak convergence of ∇dΩn to ∇dΩ in L2 (D)N and the fact that |∇dΩ | = 1 almost everywhere in D\Uh−2ε (D Ω). Therefore, since dΩn → dΩ in C0 (D) dΩn → dΩ in H01 (D)-strong, and by Theorem 4.2 (i) the convergence is true in W01,p (D) for all p ≥ 1. (iii) Properties (9.17). Consider the initial subsequence {dΩn }, which converges to dΩ in H01 (D ∪ Uh (Ω))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of dΩn to dΩ in H01 (D ∪ Uh (Ω))-weak for each Φ ∈ D 1 (Uh (Ω))N ×N , −→ −→ ∇dΩn · div Φ dx = ∇dΩ · div Φ dx. lim n→∞

Uh (Ω)

Uh (Ω)

Now each Φ ∈ D1 (Uh (Ω))N ×N has compact support in Uh (Ω) and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (Ω). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (Ωn ) ⊂ Uh−ε (Ω) ⊂ Uh (Ωn ).

For n ≥ N (ε) consider the integral −→ ∇dΩn · div Φ dx = Uh (Ω)

⇒

Uh (Ω)

∇dΩn

Uh−2ε (Ω)

∇dΩn

· div Φ dx = −→

−→

∇dΩn · div Φ dx

U (Ω )

h n · div Φ dx ≤ D 2 dΩn M 1 (Uh (D Ωn )) ΦC(Uh (D Ωn )) ≤ c ΦC(Uh−2ε (D Ω)) = c ΦC(Uh (Ω)) .

−→

By convergence of ∇dΩn to ∇dΩ in L2 (D∪Uh (Ω))-weak, for all Φ ∈ D1 (Uh (Ω))N ×N −→ ∇dΩ · div Φ dx ≤ c ΦC(Uh (Ω)) ⇒ D2 dΩ M 1 (Uh (Ω)) ≤ c. Uh (Ω)

Finally the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

9. Compactness Theorems for Sets of Bounded Curvature Proof of Lemma 9.2. The proof is similar to that of Lemma 9.1 with Uh (Ω) = Uh (D Ω) ∪ int D = Uh (D Ω)∪x∈(D)−h B(x, h), def

(D)−h = x ∈ RN : B(x, h) ⊂ D , since both dΩ and ∇dΩ are identically zero on D.

333

Chapter 7

Metrics via Oriented Distance Functions 1

Introduction

In this chapter we study oriented distance functions and their role in the description of the geometric properties and smoothness of domains and their boundary. They constitute a special family of the so-called algebraic or signed distance functions. Signed distance functions make sense only when some appropriate orientation of the underlying set is available to assign a sign, while oriented distance functions are always well-deﬁned. Our choice of terminology emphasizes the fact that, for a smooth open domain, the associated oriented distance function speciﬁes the orientation of the normal to the boundary of the set. They enjoy many interesting properties. For instance, they retain the nice properties of the distance functions but also generate the classical geometric properties associated with sets and their boundary. The smoothness of the oriented distance function in a neighborhood of the boundary of the set is equivalent to the local smoothness of its boundary. Similarly, the convexity of the function is equivalent to the convexity of the closure of the set. In addition, their respective gradient and Hessian matrix respectively coincide with the unit outward normal and the second fundamental form on the boundary of the set. Finally, they provide a framework for the classiﬁcation of domains and sets according to their degree of smoothness, much like Sobolev spaces and spaces of continuous and H¨ olderian functions do for functions. The ﬁrst part of the chapter deals with the basic deﬁnitions and constructions and the main results. The second part specializes to speciﬁc subfamilies of oriented distance functions. The last part concentrates on compact families of subsets of oriented distance functions. In the ﬁrst part, section 2 presents the basic properties, introduces the uniform metric topology, and shows its connection with the Hausdorﬀ and complementary Hausdorﬀ topologies of Chapter 6. Section 3 is devoted to the diﬀerentiability properties and the associated set of projections onto the boundary, and it completes the treatment of skeletons and cracks. Section 8 gives the equivalence of the smoothness of a set and the smoothness of its oriented distance function in a neighborhood of its boundary for sets of class C 1,1 or better. 335

336

Chapter 7. Metrics via Oriented Distance Functions

When the domain is suﬃciently smooth the trace of the Hessian matrix of secondorder partial derivatives on the boundary is the classical second fundamental form of geometry. Section 4 deals with the W 1,p -topology on the set of oriented distance functions and the closed subfamily of sets for which the volume of the boundary is zero. In the second part of the chapter, section 5 studies the subfamily of sets for which the gradient of the oriented distance function is a vector of functions of bounded variation: the sets with global or local bounded curvature. There are large classes for which general compactness theorems will be proved in section 11. Examples are given to illustrate the behavior of the norms in tubular neighborhoods as the thickness of the neighborhood goes to zero in section 5.1. Section 9 introduces Sobolev or W s,p -domains which provide a framework for the classiﬁcation of sets according to their degree of smoothness. For smooth sets this classiﬁcation intertwines with the classical classiﬁcation of C k - and H¨ olderian domains. Section 10 extends the characterization of closed convex sets by the distance function to the oriented distance function and introduces the notion of semiconvex sets. The set of equivalence classes of convex subsets of a compact holdall D is again compact, as it was for all the topologies considered in Chapters 5 and 6. Section 7 shows that sets of positive reach introduced in Chapter 6 are sets of locally bounded curvature and that the boundary of their closure has zero volume. A compactness theorem is also given. Section 12 introduces the notion of γ-density perimeter, which is a relaxation of the (N − 1)-dimensional upper Minkowski content, and a compactness theorem for a family of subsets of a bounded holdall that have a uniformly bounded γ-density perimeter. In the last part of the chapter, section 11 gives compactness theorems for sets of global and local bounded curvatures from a uniform bound in tubular neighborhoods of their boundary. They are the analogues of the theorems of Chapter 6. Section 13 gives the compactness theorem for the family of subsets of a bounded holdall that have the uniform fat segment property. Equivalent conditions are given on the local graph functions of the family in section 13.2. The results are specialized to the compactness under the uniform cusp/cone property. In particular we recover the compactness of Theorem 6.11 of section 6.4 in Chapter 5 for the associated family of characteristic functions in Lp (D). Section 14 combines the compactness under the uniform fat segment property with a bound on the De Giorgi perimeter of Caccioppoli sets in section 14.1 and the γ-density perimeter in section 14.2. Section 15 introduces the families of cracked sets and uses them in section 16 to provide an original solution to a variation of the image segmentation problem of D. Mumford and J. Shah [2]. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks, and sets that can be made up of components of diﬀerent codimensions. The Hausdorﬀ (N − 1) measure of their boundary is not necessarily ﬁnite. Yet compact families (in the W 1,p -topology) of such sets can be constructed.

2. Uniform Metric Topology

2

337

Uniform Metric Topology

2.1

The Family of Oriented Distance Functions Cb (D)

The distance function dA provides a good description of a domain A from the “outside.” However, its gradient undergoes a jump discontinuity at the boundary of A which prevents the extraction of information about the smoothness of A from the smoothness of dA in a neighborhood of ∂A. To get around this, it is natural to take into account the negative of the distance function of its complement A that cancels the jump discontinuity at the boundary of smooth sets. Deﬁnition 2.1. Given A ⊂ RN , the oriented distance function from x ∈ RN to A is deﬁned as def

bA (x) = dA (x) − dA (x).

(2.1)

This function gives a level set description of a set whose boundary coincides with the zero level set. From Chapter 6 the function bA is ﬁnite in RN if and only if ∅ = A = RN , since bA = dA = +∞ when A = ∅ and bA = −dA = −∞ when A = ∅. This condition is completely equivalent to ∂A = ∅, in which case bA coincides with the algebraic distance function to the boundary of A: x ∈ int A, dA (x) = d∂A (x), bA (x) = 0, x ∈ ∂A, −dA (x) = −d∂A (x), x ∈ int A.

(2.2)

Noting that bA = −bA , it means that we have implicitly chosen the negative sign for the interior of A and the positive sign for the interior of its complement. As bA is an increasing function from its interior to its exterior, we shall see later that for sets with a smooth boundary, the restriction of the gradient of bA to ∂A coincides with the outward unit normal to ∂A. Changing the sign of bA gives the inward orientation to the gradient and the normal. Associate with a nonempty subset D of RN the family def

Cb (D) = bA : A ⊂ D and ∂A = ∅ .

(2.3)

Theorem 2.1. Let A be a subset of RN . Then the following hold: (i) A = ∅ and A = ∅ ⇐⇒ ∂A = ∅. (ii) Given bA and bB in Cb (D), A ⊃ B ⇒ b A ≤ bB

and

A = B ⇒ b A = bB ,

bA ≤ bB in D ⇐⇒ B ⊂ A and A ⊂ B, bA = bB in D ⇐⇒ B = A and A = B ⇐⇒ B = A and ∂A = ∂B.

338

Chapter 7. Metrics via Oriented Distance Functions In particular, bint A ≤ bA ≤ bA and bA = bA ⇐⇒ ∂A = ∂A

and

bint A = bA ⇐⇒ ∂ int A = ∂A.

(2.4)

(iii) |bA | = dA + dA = max{dA , dA } = d∂A and ∂A = {x ∈ RN : bA (x) = 0}. (iv) bA ≥ 0 ⇐⇒ A ⊃ ∂A ⊃ A ⇐⇒ ∂A = A. (v) bA = 0 ⇐⇒ A = ∂A = A ⇐⇒ ∂A = RN . (vi) If ∂A = ∅, the function bA is uniformly Lipschitz continuous in RN and |bA (y) − bA (x)| ≤ |y − x|.

∀x, y ∈ RN ,

(2.5)

Moreover, bA is (Fr´echet) diﬀerentiable almost everywhere and |∇bA (x)| ≤ 1

a.e. in RN .

(2.6)

Proof. (i) The proof is obvious. (ii) By assumption + − − dA = b+ A ≤ bB = dB and dA = bA ≥ bB = dB in D.

Since A ⊂ D and B ⊂ D, then A ⊃ D and B ⊃ D and dA = dB = 0 in D

⇒ dA ≥ dB in RN

⇒ A ⊂ B.

Also since B ⊂ D for all x ∈ B, dA (x) ≤ dB (x) = 0 and x ∈ A. Therefore, A ⊂ B and B ⊂ A. Conversely, B⊂A

⇒ dA ≤ dB in RN

and

A ⊂ B

⇒ dB ≤ dA in RN

and, a fortiori, in D. The equality case follows from the fact that bA = bB if and only if bA ≥ bB and bA ≤ bB . By deﬁnition int A = A ⊂ A ⊂ A and bint A ≤ bA ≤ bA . (iii) For x in A bA (x) = −dA (x) =⇒ |bA (x)| = dA (x) ≤ d∂A (x) since A ⊃ ∂A and the inf over A is smaller than the inf over its subset ∂A. Similarly, for x in A bA (x) = dA (x) =⇒ |bA (x)| = dA (x) ≤ d∂A (x), and ﬁnally |bA (x)| ≤ max{dA (x), dA (x)} ≤ d∂A (x). Conversely, for each x in A, the set of projections ΠA (x) ⊂ A ∩ A = ∂A is not empty. Hence |bA (x)| = dA (x) =

min |x − y| ≥ inf |x − y| = d∂A (x),

y∈ΠA (x)

y∈∂A

2. Uniform Metric Topology

339

and similarly for all x in A, |bA (x)| = dA (x) ≥ d∂A (x). Therefore |bA (x)| ≥ d∂A (x). (iv) If bA = dA −dA ≥ 0, then dA ≥ dA and A ⊂ A, and necessarily A ⊂ ∂A and A = ∂A. Conversely, if x ∈ ∂A, then by deﬁnition bA (x) = 0. If x ∈ / ∂A, then x ∈ ∂A = A = int A and bA (x) = dA (x) ≥ 0. (v) bA = 0 is equivalent to bA ≥ 0 and bA = −bA ≥ 0. Then we apply (v) twice. But A = ∂A = A ⇐⇒ int A = ∅ = int A ⇐⇒ ∂A = RN . (vi) Clearly, ∀x, y ∈ A, ∀x, y ∈ A,

|bA (y) − bA (x)| = |dA (y) − dA (x)| ≤ |y − x|, |bA (y) − bA (x)| = |dA (y) − dA (x)| ≤ |y − x|.

For x ∈ A and y ∈ int A = A, dA (y) > 0 and bA (y) − bA (x) = dA (y) + dA (x) > 0

⇒ |bA (y) − bA (x)| = dA (y) + dA (x).

By assumption B(y, dA (y)) ⊂ int A. Deﬁne the point dA (y) x=y+ (x − y) ∈ A |x − y| dA (y) ⇒ dA (x) ≤ |x − x| = 1 − (y − x) = |x − y| − dA (y) |x − y| ⇒ |bA (y) − bA (x)| = dA (y) + dA (x) ≤ |x − y|. The argument is similar for x ∈ int A and y ∈ A. The diﬀerentiability follows from Theorem 2.1 (vii) in Chapter 6.

2.2

Uniform Metric Topology

From Theorem 2.1 (i) the function bA is ﬁnite at each point when ∂A = ∅. This excludes A = ∅ and A = RN . The zero function bA (x) = 0 for all x in RN corresponds to the equivalence class of sets A such that A = ∂A = A or ∂A = RN . This class of sets is not empty. For instance, choose the subset of points of RN with rational coordinates or the set of all lines parallel to one of the coordinate axes with rational coordinates. Let D be a nonempty subset of RN and associate with each subset A of D, ∂A = ∅, the equivalence class def

[A]b = B : ∀B, B ⊂ D, B = A and ∂A = ∂B

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Chapter 7. Metrics via Oriented Distance Functions

and the family of equivalence classes def

Fb (D) = [A]b : ∀A, A ⊂ D and ∂A = ∅ . The equivalence classes induced by bA are ﬁner than those induced by dA since both the closures and the boundaries of the respective sets must coincide. For Cd (D), we have an invariant closed representative A in the equivalence class [A]d and for Cdc (D) an invariant open representative int A for [A]d . For Cb (D), the deﬁciencies of Cd (D) and Cdc (D) combine dA = dA ≤ dint A

and dint A = dA = dA ≤ dA ,

and, in general, there is neither a closed nor an open representative, since bA ≤ bA ≤ bint A . One invariant in the class is ∂A, but it does not completely describes the class without one of the other two A or A. We shall see that for sets verifying a uniform segment property bA = bA = bint A and for convex sets that bA = bA . As in the case of dA we identify Fb (D) with the family Cb (D) through the embedding [A]b → bA : Fb (D) → Cb (D) ⊂ C(D). When D is bounded, the space C(D) endowed with the norm f C(D) is a Banach space. Moreover, for each A ⊂ D, bA is bounded, uniformly continuous on D, and bA ∈ C(D). This will induce the following complete metric: def

ρ([A]b , [B]b ) = bA − bB C(D)

(2.7)

on Fb (D). When D is open but not necessarily bounded, we use the space Cloc (D) deﬁned in section 2.2 of Chapter 6 endowed with the complete metric ρδ deﬁned in (2.9) for the family of seminorms {qK } deﬁned in (2.8). It will be shown below that Cb (D) is a closed subset of Cloc (D) and that this induces the following complete metric on Fb (D): def

ρδ ([A]b , [B]b ) =

∞ 1 qKk (bA − bB ) . 2k 1 + qKk (bA − bB )

(2.8)

k=1

The subfamilies def

Fb0 (D) = {[A]b : ∀A, A ⊂ D, ∂A = ∅, m(∂A) = 0} , def

Cb0 (D) = {bA ∈ Cb (D) : m(∂A) = 0} of Fb (D) and Cb (D) will be important. We now have the equivalent of Theorem 2.2 in Chapter 6.

2. Uniform Metric Topology

341

Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN . (i) The set Cb (D) is closed in Cloc (D) (resp., C(D)) and ρδ (resp., ρ) deﬁnes a complete metric topology on Fb (D). (ii) For a bounded open subset D of RN , the set Cb (D) is compact in C(D). (iii) For a bounded open subset D of RN , the map − 3 bA → (b+ A , bA , |bA |) = (dA , dA , d∂A ) : Cb (D) ⊂ C(D) → C(D)

is continuous: for all bA and bB in Cb (D),

max dB − dA C(D) , dB − dA C(D) , d∂B − d∂A C(D) ≤ bB − bA C(D) .

(2.9)

Proof. (i) It is suﬃcient to consider the case for which D is bounded. Consider a sequence {An } ⊂ D, ∂An = ∅, such that bAn → f in C(D) for some f in C(D). Associate with each g in C(D) its positive and negative parts g + (x) = max{g(x), 0},

g − (x) = max{−g(x), 0}.

Then by continuity of this operation + − and dAn = b− in C(D), d A n = b+ An → f An → f

d∂An = |bAn | → |f | in C(D). By Theorem 2.2 (i) of Chapter 6, there exists a closed subset F , ∅ = F ⊂ D, such that f + = dF in RN

and f + > 0 in D.

Moreover F = RN since D is bounded. By Theorem 2.4 and the remark at the beginning of section 2.3 of Chapter 6, there exists an open subset G ⊂ D, G = RN , such that f − = dG in RN and dG ∈ C0 (D). Therefore, f = f + − f − = dF − dG , f = f + − f − = dF in D and f > 0 in D. Deﬁne the sets

def

A+ = x ∈ RN : f (x) > 0 = x ∈ D : f (x) > 0 ∪ D,

def

A− = x ∈ RN : f (x) < 0 = x ∈ D : f (x) < 0 ,

def

A0 = x ∈ RN : f (x) = 0 = x ∈ D : f (x) = 0 .

342

Chapter 7. Metrics via Oriented Distance Functions

They form a partition of RN , RN = A− ∪ A0 ∪ A+ , and RN = A0 ∪ A− = F = ∅,

A0 ∪ A+ = G = ∅

⇒ A0 = F ∩ G = ∅,

D ⊂ A+ and F = A0 ∪ A− ⊂ D, f = dA0 ∪A− − dA0 ∪A+ . If A0 were empty, RN could be partitioned into two disjoint nonempty closed subsets. Since A0 is closed ∂A0 = A0 ∩ A0 = A0 ∩ A+ ∪ A− = A0 ∩ [A+ ∪ A− ] = A0 ∩ [A+ ∪ A− ∪ ∂A− ∪ ∂A+ ] = A0 ∩ [∂A− ∪ ∂A+ ]. Moreover since A− and A+ are open, A− ⊂ A− ∪ A0 , A+ ⊂ A+ ∪ A0 , ∂A− = A− ∩ A− ⊂ [A0 ∪ A+ ] ∩ [A0 ∪ A− ] = A0 , ∂A+ = A+ ∩ A+ ⊂ [A0 ∪ A− ] ∩ [A0 ∪ A+ ] = A0 ⇒ ∂A− ∪ ∂A+ = [∂A− ∪ ∂A+ ] ∩ A0 = ∂A0 ⇒ ∂A− ∪ ∂A+ = ∂A0

⇒ ∂A0 = ∂A− ∪ (∂A+ \∂A− ).

Let Q be the subset of points in RN with rational coordinates. Deﬁne def

B 0 = int A0 ,

def

0 B+ = B 0 ∩ Q,

def

0 B− = B 0 ∩ Q,

and notice that by density of Q and Q in RN , 0 = B0 = B0 . B+ −

Consider the following new partition of RN : 0 0 RN = A+ ∪ (∂A+ \∂A− ) ∪ A− ∪ ∂A− ∪ B+ ∪ B− .

Deﬁne 0 A = A− ∪ (∂A+ \∂A− ) ∪ B+ def

0 ⇒ A = A+ ∪ ∂A− ∪ B−

and 0 = A− ∪ ∂A− ∪ (∂A+ \∂A− ) ∪ B 0 , A = A− ∪ (∂A+ \∂A− ) ∪ B+ 0 = A+ ∪ ∂A+ ∪ ∂A− ∪ B 0 . A = A+ ∪ ∂A− ∪ B−

But ∂A− ∪ ∂A+ = ∂A− ∪ (∂A+ \∂A− ) ⊂ ∂A− ∪ (∂A+ \∂A− ) ⊂ ∂A− ∪ ∂A+ ,

2. Uniform Metric Topology

343

and since ∂A− ∪ ∂A+ = ∂A0

A = A− ∪ ∂A0 ∪ int A0 = A− ∪ A0 A = A+ ∪ ∂A0 ∪ int A0 = A+ ∪ A0

⇒ ∂A = A0 = ∅

⇒ bA = dA − dA = bA − dA = dF − dG = f. (ii) For the compactness, consider any sequence {bAn } ⊂ Cb (D) ⊂ C 0,1 (D). Since D is bounded, bAn and ∇bAn are both pointwise uniformly bounded in D. From Theorem 2.5 in Chapter 2, the injection of C 0,1 (D) into C(D) is compact and there exist f ∈ C(D) and a subsequence {bAnk } such that bAnk → f in C(D). From the proof of the closure in part (i), there exists A ⊂ D, ∂A = ∅, such that f = bA . (iii) For all bA and bB in Cb (D) and x in D, |bB (x)| ≤ |bA (x)| + |bB (x) − bA (x)| ⇒ | |bB (x)| − |bA (x)| | ≤ |bB (x) − bA (x)| ⇒ d∂B − d∂A C(D) = |bB | − |bA | C(D) ≤ bB − bA C(D) . − Moreover, dA = b+ A = (|bA | + bA )/2 and dA = bA = (|bA | − bA )/2, and necessarily ± b± B − bA C(D) ≤ bB − bA C(D) .

By combining the above three inequalities we get (2.9). We have the analogue of Theorem 2.3 of Chapter 6 and its corollary for A, A, and ∂A. In the last case it takes the following form (to be compared with T. J. Richardson [1, Lem. 3.2, p. 44] and S. R. Kulkarni, S. K. Mitter, and T. J. Richardson [1] for an application to image segmentation). Corollary 1. Let D be a nonempty open (resp., bounded open) holdall in RN . Deﬁne for a subset S of RN the sets def

Hb (S) = {bA ∈ Cb (D) : S ⊂ ∂A} , def

Ib (S) = {bA ∈ Cb (D) : ∂A ⊂ S} , def

Jb (S) = {bA ∈ Cb (D) : ∂A ∩ S = ∅} . (i) Let S be a subset of RN . Then Hb (S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then Ib (S) is closed in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then Jb (S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then Jb (S) is open in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then Ib (S) is open in Cloc (D) (resp., C(D)). (iv) For D bounded, associate with an equivalent class [A]b the number # ∂A = number of connected components of ∂A.

344

Chapter 7. Metrics via Oriented Distance Functions Then the map def

[A]b → # ∂A : F#b (D) = [A]b : ∀A ⊂ D, ∂A = ∅, and # ∂A < ∞ → R is lower semicontinuous. For a ﬁxed number c > 0, the subset {bA ∈ Cb (D) : # ∂A ≤ c} of Cb (D) is compact in C(D). In particular, the subset {bA ∈ Cb (D) : ∂A is connected} of Cb (D) is compact in C(D).

Proof. The proof is the same as for Theorem 2.3 of Chapter 6 using the Lipschitz continuity of the map bA → d∂A = |bA | from C(D) to C(D).

3

Projection, Skeleton, Crack, and Diﬀerentiability

This section is the analogue of section 3 in Chapter 6: connection between the gradient of bA and the projection onto ∂A and the characteristic functions associated with ∂A, singularities of the gradients, and the notions of skeleton1 and cracks. Let A ⊂ RN , ∅ = ∂A, and recall the notation def

(3.1) Sing (∇bA ) = x ∈ RN : ∇bA (x) for the set of singularities of the gradient of bA , def

Π∂A (x) = {z ∈ ∂A : |z − x| = d∂A (x)}

(3.2)

for the set of projections of x onto ∂A (when Π∂A (x) is a singleton, the unique element is denoted by p∂A (x)), and

def

Sk (∂A) = x ∈ RN : Π∂A (x) is not a singleton = x ∈ RN : ∇d2∂A (x) (3.3) for the skeleton of ∂A. Since Π∂A (x) is a singleton for all x ∈ ∂A we have Sk (∂A) ⊂ RN \∂A. In general, the functions bA and d∂A are diﬀerent and Sing ∇bA is usually smaller than Sing (∇d∂A ) = Sk (∂A) ∪ Ck (∂A). As a result, using the oriented distance function bA rather than d∂A requires a new deﬁnition of the set of cracks with respect to bA that will be justiﬁed in Theorem 3.1 (iv). Deﬁnition 3.1. Given A ⊂ RN , ∅ = ∂A, the set of b-cracks of A is deﬁned as def

Ck b (A) = Sing ∇bA \Sk (∂A).

(3.4)

Sk (∂A), Ck b (A), and Sing (∇bA ) have zero N -dimensional Lebesgue measure, since bA is Lipschitz continuous and hence diﬀerentiable almost everywhere. 1 Our deﬁnition of a skeleton does not exactly coincide with the one used in morphological `re [1]). mathematics (cf., for instance, G. Matheron [1] or A. Rivie

3. Projection, Skeleton, Crack, and Diﬀerentiability

345

Remark 3.1. (1) In general Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A), but Ck b (A) can be strictly smaller than Ck (∂A). Consider the open ball of unit radius A = B1 (0): x 1 x i xj bA (x) = |x| − 1, ∇bA (x) = , D2 bA (x)ij = δij − , |x| |x| |x| |x| Sing (∇bA ) = {0},

Sk (∂A) = {0},

and

Ck b (A) = ∅.

It is easy to check that for all x ∈ ∂A, ∇bA (x) is the outward unit normal, and D 2 bA (x) is the second fundamental form with eigenvalues 0 for the space spanned by the normal x and 1 for the tangent space to ∂A in x orthogonal to x. But d∂A (x) = ||x| − 1| , Sing (∇d∂A ) = {0} ∪ ∂B1 (0),

Sk (∂A) = {0},

and

Ck (∂A) = ∂B1 (0).

Using bA rather than d∂A or dA removes artiﬁcial singularities. (2) In general Sk (∂A) ⊂ RN \∂A, but if ∂A has positive reach h > 0 in the sense of Deﬁnition 6.1 in section 6 of Chapter 6 (there exists h > 0 such that b2A ∈ C 1,1 (Uh (∂A)), the skeleton will remain at a distance h from ∂A. The function bA enjoys properties similar to the ones of dA and dA since |bA | = d∂A = max{dA , dA } and we have the analogues of Theorems 3.2 and 3.3 in Chapter 6. Recall the deﬁnition of def

f∂A (x) =

1 2 |x| − d2∂A (x) . 2

Theorem 3.1. Let A be a subset of RN such that ∅ = ∂A, and let x ∈ RN . (i) For all x ∈ RN , d2∂A (x) = b2A (x), f∂A (x) =

1 2 |x| − b2A (x) 2

is convex and continuous, the function x →

|x|2 k∂A,h (x) = − |bA |(x) : RN \Uh (∂A) → R 2h 2h

is locally convex (and continuous) in RN \Uh (∂A), ∇f∂A and ∇b2A belong to N BVloc (RN )N , and ∇d∂A ∈ BVloc RN \∂(∂A) . (ii) The set Π∂A (x) is nonempty and compact, and ∀x ∈ / ∂A

Π∂A (x) ⊂ ∂(∂A)

and

∀x ∈ ∂A

Π∂A (x) = {x}.

/ A, Π∂A (x) = ΠA (x) ⊂ ∂A. For x ∈ / A, Π∂A (x) = ΠA (x) ⊂ ∂A and for x ∈ Moreover, ∂(∂A) = ∂A ∪ ∂A.

346

Chapter 7. Metrics via Oriented Distance Functions

(iii) For all x and v in RN , the Hadamard semiderivative2 of d2A always exists, b2A (x + tv) − b2A (x) = min 2(x − z) · v, t0 t z∈Π∂A (x) f∂A (x + tv) − f∂A (x) dH f∂A (x; v) = lim = σΠ∂A (x) (v) = σco Π∂A (x) (v), t0 t dH b2A (x; v) = lim

where σB is the support function of the set B, σB (v) = sup z · v, z∈B

and co B is the convex hull of B. In particular,

Sk (∂A) = x ∈ RN : ∇b2A (x) ⊂ RN \∂A.

(3.5)

Given v ∈ RN , for all x ∈ RN \∂(∂A) the Hadamard semiderivative of bA exists, dH bA (x; v) =

1 min (x − p) · v, bA (x) p∈ΠA (x)

(3.6)

and for x ∈ ∂(∂A), dH bA (x; v) exists if and only if lim

t0

bA (x + tv) t

exists.

(3.7)

(iv) The following statements are equivalent: (a) b2A (x) is (Fr´echet) diﬀerentiable at x. (b) b2A (x) is Gateaux diﬀerentiable at x. (c) Π∂A (x) is a singleton. (v) ∇f∂A (x) exists if and only if Π∂A (x) = {p∂A (x)} is a singleton. In that case 1 p∂A (x) = ∇f∂A (x) = x − ∇b2A (x). (3.8) 2 For all x and y in RN , 1 2 1 2 |y| − b2A (y) ≥ |x| − b2A (x) + p(x) · (y − x), (3.9) ∀p(x) ∈ Π∂A (x), 2 2 or equivalently, ∀p(x) ∈ Π∂A (x),

b2A (y) − b2A (x) − 2 (x − p(x)) · (y − x) ≤ |x − y|2 . (3.10)

For all x and y in RN , ∀p(x) ∈ Π∂A (x), ∀p(y) ∈ Π∂A (y), 2A

(p(y) − p(x)) · (y − x) ≥ 0.

function f : RN → R has a Hadamard semiderivative in x in the direction v if def

dH f (x; v) = lim

t0 w→v

(cf. Chapter 9, Deﬁnition 2.1 (ii)).

f (x + tw) − f (x) exists t

(3.11)

3. Projection, Skeleton, Crack, and Diﬀerentiability

347

(vi) The functions p∂A : RN \Sk (∂A) → RN

and

∇b2A : RN \Sk (∂A) → RN

are continuous. For all x ∈ RN \Sk (∂A) 1 p∂A (x) = x − ∇b2A (x) = ∇f∂A (x). 2 In particular, for all x ∈ ∂A, Π∂A (x) = {x}, ∇b2A (x) = 0,

Sk (∂A) = x ∈ RN : ∇b2A (x) and ∇bA (x) ⊂ RN \∂A,

Ck b (A) = x ∈ RN : ∇b2A (x) ∃ and ∇bA (x) ⊂ ∂(∂A),

(3.12)

(3.13)

and Sing (∇bA ) = Sk (∂A) ∪ Ck b (A). (vii) The function ∇bA : RN \ (Sk (∂A) ∪ ∂(∂A)) → RN is continuous. For all x ∈ int ∂A, ∇bA (x) = 0; for all x ∈ RN \ (Sk (∂A) ∪ ∂A), x − p∂A (x) N \A, , x ∈ R ∇b2A (x) |x − p∂A (x)| = , |∇bA (x)| = 1. ∇bA (x) = 2 bA (x) x − p∂A (x) − , x ∈ RN \A, |x − p∂A (x)| (3.14) If ∇d∂A (x) exists for some x ∈ ∂A, then ∇bA (x) = ∇d∂A (x) = 0. In particular, ∇bA (x) = 0 almost everywhere in ∂A. (viii) The functions b2A and bA are diﬀerentiable almost everywhere and m(Sk (∂A)) = m(Ck b (A)) = m(Sing (∇bA )) = 0. If ∂A = ∅, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A).

(3.15)

If ∂A = ∅, then either A = RN or A = ∅. (ix) Given A ⊂ RN , ∂A = ∅, χ∂A (x) = 1 − |∇bA (x)| in RN \Sing (∇bA ),

(3.16)

the identity holds for almost all x in RN , and the set def

∂b A = {x ∈ ∂A : ∇bA (x) exists and ∇bA (x) = 0} ⊂ ∂(∂A) has zero N -dimensional Lebesgue measure.

(3.17)

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Chapter 7. Metrics via Oriented Distance Functions

(x) Let µ be a measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1] and D ⊂ RN be bounded open such that µ(D) < ∞. The map bA → µ(∂A) : Cb (D) → R

(3.18)

is upper semicontinuous with respect to the topology of uniform convergence on Cb (D). Moreover, if bAn → bA in C(D) for bAn and bA in Cb (D), then µ(int A) ≤ lim inf µ(int An ) ≤ lim sup µ(An ) ≤ µ(A).

(3.19)

If µ(∂A) = 0, the map bA → µ(A) : Cb (D) → R is continuous in bA . def

(xi) Given x ∈ RN , α ∈ [0, 1], p ∈ Π∂A (x), and xα = p + α (x − p), then bA (xα ) = α bA (x)

∀α ∈ [0, 1] ,

and

Π∂A (xα ) ⊂ Π∂A (x).

In particular, if Π∂A (x) is a singleton, then Π∂A (xα ) is a singleton and ∂A, then for ∇b2A (xα ) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = all 0 < α ≤ 1 ∇bA (xα ) exists and ∇bA (xα ) = ∇bA (x). Proof. (i) to (v) follow from Theorems 3.2 and 3.3 of Chapter 6 using the identity b2A = d2∂A . (v) To complete the proof of identities (3.13), recall that the projection of all points of ∂A onto ∂A being a singleton, Sk (∂A ⊂ RN \∂A. Therefore, for any x ∈ Sk (∂A), we have bA (x) = 0 and if ∇b2A (x) does not exist, then ∇bA (x) cannot exist. This sharpens the characterization (3.5) of Sk (∂A). From this we readily get the characterization of Ck b (A) = Sing (∇bA )\Sk (∂A) and Ck b (A) ⊂ ∂A. This last property can be improved to Ck b (A) ⊂ ∂(∂A) by noticing that for x ∈ int ∂A, bA (x) = 0 and ∇b2A (x) = ∇bA (x) = 0. (vi) From Theorem 3.3 (iv) of Chapter 6. (vii) In RN \A, bA = dA and ∇bA is continuous in RN \(Sk (A) ∪ A); in N R \A, bA = −dA and ∇bA is continuous in RN \(Sk (A) ∪ A); in int ∂A, bA = dA = dA = 0 and ∇bA = 0. So, in view of part (v) it is continuous in RN \(Sk (A) ∪ A) ∪ RN \(Sk (A) ∪ A) ∪ int ∂A = RN \ (Sk (∂A) ∪ ∂(∂A)) . If x ∈ RN \A, then bA = −dA and repeat the proof of Theorem 3.3 (vi) in Chapter 6 to obtain −∇bA (x) = ∇dA (x) =

x − p∂A (x) x − p∂A (x) = |x − p∂A (x)| −bA (x)

and p∂A (x) ∈ ∂A is unique. Similarly, for x ∈ RN \A, bA = dA , ∇bA (x) = ∇dA (x) =

x − p∂A (x) x − p∂A (x) = |x − p∂A (x)| bA (x)

and p∂A (x) ∈ ∂A is unique. In both cases |∇bA (x) = 1.

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

349

Finally, from Theorem 3.3 (i) in Chapter 6, whenever d∂A is diﬀerentiable at a point x ∈ ∂A, ∇d∂A (x) = 0. But for all v in RN and t > 0 bA (x + tv) − bA (x) |bA (x + tv)| d∂A (x + tv) − d∂A (x) = = t t t since |bA | = d∂A . Therefore, if ∇d∂A (x) exists, ∇bA (x) exists and ∇bA (x) = ∇d∂A (x) = 0. Finally, since ∇d∂A (x) exists almost everywhere in RN , then for almost all x ∈ ∂A, ∇bA (x) = 0. (viii) From Theorem 3.3 (vii) in Chapter 6 and the fact that Ck b (A) ⊂ C(∂A). Indeed, for x ∈ ∂A, if ∇dA (x) and ∇dA (x) exist, they are 0 and for all t > 0 and v ∈ RN bA (x + tv) − bA (x) dA (x + tv) − dA (x) dA (x + tv) − dA (x) = →0 − t t t as t 0. Hence ∇bA (x) = 0. As a consequence, if ∇bA (x) does not exist, then either ∇dA (x) or ∇dA (x) does not exist and Ck b (A) ⊂ C(A) ∪ C(A) = C(∂A) ⊂ ∂(∂A) from Theorem 3.3 (vii) in Chapter 6. (ix) Since bA is a Lipschitzian function, it is diﬀerentiable almost everywhere in RN , and in view of the previous considerations, when it is diﬀerentiable |∇bA (x)| =

1, x ∈ / ∂A, 0 a.e. in ∂A

⇒ χ∂A (x) = 1 − |∇bA (x)|

a.e. in RN .

Therefore, the set ∂b A has at most a zero measure. (x) From Theorem 4.1 in Chapter 6 applied to A, A, and ∂A. (xi) Same proof as the one of Theorem 3.3 (ix) in Chapter 6. Remark 3.2. In general ∇dA (x) and ∇dA (x) do not exist for x ∈ ∂A. This is readily seen by constructing the directional derivatives for the half-space A = {(x1 , x2 ) ∈ R2 : x1 ≤ 0}

(3.20)

at the point (0, 0). Nevertheless ∇bA (0, 0) exists and is equal to (1, 0), which is the outward unit normal at (0, 0) ∈ ∂A to A. Note also that for all x ∈ ∂A, |∇bA (x)| = 1. This is possible since mN (∂A) = 0. If ∇bA (x) exists, then it is easy to check that |∇bA (x)| ≤ 1. For suﬃciently smooth domains, the subset ∂b A of the boundary ∂A coincides with the reduced boundary of ﬁnite perimeter sets.

4 4.1

W 1,p (D)-Metric Topology and the Family Cb0 (D) Motivations and Main Properties

1,p From Theorem 2.1 (vi) bA is locally Lipschitzian and belongs to Wloc (RN ) for all p ≥ 1. So the previous constructions for dA and dA can be repeated with the space

350

Chapter 7. Metrics via Oriented Distance Functions

1,p (RN ) in place of Cloc (D) to generate new W 1,p metric topologies on the family Wloc Cb (D). Moreover the other distance functions can be recovered from the map − bA → (b+ A , bA , |bA |) = (dA , dA , d∂A )

and the characteristic functions from the maps bA → b− A = dA → χint A = |∇dA |,

(4.1)

bA → = dA → χint A = |∇dA |, bA → χ∂A = 1 − |∇bA |.

(4.2) (4.3)

b+ A

One of the advantages of the function bA is that the W 1,p -convergence of sequences will imply the Lp -convergence of the corresponding characteristic functions of int A, int A, and ∂A, that is, continuity of the volume of these sets. It will be useful to introduce the following set and terminology. Deﬁnition 4.1. (i) A set A ⊂ RN , ∂A = ∅, is said to have a thin 3 boundary if mN (∂A) = 0. (ii) Denote by def

Cb0 (D) = {bA : ∀A, A ⊂ D, ∂A = ∅ and m(∂A) = 0}

(4.4)

the subset of oriented distance functions with thin boundaries. In this section we give the analogues of Theorems 4.2 and 4.3 in Chapter 6. Theorem 4.1. Let D be an open (resp., bounded open) subset of RN . 1,p (i) The topologies induced by Wloc (D) (resp., W 1,p (D)) on Cb (D) are all equivalent for p, 1 ≤ p < ∞. 1,p (D) (resp., W 1,p (D)) for p, 1 ≤ p < ∞, and (ii) Cb (D) is closed in Wloc def

ρD ([A2 ]b , [A1 ]b ) =

∞ bA2 − bA1 W 1.p (B(0,n)) 1 n 2 1 + bA2 − bA1 W 1.p (B(0,n)) n=1 def

(resp., ρD ([A2 ]b , [A1 ]b ) = bA2 − bA1 W 1,p (D) ) deﬁnes a complete metric structure on Fb (D). (iii) For p, 1 ≤ p < ∞, the map 1,p bA → χ∂A = 1 − |∇bA | : Cb (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and bA1 and bA2 in Cb (D), χ∂A2 − χ∂A1 Lp (K) ≤ ∇bA2 − ∇bA1 Lp (K) ≤ bA2 − bA1 W 1,p (K) . In particular, Cb0 (D) is a closed subset of Cb (D) for the W 1,p -topology. 3 This

terminology should not be confused with the notion of thin set in the capacity sense.

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

351

(iv) Let D be a bounded open subset of RN . For each bA ∈ Cb (D), bA W 1,p (D) = |bA | W 1,p (D) = d∂A W 1,p (D) , dA W 1,p (D) ≤ bA W 1,p (D) ,

dA W 1,p (D) ≤ bA W 1,p (D) .

(4.5)

The map − 1.p (D) → W 1.p (D)3 (4.6) bA → (b+ A , bA , |bA |) = (dA , dA , d∂A ) : Cb (D) ⊂ W

is continuous. (v) Let D be an open (resp., bounded open) subset of RN . For all p, 1 ≤ p < ∞, the map bA → (χ∂A , χint A , χint A ) 1,p (D) → Lploc (D)3 (resp.,W 1,p (D) → Lp (D)3 ) : Wloc

(4.7)

is continuous. Proof. The proofs of (i) and (ii) are essentially the same as the proof of Theorem 4.2 of Chapter 6 using the properties established for Cb (D) in C(D) of Theorem 2.2. (iii) From Theorem 3.1 (ix) for any two subsets A1 and A2 of D with nonempty boundaries and for any open U in D |∇bA2 | ≤ |∇bA1 | + |∇bA2 − ∇bA1 | + |∇bA2 − ∇bA1 | ⇒ |χ∂A1 − χ∂A2 | ≤ |∇bA2 − ∇bA1 |

⇒ χ∂A1 ≤ χ∂A2 |χ∂A2 − χ∂A1 |p dx ≤ ∇bA2 − ∇bA1 pLp (U ) ≤ bA2 − bA1 pW 1,p (U ) ⇒ U

for p, 1 ≤ p < ∞, and with the ess sup norm for p = ∞. The closure of Cb0 (D) follows from the continuity of the maps bA → m(∂A ∩ U ) = χ∂A dx : W 1,p (U ) → R ∂A∩U

for all bounded open subsets U of D. (iv) First observe that |bA (x)| = dA (x) + dA (x) and ∇|bA (x)| = ∇dA (x) + ∇dA (x), bA (x) = dA (x) − dA (x) and ∇bA (x) = ∇dA (x) − ∇dA (x) and since ∇bA = ∇dA = ∇dA = 0 almost everywhere on ∂A = A ∩ A, then |∇bA (x)| = |∇dA (x)| + |∇dA (x)| = |∇|bA (x)| | for almost all x in RN . From this we readily get (4.5). However, this is not suﬃcient to prove the continuity since the map (4.6) is not linear. To get around this consider a sequence {bAn } ⊂ Cb (D) converging to bA in W 1,p (D). From Theorem 2.2 (iii), |bAn | = d∂An , b+ An = + − dAn , and b− = d converge to |b | = d , b = d , and b = d in C(D) A ∂A A An A An A A

352

Chapter 7. Metrics via Oriented Distance Functions

and hence in W 1,p (D)-weak. To prove that the convergence is strong, consider the L2 -norm |∇dAn − ∇dA |2 + |∇dAn − ∇dA |2 dx D = |∇dAn |2 + |∇dA |2 + |∇dAn |2 + |∇dA |2 D − 2 ∇dAn · ∇dA − 2 ∇dAn · ∇dA dx |∇bAn |2 + |∇bA |2 − 2 ∇dAn · ∇dA − 2 ∇dAn · ∇dA dx = D 2 2 2 → 2 |∇bA | − 2 |∇dA | − 2 |∇dA | dx = 2 |∇bA |2 − 2 |∇bA |2 dx = 0 D

D

by weak L -convergence of ∇dAn and ∇dAn to ∇dA and ∇dA . So both sequences dAn and dAn converge in W 1,2 (D)-strong, and hence from part (i) in W 1,p (D)-strong for all p, 1 ≤ p < ∞. (v) It is suﬃcient to prove the result for D bounded open. From part (iii) it is true for χ∂A , and from part (iv) and Theorem 4.2 (ii) and (iii) of Chapter 6 for the other two. 2

4.2

Weak W 1,p -Topology

Theorem 4.2. Let D be a bounded open domain in RN . (i) If {bAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. (ii) If {bAn } converges in C(D), then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. Conversely, if {bAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, it converges in C(D). (iii) Cb (D) is compact in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Proof. (i) Recall that for D bounded there exists a constant c > 0 such that for all bA ∈ Cb (D) |bA (x)| ≤ c and |∇bA (x)| ≤ 1 a.e. in D. If {bAn } weakly converges in W 1,p (D) for some p ≥ 1, then {bAn } weakly converges in Lp (D), {∇bAn } weakly converges in Lp (D)N . By Lemma 3.1 (i) in Chapter 5 both sequences weakly converge for all p ≥ 1 and hence {bAn } weakly converges in W 1,p (D) for all p ≥ 1. (ii) If {bAn } converges in C(D), then by Theorem 2.2 (ii) there exists bA ∈ Cb (D) such that bAn → bA in C(D) and hence in Lp (D). So for all ϕ ∈ D(D)N ∇bAn · ϕ dx = − bAn div ϕ dx → − bA div ϕ dx = ∇bA · ϕ dx. D

D

D

D

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

353

By density of D(D) in L2 (D), ∇bAn → ∇bA in L2 (D)N -weak and hence bAn → bA in W 1,2 (D)-weak. From part (i) it converges in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Conversely, the weakly convergent sequence converges to some f in W 1,p (D). By compactness of Cb (D), there exist a subsequence, still indexed by n, and bA such that bAn → bA in C(D) and hence in W 1,p (D)-weak. By uniqueness of the limit, bA = f . Therefore, all convergent subsequences in C(D) converge to the same limit. So the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {bAn } in Cd (D). From Theorem 2.2 (ii) Cb (D) is compact and there exist a subsequence {bAnk } and bA ∈ Cb (D) such that bAnk → bA in C(D). From part (ii) the subsequence weakly converges in W 1,p (D) and hence Cb (D) is compact in W 1,p (D)-weak. For sets with thin boundaries the strong and weak W 1,p (D)-convergences of elements of Cb0 (D) to an element of Cb0 (D) are equivalent as in the case of Theorem 3.2 in section 3.2 of Chapter 5 for characteristic functions in Lp (D)-strong. Theorem 4.3. Let ∅ = D ⊂ RN be bounded open. (i) Let {An } be a sequence of subsets of D such that ∂An = ∅ and m(∂An ) = 0 ¯ be such that ∂A = ∅ and m(∂A) = 0. Then and A ⊂ D ⇒ bAn → bA in W 1,2 (D)-strong,

bAn bA in W 1,2 (D)-weak

and hence in W 1,p (D)-strong for all p, 1 ≤ p < ∞. (ii) The map bA → mN (int A) = mN (A) = mN (A) : Cb0 (D) → R

(4.8)

is continuous with respect to the W 1,p -topology on Cb0 (D) for all p, 1 ≤ p < ∞. Proof. (i) From Theorem 4.1 (i) on the equivalence of the W 1,p -topologies, it is suﬃcient to prove the result for p = 2. By Theorem 4.2 (ii), the weak L2 -convergence of {bAn } to bA implies the strong convergence in C(D) and hence in L2 (D)-strong. Since, for all n ≥ 1, m(∂An ) = 0 = m(∂A), |∇bA | = 1 = |∇bAn | almost everywhere in D by Theorem 3.1 (vii). As a result |∇bAn − ∇bA |2 dx = |∇bAn |2 + |∇bA |2 − 2∇bAn · ∇bA dx D D 2 χ∂A dx = 0. = 2 (1 − ∇bAn · ∇bA ) dx → 2 (1 − |∇bA | ) dx = 2 D

D

D

Therefore ∇bAn → ∇bA in L2 (D)N -strong and bAn → bA in W 1,2 (D)-strong, since the convergence bAn → bA in L2 (D)-strong follows from the weak convergence in W 1,2 (D). The convergence in W 1,p (D)-strong follows from the equivalence of the topologies on Cb (D) (cf. Theorem 4.1 (i)). (ii) From Theorem 3.1 (x) since mN (∂A) = 0 for bA ∈ Cb0 (D) and Cb0 (D) is closed in the W 1,p -topology.

354

5

Chapter 7. Metrics via Oriented Distance Functions

Boundary of Bounded and Locally Bounded Curvature

We introduce the family of sets with bounded or locally bounded curvature which are the analogues for bA of the sets of exterior and interior bounded curvature associated with dA and dA in section 5 of Chapter 6. They include C 1,1 -domains, convex sets, and the sets of positive reach of H. Federer [3]. They lead to compactness theorems for subfamilies of Cb (D) in W 1,p (D). Deﬁnition 5.1. (i) Given a bounded open nonempty holdall D in RN and a subset A of D, ∂A = ∅, its boundary ∂A is said to be of bounded curvature with respect to D if ∇bA ∈ BV(D)N .

(5.1)

This family of sets will be denoted as follows: def

BCb (D) = bA ∈ Cb (D) : ∇bA ∈ BV(D)N . (ii) Given a subset A of RN , ∂A = ∅, its boundary ∂A is said to be of locally bounded curvature if ∇bA belongs to BVloc (RN )N . The family of sets whose boundary is of locally bounded curvature will be denoted by def

BCb = bA ∈ Cb (RN ) : ∇bA ∈ BVloc (RN )N . As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, and in Theorem 5.1 of Chapter 6 for sets with a locally bounded curvature, it is suﬃcient to satisfy the BV property in a neighborhood of each point of the boundary ∂A. Theorem 5.1. Let A, ∂A = ∅, be a subset of RN . Then ∂A is of locally bounded curvature if and only ∀x ∈ ∂(∂A), ∃ρ > 0 such that ∇bA ∈ BV(B(x, ρ))N ,

(5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x. Proof. From Deﬁnition 5.1 and Theorem 3.1 (i). Theorem 5.2. (i) Let D be a nonempty bounded open Lipschitzian holdall in RN and let A ⊂ D, ∂A = ∅. If ∇bA ∈ BV(D)N , then ∇χ∂A M 1 (D) ≤ 2 D 2 bA M 1 (D) ,

χ∂A ∈ BV(D),

and ∂A has ﬁnite perimeter with respect to D. (ii) For any subset A of RN , ∂A = ∅, ∇bA ∈ BVloc (RN )N and ∂A has locally ﬁnite perimeter.

⇒

χ∂A ∈ BVloc (RN ),

5. Boundary of Bounded and Locally Bounded Curvature

355

Proof. Given ∇bA in BV(D)N , there exists a sequence {uk } in C ∞ (D)N such that uk → ∇bA in L1 (D)N

and Duk M 1 (D) → D2 bA M 1 (D)

as k goes to inﬁnity, and since |∇bA (x)| ≤ 1, this sequence can be chosen in such a way that ∀k ≥ 1, |uk (x)| ≤ 1. This follows from the use of molliﬁers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N

−

(|∇bA | − 1) div V dx =

|∇bA |2 div V dx.

2

χ∂A div V dx = D

D

D

For each uk |uk |2 div V dx = −2 [ ∗Duk ]uk · V dx = −2 uk · [Duk ]V dx, D

D

D

where ∗Duk is the transpose of the Jacobian matrix Duk and |uk |2 div V dx ≤ 2 |uk | |Duk | |V | dx D

D

≤ 2 Duk L1 V C(D) ≤ 2 Duk M 1 V C(D) since for W 1,1 (D)-functions ∇f L1 (D)N = ∇f M 1 (D)N . Therefore, as k goes to inﬁnity, χ∂A div V dx = |∇bA |2 div V dx ≤ 2D2 bA M 1 V C(D) . D

D

Therefore, ∇χ∂A ∈ M 1 (D)N .

5.1

Examples and Limit of Tubular Norms as h Goes to Zero

It is informative to compute the Laplacian of bA for a few examples. The ﬁrst three examples are illustrated in Figure 7.1. Example 5.1 (Half-plane in R2 ). (cf. Example 5.1 in Chapter 6). Consider the domain A = {(x1 , x2 ) : x1 ≤ 0},

∂A = {(x1 , x2 ) : x1 = 0}.

It is readily seen that bA (x1 , x2 ) = x1 ,

∇bA (x1 , x2 ) = (1, 0),

∆bA (x) = 0,

356

Chapter 7. Metrics via Oriented Distance Functions

and

b∂A (x1 , x2 ) = |x1 |,

∇b∂A (x1 , x2 ) = ∆b∂A , ϕ = 2 ϕ dx.

x1 ,0 , |x1 |

∂A

Figure 7.1. ∇bA for Examples 5.1, 5.2, and 5.3. Example 5.2 (Ball of radius R > 0 in R2 ). (cf. Example 5.2 in Chapter 6). Consider the domain A = {x ∈ R2 : |x| ≤ R},

∂A = {x ∈ R2 : |x| = R}.

Clearly, x bA (x) = |x| − R, ∇bA (x) = , |x| 1 ∆bA , ϕ = ϕ dx. 2 |x| R Also b∂A (x) = |x| − R ,

∇b∂A (x) =

ϕ ds −

∆b∂A , ϕ = 2 ∂A

A

|x| > R,

x , |x| x − |x| ,

1 ϕ dx + |x|

|x| < R,

A

1 ϕ dx. |x|

Again ∆b∂A contains twice the boundary measure on ∂A. Example 5.3 (Unit square in R2 ). (cf. Example 5.3 in Chapter 6). Consider the domain A = {x = (x1 , x2 ) : |x1 | ≤ 1, |x2 | ≤ 1} . Since A is symmetrical with respect to both axes, it is suﬃcient to specify bA in the ﬁrst quadrant. We use the notation Q1 , Q2 , Q3 , and Q4 for the four quadrants in

5. Boundary of Bounded and Locally Bounded Curvature

357

the counterclockwise order and c1 , c2 , c3 , and c4 for the four corners of the square in the same order. We also divide the plane into three regions: D1 = {(x1 , x2 ) : |x2 | ≤ min{1, |x1 |}} , D2 = {(x1 , x2 ) : |x1 | ≤ min{1, |x2 |}} , D3 = {(x1 , x2 ) : |x1 | ≥ 1 and |x2 | ≥ 1} . Hence for 1 ≤ i ≤ 4 |x2 | − 1, bA (x) = |x − ci |, |x1 | − 1,

x ∈ D2 ∩ Qi , x ∈ D3 ∩ Qi , x ∈ D1 ∩ Qi ,

and for the whole plane ∆bA , ϕ =

4 i=1

D3 ∩Qi

(0, 1), x−ci , ∇bA (x) = |x−c i| (1, 0),

√ 1 ϕ dx + 2 |x − ci |

x ∈ D2 ∩ Qi , x ∈ D3 ∩ Qi , x ∈ D1 ∩ Qi ,

ϕ dx, D1 ∩D2

where D1 ∩ D2 is made up of the two diagonals of the square where ∇bA has a singularity (that is, the skeleton). Moreover, for 1 ≤ i ≤ 4 x ∈ D2 ∩ Qi , ||x2 | − 1| , b∂A (x) = |x − ci |, x ∈ D3 ∩ Qi , ||x1 | − 1| , x ∈ D1 ∩ Qi , and ∆b∂A , ϕ =

4 i=1

D3 ∩Qi

√ 1 ϕ dx − 2 |x − ci |

ϕ dx + 2 D1 ∩D2

ϕ dx. ∂A

Notice that the structures of the Laplacian are similar to the ones observed in the previous examples except for the presence of a singular term along the two diagonals of the square. All C 2 -domains with a compact boundary belong to all the categories of Deﬁnition 5.1 and of Deﬁnition 5.1 of Chapter 6. The h-dependent norms D2 bA M 1 (Uh (∂A)) , D 2 dA M 1 (Uh (A)) , and D2 dA M 1 (Uh (A)) are all decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary ∂A. Example 5.4. If A ⊂ RN is of class C 2 with compact boundary, then lim D2 bA M 1 (Uh (∂A)) = 0.

h0

Example 5.5. Let A = {xi }Ii=1 be I distinct points in RN . Then ∂A = A, and lim D 2 bA M 1 (Uh (∂A)) =

h0

2 I − 1, N = 1, 0, N ≥ 2.

358

Chapter 7. Metrics via Oriented Distance Functions

Example 5.6. Let A be a closed line in RN of length L > 0. Then ∂A = A, and lim D2 bA M 1 (Uh (∂A)) =

h0

2 L, N = 2, 0, N ≥ 3.

Example 5.7. Let N = 2. For the ﬁnite square and the ball of ﬁnite radius lim ∆dA M 1 (Uh (∂A)) = H1 (∂A),

h0

where H1 is the one-dimensional Hausdorﬀ measure (cf. Examples 5.2 and 5.3 in Chapter 6). Also, looking at ∆bA in Uh (A) as h goes to zero provides information about Sk (A). Example 5.8. Let A be the unit square in R2 . Then lim ∆bA M 1 (Uh (A))

h0

√ 2 = H1 (Sk (A)), 2

where H1 is the one-dimensional Hausdorﬀ measure and Sk(A) = Skint (A) is the skeleton of A made up of the two interior diagonals (cf. Example 5.3). It would seem that in general lim ∆bA M 1 (Uh (A)) = | [∇bA ] · n| dH1 , h0

Sk(A)

where [∇bA ] is the jump in ∇bA and n is the unit normal to Sk(A) (if it exists!).

6

Approximation by Dilated Sets/Tubular Neighborhoods

We have seen in Theorem 7.1 of Chapter 6 that dA can be approximated by the distance function dAr of the dilated set Ar = {x ∈ RN : dA (x) ≤ r}, r > 0, of A in the W 1,p -topology. The approximation can also be achieved with the oriented distance function: bAr (x) → bA (x) uniformly in RN . But it requires that the measure of the boundary ∂A be zero to get the approximation in the W 1,p -topology. Lemma 6.1. Given A ⊂ RN , ∂ A¯ = ∅ if and only if A = ∅ and there exists h > 0 such that Ah = RN . Proof. If ∂ A¯ = ∅, then A = ∅ and A = ∅, A = ∅ and A = ∅, and RN = A = ∩h>0 Ah

⇒ ∃h > 0 such that Ah = RN .

Conversely, A ⊃ A = ∅ and, for some h > 0, A = ∩k>0 Ak ⊂ Ah = RN . Therefore A = ∅ and A = ∅ imply that ∂A = ∅.

6. Approximation by Dilated Sets/Tubular Neighborhoods

359

Theorem 6.1. Given A ⊂ RN such that ∂A = ∅, there exists h > 0 such that for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN . (i) For all r, 0 < r ≤ h, A ⊃ Ar = ∅, ∂Ar = ∅, bUr (A) = bAr , and Ur (A) and Ar are invariant open and closed representatives in the equivalence class [Ar ]b . Moreover, dUr (A) (x) = dAr (x) = 0,

if dA (x) ≥ r,

dUr (A) (x) = dAr (x) ≥ r − dA (x),

if 0 < dA (x) < r,

(6.1)

dUr (A) (x) = dAr (x) ≥ dA (x) + r,

if dA (x) = 0,

(6.2)

∀x ∈ R , N

0 ≤ bA (x) − bAr (x) ≤ r,

(6.3)

and, as r → 0, bUr (A) (x) = bAr (x) → bA (x) uniformly in RN , χAr → χA in Lploc (RN ),

∀p, 1 ≤ p < ∞.

(ii) The following conditions are equivalent: (a) m(∂A) = 0. 1,p (RN ) for some p, 1 ≤ p < ∞. (b) As r → 0, bUr (A) = bAr → bA in Wloc 1,p (RN ) for some p, 1 ≤ p < ∞. (c) As r → 0, dUr (A) = dAr → dA in Wloc

If (a), (b), or (c) is veriﬁed, then (b) and (c) are veriﬁed for all p, 1 ≤ p < ∞. In particular, (a) is veriﬁed for all bounded sets A = ∅ in RN such that m(∂A) = 0. Proof. The existence of the h > 0 such that Ah = RN and A = ∅ is a consequence of Lemma 6.1. From this for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN . (i) From the initial remarks, RN = A ⊃ A¯ ⊃ Ar ⊃ Ah = ∅ and ∂Ar = ∅ and ∂A = ∅, and the functions dA , dAr , bA , and bAr are well-deﬁned. From Theorem 3.1 of Chapter 6, Ar = Ur (A), and Ar = Ur (A), and ∂Ar = d−1 A {r}. Hence, dAr = dUr (A) . Hence, dAr = dUr (A) , dAr = dUr (A) , and bAr = bUr (A) . For x ∈ RN such that dA (x) ≥ r, x ∈ Ur (A) and dUr (A) (x) = 0. Inequality (6.1). For all y ∈ Ur (A) and all p ∈ A, r < dA (y) ≤ |y − p| ≤ |y − x| + |x − p|, r ≤ inf |y − x| + inf |x − p| = dUr (A) (x) + dA (x). y∈Ur (A)

p∈A

Inequality (6.2). Since A ∩ Ur (A) = ∅, for x ∈ A, there exists pr ∈ ∂Ur (A) = d−1 A {r} such that dUr (A) (x) = |pr − x| > 0. Therefore dA (pr ) = r. Either dA (x) = 0 or dA (x) > 0. If dA (x) = 0, then dUr (A) (x) = |pr − x| ≥ dA (pr ) = r = dA (x) + r.

360

Chapter 7. Metrics via Oriented Distance Functions

def If dA (x) > 0, then x ∈ / A and x ∈ int A¯ = A\∂A, Bx = B(x, dA (x)) ⊂ int A, and Bx ⊂ A. Deﬁne pr − x def p = x + dA (x) ∈ Bx ⇒ |p − x| = dA (x) and p ∈ Bx ⊂ A. |pr − x|

By construction dUr (A) (x) = |x − pr | = |x − p| + |p − pr | = dA (x) + |p − pr |, |p − pr | ≥ inf |p − pr | ≥ inf |p − pr | = dA (pr ) = r p ∈Bx

p ∈A

and dUr (A) (x) ≥ dA (x) + r. Inequality (6.3). From Theorem 2.1 (ii), A¯ ⊂ Ar implies that bA¯ (x) ≥ bAr (x). As for the second part of inequality (6.3), we use identity (7.2) from Theorem 7.1 in Chapter 6 for dUr (A) = dAr combined with (6.2) for x ∈ A and with (6.1) for x ∈ RN \A: r ≤ dA (x)

⇒ x ∈ Ur (A) ⊂ A

0 < dA (x) < r

⇒ bUr (A) (x) = dUr (A) (x) = dA (x) − r = bA (x) − r, ⇒ dUr (A) (x) = 0 and dA (x) = 0 ⇒ bUr (A) (x) = −dUr (A) (x) ≤ dA (x) − r = bA (x) − r,

⇒ dUr (A) (x) = dA (x) = 0

⇒ x ∈ A and dUr (A) (x) = 0 ⇒ bUr (A) (x) = −dUr (A) (x) ≤ −dA (x) − r = bA (x) − r.

dA (x) = 0

(ii) If m(∂A) = 0, from part (i), dUr (A) (x) → dA (x) implies that ∇dUr (A) ∇dA in L2 (D)N -weak for all bounded open sets. Consider the estimate |∇dUr (A) − ∇dA |2 dx = |∇dUr (A) |2 + |∇dA |2 − 2 ∇dUr (A) · ∇dA dx D D 1 − χUr (A) + 1 − χ − 2 ∇dUr (A) · ∇dA dx = A

D

from the identities of Theorem 3.3 (viii) in Chapter 6. As r → 0, {Ur (A)} is a family of increasing closed sets and Ur (A) ∪r>0 Ur (A) = ∩r>0 Ur (A) = A

⇒ m(D ∩ Ur (A)) → m(D ∩ A).

Going to the limit 1 − χUr (A) + 1 − χ − 2 ∇dUr (A) · ∇dA dx A D 2 − χA − χ − 2 |∇dA |2 dx = χ − χA dx = → χ∂A dx = 0, D

A

since |∇dA | = 1 − χ

A

D

A

D

, ∂A = A\A, and m(∂A) = 0. Therefore dUr (A) → dA

1,p in W (D) and hence in Wloc (RN ) for all p, 1 ≤ p < ∞. So, from Theorem 7.1 1,p in Chapter 6, bUr (A) = dUr (A) − dUr (A) → dA − dA = bA in Wloc (RN ) for all p, 1 ≤ p < ∞. 1,2

7. Federer’s Sets of Positive Reach

7

361

Federer’s Sets of Positive Reach

7.1

Approximation by Dilated Sets/Tubular Neighborhoods

We have seen in Theorem 6.1 that the oriented distance function bA can be approximated uniformly in RN by the oriented distance function bAr of its dilated set Ar , r > 0, and that the W 1,p -convergence can be achieved if and only if m(∂A) = 0. It turns out that the W 1,p -convergence can be obtained for sets with positive reach such that ∂A = ∅ without assuming a priori that m(∂A) = 0. Theorem 7.1. Let A ⊂ RN such that ∂A = ∅ and reach (A) > 0. (i) A = ∅ and there exists h such that 0 < h ≤ reach (A) and Ah = RN . For all r, 0 < r < h, the sets Ur (A) are of class C 1,1 , bUr (A) = bAr , and bAr (x) = bA (x) − r in RN .

(7.1)

As r goes to zero, bAr (x) → bA (x) uniformly in RN , 1,p bAr → bA in Wloc (RN ),

dAr → d A ,

dAr → dA ,

d∂Ar → d∂A in

(7.2) 1,p Wloc (RN ),

(7.3)

m(∂A) = 0, and χAr → χA = χint A in Lploc (RN ) for all p, 1 ≤ p < ∞. (ii) Furthermore, ∂A is of locally bounded curvature and of locally ﬁnite perimeter: ∇bA ∈ BVloc (RN )N

and

χ∂A ∈ BVloc (RN ).

Moreover, for all r, 0 < r < h, / 1 1. 2 bA (x) = f∂Ar (x) − f∂A (x) + r = bA (x) − b2Ar (x) + r, r r ∀x ∈ RN , ∀v ∈ RN , dH bA (x; v) exists.

(7.4)

(7.5) (7.6)

Remark 7.1. Recall from Theorem 3.1 (iii) that the Hadamard semiderivative of bA always exists at points oﬀ the boundary. The last statement in part (ii) says that for a closed set with nonempty boundary and positive reach, the Hadamard semiderivative exists in all points of its boundary and there is an explicit expression for it. In particular, this is true for all closed convex sets. Proof. (i) The uniform convergence of bAr to bA in RN was proved in Theorem 6.1 (i), and it remains to prove the W 1,p -convergence without the a priori assumption that m(∂A) = 0 that will come as a consequence rather than as a hypothesis. To do that 1,p (RN ). we ﬁrst prove the convergence dUr (A) → dA in Wloc From Theorem 6.1 (i), it suﬃcient to show that inequalities (6.1) and (6.2) become equalities for x ∈ Uh (A): for 0 < r < h dUr (A) (x) = 0,

if dA (x) ≥ r,

dUr (A) (x) ≥ r − dA (x),

if 0 < dA (x) < r,

dUr (A) (x) ≥ dA (x) + r, if dA (x) = 0.

(7.7) (7.8)

362

Chapter 7. Metrics via Oriented Distance Functions

For x ∈ RN such that dA (x) ≥ r, dA (x) = dUr (A) (x) = 0 and, from identity (7.2) of Chapter 6 in Theorem 6.1, bAr (x) = dAr (x) = dA (x) − r = bA (x) − r. For x ∈ RN such that 0 < dA (x) < r, dA (x) = 0 = dAr (x). From Theorem 6.2 (iii) of Chapter 6 for all r, 0 < dA (x) < r < h ≤ reach (A), def

x(r) = pA (x) + r

x − pA (x) dA (x)

⇒ dA (x(r)) = r and pA (x(r)) = pA (x)

/ Ur (A), dUr (A) (x) = inf p∈∂Ar |x − p| and and x(r) ∈ ∂Ar = ∂Ur (A). Since x ∈ dUr (A) (x) = dAr (x) = d∂Ar (x) ≤ |x(r) − x| = r − dA (x). Combining this with inequality (7.7), bAr (x) = bA (x) − r. Finally, for x ∈ RN such that 0 = dA (x), x ∈ A, and since ∂A = ∅, there exists p¯ ∈ ∂A such that p − x|. There exists a sequence {yn } ⊂ Ur (A)\A such that yn → p and dA (x) = |¯ cr such that p)| ≤ cr |yn − p¯| |pA (yn ) − pA (¯

⇒ pA (yn ) → pA (¯ p) = p¯.

Associate with each yn the points def

qn = pA (yn ) + r ∇dA (yn ). Since 0 < dA (qn ) ≤ r < h ≤ reach (A), by Theorem 6.2 (iii) of Chapter 6 pA (qn ) = pA (yn ),

dA (qn ) = r,

qn ∈ ∂Ar .

The sequence {qn } is bounded since p)| |qn − p¯| ≤ |qn − pA (qn )| + |pA (qn ) − p¯| = |qn − pA (qn )| + |pA (yn ) − pA (¯ ≤ r + cr |yn − p¯|. There exist q and a subsequence, still indexed by n, such that qn → q ⇒ r = dA (qn ) → dA (q) ⇒ pA (yn ) = pA (qn ) → pA (q) ⇒ dA (q) = r and pA (q) = p¯, r = dA (q) = |q − pA (q)| = |q − p¯|. Finally, since x ∈ A, A ∩ Ar = ∅, and ∂Ar = ∂Ar = d−1 A {r} dAr (x) =

inf dA (pr )=r

|x − pr | ≤ |x − q| ≤ |x − p¯| + |¯ p − q| = dA (x) + r.

Hence, from inequality (7.8), dAr (x) = dA (x) + r, and since dA (x) = 0 = dAr (x), bAr (x) = bA (x) − r. Since ∇bAr = ∇bA , for all bounded open subsets D of RN as r → 0, bAr − bA W 1,p (D) = bAr − bA Lp (D) ≤ r m(D)1/p → 0.

7. Federer’s Sets of Positive Reach

363

From Theorem 7.1 of Chapter 6 1,p (RN ) dAr → dA in Wloc

and χAr → χA in Lploc (RN )

for all p, 1 ≤ p < ∞. For all 0 < r < h ≤ reach (A), dAr = dAr − bAr → dA − bA = 1,p 1,p (RN ) and d∂Ar = dAr + dAr → dA + dA = d∂A in Wloc (RN ). Finally, dA in Wloc 1,p N since bAr → bA in Wloc (R ), 0 = χ∂Ar = 1 − |∇bAr | → 1 − |∇bA | = χ∂A in Lploc (RN ) and χ∂A = 0 and m(∂A) = 0. (ii) For 0 < r < h < reach (A), the open set Ur (A) is deﬁned via the r-level N set d−1 A {r} = {x ∈ R : dA (x) = r} of the function dA . From Theorem 6.2 (vi) in Chapter 6, for each x ∈ d−1 A {r} = ∂Ur (A) there exists a neighborhood Bρ (x), ρ = min{r, h − r}/2 > 0, of x in Uh (A)\A such that dA ∈ C 1,1 (Bρ (x)). From part (i) for all 0 < r < h and y ∈ Bρ (x) ⊂ Uh (A)\A, dA (y) = 0, bAr (y) = bA (y)−r = dA (y)−r. Therefore, bAr ∈ C 1,1 (Bρ (x)), and, a fortiori, for all x ∈ ∂Ar , there exists Bρ (x) such that ∇bAr = ∇dA ∈ BV(Bρ (x)). By Theorem 5.1, ∂Ar is of locally bounded curvature. By Deﬁnition 5.1, ∇bAr ∈ BVloc (RN )N and, since bAr = bA − r in RN , we also have ∇bA = ∇bAr ∈ BVloc (RN )N . By Theorem 5.2 (ii), χ∂A ∈ BVloc (RN ). Finally, since bAr (x) = bA (x) − r in RN from part (i) |x|2 − b2Ar (x) = |x|2 − b2A (x) − r2 + 2r bA (x), f∂Ar (x) − f∂A (x) + r2 = r bA (x) and bA is the diﬀerence of two convex continuous functions for which the Hadamard semiderivative exists in all points x ∈ RN and for all directions v ∈ RN .

7.2

Boundaries with Positive Reach

The next theorem is a specialization of Theorem 6.2 of Chapter 6 to the oriented distance function. Theorem 7.2. Let A ⊂ RN , ∂A = ∅. (i) Associate with a ∈ ∂A the sets def

P (a) = v ∈ RN : Π∂A (a + v) = {a} ,

def

Q(a) = v ∈ RN : d∂A (a + v) = |v| .

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta ∂A)∗ . (ii) Given a ∈ ∂A and v ∈ RN , assume that def

0 < r(a, v) = sup {t > 0 : Π∂A (a + tv) = {a}} . Then for all t, 0 ≤ t < r(a, v), Π∂A (a + tv) = {a} and d∂A (a + tv) = t |v|. Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (∂A). (iii) If there exists h such that 0 < h ≤ reach (∂A), then for all x ∈ Uh (∂A)\∂A and 0 < |t| < h, Π∂A (x(t)) = {p∂A (x)} and bA (x(t)) = t, where def

x(t) = p∂A (x) + t ∇bA (x) = p∂A (x) + t

x − p∂A (x) . bA (x)

(7.9)

364

Chapter 7. Metrics via Oriented Distance Functions

(iv) Given b ∈ ∂A, x ∈ RN \Sk (∂A), a = p∂A (x), such that reach (∂A, a) > 0, then (x − a) · (a − b) ≥ −

|a − b|2 |x − a| . 2 reach (∂A, a)

(7.10)

(v) Given 0 < r < q < ∞ and x ∈ RN \Sk (∂A) such that |bA (x)| ≤ r and reach (∂A, p∂A (x)) ≥ q, y ∈ RN \Sk (∂A) such that |bA (y)| ≤ r and reach (∂A, p∂A (y)) ≥ q, then |p∂A (y) − p∂A (x)| ≤

q |y − x|. q−r

(7.11)

(vi) If 0 < reach (∂A) < +∞, then for all h, 0 < h < reach (∂A), and all x, y ∈ (∂A)h = {x ∈ RN : |bA (x)| ≤ h} reach (∂A) |p∂A (y) − p∂A (x)| ≤ |y − x|, reach (∂A) − h 2 ∇bA (y) − ∇b2A (x) ≤ 2 1 + reach (∂A) |y − x|; (7.12) reach (∂A) − h

for all s, 0 < s < h < reach (∂A), x, y ∈ ∂As,h = x ∈ RN : s ≤ |bA (x)| ≤ h , reach (∂A) 1 2+ |y − x| . |∇bA (y) − ∇bA (x)| ≤ s reach (∂A) − h If reach (∂A) = +∞, ∀x, y ∈ RN , |p∂A (y) − p∂A (x)| ≤ |y − x|, 2 ∇bA (y) − ∇b2A (x) ≤ 4 |y − x|;

(7.13)

for all 0 < s and all x, y ∈ {x ∈ RN : s ≤ d∂A (x)} |∇bA (y) − ∇bA (x)| ≤

3 |y − x| . s

(vii) If reach (∂A) > 0, then, for all 0 < r < reach (∂A), Ur (∂A) is a set of class C 1,1 , Ur (∂A) = (∂A)r , ∂Ur (∂A) = {x ∈ RN : |bA (x)| = r}, and (∂A)r = Ur (∂A) = {x ∈ RN : |bA (x)| ≥ r}. (viii) For all a ∈ ∂A,

Ta ∂A =

|bA (a + th)| =0 . h ∈ R : lim inf t0 t N

In particular, if a ∈ int ∂A, Ta ∂A = RN .

8. Boundary Smoothness and Smoothness of bA

365

The next theorem is a specialization of Theorem 6.3 of Chapter 6 to the oriented distance function. Theorem 7.3. Given A ⊂ RN such that ∂A = ∅, the following conditions are equivalent: (i) ∂A has positive reach, that is, reach (∂A) > 0. (ii) There exists h > 0 such that b2A ∈ C 1,1 (Uh (∂A)). (iii) There exists h > 0 such that for all s, 0 < s < h, bA ∈ C 1,1 (Us,h (∂A)). (iv) There exists h > 0 such that bA ∈ C 1 (Uh (∂A)\∂A). (v) There exists h > 0 such that for all x ∈ Uh (∂A), Π∂A (x) is a singleton. Recall from Theorem 3.1 (viii) and (3.15) the following properties. If ∂A = ∅, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A). If ∂A has positive reach, then both A and A have positive reach.

8

Boundary Smoothness and Smoothness of bA

In this section the smoothness of the boundary ∂A is related to the smoothness of the function bA in a neighborhood of ∂A. The next two theorems give a complete characterization of sets A such that m(∂A) = 0 and bA ∈ C k, in a neighborhood of ∂A. We ﬁrst present two equivalence theorems that will follow from the more detailed Theorems 8.3 and 8.4. It is useful to recall that for a set that is locally of class C k, in a point of ∂A, ∂A is locally a C k, -submanifold of dimension N − 1 in that point, but the converse is not true (cf. Deﬁnitions 3.1 and 3.2 in Chapter 2)4 . Theorem 8.1 (Local version). Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤ ≤ 1 be a real number, and let x ∈ ∂A. 4 It is important to note that Theorem 8.1 is not true when b is replaced by d A ∂A since its gradient ∇d∂A is discontinuous across ∂A. Part (iii) of Theorem 8.1 in the direction (⇒) was asserted by J. Serrin [1] in 1969 for N = 3 and proved in 1977 by D. Gilbarg and N. S. Trudinger [1] for k ≥ 2 (provided that d∂A is replaced by bA in Lemma 1, p. 382 in [1], and Lemma 14.16, p. 355 in [2]) with a diﬀerent proof by S. G. Krantz and H. R. Parks [1] in 1981. Another proof with the ´sio [17], who extended the result function bA was given in 1994 by M. C. Delfour and J.-P. Zole from (iii) to (ii) in the direction (⇒) down to the C 1,1 case and established the equivalence (⇔) in the whole range from C ∞ to C 1,1 . The counterexample of Example 6.2 for domains of class C 1,1− is the same as the one provided earlier in S. G. Krantz and H. R. Parks [1], where they observe only that the domain is C 2−ε , leaving the reader under the misleading impression that part (ii) of Theorem 8.1 would not be true for domains ranging from class C 1,1 to C 2 . Part (i) of Theorem 8.1 in the direction (⇒) was proved in the C 1 case by S. G. Krantz and H. R. Parks [1] in 1981. The equivalence (⇔) here for C 1,λ , 0 ≤ λ < 1, was given in 2004 by M. C. Delfour and ´sio [40]. J.-P. Zole

366

Chapter 7. Metrics via Oriented Distance Functions

(i) (k = 1, 0 ≤ < 1). A is a set of class C 1, in a bounded open neighborhood V (x) of x and Sk (∂A) ∩ V (x) = ∅ if and only if ∃ρ > 0 such that bA ∈ C 1, (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.1)

(ii) (k = 1, = 1). A is locally a set of class C 1,1 at x if and only if ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.2)

(iii) (k ≥ 2, 0 ≤ ≤ 1). A is locally a set of class C k, at x if and only if ∃ρ > 0 such that bA ∈ C k, (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.3)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A, and ∂A is locally a C k, -submanifold of dimension N − 1 at x. Remark 8.1. The condition Sk (∂A) ∩ V (x) = ∅ in part (i) is really necessary. Example 6.2 in Chapter 6 gives a family of two-dimensional sets of class C 1,λ , 0 ≤ λ < 1, for which bA ∈ C 1,λ and d2∂A = b2A ∈ / C 1 in any neighborhood of the point (0, 0) of its boundary ∂A. From Theorem 8.1 we get the global version that has had and still has an interesting history of inaccurate statements or incomplete proofs Theorem 8.2 (Global version). Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, and let 0 ≤ ≤ 1 be a real number. (i) (k = 1, 0 ≤ < 1). A is a set of class C 1, and ∂A ∩ Sk (∂A) = ∅ if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C 1, (Bρ (x)).

(8.4)

(ii) (k = 1, = 1). A is a set of class C 1,1 if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x)).

(8.5)

(iii) (k ≥ 2, 0 ≤ ≤ 1). A is a set of class C k, if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C k, (Bρ (x)).

(8.6)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A, and ∂A is a C k, -submanifold of dimension N − 1. Remark 8.2. Theorems 8.1 and 8.2 deal only with sets of class C 1, whose boundary is a C 1, submanifold of dimension (N − 1). For arbitrary closed submanifolds M of RN of codimension greater than or equal to 1, ∇bM generally does not exist on M . In that case the smoothness of M is related to the existence and the smoothness of

8. Boundary Smoothness and Smoothness of bA

367

∇d2M in a neighborhood of M that implies that M is locally of positive reach. The reader is referred to Theorem 6.5 in section 6.2 of Chapter 6. The above two equivalence theorems follow from the following two local theorems in each direction. Theorem 8.3. Let A ⊂ RN be such that ∂A = ∅. Given x ∈ ∂A, assume that ∃ a bounded open neighborhood V (x) of x such that bA ∈ C k, (V (x))

(8.7)

for some integer k ≥ 1 and some real , 0 ≤ ≤ 1, and that m(∂A ∩ V (x)) = 0. (i) Then A is locally a set of class C k, in V (x) and bA = bint A = bA in V (x). (ii) Π∂A (y) = {p∂A (y)} is a singleton for all y ∈ V (x) and 1,1 k, (V (x)) ∩ Cloc (V (x)) b2A ∈ Cloc

and

0,1 k−1, p∂A ∈ Cloc (V (x))N ∩ Cloc (V (x))N .

Remark 8.3. If condition (8.7) is veriﬁed, the condition m(∂A) = 0 is equivalent to ∂A = RN . The condition bA ∈ Cb0 (RN ) is necessary to rule out the case ∅ = ∂A = RN that yields bA = 0 and A = A = RN that trivially satisﬁes condition (8.7). Proof. (i) Since ∇bA exists and is continuous everywhere in V (x), |∇bA | is continuous in V (x). But |∇bA (x)| = 1 in V (x)\∂A and |∇bA (x)| = 0 almost everywhere in V (x) ∩ ∂A. By continuity, either |∇bA | = 1 or |∇bA | = 0 in V (x). If ∇bA = 0 in V (x), then bA is constant in V (x) and equal to bA (x) = 0. Therefore, V (x) ⊂ ∂A and this yields the contradiction 0 < m(V (x)) ≤ m(∂A ∩ V (x)) = 0. Consider def the set Ω = {x ∈ RN : −bA (x) > 0}. By Theorem 4.2 in Chapter 2, Ω is a set of class C k, since bA ∈ C k, (V (x)) and ∇bA = 0 on b−1 A {0} ∩ V (x). In particular, ∂Ω = b−1 {0} = ∂A, int A = Ω, and int A = int Ω in V (x). Finally A bA = bΩ = bint A = bA . k, k, (ii) By assumption, bA ∈ Cloc (V (x)) and hence b2A belongs to Cloc (V (x)). By Theorem 3.1 (iv), Π∂A (y) = {p∂A (y)} is a singleton in V (x). By Theorem 7.3, 1,1 k, 0,1 (V (x)) and, a fortiori, in Cloc (V (x)). This yields p∂A ∈ Cloc (V (x)) ∩ b2A ∈ Cloc k−1, Cloc (V (x)). In the other direction the smoothness of the domain implies the smoothness of bA in a neighborhood of ∂A when the domain is at least of class C 1,1 . A counterexample will be given in dimension 2. Theorem 8.4. Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤ ≤ 1 be a real number, and let x ∈ ∂A if it is not. (i) (k = 1, 0 ≤ < 1). If A is locally a set of class C 1, in a bounded open neighborhood V (x) of x and Sk (∂A) ∩ V (x) = ∅, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C 1, (Bρ (x))

and

p∂A ∈ C 0,1 (Bρ (x)).

(8.8)

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Chapter 7. Metrics via Oriented Distance Functions

(ii) (k = 1, = 1). If A is locally a set of class C 1,1 in a bounded open neighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x))

and

p∂A ∈ C 0,1 (Bρ (x)).

(8.9)

(iii) (k ≥ 2, 0 ≤ ≤ 1). If A is locally a set of class C k, in a bounded open neighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C k, (Bρ (x))

and

p∂A ∈ C k−1, (Bρ (x)).

(8.10)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A. Proof. (i) From Deﬁnition 3.2 of Chapter 2 of a locally a C 1, -submanifold of dimension N − 1, m(∂A ∩ V (x)) = 0 in a bounded open neighborhood V (x) and the set A can be locally described by the level sets of the C 1, -function def

f (y) = gx (y) · eN ,

(8.11)

since by deﬁnition int A ∩ V (x) = {y ∈ V (x) : f (y) > 0} , ∂A ∩ V (x) = {y ∈ V (x) : f (y) = 0} . The boundary ∂A in V (x) is the zero level set of f and the gradient ∇f (y) = Dgx (y)∗ eN = 0 in V (x) is normal to that level set. Thus the outward unit normal n to A on V (x) ∩ ∂A is given by n(y) = −

∇f (y) Dgx (y)∗ eN Dhx (gx (y))−∗ eN =− = − . |∇f (y)| |Dgx (y)∗ eN | |Dhx (gx (y))−∗ eN |

(8.12)

Since V (x) ∩ Sk (∂A) = ∅, for all y ∈ V (x), ∃ unique p∂A (y) ∈ ∂A,

d∂A (y) = inf |y − p| = |y − p∂A (y)|, p∈∂A

1 1 p∂A (y) = y − ∇d2∂A (y) = y − ∇b2A (y). 2 2 Choose ρ > 0 such that B3ρ (x) ⊂ V (x). Then for all y ∈ Bρ (x) |p∂A (y) − x| ≤ |p∂A (y) − y| + |y − x| ≤ 2 |y − x| ≤ 2r ⇒ p∂A (y) ∈ B2ρ (x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f −1 {0}.

(8.13)

8. Boundary Smoothness and Smoothness of bA

369

Therefore, for each y ∈ Bρ (x), there exists a unique p∂A (y) ∈ V (x) ∩ f −1 {0} such that d2∂A (y) =

inf p∈V (x), f (p)=0

|p − y|2 = |y − p∂A (y)|2 .

By using the Lagrange multiplier theorem for the Lagrangian |z − y|2 + λ f (z) ˆ ∈ R such that there exists a λ ˆ ∇f (p∂A (y)) = 0 and f (p∂A (y)) = 0 2(p∂A (y) − y) + λ ˆ |∇f (p∂A (y))| ⇒ 2d∂A (y) = |λ| ⇒ p∂A (y) = y + bA (y)

∇f (p∂A (y)) = y − bA (y) n(p∂A (y)) |∇f (p∂A (y))|

since ∇f (p∂A (y)) = 0 on ∂A ∩ V (x). Combining this with (8.13) 1 y − ∇b2A (y) = p∂A (y) = y − bA (y) n(p∂A (y)) 2 ⇒ ∀y ∈ Br (x)\∂A, ∇bA (y) = n(p∂A (y))

∀y ∈ Br (x),

(8.14)

since V (x)∩Sk (∂A) = ∅ implies that ∇bA (y) exists in V (x)\∂A by Theorem 3.1 (vii). Consider the normal n(z) = −∇f (z)/|∇f (z)| in V (x)\∂A. For z2 and z1 in ∂A ∩ B2ρ (x) the diﬀerence ∇f (z2 ) ∇f (z1 ) − |n(z2 ) − n(z1 )| = |∇f (z2 )| |∇f (z1 )| 1 1 1 = (∇f (z2 ) − ∇f (z1 )) + − ∇f (z1 ) |∇f (z2 )| |∇f (z2 )| |∇f (z1 )| 1 ∇f (z2 ) − ∇f (z1 ) + (|∇f (z1 )| − |∇f (z2 )|) ∇f (z1 ) = |∇f (z2 )| |∇f (z1 )| 1 1 |∇f (z2 ) − ∇f (z1 )| ≤ 2 |∇f (z2 ) − ∇f (z1 )| . ≤2 |∇f (z2 )| inf z∈∂A∩B2ρ (x) |∇f (z)| But f ∈ C 1, (B2ρ (x)) and since |∇f | = 0 is continuous on V (x), |∇f (z2 ) − ∇f (z1 )| ≤ cx |z2 − z1 |

def

and α =

inf

|∇f (z)| > 0

z∈∂A∩B2ρ (x)

and the outward unit normal is -H¨olderian, ∀z2 , z1 ∈ B2ρ (x) ∩ ∂A,

|n(z2 ) − n(z1 )| ≤

2 cx |z2 − z1 | , α

where cx is the Lipschitz constant of f in B2ρ (x). By construction, B3ρ (x) ∩ Sk (∂A) ⊂ V (x) ∩ Sk (∂A) = ∅ and for all z ∈ Bρ (x), B2ρ (z) ∩ Sk (∂A) = ∅

370

Chapter 7. Metrics via Oriented Distance Functions

implies that reach (∂A, z) ≥ 2r > 0. By Theorem 6.2 (v) in Chapter 6 applied to ∂A with 0 < r = ρ < q = 2ρ, ∀z1 , z2 ∈ Bρ (x),

|p∂A (z1 ) − p∂A (z2 )| ≤ 2 |z1 − z2 |

(8.15)

and hence p∂A ∈ C 0,1 (Bρ (x))N . So the composition n◦p∂A belongs to C 0, (Bρ (x))N since we have already shown that n ∈ C 0, (B2ρ (x) ∩ ∂A)N . Finally, ∇bA ∈ L∞ (Bρ (x))N , n(p∂A ) ∈ C 0, (Bρ (x))N , m(∂A) = 0, and from (8.14) ∇bA (y) = n(p∂A (y))

a.e. in Bρ (x)

imply that ∇bA ∈ C 0, (Bρ (x))N , bA ∈ C 1, (Bρ (x)), and ∇bA = n ◦ p∂A in Bρ (x). (ii) The ﬁrst steps are the same as in part (i) up to (8.12) with C 1,1 in place 1, of C . For each y ∈ V (x), Π∂A (y) is compact and nonempty and ∀p ∈ Π∂A (y),

d2∂A (y) = inf |a − y|2 = |p − y|2 . a∈∂A

Choose r > 0 such that B3r (x) ⊂ V (x). Then for all y ∈ Br (x) and p ∈ Π∂A (y) |p − x| ≤ |p − y| + |y − x| ≤ 2 |y − x| ≤ 2r ⇒ p ∈ B2r (x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f −1 {0}. Therefore, for each y ∈ Br (x) and p ∈ Π∂A (y), p ∈ V (x) ∩ f −1 {0} such that d2∂A (y) =

inf p∈V (x), f (p)=0

|p − y|2 = |y − p∂A (y)|2 .

By using the Lagrange multiplier theorem for the Lagrangian |a − y|2 + λ f (a) ˆ ∈ R and p ∈ Π∂A (y) such that there exist λ ˆ ∇f (p) = 0 and f (p) = 0 2(p − y) + λ ⇒ p = y + bA (y)

ˆ |∇f (p)| ⇒ 2d∂A (y) = |λ|

∇f (p) = y − bA (y) n(p) |∇f (p)|

⇒ ∀y ∈ Br (x), ∀p ∈ Π∂A (y),

p = y − bA (y) n(p),

(8.16)

since ∇f (p) = 0 on ∂A ∩ V (x). By the same proof as in part (i) with = 1 2 ∀z2 , z1 ∈ B2r (x) ∩ ∂A, |n(z2 ) − n(z1 )| ≤ cx |z2 − z1 |, α where cx is the Lipschitz constant associated with f in B2r (x). Therefore, n ∈ C 0,1 (B2r (x) ∩ ∂A). From (8.16) for y ∈ Br (x) and for all p1 , p2 ∈ Π∂A (y), |p2 − p1 | ≤ |bA (y)| |n(p2 ) − n(p1 )| ≤ |bA (y)|

2 cx |p2 − p1 |. α

8. Boundary Smoothness and Smoothness of bA

371

By choosing h such that 0 < h < min{r, α/(2 cx )}, h 2 cx /α < 1, for y ∈ Bh (x) ⊂ Br (x) ∩ {z ∈ RN : |bA (z)| ≤ h} and all p1 , p2 ∈ Π∂A (y), p2 = p1 , Π∂A (y) is a singleton, and p∂A (y) = y − bA (y) n(p∂A (y)). By construction, Bh (x)∩Sk (∂A) = ∅ and for all z ∈ Bh/3 (x), B2h/3 (z)∩Sk (∂A) = ∅ implies that reach (∂A, z) ≥ 2h/3 > 0. By Theorem 6.2 (v) in Chapter 6 applied to ∂A with 0 < r = h/3 < q = 2h/3 ∀z1 , z2 ∈ Bh/3 (x),

|p∂A (z1 ) − p∂A (z2 )| ≤ 2 |z1 − z2 |

(8.17)

and hence p∂A ∈ C 0,1 (Bh/3 (x))N . Thus n ◦ p∂A belongs to C 0,1 (Bh/3 (x))N since we have already shown that n ∈ C 0,1 (B2r (x) ∩ ∂A)N . The remainder and the conclusion of the proof is the same as in part (i) with ρ = h/3. (iii) When A is of class C k, in V (x) for an integer k, k ≥ 2, and a real number , 0 ≤ ≤ 1, it is C 1,1 and the previous constructions and conclusions of (ii) are veriﬁed. There exists ρ > 0 such that for each y ∈ Bρ (x) there exists a unique p∂A (y) ∈ B2ρ (x) such that p∂A (y) = y − bA (y)

∇f (p∂A (y)) . |∇f (p∂A (y))|

Since ∇f is continuous and ∇f = 0 on ∂A ∩ V (x), there exists r, 0 < 2r < ρ, such that inf

|∇f (z)| > 0

z∈B2r (x)

and deﬁning N (z) = ∇f (z)/|∇f (z)|, ∇f (z) 1 D2 f (z) − ∇N (z) = |∇f (z)| |∇f (z)|

∗

∇f (z) D f (z) |∇f (z)|

2

belongs to C k−1, (B2r (x)), where ∗v denotes the transpose of a vector v ∈ RN , so that (u ∗v)ij = ui vj is the tensor product. Consider the map def

(y, z) → G(y, z) = z − y − bA (y)

∇f (z) : Br (x) × B2r (x) → RN . |∇f (z)|

From part (ii), there exists a unique p(y) = p∂A (y) ∈ B2r (x) ∩ ∂A such that p∂A ∈ C 0,1 (B2r (x))N , bA ∈ C 1,1 (Br (x)), and G(y, p(y)) = 0 where DGz (y, z) = I −

⇒ DGy (y, p(y)) + DGz (y, p(y)) Dp(y) = 0,

bA (y) ∇f (z) D2 f (z) − |∇f (z)| |∇f (z)|

∗

D2 f (z)

DGy (y, z)) = −I − N (z) ∗∇bA (y).

∇f (z) |∇f (z)|

,

372

Chapter 7. Metrics via Oriented Distance Functions

DGz (y, z) is at least C 1,1 in (y, z) since bA ∈ C 1,1 and N is at least C 1,1 for k ≥ 2. Since f is at least C 2 in V (x), f and its ﬁrst and second derivatives are bounded in some smaller neighborhood Br (x) of x. By reducing the upper bound on |bA (y)|, there exists a new smaller neighborhood Br (x) of x such that Dz G(y, z) is invertible for all (y, z) ∈ Br (x) × B2r (x). So the conditions of the implicit function theorem are met. There exist a neighborhood W (x) ⊂ Br (x) of x and a unique C 1 -mapping p : W (x) → RN such that ∀y ∈ W (x),

G(y, p(y)) = 0

that coincides with p∂A (cf., for instance, L. Schwartz [3, Thm. 26, p. 283]). Therefore Dp∂A (y) = (DGz (y, p∂A (y)))−1 [I + N (p∂A (y)) ∗∇bA (y)] and Dp∂A is C 0 and p∂A is C 1 . Thus ∇bA = N ◦ p∂A ∈ C 1 and bA is now C 2 . Since N ∈ C k−1, and D2 f ∈ C k−2, , we can repeat this argument until we get Dp∂A in C k−2, , and ﬁnally p∂A ∈ C k−1, , ∇bA = N ◦ p∂A ∈ C k−1, , and bA ∈ C k, . From Theorem 8.1 (ii) additional information is available on the ﬂow of ∇bΩ . def

Theorem 8.5. Let Ω ⊂ RN be of class C 1,1 . Denote its boundary by Γ = ∂Ω. (i) For each x ∈ Γ there exists a bounded open neighborhood W (x) of x such that bΩ ∈ C 1,1 (W (x)),

∇bΩ = n ◦ pΓ = ∇bΩ ◦ pΓ ,

|∇bΩ | = 1 in W (x), (8.18)

where n is the unit exterior normal to Ω on Γ. (ii) Associate with x ∈ Γ and y in W (x) the ﬂow Tt (y) of ∇bΩ in W (x), that is, dx (t) = ∇bΩ (x(t)), dt

x(0) = y,

def

Tt (y) = x(t).

(8.19)

Then for t in a neighborhood of 0 we have the following properties: pΓ ◦ Tt = pΓ ,

∇bΩ ◦ Tt = ∇bΩ ,

Tt (y) = y + t ∇bΩ (y).

(8.20)

(iii) Almost everywhere in W (x) d DTt = D 2 bΩ dt

−1 D2 bΩ ◦ Tt = D2 bΩ I + tD2 bΩ ,

DTt = I + t D 2 bΩ , ⇒ D2 bΩ ◦ Tt DTt = D 2 bΩ ,

(DTt )−1 ∇bΩ = ∇bΩ = ∗(DTt )−1 ∇bΩ .

(8.21) (8.22) (8.23)

Proof. For simplicity, we use the notation b = bΩ and p = pΓ . (i) The result follows from Theorem 8.4 (i). (ii) By assumption from part (i) the function b belongs to C 1,1 (W (x)) and hence p ∈ C 0,1 (W (x); RN ). Thus almost everywhere in W (x), Dp = I − ∇b ∗∇b − bD2 b

⇒ Dp∇b = 0.

9. Sobolev or W m,p Domains

373

Now dTt d (p ◦ Tt ) = Dp ◦ Tt = Dp ◦ Tt ∇b ◦ Tt = (Dp ∇b) ◦ Tt = 0 dt dt ⇒ p ◦ Tt = p in W (x). But since p and Tt belong to C 0,1 (W (x); RN ), the identity necessarily holds everywhere in W (x). Moreover, ∇b ◦ Tt = n ◦ p ◦ Tt = n ◦ p = ∇b in W (x). Finally dTt = ∇b ◦ Tt = ∇b dt

⇒ Tt (y) = y + t∇b(y) in W (x).

In particular, almost everywhere in W (x), d DTt = D2 b, dt ⇒ (DTt )−1 ∇b = ∇b = ∗(DTt )−1 ∇b.

DTt = I + t D2 b DTt ∇b = ∇b

9

⇒

Sobolev or W m,p Domains

The N -dimensional Lebesgue measure of the boundary of a Lipschitzian subset of RN is zero. However, this is generally not true for sets of locally bounded curvature, as can be seen from the following example. Example 9.1. Let B be the open unit ball centered in 0 of R2 and deﬁne A = {x ∈ B : x with rational coordinates}. Then ∂A = B, bA = dB , and for all h > 0 ∇bA ∈ BV(Uh (∂A))2 , 1 ∆bA , ϕ = ϕ dx. ϕ dx + |x| ∂B B The next natural question that comes to mind is whether the boundary or the skeleton has locally ﬁnite (N − 1)-Hausdorﬀ measure. Then come questions about the mean curvature of the boundary. For the function bA , ∆bA = tr D2 bA is proportional to the mean curvature of the boundary, which is only a measure in RN for sets of bounded local curvature. This calls for the introduction of some classiﬁcation of sets that would complement the classical H¨olderian terminology introduced in Chapter 2, would ﬁll the gap in between, and would possibly say something about sets whose boundary is not even continuous. The following deﬁnition seems to have ´sio [32]. been ﬁrst introduced in M. C. Delfour and J.-P. Zole

374

Chapter 7. Metrics via Oriented Distance Functions

Deﬁnition 9.1 (Sobolev domains). Given m > 1 and p ≥ 1, a subset A of RN is said to be an (m, p)-Sobolev domain or simply a W m,p -domain if ∂A = ∅ and there exists h > 0 such that m,p bA ∈ Wloc (Uh (∂A)).

This classiﬁcation gives an analytical description of the smoothness of boundaries in terms of the derivatives of bA . The deﬁnition is obviously vacuous for m = 1 1,∞ since bA ∈ Wloc (RN ) for any set A such that ∂A = ∅. For 1 < m < 2 we shall see that it encompasses sets that are less smooth than sets of locally bounded curvature. For m = 2 the W 2,p -domains are of class C 1,1−N/p for p > N . For m > 2 they are intertwined with sets of class C k, . Theorem 9.1. Given any subset A of RN , ∂A = ∅, ∇bA ∈ BVloc (RN )N 1 1+η,p ⇒ ∀p, 1 ≤ p < ∞, ∀η, 0 ≤ η < , bA ∈ Wloc (RN ). p N N Proof. Since |∇bA (x)| ≤ 1 almost everywhere, ∇bA ∈ BVloc (RN )N ∩ L∞ loc (R ) and the theorem follows directly from Theorem 6.9 in Chapter 5.

It is quite interesting that a domain of locally bounded curvature is a W 2−ε,1 domain for any arbitrary small ε > 0. So it is almost a W 2,1 -domain, and domains of class W 2−ε,1 seem to be a larger class than domains of locally bounded curvature. In addition, their boundary does not generally have zero measure. Now consider the “threshold case” m = 2. Theorem 9.2. Given an integer N ≥ 1, let A be a subset of RN such that ∂A = ∅. (i) If m(∂A) = 0 and there exist p > N and h > 0 such that 2,p bA ∈ Wloc (Uh (∂A)),

(9.1)

then reach (∂A) ≥ h > 0,

1,1−N/p

bA ∈ Cloc

(Uh (∂A)),

and A is a H¨ olderian set of class C 1,1−N/p . 2,p (Uh (∂A)) is equivalent to ∆bA ∈ (ii) In dimension N = 2 the condition bA ∈ Wloc p Lloc (Uh (∂A)).

Proof. (i) Consider the function |∇bA |2 . Since |∇bA | ≤ 1, then 2,p 1,∞ (Uh (∂A)) ∩ Wloc (Uh (∂A)) bA ∈ Wloc

2,p 1,∞ ⇒ b2A ∈ Wloc (Uh (∂A)) ∩ Wloc (Uh (∂A)).

In particular, for all x ∈ ∂A, bA , b2A ∈ W 2,p (B(x, h)) ∩ W 1,∞ (B(x, h)).

10. Characterization of Convex and Semiconvex Sets

375

For p > N , from R. A. Adams [1, Thm. 5.4, Part III, and Rem. 5.5 (3), p. 98], bA , b2A ∈ C 1,λ (B(x, h)), ⇒

bA , b2A

∈

1,λ Cloc (Uh (A)),

0 < λ ≤ 1 − N/p 0 < λ ≤ 1 − N/p.

From Theorem 8.2 (i), if m(∂A) = 0, the set A is a set of class C 1,1−N/p , ∂A ∩ Sk (∂A) = ∅, and reach (∂A) ≥ h. (ii) From part (i), |∇bA (x)|2 = 1 in Uh (∂A), and D2 bA (x) ∇bA (x) = 0 almost everywhere. Hence in dimension N = 2, 2 2 bA (x) = ∂21 bA (x) = −∂1 bA (x) ∂2 bA (x) ∆bA (x), ∂12 2 ∂11 bA (x) = (∂2 bA (x))2 ∆bA (x), 2 ∂22 bA (x) = (∂1 bA (x))2 ∆bA (x).

As can be seen, a certain amount of work would be necessary to characterize all the Sobolev domains and answer the many associated open questions.

10 10.1

Characterization of Convex and Semiconvex Sets Convex Sets and Convexity of bA

We have seen in Chapter 6 that the convexity of dA is equivalent to the convexity of A. This characterization remains true with bA in place of dA . However, if in both cases the convexity of dA or bA is suﬃcient to get the convexity of A, it is not the case for A as can be seen from the following example. Example 10.1. Let B be the open unit ball in RN and A be the set B minus all the points in B with rational coordinates. By deﬁnition, A¯ = B, A = RN , dA = dB = dA ,

∂A = B,

∂A = ∂B = ∂A,

dA = dA = dRN = 0

⇒ bA = dB = bB = bA . By Theorem 8.1 of Chapter 6 the functions dB and, a fortiori, bA are convex, but A is not convex and ∂A = ∂A. Theorem 10.1. (i) Let A be a convex subset of RN . Then bA = bA

and

[A]b = [A]b .

(10.1)

Hence, the equivalence class [A]b can be identiﬁed with the unique closed set A in the class.

376

Chapter 7. Metrics via Oriented Distance Functions

(ii) Let A be a convex subset of RN such that int A = ∅. Then bint A = bA = bA

and

[int A]b = [A]b = [A]b .

(10.2)

Hence, the equivalence class [A]b can be identiﬁed with either the unique closed set A or the unique open set int A = A = ∅ in the class. If, in addition, A = RN , then ∂A = ∅ and int A = ∅. (iii) Let A be a subset of RN such that ∂A = ∅.5 Then bA is convex

⇐⇒

A is convex.

(10.3)

Proof. (i) If A = ∅, then A = ∅, A = RN , and A = RN . Hence bA = bA . If A = RN , we use the identity bA = −bA . If ∂A = ∅, then dA = dA and, by Theorem 3.5 (ii) in Chapter 5, A = A, and bA = bA . (ii) From Theorem 3.5 (ii) in Chapter 5, int A = A and int A = A = A. Hence bA = bint A . From part (i), bA = bA . If, in addition, A = RN , always from Theorem 3.5 (ii) in Chapter 5, ∂A = ∅ and int A = ∅. (iii) (⇒) Denote by xλ the convex combination λx + (1 − λ)y of two points x and y in A for some λ ∈ [0, 1]. By convexity of bA , bA (xλ ) ≤ λbA (x) + (1 − λ)bA (y) = − [λdA (x) + (1 − λ)dA (y)] ≤ 0 ⇒ dA (xλ ) = b+ A (xλ ) = max{bA (xλ ), 0} = 0

⇒ xλ ∈ A

and A is convex. (⇐) We ﬁrst prove that for A closed the function bA is convex. From this we conclude that if A is convex, then bA is convex and from part (i) bA = bA is convex. So for A closed, consider three cases. The ﬁrst one deserves a lemma. Lemma 10.1. Let A be a subset of RN such that ∂A = ∅ and let A be convex. Then bA = −dA is convex in A. Proof. Again we can assume that A is closed. Associate with x and y in A the radii rx = dA (x), ry = dA (y), and rλ = λrx + (1 − λ)ry and the closed balls B x of center x and radius rx , B y of center y and radius ry , and B λ of center xλ and radius rλ . By the deﬁnition of dA , B x ⊂ A and B y ⊂ A since A is closed. Associate with 5 In 1985 D. H. Armitage and U. ¨ Kura [1] showed that a proper closed domain Ω is convex if and only if −bA is superharmonic and that, for N = 2, the result still holds if −bA is superharmonic only on A. They also showed that, for A compact, d∂A is subharmonic on A if and only if A is convex. This work was pursued in 1987 by M. J. Parker [1] and in 1988 by M. J. Parker [2]. ´sio [17] that the property that A is convex In 1994 it was shown in M. C. Delfour and J.-P. Zole if and only if dA is convex remains true with bA in place of dA .

10. Characterization of Convex and Semiconvex Sets

377

each z ∈ B λ the points zx = x and zy = y if rλ = 0, and if rλ > 0 the points rx (z − xλ ) ⇒ zx ∈ B x , rλ ry def zy = y + (z − xλ ) ⇒ zy ∈ B y rλ λrx + (1 − λ)ry ⇒ λzx + (1 − λ)zy = xλ + (z − xλ ) = z. rλ def

zx = x +

Obviously z = xλ if rλ = 0. Therefore, B λ ⊂ λB x + (1 − λ)B y ⊂ A, since A is closed and convex. In particular dA (xλ ) ≥ dB λ (xλ ) ≥ rλ = λdA (x) + (1 − λ)dA (y), and this proves the lemma. For the second case consider x and y in A. By deﬁnition bA (x) = dA (x)

and bA (y) = dA (y).

By Theorem 8.1 of Chapter 6, dA is convex when A is convex and bA (xλ ) ≤ dA (xλ ) ≤ λdA (x) + (1 − λ)dA (y) = λbA (x) + (1 − λ)bA (y). The third and last case is the mixed one for x ∈ A and y ∈ A. Deﬁne def

xλ = λx + (1 − λ)y,

def

pλ = p∂A (xλ ).

Since A is convex, denote by H the tangent hyperplane to A through pλ , by H + the closed half-space containing A, and by H − the other closed subspace associated with H. By the deﬁnition of H, d∂A (xλ ) = dH (xλ ),

H ⊂ H ±,

and by convexity of A, A ⊂ H + and H − ⊂ A. The projection onto H is a linear operator and pλ = pH (xλ ) = λpH (x) + (1 − λ)pH (y), pλ − xλ = λ(pH (x) − x) + (1 − λ)(pH (y) − y). If y ∈ H + and x ∈ H + , dH (xλ ) = λdH (x) + (1 − λ)dH (y),

(10.4)

378

Chapter 7. Metrics via Oriented Distance Functions

and if y ∈ H + and x ∈ H − ,

dH (xλ ) =

λdH (x) − (1 − λ)dH (y), (1 − λ)dH (y) − λdH (x),

if xλ ∈ H − , if xλ ∈ H + .

(10.5)

By convexity any y ∈ A belongs to H + and since H − ⊂ A, we readily have dH (y) = dH − (y) ≥ dA (y) = −bA (y). First consider the case xλ ∈ A. Then dA (xλ ) > 0, xλ ∈ H − , and necessarily x ∈ H − . From (10.5) bA (xλ ) = dA (xλ ) = dH (xλ ) = λdH (x) − (1 − λ)dH (y). But since x ∈ H − and A ⊂ H + dH (x) = dH + (x) ≤ dA (x) and bA (xλ ) = dH (xλ ) ≤ λdA (x) − (1 − λ)dA (y) = λbA (x) + (1 − λ)bA (y). Next consider the case xλ ∈ A for which xλ ∈ H + and −bA (xλ ) = dA (xλ ) = dH (xλ ). If x ∈ H − , then since A ⊂ H + , dH (x) = dH + (x) ≤ dA (x), and from (10.5) −bA (xλ ) = dH (xλ ) = (1 − λ)dH (y) − λdH (x) ≥ (1 − λ)dA (y) − λdA (x) = −[(1 − λ)bA (y) + λbA (x)]. If, on the other hand, x ∈ H + , then x, y ∈ H + and xλ ∈ A ⊂ H + . So from (10.4) −bA (xλ ) = dH (xλ ) = (1 − λ)dH (y) + λdH (x) ≥ (1 − λ)dA (y) − λdA (x) = −[(1 − λ)bA (y) + λbA (x)]. This covers all cases and concludes the proof.

10. Characterization of Convex and Semiconvex Sets

10.2

379

Families of Convex Sets Cb (D), Cb (E; D), and Cb,loc (E; D)

We have seen in Theorem 10.1 (i) that for a convex subset A of D, bA = bA and that A is the unique closed representative in the equivalence class. To work with open convex subsets is more delicate. Always from Theorem 10.1 (ii), int A becomes the unique open representative in the equivalence class when the convex set A has a nonempty interior. In order to work with families of open rather than closed convex subsets, it will be necessary to add a constraint on the interior of the subsets to prevent the occurrence of subsets with an empty interior. Theorem 10.2. Let D be a nonempty open (resp., bounded open) subset of RN . (i) The family def

Cb (D) = bA : A ⊂ D, ∂A = ∅, A convex

(10.6)

is closed in Cloc (D) (resp., compact in C(D)). It coincides with the family bA : A ⊂ D, ∂A = ∅, A closed and convex .

(ii) Let E be a nonempty open subset of D. Then the families def

Cb (E; D) = {bΩ : E ⊂ Ω ⊂ D, ∂Ω = ∅, Ω open and convex} , def

Cb,loc (E; D) =

bΩ :

∃x ∈ RN , ∃A ∈ O(N) such that x + AE ⊂ Ω and Ω is open and convex ⊂ D

&

(10.7) (10.8)

are closed in Cloc (D) (resp., compact in C(D)). 1,p (D)- (resp., (iii) The conclusions of parts (i) and (ii) remain true with Wloc 1,p W (D)-) strong in place of Cloc (D) (resp., C(D)).

Proof. It is suﬃcient to prove the result for D bounded. Furthermore, by compactness of Cb (D) in C(D), it is suﬃcient to prove that Cb (D), Cb (E; D), and Cb,loc (E; D) are closed in C(D). (i) Let {bAn } be a Cauchy sequence in Cb (D). It converges to some bA ∈ Cb (D). The function bA is convex as the pointwise limit of a family of convex functions. By Theorem 10.1 (iii), A is convex, and by Theorem 10.1 (i) bA = bA is convex. Therefore the Cauchy sequence converges to bA in Cb (D) and Cb (D) is closed in C(D). (ii) From part (i), given a Cauchy sequence {bΩn } in Cb (E; D), there exists an open convex subset Ω of D such that bΩn → bΩ . Moreover, since dΩn → dΩ , E ⊂ Ωn ⇒ E ⊃ Ωn ⇒ dE ≤ dΩn ⇒ dE ≤ dΩ ⇒ E ⊃ Ω ⇒ E ⊂ Ω. c The proof for Cb,loc (E; D) is similar to the proof for Cd,loc (E; D) in Theorem 2.6 of Chapter 6. (iii) Let {bAn } ⊂ Cb (D) be a Cauchy sequence in W 1,p (D). From Theorem 4.2 (ii), it is also Cauchy in C(D). Hence from part (i) there exists a convex set

380

Chapter 7. Metrics via Oriented Distance Functions

A, ∂A = ∅, such that bAn → bA in C(D) and, a fortiori, in W 1,p (D). This shows that Cb (D) is closed in W 1,p (D). To show that it is compact consider a sequence {bAn } ⊂ Cb (D). From part (i) there exist a subsequence, still indexed by n, and a convex subset A, ∂A = ∅, of D such that bAn → bA in C(D). Since D is bounded, it also converges in Lp (D). Since |∇bAn | is pointwise bounded by one, there exists another subsequence of {bAn }, still indexed by n, such that ∇bAn converges to ∇bA in W 1.p (D)N -weak. But An and A are convex. Thus m(∂A) = 0 = m(∂An ) and |∇bAn | = 1 = |∇bA | almost everywhere in RN . As a result for p = 2 2 |∇bAn − ∇bA | dx = |∇bAn |2 + |∇bA |2 dx − 2∇bAn · ∇bA dx D D 2 − 2∇bAn · ∇bA dx → 0. = D

Hence we get the compactness in W 1,2 (D)-strong and by Theorem 4.1 (i) in W 1,p (D)strong, 1 ≤ p < ∞.

10.3

BV Character of bA and Semiconvex Sets

The next theorem is a consequence of the fact that the gradient of a convex function is locally of bounded variations. Theorem 10.3. For all convex sets A such that ∂A = ∅, (i) ∇bA ∈ BVloc (RN )N , ∇χ∂A ∈ BVloc (RN ), ∇dA ∈ BVloc (RN )N , ∇χA ∈ BVloc (RN ), and for all x ∈ RN and v ∈ RN , dH bA (x, v) exists. (ii) The Hessian matrix D2 bA of second-order derivatives is a matrix of signed Radon measures which are nonnegative on the diagonal. (iii) bA has a second-order derivative almost everywhere, and for almost all x and y in RN , bA (y) − bA (x) − ∇bA (x) · (y − x) − 1 (y − x) · D 2 bA (x) (y − x) = o(|y − x|2 ) 2 as y → x. Proof. Same argument as in the proof of Theorem 8.2 of Chapter 6. When A is convex and of class C 2 , then for any x0 ∈ ∂A, there exists a strictly convex neighborhood N (x0 ) of x0 such that bA ∈ C 2 (N (x0 )) and ∀x ∈ N (x0 ), ∀ξ ∈ RN ,

D 2 bA (x)ξ · ξ ≥ 0,

(10.9)

or, since D 2 bA (x) ∇bA (x) = 0 in N (x0 ), then for all x ∈ N (x0 ), ∀ξ ∈ RN such that ξ · ∇bA (x) = 0,

D2 bA (x)ξ · ξ ≥ 0.

(10.10)

This is related to the notion of strong elliptic midsurface in the theory of shells: there exists c > 0 ∀x ∈ ∂A, ∀ξ ∈ RN such that ξ · ∇bA (x) = 0,

D 2 bA (x) ξ · ξ ≥ c|ξ|2 .

All this motivates the introduction of the following notions.

11. Compactness and Sets of Bounded Curvature

381

Deﬁnition 10.1. Let A be a closed subset of RN such that ∂A = ∅. (i) The set A is locally convex (resp., locally strictly convex ) if for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 such that bA is convex (resp., strictly convex) in N (x0 ). (ii) The set A is semiconvex if ∃α ≥ 0,

bA (x) + α|x|2 is convex in RN .

(iii) The set A is locally semiconvex if for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 and ∃α ≥ 0,

bA (x) + α|x|2 is convex in N (x0 ).

Remark 10.1. When A is a C 2 domain with a compact boundary, D 2 bA is bounded in a bounded neighborhood of ∂A and A is necessarily locally semiconvex. When A is a locally semiconvex set, for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 such that ∇bA ∈ BV(N (x0 ))N . If, in addition, ∂A is compact, then there exists h > 0 such that ∇bA ∈ BV(Uh (∂A))N . Remark 10.2. Given a ﬁxed constant β > 0, consider all the subsets of D that are semiconvex with constant 0 ≤ α ≤ β. Then this set is closed for the uniform and the W 1,p -topologies, 1 ≤ p < ∞.

11

Compactness and Sets of Bounded Curvature

For the family of sets with bounded curvature, the key result is the compactness of the embedding

(11.1) BCb (D) = bA ∈ Cb (D) : ∇bA ∈ BV(D)N → W 1,p (D) for bounded open Lipschitzian subsets of RN and p, 1 ≤ p < ∞. It is the analogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets BX(D) = {χ ∈ X(D) : χ ∈ BV(D)} → Lp (D),

(11.2)

which is a consequence of the compactness of the embedding BV(D) → L1 (D)

(11.3)

for bounded open Lipschitzian subsets of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]). As in the case of characteristic functions in Chapter 5, we give a ﬁrst version involving global conditions on a ﬁxed bounded open Lipschitzian holdall D. In the second version the sets are contained in a bounded open holdall D with local conditions in the tubular neighborhood of their boundary.

382

11.1

Chapter 7. Metrics via Oriented Distance Functions

Global Conditions on D

Theorem 11.1. Let D be a nonempty bounded open Lipschitzian subset of RN . The embedding (11.1) is compact. Thus for any sequence {An }, ∂An = ∅, of subsets of D such that ∃c > 0, ∀n ≥ 1, D2 bAn M 1 (D) ≤ c, (11.4) there exist a subsequence {Ank } and a set A, ∂A = ∅, such that ∇bA ∈ BV(D)N and bAnk → bA in W 1,p (D)-strong for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D), lim ∂ij bAnk , ϕ = ∂ij bA , ϕ,

n→∞

1 ≤ i, j ≤ N,

D2 bA M 1 (D) ≤ c.

(11.5)

def

Proof. Given c > 0 consider the set Sc = bA ∈ Cb (D) : D2 bA M 1 (D) ≤ c . By compactness of the embedding (11.3), given any sequence {bAn } there exist a subsequence, still denoted by {bAn }, and f ∈ BV(D)N such that ∇bAn → f in L1 (D)N . But by Theorem 2.2 (ii), Cb (D) is compact in C(D) for bounded D and there exist another subsequence {bAnk } and bA ∈ Cb (D) such that bAnk → bA in C(D) and, a fortiori, in L1 (D). Therefore, bAnk converges in W 1,1 (D) and also in L1 (D). By uniqueness of the limit, f = ∇bA and bAnk converges in W 1,1 (D) to bA . For Φ ∈ D1 (D)N ×N as k goes to inﬁnity −→ −→ ∇bAnk · div Φ dx → ∇bA · div Φ dx D D −→ −→ ∇bA · div Φ dx = lim ∇bAnk · div Φ dx ≤ cΦC(D) , ⇒ k→∞

D

D

D bA M 1 (D) ≤ c, and ∇bA ∈ BV(D) . This proves the sequential compactness of the embedding (11.1) for p = 1 and properties (11.5). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cb (D) in Theorem 4.1 (i). 2

11.2

N

Local Conditions in Tubular Neighborhoods

The global condition (11.4) is now weakened to a local one in a neighborhood of each set of the sequence. Simultaneously, the Lipschitzian condition on D is removed since only the uniform boundedness of the sets of the sequence is required. ∅, be a Theorem 11.2. Let ∅ = D ⊂ RN be bounded open and {An }, ∂An = sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that ∀n,

D2 bAn M 1 (Uh (∂An )) ≤ c.

(11.6)

Then, there exist a subsequence {Ank } and A ⊂ D, ∅ = ∂A, such that ∇bA ∈ BVloc (RN )N and for all p, 1 ≤ p < ∞, bAnk → bA in W 1,p (Uh (D))-strong.

(11.7)

11. Compactness and Sets of Bounded Curvature

383

Moreover, for all ϕ ∈ D0 (Uh (∂A)), lim ∂ij bAnk , ϕ = ∂ij bA , ϕ,

1 ≤ i, j ≤ N,

k→∞

D2 bA M 1 (Uh (∂A)) ≤ c,

(11.8)

and χ∂A belongs to BVloc (RN ). Proof. (i) By assumption, An ⊂ D implies that Uh (An ) ⊂ Uh (D). Since Uh (D) is bounded, there exist a subsequence, still indexed by n, and a subset A of D, ∂A = ∅, such that bAn → bA

in C(Uh (D))-strong

and another subsequence, still indexed by n, such that bAn → bA

in H 1 (Uh (D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and all x ∈ Uh (D), d∂An (x) ≤ d∂A (x) + ε,

d∂A (x) ≤ d∂An (x) + ε.

(11.9)

Therefore, ∂An ⊂ Uh−2 ε (∂An ) ⊂ Uh−ε (∂A) ⊂ Uh (∂An ), Uh−ε (∂A) ⊂ Uh−2 ε (∂An ) ⊂ ∂An .

(11.10) (11.11)

From (11.6) and (11.10) ∀n ≥ N,

D 2 bAn M 1 (Uh−ε (∂A)) ≤ c.

In order to use the compactness of the embedding (11.3) as in the proof of Theorem 11.1, we would need Uh−ε (∂A) to be Lipschitzian. To get around this we construct a bounded Lipschitzian set between Uh−2ε (∂A) and Uh−ε (∂A). Indeed by deﬁnition, Uh−ε (∂A) = ∪x∈∂A B(x, h − ε)

and

Uh−2ε (∂A) ⊂ Uh−ε (∂A),

and by compactness there exists a ﬁnite sequence of points {xi }ni=1 in ∂A such that def

Uh−2ε (∂A) ⊂ UB = ∪ni=1 B(xi , h − ε) ⊂ Uh (D). Since UB is Lipschitzian as the union of a ﬁnite number of balls, it now follows by compactness of the embedding (11.3) for UB that there exist a subsequence, still denoted by {bAn }, and f ∈ BV(UB )N such that ∇bAn → f in L1 (UB )N . Since Uh (D) is bounded, Cb (Uh (D)) is compact in C(Uh (D)) and there exist another subsequence, still denoted by {bAn }, and A ⊂ D, ∂A = ∅, such that bAn → bA in C(Uh (D)) and, a fortiori, in L1 (Uh (D)). Therefore, bAn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇bA on UB and bAn converges to bA in W 1,1 (UB ). By Deﬁnition 5.1 and Theorem 3.1 (i), ∇bA and ∇bAn all belong

384

Chapter 7. Metrics via Oriented Distance Functions

to BVloc (RN )N since they are of bounded variation in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), χ∂A ∈ BVloc (RN ). The above conclusions also hold for the subset Uh−2ε (∂A) of UB . (ii) Convergence in W 1,p (Uh (D)). Consider the integral |∇bAn − ∇bA |2 dx Uh (D)

|∇bAn − ∇bA |2 dx +

=

|∇bAn − ∇bA |2 dx.

Uh−2ε (∂A)

Uh (D)\Uh−2ε (∂A)

From part (i), the ﬁrst integral on the right-hand side converges to zero as n goes to inﬁnity. The second integral is on a subset of Uh−2ε (∂A). From (11.11) for all n ≥ N, |∇bAn (x)| = 1 a.e. in ∂An ⊃ Uh−3ε (∂An ) ⊃ Uh−2ε (∂A), |∇bA (x)| = 1 a.e. in ∂A ⊃ Uh−2ε (∂A). The second integral reduces to

|∇bAn − ∇bA |2 dx =

Uh (D)\Uh−2ε (∂A)

2 (1 − ∇bAn · ∇bA ) dx,

Uh (D)\Uh−2ε (∂A)

which converges to zero by weak convergence of ∇bAn to ∇bA in the space L2 (Uh (D))N in part (i) and the fact that |∇bA | = 1 almost everywhere in Uh (D)\ Uh−2ε (∂A). Therefore, since bAn → bA in C(Uh (D)) bAn → bA in H 1 (Uh (D))-strong, and by Theorem 4.1 (i) the convergence is true in W 1,p (Uh (D)) for all p ≥ 1. (iii) Properties (11.8). Consider the initial subsequence {bAn } which converges to bA in H 1 (Uh (D))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of bAn to bA in H 1 (Uh (D))-weak for each Φ ∈ D1 (Uh (∂A))N ×N , lim

n→∞

Uh (∂A)

−→

∇bAn · div Φ dx =

−→

∇bA · div Φ dx. Uh (∂A)

Each such Φ has compact support in Uh (∂A), and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (∂A). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (∂An ) ⊂ Uh−ε (∂A) ⊂ Uh (∂An ).

12. Finite Density Perimeter and Compactness For n ≥ N (ε) consider the integral −→ ∇bAn · div Φ dx = Uh (∂A)

∇bAn

385

· div Φ dx = −→

Uh−2ε (∂A)

⇒

∇bAn

Uh (∂A)

−→

∇bAn · div Φ dx

U (∂A )

n h · div Φ dx ≤ D2 bAn M 1 (Uh (∂An )) ΦC(Uh (∂An )) ≤ c ΦC(Uh−2ε (∂A)) = c ΦC(Uh (∂A)) .

−→

By convergence of ∇bAn to ∇bA in the space L2 (Uh (D))-weak, then for all Φ ∈ D1 (Uh (∂A))N ×N −→ ∇bA · div Φ dx ≤ c ΦC(Uh (∂A)) ⇒ D2 bA M 1 (Uh (∂A)) ≤ c. Uh (∂A)

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

12

Finite Density Perimeter and Compactness

´sio [14] is a reThe density perimeter introduced by D. Bucur and J.-P. Zole laxation of the (N − 1)-dimensional upper Minkowski content which leads to the compactness Theorem 12.1. Deﬁnition 12.1. Let h > 0 be a ﬁxed real number and let ∅ = A ⊂ RN , ∂A = ∅. Consider the quotient mN (Uk (∂A)) def Ph (∂A) = sup , (12.1) 2k 0 0 such that ∀n,

Ph (∂An ) ≤ c.

(12.2)

Then there exist a subsequence {Ank } and A ⊂ D, ∂A = ∅, such that Ph (∂A) ≤ lim inf Ph (∂An ) ≤ c, n→∞

∀p, 1 ≤ p < ∞,

bAnk → bA in W

1,p

(Uh (D))-strong.

(12.3) (12.4)

386

Chapter 7. Metrics via Oriented Distance Functions

Proof. The proof essentially rests on Theorem 4.3 since Ph (∂A) ≤ c implies mN (∂A) = 0. Since D is bounded, the family of oriented distance functions Cb (D) is compact in C(D) and W 1,p (D)-weak for all p, 1 ≤ p < ∞ (cf. Theorems 2.2 (ii) and 4.2 (iii)). So there exist bA ∈ Cb (D) and a subsequence, still indexed by n, such that bAn → bA in the above topologies. Moreover, for all k, 0 < k < h, and all ε, 0 < ε < h − k, ∃N (ε) > 0 such that ∀n ≥ N (ε),

Uk−ε (∂An ) ⊂ Uk (∂A) ⊂ Uk+ε (∂An )

(cf. proof of part (i) of Theorem 11.2). As a result for all n ≥ N (ε), mN (Uk−ε (∂An )) k − ε mN (Uk (∂A)) mN (Uk+ε (∂An )) k + ε k+ε ≤ ≤ ≤c 2(k − ε) k 2k 2(k + ε) k k mN (Uk+ε (∂An )) k + ε mN (Uk (∂A)) k+ε ≤ ≤ Ph (∂An ) ⇒ 2k 2(k + ε) k k mN (Uk (∂A)) k+ε ≤ lim inf Ph (∂An ) . (12.5) ⇒ n→∞ 2k k Going to the limit as ε goes to zero in the second and fourth terms mN (Uk (∂A)) ≤ lim inf Ph (∂An ) ≤ c n→∞ 2k ⇒ Ph (∂A) ≤ lim inf Ph (∂An ) ≤ c ⇒ mN (∂A) = 0.

∀k, 0 < k < h,

n→∞

The theorem now follows from the fact that mN (∂A) = 0 and Theorem 4.3. Corollary 1. Let D = ∅ be a bounded open subset of RN and A, ∂A = ∅, be a subset of D such that ∃h > 0 and c > 0 such that Ph (∂A) ≤ c.

(12.6)

Then the mapping bA → Ph (∂A ) : Cb (D) → R ∪{+∞} is lower semicontinuous in A for the W 1,p (D)-topology. Proof. Since we have a metric topology, it is suﬃcient to prove the property for W 1,p (D)-converging sequences {bAn } to bA . From that point on the argument is the same as the one used to get (12.5) in the proof of Theorem 12.1 after the extraction of the subsequence. Remark 12.1. It is important to notice that even if {An } is a W 1,p -convergent sequence of bounded open subsets of RN with a uniformly bounded perimeter, the limit set A need not be an open set or have a nonempty interior int A such that bA = bint A . It would be tempting to say that bAn → bA implies dAn → dA and use the open set int A = A for which dA = dint A to conclude that bA = bint A . This is incorrect as can be seen in the following example shown in Figure 7.2. Consider a family {An } of open rectangles in R2 of width equal to 1 and height 1/n, n ≥ 1, an integer. Their density perimeter is bounded by 4, and the bAn ’s converge to bA for A equal to the line of length 1 which has an empty interior.

13. Compactness and Uniform Fat Segment Property

An

1 n

1

387

A

1

Figure 7.2. W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R with uniformly bounded density perimeter to a set with empty interior. 2

13 13.1

Compactness and Uniform Fat Segment Property Main Theorem

In Theorem 6.11 of section 6.4 of Chapter 5, we have seen a compactness theorem for the family of subsets of a bounded holdall D satisfying the uniform cone property. This compactness theorem is no longer true when the uniform cone property is replaced by a uniform segment property. This is readily seen by considering the following example. Example 13.1. Given an integer n ≥ 1, consider the following sequence of open domains in R2 : def Ωn = (x, y) ∈ R2 : |x| < 1 and |x|1/n < y < 2 . They satisfy the uniform segment property of Deﬁnition 6.1 (ii) of Chapter 2 by choosing λ = r = 1/4. The sequence {Ωn } converges to the closed set def

A = (x, y) ∈ R2 : |x| ≤ 1 and 1 ≤ y ≤ 2 ∪ L, def

L = (0, y) ∈ R2 : 0 ≤ y ≤ 1 in the uniform topologies associated with dΩn and bΩn or in the Lp -topologies associated with χΩn and χΩn . However, the segment property is not satisﬁed along the line L and the corresponding family of subsets of the holdall D = B(0, 4) satisfying the uniform segment property with r = λ = 1/4 is not closed and, a fortiori, not compact. This example shows that a uniform segment property is too meager to make the corresponding family compact. Looking back at the proof of Theorem 6.11 of section 6.4 in Chapter 5, the fact that the cone is an open set around the segment is used critically. That was the motivation to introduce the more general uniform fat segment property (Deﬁnition 6.1 (iii) in section 6 of Chapter 2) that includes as special cases both the uniform cone and the uniform cusp properties (cf. Deﬁnitions 6.3 and 6.4 in section 6.4 of Chapter 2). However, the proof of Theorem 6.11 in Chapter 5 was based on the fact that the perimeter of all the sets in the family

388

Chapter 7. Metrics via Oriented Distance Functions

were uniformly bounded. There is no hope of getting that property for general H¨ olderian domains, and a completely diﬀerent proof is necessary. Given a bounded open subset D of RN consider the family & Ω satisﬁes the uniform fat def L(D, r, O, λ) = Ω ⊂ D : . (13.1) segment property for (r, O, λ) We ﬁrst give a general proof of the compactness of L(D, r, O, λ) in the C(D)- and W 1,p (D)-topologies associated with the oriented distance function bΩ . As a consequence, we get the compactness for the C(D)- and W 1,p (D)-topologies associated with dΩ and dΩ and the Lp (D)-topologies associated with χΩ and χΩ . We recover the compactness Theorem 6.11 of Chapter 5 under the uniform cone property for χΩ in Lp (D) as a special case. In the last part of this section, we give several other ways to specify the compact families by using the equivalent conditions of Chapter 2 of the uniform fat segment property in terms of conditions on the local epigraphs via the dominating function and equicontinuity of the local graph functions. Theorem 13.1. Let D be a nonempty bounded open subset of RN and let 1 ≤ p < ∞. Assume that L(D, r, O, λ) is not empty for r > 0, λ > 0, and an open / O. Then the family subset O of RN such that (0, eN ) ⊂ O and 0 ∈ def

B(D, r, O, λ) = {bΩ : ∀Ω ∈ L(D, r, O, λ)} is compact in C(D) and W 1,p (D). As a consequence the families def

Bd (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)}, def Bdc (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)}, def

Bd∂ (D, r, O, λ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ)} are compact in C(D) and W 1,p (D), and the families def

X(D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)}, def X c (D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)} are compact in Lp (D). From Lemma 6.1 of Chapter 2, Theorem 13.1 can be specialized to structured open sets O. Corollary 1. The conclusions of Theorem 13.1 hold when O is specialized to open regions of the form

(13.2) O(h, ρ, λ) = ξ ∈ RN : ξ ∈ BH (0, ρ) and h(|ξ |) < ξN < λ for dominating functions h ∈ H. Theorem 13.1 can be further specialized to Lipschitzian and H¨ olderian sets.

13. Compactness and Uniform Fat Segment Property

389

Corollary 2. The conclusions of Theorem 13.1 hold under the uniform cone property of Deﬁnition 6.3 (ii) in section 6.4 of Chapter 2 with O = O(h, ρ, λ) with h(θ) = θ/ tan ω and ρ = λ tan ω,

0 < ω < π/2,

(13.3)

and the uniform cusp property of Deﬁnition 6.4 (ii) of Chapter 2 with

O = O(h , ρ, λ) with h (θ) = λ (θ/ρ) ,

0 < < 1.

The proof of Theorem 13.1 will require Theorem 4.3 and the following lemma. Lemma 13.1. Given a sequence {bΩn } ⊂ Cb (D) such that bΩn → bΩ in C(D) for some bΩ ∈ Cb (D), we have the following properties: ∀x ∈ Ω, ∀R > 0,

∃N (x, R) > 0, ∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅,

and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅.

(13.4)

Moreover, ∀x ∈ ∂Ω, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), B(x, R) ∩ Ωn = ∅ and B(x, R) ∩ Ωn ) = ∅, and B(x, R) ∩ ∂Ωn = ∅. Proof. We proceed by contradiction. Assume that ∃x ∈ Ω,

∃R > 0,

∀N > 0,

∃n ≥ N,

B(x, R) ∩ Ωn = ∅.

So there exists a subsequence {Ωnk }, nk → ∞, such that bΩnk (x) = dΩnk (x) ≥ R = 0 = dΩ (x) ≥ bΩ (x), which contradicts the fact that bΩnk → bΩ . Of course, the same assertion is true

for the complements and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅.

When x ∈ ∂Ω, x ∈ Ω ∩ Ω and we combine the two results. For the last result use the fact that the open ball cannot be partitioned into two nonempty disjoint open subsets. Proof of Theorem 13.1. (i) Compactness in C(D). Since for all Ω in L(D, r, O, λ), Ω is locally a C 0 -epigraph, ∂Ω = ∅, bΩ ∈ Cb (D), and mN (∂Ω) = 0. Consider an arbitrary sequence {Ωn } in L(D, r, O, λ). For D compact, Cb (D) is compact in C(D) and there exist Ω ⊂ D, ∂Ω = ∅, and a subsequence {Ωnk } such that bΩnk → bΩ in C(D). It remains to prove that Ω ∈ L(D, r, O, λ), that is, ∀x ∈ ∂Ω, ∃A ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + λAO ⊂ int Ω.

390

Chapter 7. Metrics via Oriented Distance Functions

From Lemma 13.1, for each x ∈ ∂Ω, for all k ≥ 1, there exists nk ≥ k such that B x, r/2k ∩ ∂Ωnk = ∅. Denote by xk an element of that intersection: ∀k ≥ 1, xk ∈ B x, r/2k ∩ ∂Ωnk . By construction xk → x. Next consider y ∈ B(x, r) ∩ Ω. From the ﬁrst part of the lemma, there exists a subsequence of {Ωnk }, still denoted by {Ωnk }, such that for all k ≥ 1, B(y, r/2k ) ∩ Ωnk = ∅. For each k ≥ 1 denote by yk a point of that intersection. By construction yk ∈ Ωnk → y ∈ Ω ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this, note that y ∈ B(x, r) and that ∃ρ > 0,

B(y, ρ) ⊂ B(x, r)

and |y − x| + ρ/2 < r.

Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | . r ρ ρ/ r r ≤ k + r − + k ≤ r + k−1 − < r. 2 2 2 2 2 Since r/ρ > 1 the result is true for r 2k−1

−

ρ 2 + log

r . ρ

So we have constructed a subsequence {Ωnk } such that for k ≥ K xk ∈ ∂Ωnk → x ∈ ∂Ω

and yk ∈ Ωnk ∩ B(xk , r) → y ∈ Ω ∩ B(x, r).

For all k, there exists Ak ∈ O(N), Ak ∗Ak = ∗Ak Ak = I, such that yk + λAk O ⊂ int Ωnk . Pick another subsequence of {Ωnk }, still denoted by {Ωnk }, such that ∃A ∈ O(N), A ∗A = ∗A A = I,

Ak → A.

Now consider z ∈ y + λAO. Since y + λAO is open there exists ρ > 0 such that B(z, ρ) ⊂ y + λAO, and there exists K ≥ K such that ∀k ≥ K ,

B(z, ρ/2) ⊂ yk + λAk O ⊂ int Ωnk = Ωnk

⇒ B(z, ρ/2) ⊃ Ωnk ⇒ 0 < ρ/2 = dB(z,ρ/2) (z) ≤ dΩn (z) → dΩ (z) k

⇒ 0 < ρ/2 ≤ dΩ (z)

⇒ z ∈ Ω = int Ω

⇒ y + λAO ⊂ int Ω.

Hence Ω ⊂ D satisﬁes the uniform fat segment property.

13. Compactness and Uniform Fat Segment Property

391

(ii) W 1,p (D)-compactness. From Theorem 4.1 (i), it is suﬃcient to prove the result for p = 2. Consider the subsequence {Ωnk } ⊂ L(D, λ, O, r) and let Ω ∈ L(D, λ, O, r) be the set previously constructed such as bΩnk → bΩ in C(D). Hence bΩnk → bΩ in L2 (D). Since D is compact |bΩnk |2 dx ≤ diam (D)2 dx ≤ diam (D)2 mN (D), ∀Ω ⊂ D, D D 2 |∇bΩnk | dx ≤ dx = mN (D), D

D

and there exists a subsequence, still denoted by {bΩnk }, which converges weakly to bΩ . Since all the sets in L(D, r, O, λ) verify a uniform segment property, they are locally C 0 -epigraphs, mN (∂Ωnk ) = 0 = mN (∂Ω) and, by Theorem 4.3, bΩnk → bΩ in W 1,p (D)-strong, 1 ≤ p < ∞. (iii) The other compactness follows by continuity of the maps (4.6) and (4.7) in Theorem 4.1 and the fact that mN (∂Ω) = 0 implies χint Ω = χΩ and χint Ω = χΩ almost everywhere for Ω ∈ L(D, λ, O, r).

13.2

Equivalent Conditions on the Local Graph Functions

We have shown in Theorem 6.2 of Chapter 2 that the uniform fat segment property is equivalent to the equi-C 0 epigraph property. This means that we can equivalently characterize the compact family L(D, r, O, λ) in terms of conditions on the local graph functions such as the ones introduced in Theorem 5.1 of Chapter 2. Recall the following equivalent conditions: (i) Ω veriﬁes the fat segment property. (ii) Ω is an equi-C 0 -epigraph; that is, Ω is a C 0 -epigraph and the local graph functions are uniformly bounded and equicontinuous. (iii) Ω is a C 0 -epigraph and there exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ζ , ξ ∈ BH (0, ρ) such that |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |). (13.5)

(iv) Ω is a C 0 -epigraph and there exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ξ ∈ BH (0, ρ),

ax (ξ ) ≤ h(|ξ |).

(13.6)

Recall also from Deﬁnition 5.1 (ii) in Chapter 2 that Ω is said to be a C 0 epigraph if it is locally a C 0 -epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

392

Chapter 7. Metrics via Oriented Distance Functions

By applying each of the last three characterizations to all members of a family of subsets of a bounded holdall D, we get the analogue of the compactness Theorem 13.1 under the uniform fat segment property. Theorem 13.2. Let D be a bounded open nonempty subset of RN and consider the family L(D, ρ, U ) of all subsets Ω of D that verify the C 0 -epigraph property: there exist ρ > 0 and a bounded open neighborhood U of 0 such that

def (13.7) U ⊂ ζ ∈ RN : PH (ζ) ∈ BH (0, ρ) , V = BH (0, ρ), and for each Ω ∈ L(D, ρ, U ) and each x ∈ ∂Ω, there exists AΩ x ∈ O(N) such that def

U Ω (x) = x + AΩ xU

(13.8)

Ω and a C 0 -graph function aΩ x : V → R such that ax (0) = 0 and

U Ω (x) ∩ ∂Ω = U Ω (x) ∩ x + Ax (ζ + ζN eN ) : ζ ∈ V, ζN = aΩ x (ζ ) ,

U Ω (x) ∩ int Ω = U Ω (x) ∩ x + Ax (ζ + ζN eN ) : ζ ∈ V, ζN > aΩ x (ζ ) .

Then, for all h ∈ H, the subfamilies {aΩ x : Ω ∈ L(D, ρ, U ), x ∈ ∂Ω} def L0 (D, ρ, U ) = Ω ∈ L(D, ρ, U ) : is uniformly bounded , and equicontinuous def

L1 (D, ρ, U, h) = Ω ∈ L(D, ρ, U ) : ∀x ∈ ∂Ω, ∀ζ ∈ V, aΩ x (ζ ) ≤ h(|ζ |) , ∀x ∈ ∂Ω, ∀ζ , ξ ∈ V def L2 (D, ρ, U, h) = Ω ∈ L(D, ρ, U ) : such that |ξ − ζ | < ρ |ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |)

(13.9)

(13.10)

(13.11) (13.12)

are compact (possibly empty) in C(D) and W 1,p (D), 1 ≤ p < ∞, and the conclusions of Theorem 13.1 hold. Remark 13.1. The geometric fat segment property of Corollary 1 was introduced in the form (13.2) ´sio [37] under the name uniform in the book of M. C. Delfour and J.-P. Zole cusp property in 2001, where the condition was specialized to (13.3) for Lipschitzian and H¨ olderian domains as stated in Corollary 2. Condition (13.10) is a streamlined version of the suﬃcient condition given in an unpublished report by D. Tiba [2], announced without proof in the note ¨ ki, and D. Tiba [1] in 2000, and later documented by W. Liu, P. Neittaanma in D. Tiba [3] in 2003. He gets a compactness result by introducing the following conditions on the local C 0 -epigraphs: the boundaries of the domains belong (after a rotation and a translation) to a family F of uniformly equicontinuous functions deﬁned in a ﬁxed neighborhood V of 0 in RN −1 . This means that there exists a modulus of continuity µ(ε) > 0 such that ∀ε > 0, ∀f ∈ F, ∀x, y ∈ V,

|x − y| < µ(ε)

⇒ |f (y) − f (x)| < ε.

Then the compactness follows by Ascoli–Arzel`a Theorem 2.4 of Chapter 2.

14. Compactness under Uniform Fat Segment Property and Bound on a Perimeter 393 The connection between that condition and our geometric condition was ´sio [1, 2] in ﬁrst established in M. C. Delfour, N. Doyon, and J.-P. Zole 2005. Condition (13.11) that relaxes the equicontinuity condition (13.10) to the simpler dominating function condition and condition (13.12) were both introduced ´sio [43] in 2007 along with the general form in M. C. Delfour and J.-P. Zole of the uniform fat segment property that involves only the unparametrized open set O.

14

Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter

Domains that are locally Lipschitzian epigraphs or equivalently satisfy the local uniform cone property enjoy the additional property that for any bounded open set D the (N − 1)-Hausdorﬀ measure of D ∩ ∂Ω is ﬁnite. In general this is no longer true for domains verifying a uniform cusp property for some function h(θ) = |θ/ρ|α , 0 < α < 1 (cf., for instance, Example 6.2 given6 in section 6.5 of Chapter 2 of a bounded domain Ω in RN for which HN −1 (∂Ω) = +∞ and the Hausdorﬀ dimension of ∂Ω is exactly N − α). Yet, there are many domains with cusps whose boundary has ﬁnite HN −1 measure.

14.1

De Giorgi Perimeter of Caccioppoli Sets

One of the classical notions of perimeter is the one introduced in Deﬁnition 6.2 in section 6 of Chapter 5 in the context of the problem of minimal surfaces for Caccioppoli sets. Theorem 13.1 extends to the following subfamily of L(D, r, O, λ) for a bounded open subset D of RN and a constant c : Ω satisﬁes the uniform fat def L(D, r, O, λ, c ) = Ω ⊂ D : segment property for (r, O, λ) . (14.1) and PD (Ω) ≤ c Theorem 14.1. Let D be a nonempty bounded open subset of RN and let 1 ≤ p < ∞. Assume that L(D, r, O, λ, c ) is not empty for r > 0, λ > 0, c > 0, and a / O. Then the family bounded open subset O of RN such that (0, eN ) ⊂ O and 0 ∈ B(D, r, O, λ, c ) = {bΩ : ∀Ω ∈ L(D, r, O, λ, c )} def

is compact in C(D) and W 1,p (D). As a consequence the families Bd (D, r, O, λ, c ) = {dΩ : ∀Ω ∈ L(D, r, O, λ, c )}, def Bdc (D, r, O, λ, c ) = {dΩ : ∀Ω ∈ L(D, r, O, λ, c )}, def Bd∂ (D, r, O, λ, c ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ, c )} def

6 This

´sio [1]. was originally in M. C. Delfour, N. Doyon, and J.-P. Zole

394

Chapter 7. Metrics via Oriented Distance Functions

are compact in C(D) and W 1,p (D), and the families X(D, r, O, λ, c ) = {χΩ : ∀Ω ∈ L(D, r, O, λ, c )} , def X c (D, r, O, λ, c ) = {χΩ : ∀Ω ∈ L(D, r, O, λ, c )} def

are compact in Lp (D).

14.2

Finite Density Perimeter

It is also possible to use the density perimeter introduced in Deﬁnition 12.1 in place of the De Giorgi perimeter. The proof of the next result combines Theorem 14.1, which says that the family L(D, r, O, λ, c ) is compact, with Theorem 12.1, which says that the family of sets verifying (12.2) is compact in W 1,p (D). The intersection of the two families of oriented distance functions is compact in W 1,p (D). Theorem 14.2. For ﬁxed h > 0 and an open subset Ω of RN , Theorem 14.1 remains true when PD (Ω) is replaced by the density perimeter Ph (∂Ω).

15

The Families of Cracked Sets

In this section we introduce new families of thin sets, that is, sets A such that mN (∂A) = 0, by introducing conditions on the semiderivative of the distance function at points of the boundary. They have been used in M. C. Delfour and ´sio [38] in the context of the image segmentation problem of D. MumJ.-P. Zole ford and J. Shah [2]. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks and sets that can be made up of components of diﬀerent codimensions. The Hausdorﬀ (N − 1) measure of their boundary is not necessarily ﬁnite. Yet, compact families (in the W 1,p -topology) of such sets can be constructed. First recall lower and upper semiderivatives7 of Dini type for the diﬀerential quotient of a function f : V (x) ⊂ RN → R deﬁned in a neighborhood V (x) of a point x ∈ RN in the direction d ∈ RN : f (x + td) − f (x) def f (x + td) − f (x) , = lim inf δ0 0 xk,j−1 + 2δk .

406

Chapter 7. Metrics via Oriented Distance Functions

Ω2 f = 0

frame D

Ω1 f = 1

Figure 7.5. The two open components Ω1 and Ω2 of the open domain Ω for N = 2. Note that gk,j (xk,j−1 + δk ) = (δk )α is independent of j and is the maximum of the function gk,j . The uniform cusp property is veriﬁed for ρ = 1/8, λ = h(ρ), and h(θ) = θα . The boundary of Ω1 is made up of straight lines of total length 9 plus the length of the curve def

C = {(x, f (x)) : 0 ≤ x < 1} ,

C = ∪∞ k=0 Ck ,

η /2

k Ck = ∪j=1 Ck,j ,

def

Ck,j = {(x, f (x)) : xk,j−1 ≤ x < xk,j−1 + δk } . The length of the curve Ck,j is bounded below by HN −1 (Ck,j ) ≥ 2

'

ηk /2

(δk )2 + (δk )2α ≥ 2 (δk )α

ηk 2 (δk )α = ηk (δk )α = ηk HN −1 (Ck ) ≥ 2 HN −1 (Ck ) ≥ ηk 1−α

⇒ HN −1 (C) =

α

1 2k+1

∞

⇒ HN −1 (Ck ) ≥

1 ηk 2k+1

HN −1 (Ck,j ),

j=1

α

= ηk

1−α

1 2k+1

α ,

α 1−α α 1 = 2[(2k+1 ) 1−α ] 2k+1 α 1 ≥ 21−α (2k+1 )α = 21−α 2k+1

HN −1 (Ck ) ≥ +∞ 21−α = +∞.

k=0

When at least one solution has a bounded perimeter, it is possible to show that there is one that minimizes the h-density perimeter.

16. A Variation of the Image Segmentation Problem of Mumford and Shah

407

Theorem 16.3. Assume that the assumptions of Theorem 16.1 are veriﬁed. There exists an Ω∗ in F ∗ (D, h, α) which minimizes the h-density perimeter. Proof. If for all Ω in F ∗ (D, h, α) the h-density perimeter is +∞, the theorem is true. If for some Ω ∈ F ∗ (D, h, α), Ph (Γ) ≤ c, then there exists a sequence {Ωn } in F ∗ (D, h, α) such that Ph (Γn ) →

inf

Ω∈F ∗ (D,h,α)

Ph (Γ).

By the compactness Theorems 12.1 and 16.2, there exist a subsequence and Ω∗ , Γ∗ = ∅, such that bΩn → bΩ∗ in W 1,p (D), Ω∗ ∈ F (D, h, α), mN (Ω) = mN (D), and Ph (Γ∗ ) ≤ lim inf Ph (Γn ) ≤ c. n→∞

Finally by going back to the proof of Theorem 16.1 and using the fact that all the Ωn ’s are already minimizers in F ∗ (D, h, α), it can be shown that Ω∗ is indeed one of the minimizers in the set F ∗ (D, h, α).

16.4

Uniform Bound or Penalization Term in the Objective Function on the Density Perimeter

To complete the results on the segmentation problem, we turn to the existence of a segmentation for a family of sets with a uniform bound or with a penalization term in the objective function on the h-density perimeter. Theorem 16.4. Given a bounded open frame ∅ = D ⊂ RN with a Lipschitzian boundary and real numbers h > 0 and c > 0,10 there exists an open subset Ω∗ of D, Γ∗ = ∅, with ﬁnite h-density perimeter such that mN (Ω∗ ) = mN (D) (Ph (Γ∗ ) ≤ c for (16.8)), and y ∈ H 1 (Ω∗ ) solutions of the respective problems inf inf1 ε |∇ϕ|2 + |ϕ − f |2 dx, (16.8) Ω open ⊂D, Ph (Γ)≤c ϕ∈H (Ω) mN (Ω)=mN (D)

Ω

inf

ε |∇ϕ|2 + |ϕ − f |2 dx + c Ph (Γ).

inf

Ω open ⊂D ϕ∈H 1 (Ω) mN (Ω)=mN (D)

(16.9)

Ω

Proof. The proof for the objective function (16.8) is exactly the same as the one of Theorem 16.1. It uses Lemma 16.2 to show that the minimizing set has a thin boundary and the compactness Theorem 12.1. The proof for the objective function (16.9) uses the fact that there is a minimizing sequence for which the h-density perimeter is uniformly bounded and the lower semicontinuity of the density perimeter in the W 1,p -topology is given by Corollary 1. ´sio [8]. Problem (16.9) was originally considered in D. Bucur and J.-P. Zole The above two identiﬁcation problems can be further specialized to the family of cracked sets F(D, h, α). 10 Note that the constant c must be large enough to take into account the contribution of the boundary of D.

408

Chapter 7. Metrics via Oriented Distance Functions

Corollary 1. Given a bounded open frame D ⊂ RN with a Lipschitzian boundary and real numbers h > 0, α > 0, and c > 0, there exists an open subset Ω∗ of D in F(D, h, α) such that mN (Ω∗ ) = mN (D) (Ph (Γ∗ ) ≤ c for (16.10)), and y ∈ H 1 (Ω∗ ) solutions of the problem inf

inf1

ε |∇ϕ|2 + |ϕ − f |2 dx,

(16.10)

ε |∇ϕ|2 + |ϕ − f |2 dx + c Ph (Γ).

(16.11)

Ω open ⊂D, Ω∈F (D,h,α) ϕ∈H (Ω) Ph (Γ)≤c, mN (Ω)=mN (D)

Ω

inf

inf1

Ω open ⊂D, Ω∈F (D,h,α) ϕ∈H (Ω) mN (Ω)=mN (D)

Ω

Proof. Since the minimizing sequence {bΩn } constructed in the proof of Theorem 16.1 strongly converges to bΩ∗ in W 1,p (D) for all p, 1 ≤ p < ∞, from property (12.3) in Theorem 12.1, we have Ph (Γ∗ ) ≤ lim inf Ph (Γn ) ≤ c n→∞

and the optimal Ω∗ constructed in the proof of the theorem satisﬁes the additional constraint on the density perimeter.

Chapter 8

Shape Continuity and Optimization 1

Introduction and Generic Examples

The underlying philosophy behind Chapters 5, 6, and 7 was to introduce metric topologies on sets associated with families of functions parametrized by sets rather than parametrize sets by functions. In the case of the characteristic function we have seen several examples where it appears explicitly in the modeling of the problem. The distance functions provide other topologies and constructions. For instance the oriented distance function gives a direct analytic access to geometric entities such as the normal and the fundamental forms on the boundary of a geometric domain without ad hoc local bases or Christoﬀel symbols. Many of the advantages of working in a truly intrinsic framework for shape derivatives will be illustrated in Chapters 9 and 10. Now that we have meaningful topologies on sets, we can consider the continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc. In this chapter we concentrate on continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the objective function is deﬁned as an inﬁmum over a family of functions over a ﬁxed domain or set such as eigenvalue and compliance problems. What is nice about them is that no adjoint system is necessary to characterize the minimizing function. Other problems have the structure of optimal control theory, that is, a state equation depending on the control that describes the evolution of the state and an objective functional that depends on the state and the control. In that case, the characterization of the optimal control involves an adjoint state equation coupled with the state equation. In the context of shape and geometric optimization the control will be the underlying geometry. The state equation will be a static or dynamical partial differential equation on the domain, and the objective functional will depend on the domain and the state that itself depends on the domain. As in control theory, we need continuity of the objective functional and the state with respect to the geometry and compactness of the family of domains over which the optimization takes 409

410

Chapter 8. Shape Continuity and Optimization

place. Since it is not possible to give a complete general theory of shape optimization problems within this book, we choose to concentrate on a few simple generic examples that are illustrative of the underlying technicalities involved. Several examples of optimization problems involving the Lp -topology on measurable sets via the characteristic function were given in Chapter 5. In this chapter, we ﬁrst characterize the continuity of the transmission problem and the upper semicontinuity of the ﬁrst eigenvalue of the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of deﬁnition since they require diﬀerent constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial diﬀerential equations. In the above problems the strong continuity of solutions of the elliptic equation or the eigenvector equation with respect to the underlying domain is the key element in the proof of the existence of optimal domains. To get that continuity, some extra conditions have to be imposed on the family of open domains. Those issues have received a lot of attention for the Laplace equation with homogeneous Dirichlet boundary conditions. With a sequence of domains in a ﬁxed holdall D associate a sequence of extensions by zero of the solutions in the ﬁxed space H01 (D). The Poincar´e inequality is uniform for that sequence, as the ﬁrst eigenvalue of the Laplace equation in each domain is dominated by the one associated with the larger holdall D. By a classical compactness argument, the sequence of extensions converges to some limiting element y in H01 (D). To complete the proof of the continuity, two more fundamental questions remain. Is y the solution of the Laplace equation for the limit domain? Does y satisfy the Dirichlet boundary condition on the boundary of the limit domain? The ﬁrst issue can be resolved by assuming that the Sobolev spaces associated with the moving domains converge in the Kuratowski sense, i.e., with the property that any element in the Sobolev space associated with the limit domain can be approached by a sequence of elements in the moving Sobolev spaces. That property is obtained in examples where the compactivorous property 1 follows from the choice of the deﬁnition of convergence of the domains. If the domains are open subsets of D, then the complementary Hausdorﬀ topology has that property. The family of subsets of D satisfying a uniform fat segment property has that property. For the second issue it is necessary to impose some constraints at least on the limit domain. The stability condition introduced by J. Rauch and M. Taylor [1] and used by E. N. Dancer [1] and D. Daners [1] precisely assumes that the limit domain is suﬃciently smooth, just enough to have the solution in H01 (Ω). Of course, assuming such a regularity on the limit domain makes things easier but is a little bit artiﬁcial. The more fundamental issue is to identify the families of domains for which this stability property is preserved in the limit. The uniform cone property was used in the context of shape optimization by D. Chenais [1, 2] in ´sio [3, 5, 6, 8, 9] showed that this is true 1973. In 1994 D. Bucur and J.-P. Zole under general capacitary conditions and have constructed new compact subfamilies 1 A sequence of open sets converging to a limit open set in some topology has the compactivorous property if any compact subset of the limit set is contained in all sets of the sequence after a certain rank: the sequence eats up the compact set after a certain rank (cf. Theorem 2.4 (iii) in Chapter 6).

1. Introduction and Generic Examples

411

of domains with respect to the Hausdorﬀ complementary topology. Furthermore ˇ ´ k [1, 2] in D. Bucur [1] proved that the condition given in 1993 by V. Sver a dimension 2 involving a bound on the number of connected components of the complement of the domain can be recovered from the more general capacity conditions that are suﬃcient in the case of the Laplacian with homogeneous Dirichlet boundary conditions. In a more recent paper D. Bucur [5] proved that they are almost necessary. Intuitively, those capacity conditions are such that, locally, the complement of the domains in the family under consideration has enough capacity to preserve and retain the homogeneous Dirichlet boundary condition in the limit.

1.1

First Generic Example

The ﬁrst example is the optimization of the ﬁrst eigenvalue of an associated elliptic diﬀerential operator: sup λA (Ω) A∇ϕ · ∇ϕ dx Ω∈A(D) def A Ω , (1.1) inf 1 λ (Ω) = A |ϕ|2 dx 0 = ϕ∈H (Ω) inf λ (Ω) 0 Ω Ω∈A(D) where A(D) is a family of admissible open subsets of D.

1.2

Second Generic Example

As a second generic example, consider the transmission problem over the ﬁxed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D: ﬁnd y = y(χΩ ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D), (k1 χΩ + k2 (1 − χΩ )) ∇y · ∇ϕ dx = f, ϕ (1.2) D

for some strictly positive constants k1 and k2 , α = max{k1 , k2 } > 0, f ∈ H −1 (D), H −1 (D) the topological dual of H01 (D), and ·, · denotes the duality pairing between H −1 (D) and H01 (D). This problem was studied in section 4 of Chapter 5 in the context of the compliance problem with two materials.

1.3

Third Generic Example

As a third example, consider the minimization of the objective function def 1 |uΩ − g|2 dx J(Ω) = 2 Ω

(1.3)

over the family of open subsets Ω of a bounded open holdall D of RN , where g in L2 (D), uΩ ∈ H01 (Ω) is the solution of the variational problem ∀ϕ ∈ H01 (Ω), A∇uΩ · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω) , (1.4) Ω

A ∈ L∞ (D; L(RN , RN )) is a matrix function on D such that ∗A = A and αI ≤ A ≤ βI for some coercivity and continuity constants 0 < α ≤ β, and f |Ω denotes the restriction of f in H −1 (D) to H −1 (Ω).

412

1.4

Chapter 8. Shape Continuity and Optimization

Fourth Generic Example

As a fourth generic example, consider the previous example but with the homogeneous Dirichlet boundary value problem replaced by the homogeneous Neumann boundary value problem of the mathematician for the Laplacian ∂y = 0 on ∂Ω. ∂n

−∆y + y = f in Ω,

(1.5)

The classical (physical) homogeneous Neumann boundary value problem ∂y = 0 on ∂Ω ∂n

−∆y = f in Ω, has a unique solution up to a constant if

2

D

(1.6)

f dx = 0.

Upper Semicontinuity and Maximization of the First Eigenvalue

Given a bounded open nonempty domain D ⊂ RN , consider the minimization of the Raleigh quotient for an open nonempty subset Ω of D: A∇ϕ · ∇ϕ dx def D inf1 A∇ϕ · ∇ϕ dx, (2.1) λA (Ω) = = inf |ϕ|2 dx 0=ϕ∈H0 (Ω;D) ϕ∈H01 (Ω;D) D D ϕL2 =1

where A ∈ L∞ (D; L(RN , RN )) is a matrix function on D such that ∗

and αI ≤ A(x) ≤ βI

A(x) = A(x)

(2.2)

for some coercivity and continuity constants 0 < α ≤ β. We set λA (∅) = +∞. The minimizers are characterized as follows. Theorem 2.1. (i) Given a bounded open nonempty domain Ω in RN , there exist uΩ ∈ H01 (Ω; D), uΩ = 0, and λA (Ω) > 0 such that A∇uΩ · ∇ϕ − λA (Ω) uΩ ϕ dx = 0 (2.3) ∀ϕ ∈ H01 (Ω; D), Ω

and ∀ϕ ∈

H01 (Ω; D),

A

λ (Ω) =

D

A∇uΩ · ∇uΩ dx ≤ |uΩ |2 dx D

D

A∇ϕ · ∇ϕ dx . |ϕ|2 dx D

Conversely, if there exist u ∈ H01 (Ω; D), u = 0, and λ such that A∇u · ∇ϕ − λ u ϕ dx = 0 ∀ϕ ∈ H01 (Ω; D), Ω

(2.4)

(2.5)

2. Upper Semicontinuity and Maximization of the First Eigenvalue and

∀ϕ ∈ H01 (Ω; D),

then

λ=

D

λ≤

D

A∇ϕ · ∇ϕ dx , |ϕ|2 dx D

A∇u · ∇u dx = inf |u|2 dx ϕ∈H01 (Ω;D) D

D

413

(2.6)

A∇ϕ · ∇ϕ dx , |ϕ|2 dx D

u is a nonzero solution of the minimization problem (2.1), and λ is equal to the ﬁrst eigenvalue λA (Ω) associated with Ω. (ii) Given a bounded open nonempty domain D, there exists λA D > 0 (that depends only on the diameter of D and A) such that for all open subsets Ω of D, λA (Ω) ≥ λA (D) ≥ λA D > 0, where λD (Ω) is the ﬁrst eigenvalue of the symmetrical linear operator ϕ → −div (A ∇ϕ) on D. Proof. (i) For any bounded open Ω, the inﬁmum is bounded below by 0 and hence ﬁnite. Let {ϕn } be a minimizing sequence such that ϕn L2 (Ω) = 1. By coercivity, α ∇ϕn 2L2 (Ω) is bounded. Hence {ϕn } is a bounded sequence in H01 (Ω) and there exist ϕ ∈ H01 (Ω) and a subsequence, still indexed by n, such that ϕn ϕ in H 1 (Ω)-weak. By the Rellich–Kondrachov compactness theorem, the subsequence strongly converges in L2 (Ω). Then ϕ2n dx → ϕ2 dx and A∇ϕ · ∇ϕ dx ≤ lim inf A∇ϕn · ∇ϕn dx 1= n→∞ Ω Ω Ω Ω A∇ϕ · ∇ϕ dx A∇ϕn · ∇ϕn dx ⇒ Ω 2 ≤ lim inf Ω 2 = λA (Ω). n→∞ ϕ dx ϕ dx Ω Ω n By deﬁnition of λA (Ω), 0 = ϕ ∈ H01 (Ω) is a minimizing element and inequality (2.4) is veriﬁed. Since ϕ = 0, the Rayleigh quotient is diﬀerentiable in ϕ and its directional derivative in the direction ψ is given by A∇ϕ · ∇ψ dx ϕ ψ dx Ω 2 −2 A∇ϕ · ∇ϕ dx Ω 4 . 2 ϕL2 (Ω) ϕL2 (Ω) Ω A nonzero solution ϕ of the minimization problem on Ω is necessarily a stationary point, and for all ψ ∈ H01 (Ω), A∇ϕ · ∇ϕ dx A Ω A∇ϕ · ∇ψ dx = ϕ ψ dx = λ (Ω) ϕ ψ dx ϕ2L2 (Ω) Ω Ω Ω and we get (2.3). Conversely, any λ and nonzero u ∈ H01 (Ω; D) that verify (2.5) yield λ ≥ λA (Ω). But, from inequality (2.6), λ ≤ λA (Ω) and λ = λA (Ω). Hence, u is a minimizer of the Rayleigh quotient over H01 (Ω; D).

414

Chapter 8. Shape Continuity and Optimization

(ii) Let diam (D) be the diameter of D and x ∈ RN be a point such that D ⊂ Bx = B(x, diam (D)). Therefore, from Theorem 2.3 in Chapter 2, H01 (Ω; Bx ) ⊂ H01 (D; Bx ) ⊂ H01 (Bx ),

H01 (Ω; D) ⊂ H01 (D)

with the associated isometries. Hence, by deﬁnition of the minimum of a functional over an increasing sequence of sets, λA (Ω) ≥ λA (D) ≥ λA (Bx ). If λA (Bx ) = 0, we repeat the above construction with H01 (Bx ) in place of H01 (Ω; Bx ) and end up with an element ϕ ∈ H01 (Bx ) such that ϕ2 dx = 1 and α |∇ϕ|2 dx ≤ A∇ϕ · ϕ dx = 0, Bx

Bx

Bx

which is impossible in H01 (Bx ). Finally λA (Bx ) is independent of the choice of x since the eigenvalue is invariant under a translation of the domain: it depends only on the diameter of D. The case λA (∅) = +∞ is compatible with the results. Remark 2.1. For A equal to the identity operator I, denote by λ(Ω), λ(D), and λD the quantities 1 λA (Ω), λA (D), and λA D of Theorem 2.1. For any open Ω ⊂ D and any ϕ ∈ H0 (Ω; D) 1 ϕL2 (D) ≤ √ ∇ϕL2 (D) λD 4 1 ⇒ ∀Ω open ⊂ D, ϕH 1 (Ω) ≤ 1 + ∇ϕL2 (Ω) λD

(2.7) (2.8)

and the spaces H01 (Ω) and H01 (D) will be endowed with the respective equivalent norms ∇ϕL2 (Ω) and ∇ϕL2 (D) . Before considering the existence of minimizers/maximizers, we investigate the question of the continuity of the ﬁrst eigenvalue λA (Ω) with respect to Ω for the uniform complementary Hausdorﬀ topology on def

Cdc (D) = dΩ : ∀Ω open subset in D and Ω = RN (cf. (2.11) of Deﬁnition 2.2 in Chapter 6). Theorem 2.2. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). The mapping dΩ → λA (Ω) : Cdc (D) ⊂ C0 (D) → R ∪{+∞}

(2.9)

is upper semicontinuous. Proof. Since λA (Ω) is ﬁnite in Ω = ∅ and +∞ in Ω = ∅, it is upper semicontinuous in ∅. For Ω = ∅, consider a sequence {Ωn } of open subsets of D such that dΩn → dΩ in C(D) and the associated sequence {λA (Ωn )}. We want to show that lim sup λA (Ωn ) ≥ λA (Ω).

2. Upper Semicontinuity and Maximization of the First Eigenvalue

415

Consider the new sequence {Ωn ∩Ω}. Since Ω ⊂ D, for all n, (Ωn ∩Ω) = Ωn ∪Ω ⊃ D = ∅ and d(Ωn ∩Ω) = dΩn ∪Ω = min{dΩn , dΩ } → dΩ in C(D). Let un ∈ H01 (Ωn ∩ Ω), un L2 (D) = 1, be an eigenvector for the eigenvalue λn = λA (Ωn ∩ Ω). From Theorem 2.1 un is solution of the variational equation (2.3): A∇un · ∇ϕn dx = λn un ϕn dx. (2.10) ∀ϕn ∈ H01 (Ωn ; D), D

D

Since Ω = ∅, there exists a compact ball Kρ = B(x, ρ) ⊂ Ω for some x ∈ Ω and ρ > 0. By the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6, there exists an integer N (Kρ ) such that for all n ≥ N (Kρ ), Kρ ⊂ Ωn ∩ Ω and by Theorem 2.1 (ii), 0 < λn ≤ λA (B(x, ρ)). Therefore, the sequence of eigenvalues {λn } is bounded. By coercivity α∇un }2L2 (D) ≤ λn ∇un }2L2 (D) = λn and {un } is bounded in H01 (Ω; D) since Ωn ∩ Ω ⊂ Ω. Letting λ∗ = lim sup λn , there exist subsequences, still indexed by n, and u ∈ H01 (Ω; D) such that λn → λ ∗ ,

un L2 (D) = 1,

un u in H01 (Ω; D)-weak

⇒ uL2 (D) = 1.

Since for all n, λA (Ωn ) ≤ λA (Ωn ∩ Ω), it is readily seen that lim sup λA (Ωn ) ≤ lim sup λA (Ωn ∩ Ω) = λ∗ and that it is suﬃcient to show that λ∗ = λA (Ω) to get the upper semicontinuity. We cannot directly go to the limit in (2.10), since the test functions ϕn depend on Ωn . To get around this diﬃculty, recall the density of D(Ω) in H01 (Ω; D) and use the compactivorous property a second time. For any ϕ ∈ D(Ω), K = supp ϕ ⊂ Ω is compact and there exists N (K) such that for all n ≥ N (K), supp ϕ ⊂ Ωn ∩ Ω. Therefore, for all n ≥ N (K) 1 1 un ∈ H0 (Ωn ∩ Ω; D) ⊂ H0 (Ω; D), A∇un · ∇ϕ dx = λn un ϕ dx. D

By letting n → ∞

A∇u · ∇ϕ dx = λ∗

∀ϕ ∈ D(Ω), D

D

u ϕ dx, D

and, by density of D(Ω) in H01 (Ω; D), the variational equation is veriﬁed in H01 (Ω; D). Hence, u ∈ H01 (Ω; D), uL2 (Ω) = 1, satisﬁes the variational equation A∇u · ∇ϕ dx = λ∗ u ϕ dx. ∀ϕ ∈ H01 (Ω; D), D

D

416

Chapter 8. Shape Continuity and Optimization

Therefore, since λA (Ω) is the smallest eigenvalue with respect to H01 (Ω; D), λA (Ω) ≤ λ∗ . To show that λ∗ = λA (Ω), it remains to prove that λ∗ ≤ λA (Ω). For each n A∇ϕn · ∇ϕn dx 1 A . ∀ϕn ∈ H0 (Ωn ∩ Ω; D), λ (Ωn ∩ Ω) ≤ D ϕn 2L2 (D) By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N such that supp ϕ ⊂ Ω ∩ Ωn for all n ≥ N . Therefore for all n ≥ N A∇ϕ · ∇ϕ dx A λ (Ωn ∩ Ω) ≤ D ϕ2L2 (D) and by taking the lim sup

∀ϕ ∈ D(Ω),

∗

λ ≤

D

A∇ϕ · ∇ϕ dx . ϕ2L2 (D

By density this is true for all ϕ ∈ H01 (Ω; D), ϕ = 0, and λ∗ ≤ λA (Ω). As a result λ∗ = λA (Ω). Finally, λA (Ωn ) ≤ λA (Ωn ∩ Ω)

⇒ lim sup λA (Ωn ) ≤ lim sup λA (Ωn ∩ Ω) = λ∗ = λA (Ω),

and we have the upper semicontinuity of the map dΩ → λA (Ω). In view of the previous theorem, we now deal with the maximization of the ﬁrst eigenvalue λA (Ω) associated with the operator A with respect to a family of open subsets Ω of D. Theorem 2.3. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). (i) Then λA (Ω) = λA (∅) = +∞.

sup

(2.11)

Ω∈Cdc (D)

The conclusion remains true when Cdc (D) is replaced by Cdc (D) = {dΩ : Ω convex and open ⊂ D}

(2.12)

(cf. Theorem 8.6 in Chapter 6). (ii) Given an open set E, ∅ = E ⊂ RN , consider the compact family c Cd,loc (E; D)

=

dΩ

∃x ∈ RN , ∃A ∈ O(N) such that : x + AE ⊂ Ω open ⊂ D

&

of Theorem 2.6 in Chapter 6. Then, there exists an open Ω∗ such that dΩ∗ ∈ c Cd,loc (E; D) and λA (Ω∗ ) =

sup c Ω∈Cd,loc (E;D)

λA (Ω) ≤ β λI (E).

(2.13)

3. Continuity of the Transmission Problem

417

c (E; D) is replaced by Cdc (E; D) or The conclusion remains true when Cd,loc c c the subfamilies of convex subsets Cd (E; D) and Cd,loc (E; D) of Theorem 8.6 in Chapter 6.

(iii) There exists Ω∗ ∈ L(D, r, O, λ) such that λA (Ω∗ ) =

sup

λ(Ω) ≤ βλI (O).

(2.14)

Ω∈L(D,r,O,λ)

Proof. (i) From Theorem 2.2 and the compactness of Cdc (D) in C(D). (ii) All the families are compact by Theorems 2.6 and 8.6 in Chapter 6. So there is a maximizer dΩ∗ of λA (Ω). The upper bound is obvious. (iii) By compactness of L(D, r, O, λ) in W 1,p (D) and hence in Cdc (D) for the C(D)-topology.

3

Continuity of the Transmission Problem

As a second generic example, consider the transmission problem over the ﬁxed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D: ﬁnd y = y(χΩ ) ∈ H01 (D) such that (k1 χΩ + k2 (1 − χΩ )) ∇y · ∇ϕ dx = f, ϕ (3.1) ∀ϕ ∈ H01 (D), D

for some strictly positive constants k1 and k2 , α = max{k1 , k2 } > 0, f ∈ H −1 (D), H −1 (D) the topological dual of H01 (D), and ·, · denotes the duality pairing between H −1 (D) and H01 (D). This problem was studied in section 4 of Chapter 5 in the context of the compliance problem with two materials. Theorem 3.1. Let D be a bounded open domain in RN and k1 and k2 be some strictly positive constants such that α = max{k1 , k2 } > 0 and f ∈ H −1 (D). Denote by y(χ) the solution of the transmission problem (3.1) associated with χ ∈ X(D). The mapping χ → y(χ) : X(D) → H01 (D) is continuous when X(D) is endowed with the strong Lp (D)-topology, 1 ≤ p < ∞. The same result is true with co X(D) in place of X(D). Remark 3.1. This theorem applies to the compact families of sets of Theorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat segment property and for the conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 of Chapter 2) since the convergence of bΩn → bΩ in W 1,p (D) implies the convergence of χΩn → χΩ in Lp (D). It also applies to the compact family of convex sets C(D) = {χΩ : Ω convex subset of D} introduced in (3.24) in section 3.4 of Chapter 5 (cf. Theorem 3.5 (iii) and Corollary 1 to Theorem 6.3 in section 6.1 of Chapter 5).

418

Chapter 8. Shape Continuity and Optimization

´sio [2], [24, Proof of Theorem 3.1. The proof follows the arguments of J.-P. Zole sect. 4.2.1, pp. 446–448]. It is suﬃcient to prove that for any convergent sequence χn → χ in L2 (D)-strong, the corresponding solutions yn = y(χn ) ∈ H01 (D) converge to y = y(χ) in H01 (D)-strong. First, the sequence {yn } is bounded in H01 (D): α∇yn 2L2 (D) ≤ (k1 χn + k2 (1 − χn )) ∇yn · ∇yn dx D

⇒ α∇yn 2L2 (D)

= f, yn ≤ f H −1 (D) yn H01 (D) 4 1 ≤ f H −1 (D) yn H01 (D) ≤ f H −1 (D) 1 + ∇yn L2 (Ω) λD

since we have the equivalence of norms from (2.7) in Remark 2.1. So there is a subsequence, still denoted by {yn }, that weakly converges to some y ∈ H01 (D). To show that y is solution of (3.1) corresponding to χ, we go to the limit in ∀ϕ ∈ H01 (D), (k1 χn + k2 (1 − χn )) ∇ϕ · ∇yn dx = f, ϕ. D

Since (k1 χn +k2 1−χn ))∇ϕ → (k1 χ+k2 (1−χ))∇ϕ in L2 (D)-strong and ∇yn → ∇y in L2 (D)-weak, we get that y ∈ H01 (D) is the unique solution of ∀ϕ ∈ H01 (D), (k1 χ + k2 (1 − χ)) ∇y · ∇ϕ dx = f, ϕ. D

Finally, we prove the strong convergence. Indeed (k1 χn + k2 (1 − χn )) ∇(yn − y) · ∇(yn − y) dx α ∇(yn − y)2L2 (D) ≤ D ≤ (k1 χn + k2 (1 − χn )) ∇yn · ∇yn dx D (k1 χn + k2 (1 − χn )) ∇yn · ∇y dx −2 D (k1 χn + k2 (1 − χn )) ∇y · ∇y dx + D (k1 χn + k2 (1 − χn )) ∇y · ∇y dx =f, yn − 2 f, y + D (k1 χ + k2 (1 − χ)) ∇y · ∇y dx = 0. → −f, y + D

Since every strongly convergent subsequence converges to the same limit, the whole sequence strongly converges to y in H01 (D).

4 4.1

Continuity of the Homogeneous Dirichlet Boundary Value Problem Classical, Relaxed, and Overrelaxed Problems

We have seen in section 3.5 of Chapter 5 two relaxations of the homogeneous Dirichlet boundary value problem for the Laplacian to measurable subsets Ω of a bounded

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

419

open holdall D by introducing the following closed subspaces of H01 (D): def

H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D ,

def H•1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )∇ϕ = 0 a.e. in D

(4.1) (4.2)

(cf. (3.27) and (3.28) in Theorem 3.6 of Chapter 5). It is readily seen that H1 (Ω; D) ⊂ H•1 (Ω; D). For Ω open, H01 (Ω; D) ⊂ H1 (Ω; D) ⊂ H•1 (Ω; D) and the three closed subspaces of H01 (D) are generally not equal as can be seen from Examples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5. The choice of a relaxation is very much problem dependent and should be guided by the proper modeling of the underlying physical or technological phenomenon at hand. In general, there is a price to pay for the above relaxations. Both H1 (Ω; D) and H•1 (Ω; D) depend on the equivalence class of measurable sets [Ω] and not speciﬁcally on Ω. In general, there is not even an open representative of Ω in the class. From section 3.3 in Chapter 5, [Ω] = [I], where I is the measure theoretic interior of Ω, m(B(x, r) ∩ Ω) N I = x ∈ R : lim =1 , r0 m(B(x, r)) int I = Ω1 , where

Ω1 = x ∈ RN : ∃ρ > 0 such that m Ω ∩ B(x, ρ) = m B(x, ρ) is open, and int Ω ⊂ int I and ∂I ⊂ ∂Ω (cf. Deﬁnition 3.3 and Theorem 3.3 in Chapter 5). In view of the cleaning operation in the deﬁnition of I, all cracks of zero measure will be deleted or not seen by the functions of the spaces H1 (Ω; D) and H•1 (Ω; D). When mN (∂Ω) = 0, [int I] = [Ω] and int I = Ω1 is an open representative in the class [Ω]. Let D be a bounded open subset of RN and let f ∈ H −1 (D), where H −1 (D) is the topological dual of H01 (D). For all open subsets Ω of D ϕ → f, ϕH −1 (D)×H01 (D) : D(Ω) → R

(4.3)

is well-deﬁned and continuous with respect to the H 1 (Ω)-topology and extends to an element f |Ω ∈ H −1 (Ω). The map ϕ → f, ϕH −1 (D)×H01 (D) : H1 (Ω; D) → R is also well-deﬁned and continuous with respect to the closed subspace H1 (Ω; D) of H01 (D) and deﬁnes an element f |Ω of the dual H1 (Ω; D) of H1 (Ω; D). Similarly, the map ϕ → f, ϕH −1 (D)×H01 (D) : H•1 (Ω; D) → R is also well-deﬁned and continuous with respect to the closed subspace H•1 (Ω; D) of H01 (D) and deﬁnes an element f |•Ω of the dual H•1 (Ω; D) of H•1 (Ω; D). Let A ∈ L∞ (D; L(RN , RN )) be a symmetrical matrix function on D such that ∗

A=A

and αI ≤ A ≤ βI

(4.4)

420

Chapter 8. Shape Continuity and Optimization

for some coercivity and continuity constants 0 < α ≤ β. Consider the classical homogeneous Dirichlet boundary value problem for the generalized Laplacian ∃y(Ω) ∈ H01 (Ω; D) such that, A∇y(Ω) · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω) , ∀ϕ ∈ H01 (Ω; D),

(4.5)

D

the relaxed homogeneous Dirichlet boundary value problem ∃y(χΩ ) ∈ H1 (Ω; D) such that, 1 A∇y(χΩ ) · ∇ϕ dx = f |Ω , ϕH1 (Ω;D) ×H1 (Ω;D) , ∀ϕ ∈ H (Ω; D),

(4.6)

D

and the overrelaxed homogeneous Dirichlet boundary value problem ∃y(χΩ ) ∈ H•1 (Ω; D) such that, A∇y(χΩ ) · ∇ϕ dx = f |•Ω , ϕH•1 (Ω;D) ×H•1 (Ω;D) . ∀ϕ ∈ H•1 (Ω; D),

(4.7)

D

By convention, set y(∅) = 0 and y(χ∅ ) = 0 as elements of H01 (D). In addition we shall keep the terminology homogeneous Dirichlet boundary value problem for what we just called the classical homogeneous Dirichlet boundary value problem for the generalized Laplacian. Since, in general, H01 (Ω; D) is a distinct closed subspace of H1 (Ω; D), a solution of (4.6) is not necessarily a solution of (4.5). In the other direction, a solution y = y(Ω) ∈ H01 (Ω; D) of (4.5) belongs to H1 (Ω; D) and is a solution of ∃y ∈ H1 (Ω; D) such that, A∇y · ∇ϕ dx = f |Ω , ϕH1 (Ω;D) ×H1 (Ω;D) , ∀ϕ ∈ H01 (Ω; D),

(4.8)

D

but y is not necessarily unique in H1 (Ω; D) since this variational equation is veriﬁed only on the closed subspace H01 (Ω; D) of H1 (Ω; D). However, the space H1 (Ω; D) is very close to the space H01 (Ω; D) and the two spaces will coincide on smooth sets and more generally on crack-free sets that will be introduced in section 7 (cf. Deﬁnition 7.1 (ii) and Theorem 7.3 (ii)). The following terminology has been introduced by J. Rauch and M. Taylor [1]. Deﬁnition 4.1. An open subset Ω of D is said to be stable with respect to D if H01 (Ω; D) = H1 (Ω; D). The same considerations apply to the overrelaxed problem (4.7) in H•1 (Ω; D) with the additional observation that even for sets with a very smooth boundary, the solution does not coincide with the one of the (classical) Dirichlet problem when the Ω has more than one connected component as seen from Examples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5. The problem in H•1 (Ω; D) is associated with diﬀerent physical phenomena.

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

4.2

421

Classical Dirichlet Boundary Value Problem

We have the following general results. Theorem 4.1. Let D be a bounded open nonempty subset in RN . Associate with each open subset Ω of D the solution y(Ω) ∈ H01 (Ω; D) of the classical Dirichlet problem (4.5) on Ω. (i) Let Ω and the sequence {Ωn } be open subsets of D such that dΩn → dΩ in C(D). Then y(Ωn ∩ Ω) → y(Ω) in H01 (D)-strong

(4.9)

and there exist a subsequence {Ωnk } and y0 ∈ H1 (Ω; D) such that y(Ωnk ) y0 in H01 (D)-weak

(4.10)

and y0 is a (not necessarily unique) solution of (4.8). (ii) The function dΩ → y(Ω) : Cdc (D) → H01 (D)-strong

(4.11)

is continuous for each Ω such that H01 (Ω; D) = H1 (Ω; D). Remark 4.1. We have the additional information that y0 ∈ H1 (Ω; D) ⊂ H01 (int Ω; D) in part (i) of the theorem (cf. Theorem 7.3 (i)). Remark 4.2. The projection operator PΩ : H01 (D) → H01 (Ω; D)

(4.12)

is deﬁned for u ∈ H01 (D) as the solution of the variational equation ∃PΩ u ∈ H01 (Ω; D), ∀ϕ ∈ H01 (Ω; D), ∇(PΩ u) · ∇ϕ dx = ∇u · ∇ϕ dx. D

D

Applying the theorem to a sequence of open subsets {Ωn } of D converging to Ω in the complementary Hausdorﬀ topology with def f, ϕ = ∇u · ∇ϕ dx, D

there exist u0 ∈ H1 (Ω; D) and a subsequence {Ωnk } such that dΩn → dΩ in C(D),

PΩnk u u0 in H01 (D)-weak.

The projection operator PΩ is continuous at Ω if H01 (Ω; D) = H1 (Ω; D), but this is only a suﬃcient condition.

422

Chapter 8. Shape Continuity and Optimization

Proof. (i) The proof of the continuity of (4.9) follows the same steps as the proof of the upper semicontinuity of the ﬁrst eigenvalue of the Laplacian in Theorem 2.2. As for the convergence (4.10), it is readily seen that the sequence {yn = y(Ωn )} of elements of H01 (Ωn ; D) is bounded in the bigger space H01 (D): by coercivity 4 1 2 α ∇yn L2 (D) ≤ f H −1 (D) yn H01 (D) ≤ 1 + f H −1 (D) ∇yn L2 (D) λD from Remark 2.1, where λD > 0 is the ﬁrst eigenvalue of the Laplacian on D. There exist a subsequence, still indexed by n, and some y0 in H01 (D) such that yn y0 in H01 (D)-weak and hence yn → y0 in L2 (D)-strong. From Corollary 1 to Theorem 4.4 in Chapter 6 ∀x ∈ RN , ⇒ ∀ a.a. x ∈ RN ,

lim inf χΩn (x) ≥ χΩ (x) n→∞

0 = lim inf (1 − χΩn (x))|yn (x)|2 ≥ (1 − χΩ (x))|y0 (x)|2 n→∞

since (1 − χΩn (x)) ≥ 0, |yn | ≥ 0, and |yn |2 → 0 a.e. in D. By deﬁnition, y0 ∈ H1 (Ω; D). Finally, by the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6, for each ϕ ∈ D(Ω) there exists N such that for all n > N , supp ϕ ⊂ Ωn . Therefore, by substituting ϕ in (4.5) on Ωn , for all n > N A∇yn · ∇ϕ dx = f |Ωn , ϕH −1 (Ωn )×H01 (Ωn ) = f |Ω , ϕH −1 (Ω)×H01 (Ω) (4.13) 2

D

and by going to the limit

∀ϕ ∈ D(Ω), D

A∇y0 · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω)

(4.14)

and, by density of D(Ω) in H01 (Ω; D), y0 ∈ H1 (Ω; D) is a solution of (4.8). (ii) If, in addition, H1 (Ω; D) = H01 (Ω; D), then y0 ∈ H01 (Ω; D) is the unique solution of (4.5). Hence y0 = y(Ω), solution of the classic problem (4.5), is independent of the choice of the converging subsequence. Therefore the whole sequence {yn } converges to y0 = y(Ω) in H01 (D)-weak. The strong continuity in H01 (D) now follows from the convergence of un → u0 2 in L (D)-strong: α∇(un − u0 )2L2 (D) ≤ A∇(un − u0 ) · ∇(un − u0 ) dx D = A∇un · ∇un dx + A∇u0 · ∇u0 dx − 2 A∇un · ∇u0 dx D D D A∇un · ∇u0 dx = f, un + f, u0 − 2 D A∇u0 · ∇u0 dx = 0 → 2 f, u0 − 2 D

and the whole sequence {yn } converges to y0 = y(Ω) in H01 (D)-strong. Therefore, the function (4.11) is continuous in Ω.

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

4.3

423

Overrelaxed Dirichlet Boundary Value Problem

In this section we concentrate on the overrelaxation (4.7) of the homogeneous Dirichlet boundary value problem. To show the continuity with respect to the characteristic function of domains, we ﬁrst approximate its solution by transmission problems. 4.3.1

Approximation by Transmission Problems

The overrelaxed Dirichlet problem (4.7) corresponding to χ = χΩ ∈ X(D) is ﬁrst approximated by a transmission problem over D indexed by ε > 0 going to zero: ﬁnd yε = yε (χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D) 1 χ + (1 − χ) ∇yε · ∇ϕ dx = χf ϕ dx. ε D D

(4.15)

They are special cases of the general transmission problem (3.1) over the ﬁxed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D for k1 = 1 and k2 = 1/ε and χ = χΩ . Theorem 4.2. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H•1 (Ω; D) be the solution of the overrelaxed Dirichlet problem (4.7). For ε, 0 < ε < 1, let yε be the solution of problem (4.15) in H01 (D). Then y(χΩ ) = lim yε in H01 (D)-strong. ε→0

Proof. For ε, 0 < ε < 1, substitute ϕ = yε in (4.15) to get the bounds √ ∇yε L2 (D) ≤ c(D) f L2 (D) , (1 − χΩ ) ∇yε L2 (D) ≤ ε c(D) f L2 (D) . There exist a sequence εn 0 and y0 ∈ H01 (D) such that yn = yεn y0 in H01 (D)weak and (1 − χΩ ) ∇yn → 0 in L2 (D)-strong. Therefore, (1 − χΩ ) ∇y0 = 0 and y0 ∈ H•1 (Ω; D). For ϕ ∈ H•1 (Ω; D), (4.15) reduces to ∇yε · ∇ϕ dx = χΩ f ϕ dx ∀ϕ ∈ H•1 (Ω; D), D

and as n → ∞

D

∀ϕ ∈

H•1 (Ω; D),

∇y0 · ∇ϕ dx =

D

χΩ f ϕ dx. D

Hence y0 ∈ H•1 (Ω; D) is the solution of (4.7). Since the limit is the same for all convergent sequences, yε → y0 in H01 (D)-weak. By subtracting the last two equations ∀ϕ ∈ H•1 (Ω; D), [∇yε − ∇y0 ] · ∇ϕ dx = 0 ⇒ [∇yε − ∇y0 ] · ∇y0 dx = 0. D

D

424

Chapter 8. Shape Continuity and Optimization

Finally, we estimate the following term: 2 |∇(yε − y0 )| dx = [∇yε − ∇y0 ] · ∇yε dx − [∇yε − ∇y0 ] · ∇y0 dx D D D [∇yε − ∇y0 ] · ∇yε dx = D ∇yε · ∇yε dx − ∇y0 · ∇yε dx = D D = χΩ f yε dx − ∇y0 · ∇yε dx D D χΩ f y0 dx − ∇y0 · ∇y0 dx = 0. → D

As a result yε → y0 in 4.3.2

D

H01 (D)-strong.

Continuity with Respect to X(D) in the Lp (D)-Topology

Theorem 4.3. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H•1 (Ω; D) be the solution of the overrelaxed Dirichlet problem (4.7). Then the function χΩ → y(χΩ ) : X(D) → H01 (D)-strong

(4.16)

is continuous for all Lp (D)-topologies, 1 ≤ p < ∞, on X(D). Proof. Given χ = χΩ ∈ X(D), let {χΩn } in X(D) be a converging sequence to χ = χΩ . In view of Theorem 4.2, let εn > 0 be such that the approximating solution yn ∈ H01 (D) of (4.15), 1 1 χΩ n + ∀ϕ ∈ H0 (D), (1 − χΩn ) ∇yn · ∇ϕ dx = χΩn f ϕ dx, εn D D is such that yn − y(χΩn )H 1 (D) < 1/n and recall the bounds ∇yn L2 (D) ≤ c(D) f L2 (D) , √ (1 − χΩn ) ∇yn L2 (D) ≤ εn c(D) f L2 (D) ≤ c(D) f L2 (D) .

(4.17) (4.18)

Therefore the sequence {y(χΩn } is also bounded in H01 (D) and there exist subsequences, still indexed by n, and y0 ∈ H01 (D) such that yn y0 and y(χΩn ) y0 in H01 (D)-weak. Moreover, (1 − χΩn ) ∇y(χΩn ) = 0. Hence (1 − χΩ ) ∇y0 = 0 and y0 ∈ H•1 (Ω; D). For ϕ ∈ H•1 (Ω; D) 1 χΩ n + ∀ϕ ∈ H01 (D), (1 − χΩn ) ∇yn · (χΩ ∇ϕ) dx = χΩn f ϕ dx. εn D D

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

425

χΩ f ϕ dx. The left-hand side can be rewritten as 1 χΩ χΩn ∇yn · ∇ϕ + χΩ (1 − χΩn ) (1 − χΩn ) ∇yn · ∇ϕ dx εn D χΩ ∇y0 · ∇ϕ dx = ∇y0 · ∇ϕ dx →

The right-hand side converges to

D

D

D

since χΩ χΩn → χΩ and χΩ (1 − χΩn ) → 0 in L2 (D)-strong and the bound (4.18). Finally, y0 ∈ H•1 (Ω; D) is solution of 1 ∇y0 · ∇ϕ dx = χΩ f ϕ dx. ∀ϕ ∈ H• (Ω; D), D

D

Since the limit point y0 = y(χΩ ) is independent of the choice of the sequence, y(χΩn ) y(χΩ ) in H01 (D)-weak. Finally, 2 |∇ (y(χΩn ) − y(χΩ ))| dx D 2 2 = |∇y(χΩn )| dx + |∇y(χΩ )| dx − 2 ∇y(χΩn ) · ∇y(χΩ ) dx D D D χΩn f y(χΩn ) dx + χΩ f y(χΩ ) dx − 2 ∇y(χΩn ) · ∇y(χΩ ) dx = D D D χΩ f y(χΩ ) dx − 2 ∇y(χΩ ) · ∇y(χΩ ) dx = 0 →2 D

D

and y(χΩn ) → y(χΩ ) in

H01 (D)-strong.

Remark 4.3. The penalization technique used in this section can be applied to higher-order elliptic, and even parabolic, problems and the Navier–Stokes equation (cf. R. Dziri ´sio [6]) provided that the relaxation is acceptable in the modeling and J.-P. Zole of the physical phenomenon. The continuity of the classical homogeneous Dirichlet problem for the Laplacian will be given in section 6 under capacity conditions.

4.4

Relaxed Dirichlet Boundary Value Problem

In this section we consider the relaxation (4.6) of the homogeneous Dirichlet boundary value problem for the Laplacian to measurable subsets of a bounded open holdall D for the space H1 (Ω; D). Theorem 4.4. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H1 (Ω; D) be the solution of the relaxed Dirichlet problem (4.1). Then the function χΩ → y(χΩ ) : X(D) → H01 (D)-strong is continuous for all Lp (D)-topology, 1 ≤ p < ∞, on X(D).

(4.19)

426

Chapter 8. Shape Continuity and Optimization

Proof. Given χΩ ∈ X(D), let {χΩn } in X(D) be a converging sequence to χΩ and let y(χΩ ) ∈ H1 (Ω; D) and y(χΩn ) ∈ H1 (Ωn ; D) be the corresponding solutions of (4.1). Since H1 (Ω; D) ⊂ H•1 (Ω; D), we have the continuity χΩ → y(χΩ ) from Theorem 4.3. To complete the proof we need to show that the limiting element y(χΩ ) in the proof of Theorem 4.3 is solution of the variational equation (4.1) in the smaller space H1 (Ω; D). Indeed, for each n, (1 − χΩn )y(χΩn ) = 0 implies (1 − χΩ )y(χΩ ) = 0 and y(χΩ ) ∈ H1 (Ω; D). Finally, since y(χΩ ) is solution of the variational equation (4.7) for all ϕ ∈ H•1 (Ω; D), it still is solution for ϕ in the smaller subspace H1 (Ω; D). Hence y(χΩ ) ∈ H1 (Ω; D) is the solution of (4.1).

5

Continuity of the Homogeneous Neumann Boundary Value Problem

As a fourth generic example, consider the homogeneous Neumann boundary value problem of mathematicians for the Laplacian ∂y = 0 on ∂Ω, −∆y + y = f in Ω, ∂n ∇y · ∇ϕ + y ϕ dx = f ϕ dx. ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), Ω

(5.1) (5.2)

Ω

The classical homogeneous Neumann boundary value problem −∆y = f in Ω, ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω),

∂y = 0 on ∂Ω, ∂n ∇y · ∇ϕ dx = f ϕ dx Ω

(5.3) (5.4)

Ω

has a unique solution up to a constant if D f dx = 0. One way to get around the nonuniqueness is to introduce the closed subspace def Ha1 (Ω) = ϕ ∈ H 1 (Ω) : ϕ dx = 0 Ω

of functions with zero average. Then there exists a unique y ∈ Ha1 (Ω) such that ∀ϕ ∈ Ha1 (Ω), ∇y · ∇ϕ dx = f ϕ dx. (5.5) Ω

Ω

Moreover, the solution of (5.5) can be approximated by the ε-problems ∇yε · ∇ϕ + ε yε ϕ dx = f ϕ dx ∃yε ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), Ω

(5.6)

Ω

and yε → y in Ha1 (Ω)-strong. Note that, by assumption on f , yε ∈ Ha1 (Ω). One of the key conditions to get the continuity with respect to the domain Ω is the following density property.

5. Continuity of the Homogeneous Neumann Boundary Value Problem

427

Deﬁnition 5.1. An open set Ω is said to have the H 1 -density property if the set {ϕ|Ω : ϕ ∈ C01 (RN )}

(5.7)

is dense in H 1 (Ω). This property is not veriﬁed for a domain Ω with a crack in R2 , since elements of H 1 (Ω) have diﬀerent traces on each side of the crack. Recall from section 6.1 in Chapter 2 that this condition is veriﬁed for a large class of sets Ω that can be characterized by the geometric segment property that is equivalent to the property that the set be locally a C 0 -epigraph (cf. section 6.2 of Chapter 2). We recall Theorem 6.3 of Chapter 5. Theorem 5.1. If the open set Ω has the segment property, then the set

f |Ω : ∀f ∈ C0∞ (RN ) of restrictions of functions of C0∞ (RN ) to Ω is dense in W m,p (Ω) for 1 ≤ p < ∞ and m ≥ 1. In particular C k (Ω) is dense in W m,p (Ω) for any m ≥ 1 and k ≥ m. Lemma 5.1. Let D be a bounded open subset of RN , let f ∈ L2 (D), and let Ω and the sequence {Ωn } be open subsets of D. Denote by yn ∈ H 1 (Ωn ) the solution of (5.1) in Ωn and by y ∈ H 1 (Ω) the solution of (5.1) in Ω. Assume the following: (i) χΩn → χΩ in L2 (D); (ii) dΩn → dΩ in C(D); (iii) Ω has the H 1 -density property. Then 9n ) → (˜ : in (L2 (D) × L2 (D))-weak, y , ∇y) (˜ yn , ∇y where z˜ denotes the extension by zero of a function z from Ω or Ωn to D. Proof. The proof of the lemma follows the approach of W. Liu and J. E. Rubio [1] in 1992. Since f ∈ L2 (D), we readily have the following estimates: 2 2 2 9n 2 2 ˜ yn 2L2 (D) + ∇y L (D) = yn L2 (Ωn ) + ∇yn L2 (Ωn ) ≤ f L2 (D) .

There exist subsequences, still indexed by n, z ∈ L2 (D), and Z ∈ L2 (D)N such that y˜n z weakly in L2 (RN )

9n Z weakly in L2 (RN )N . and ∇y

By the H 1 -density property of Ω, the variational equation (5.2) on Ωn can be rewritten in the form / . 9n · ∇ϕ + y˜n ϕ dx = χΩn ∇y χΩn f ϕ dx (5.8) ∀ϕ ∈ C01 (RN ), RN

RN

428

Chapter 8. Shape Continuity and Optimization

and we can go to the limit as n → ∞: ∀ϕ ∈ C01 (RN ), χΩ [Z · ∇ϕ + z ϕ] dx = RN

χΩ f ϕ dx.

(5.9)

RN

By deﬁnition of the distributional derivative, for all n ∂i yn ϕn dx = − yn ∂i ϕn dx, ∀ϕn ∈ D(Ωn ). Ωn

Ωn

By the compactivorous property (cf. Theorem 2.4 (iii) in Chapter 6) and the convergence dΩn → dΩ in C(D), for each ϕ ∈ D(Ω), there exists N such that for all n > N , supp ϕ ⊂ Ωn . Hence for all n > N y ϕ dx = − y˜n ∂i ϕ dx, ∀ϕ ∈ D(Ω) ∂9 i n Ω

Ω

and by going to the limit Zi ϕ dx = − z ∂i ϕ dx, Ω

∀ϕ ∈ D(Ω),

⇒ ∇z = Z ∈ L2 (Ω)

Ω

and from (5.9), z ∈ H 1 (Ω) is solution of the variational equation 1 N ∀ϕ ∈ C0 (R ), ∇z · ∇ϕ + z ϕ dx = f ϕ dx. Ω

Ω

By the H 1 -density property of Ω, this extends to all ϕ ∈ H 1 (Ω), which implies that z = y. Since all weakly convergent subsequences converge to the same limit, the whole sequence weakly converges and this completes the proof. Remark 5.1. We know that the H 1 -density condition holds for the family of domains satisfying the segment property of Deﬁnition 6.1 in Chapter 2 and not only for H 1 -spaces but also W m,p -spaces (cf. Theorem 6.3 in Chapter 2), thus extending the property and the results to a broader class of partial diﬀerential equations. We have shown in Theorem 6.5 of Chapter 2 that a domain satisfying the segment property is locally a C 0 -epigraph in the sense of Deﬁnition 5.1 of Chapter 2. Theorem 5.2. Let D be a bounded open subset of RN , let f ∈ L2 (D), and let L(D, r, O, λ) be the compact family of sets in W 1,p (D), 1 ≤ p < ∞, verifying the uniform fat segment property. Then the function : L(D, r, O, λ) → (L2 (D) × L2 (D))-weak, Ω → (˜ y (Ω), ∇y(Ω)) where z˜ denotes the extension by zero of a function z from Ω to D, is continuous. It is also true when L(D, r, O, λ) is replaced by the family of convex sets def

Cb (D) = bΩ : Ω ⊂ D, ∂Ω = ∅, Ω convex (cf. Theorem 10.2 in Chapter 7).

(5.10)

6. Elements of Capacity Theory

429

Proof. In L(D, r, O, λ) endowed with the W 1,p (D)-topology, 1 ≤ p < ∞, the three assumptions of the lemma are veriﬁed. Remark 5.2. Therefore the previous arguments are veriﬁed for the compact families of sets of Theorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat segment property and for the conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 of Chapter 2). However, the subsets of a bounded holdall verifying a uniform segment property do not include the very important domains with cracks ´sio for which some speciﬁc results have been obtained by D. Bucur and J.-P. Zole [2, 4] (and D. Bucur and N. Varchon [1] in dimension 2). An issue that was not tackled is the nonhomogeneous boundary condition that involves a boundary integral in the variational formulation: ∂y = g on ∂Ω, (5.11) −∆y + y = f in Ω, ∂n ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), ∇y · ∇ϕ + y ϕ dx = f ϕ dx + g ϕ dΓ. (5.12) Ω

6

Ω

Γ

Elements of Capacity Theory

Recall the relaxation of the homogeneous Dirichlet boundary value problem for the Laplacian (4.6) in section 4.4 by using the space def

H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D .

(6.1)

The classical solution lies in the closed subspace H01 (Ω; D) of H1 (Ω; D), and we know that they are not equal when Ω is not stable with respect to D. The diﬀerence between the two spaces is that a function in H01 (Ω; D) is zero in D\Ω as an H01 (D)function and not just almost everywhere as an L2 (D)-function. Some information is lost in the relaxation, and this requires the ﬁner notion of quasi-everywhere that will characterize the set over which an H01 (D)-function is zero. This section introduces the basic elements and results from capacity theory that will play an essential role in establishing the continuity of the classical homogeneous Dirichlet problem (4.5) with respect to its underlying domain. The characterization of the space H01 (Ω; D) in terms of capacity will be given in Theorem 6.2. By introducing capacity constraints on sets, it will be possible to construct larger compact families that include domains with cracks.

6.1

Deﬁnition and Basic Properties

For p > N the elements of W 1,p (RN ) can be redeﬁned on sets of measure zero to be continuous functions (cf. R. A. Adams [1, Thm. 5.4, p. 97]): W 1,p (RN ) → C 1,λ (RN ),

0 < λ ≤ 1 − N/p.

(6.2)

430

Chapter 8. Shape Continuity and Optimization

But for p ≤ N this is no longer the case. We ﬁrst recall the deﬁnition of (1, p)capacity with respect to RN and with respect to an open subset G of RN along with a number of basic technical results. Deﬁnition 6.1. Let p, 1 < p < ∞, be a real number and N ≥ 1 be an integer. (i) (L. I. Hedberg [2, p. 239]) The capacity is deﬁned as follows: for a compact subset K ⊂ RN def Cap1,p (K) = inf |∇ϕ|p dx : ϕ ∈ C0∞ (RN ), ϕ > 1 on K ; RN

for an open subset G ⊂ RN

def Cap1,p (G) = sup Cap1,p (K) : ∀K ⊂ G, K compact ; for an arbitrary subset E ⊂ D

def Cap1,p (E) = inf Cap1,p (G) : ∀G ⊃ E, G open . In what follows, we shall use the simpler notation Capp as in L. C. Evans and R. F. Gariepy [1, pp. 147] instead of Cap1,p . (ii) (L. C. Evans and R. F. Gariepy [1, pp. 160] and J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 87]) A function f is said to be quasi-continuous on D ⊂ RN if, for each ε > 0, there exists an open set Gε such that Cap1,p (Gε ) < ε and f is continuous on D\Gε . (iii) A set E in RN is said to be quasi-open if, for all ε > 0, there exists an open set Gε such that Cap1,p (Gε ) < ε and E ∪ Gε is open. In the above deﬁnitions the capacity is deﬁned with respect to RN . In J. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 2, p. 27], the capacity is deﬁned relative to an open subset D of RN instead of the whole space RN . Deﬁnition 6.2. Let N ≥ 1 be an integer, let p, 1 < p < ∞, and let D be a ﬁxed open subset of RN . (i) The (1, p)-capacity is deﬁned as follows: for a compact subset K ⊂ D def p ∞ |∇ϕ| dx : ϕ ∈ C0 (D), ϕ > 1 on K ; Cap1,p (K, D) = inf D

for an open subset G ⊂ D

def Cap1,p (G, D) = sup Cap1,p (K, D) : ∀K ⊂ G, K compact ; for an arbitrary subset E ⊂ D

def Cap1,p (E, D) = inf Cap1,p (G, D) : ∀G open such that E ⊂ G ⊂ D .

6. Elements of Capacity Theory

431

(ii) A function f is said to be quasi-continuous in D if, for all ε > 0, there exists an open set Gε ⊂ D such that capD (Gε ) < ε and f is continuous on D\Gε . (iii) A set E in D is said to be quasi-open if, for all ε > 0, there exists an open set Gε ⊂ D such that capD (Gε ) < ε and E ∪ Gε is open. Clearly, by deﬁnition, ∀E ⊂ D,

Cap1,p (E) ≤ Cap1,p (E, D).

(6.3)

The number Cap1,p (E, D) ∈ [0, ∞] is called the (variational) (p, mN )-capacity of the condenser (E, G), where mN is the N -dimensional Lebesgue measure. It can easily be shown that for a quasi-open set there exists a decreasing sequence {Ωn } of open sets such that Ωn ⊃ E and capD (Ωn \E) goes to zero as n goes to inﬁnity. We say that a property holds quasi-everywhere (q.e.) in D if it holds in the complement D\E of a set E of zero capacity. A set of zero capacity has zero measure, but the converse is not true. The capacity is a countably subadditive set function, but it is not additive even for disjoint sets. Hence the union of a countable number of sets of zero capacity has zero capacity.

6.2

Quasi-continuous Representative and H 1 -Functions

The following theorem is from L. I. Hedberg [2] (sect. 2, p. 241 and Thm. 1.1, p. 237, and footnote, p. 238 referring to T. Wolﬀ and the paper by L. I. Hedberg and Th. H. Wolff [1]). See also J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.4, p. 89] Theorem 6.1. Let p, 1 < p < ∞, be a real number, N ≥ 1 be an integer, and D ⊂ RN be open. (i) Every f in W 1,p (D) has a quasi-continuous representative: there exists a quasi-continuous function f1 deﬁned on D such that f1 = f almost everywhere in D (hence f1 is a representative of f in W 1,p (D) ). Any two quasicontinuous representatives f1 and f2 of f ∈ W 1,p (D) such that f1 = f2 almost everywhere are equal quasi-everywhere in D. (ii) Let f ∈ W 1,p (RN ). Let K ⊂ RN be closed, and suppose that f |K = 0 (trace deﬁned quasi-everywhere). Then f |K ∈ W01,p (K). The following key lemma completes the picture (cf. J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.5, p. 90]). See also L. I. Hedberg [2, p. 241]. Lemma 6.1. (i) Let Ω be an open subset of RN , let p, 1 ≤ p < ∞, and consider an element u of W 1.p (Ω). Then u ∈ W01.p (Ω) if and only if there exists a quasicontinuous representative u1 of u in RN such that u1 = 0 quasi-everywhere in Ω and u1 = u almost everywhere in Ω.

432

Chapter 8. Shape Continuity and Optimization

(ii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consider an element u of W01.p (D). Then u|Ω ∈ W01.p (Ω) if and only if there exists a quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere in D\Ω and u1 = u almost everywhere in Ω. (iii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consider an element u of W 1.p (D). Then u|Ω ∈ W01.p (Ω) if and only if there exists a quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere in D\Ω and u1 = u almost everywhere in Ω. A function ϕ in W 1,p (D) is said to be zero quasi-everywhere in a subset E of D if there exists a quasi-continuous representative of ϕ which is zero quasi-everywhere in E. This makes sense since any two quasi-continuous representatives of an element ϕ of W 1,p (D) are equal quasi-everywhere. For any ϕ ∈ W01,p (D) and t ∈ R, the set {x ∈ D : ϕ(x) > t} is quasi-open. Moreover, the subspace W01,p (Ω; D) introduced in section 2.5.3 in Chapter 2 can now be characterized by the capacity. Theorem 6.2. Given two bounded open subsets Ω and D of RN , W01,p (Ω; D) = ϕ ∈ W01,p (D) : (1 − χΩ )ϕ = 0 q.e. in D , if Ω ⊂ D,

W01,p (Ω; D) = ϕ ∈ W 1,p (D) : (1 − χΩ )ϕ = 0 q.e. in D , if Ω ⊂ D.

(6.4) (6.5)

The characterization of W01,p (Ω; D) requires the notion of capacity in a very essential way. It cannot be obtained by saying that the function and its derivatives are zero almost everywhere in D\Ω. Recall the deﬁnition of the other extension H1 (Ω; D) of H 1 (Ω) to a measurable set D containing Ω (cf. Chapter 5, section 3.5, Theorem 3.6, identity (3.27)). Its deﬁnition readily extends from H 1 to W 1,p as follows: def W1,p (Ω; D) = ψ ∈ W01,p (D) : (1 − χΩ )ψ = 0 a.e. in D . By deﬁnition W01,p (Ω; D) ⊂ W1,p (Ω; D), but the two spaces are generally not equal, as can be seen from the following example. Denote by Br , r > 0, the open ball of radius r > 0 in R2 . Deﬁne Ω = B2 \∂B1 and D = B3 . The circular crack ∂B1 in Ω has zero measure but nonzero capacity. Since ∂B1 has zero measure, W1,p (Ω; D) contains functions ψ ∈ W01,p (B2 ) whose restrictions to B2 are not zero on the circle ∂B1 and hence do not belong to W01,p (Ω). Yet for Lipschitzian domains and, more generally, for domains Ω that are stable with respect to D in the sense of Deﬁnition 4.1, the two spaces are indeed equal.

6.3

Transport of Sets of Zero Capacity

Any Lipschitz continuous transformation of RN which has a Lipschitz continuous inverse transports sets of zero capacity onto sets of zero capacity. Lemma 6.2. Let D be an open subset of RN and let T be an invertible transformation of D such that both T and T −1 are Lipschitz continuous. For any E ⊂ D Cap1,2 (E, D) = 0 ⇐⇒ Cap1,2 (T (E), D) = 0.

6. Elements of Capacity Theory

433

Proof. (a) E = K, K compact in D. Observe that, in the deﬁnition of the capacity, it is not necessary to choose the functions ϕ in C0∞ (D). They can be chosen in a larger space as long as |∇ϕ|2 dx < ∞ and ϕ > 1 on K. D

So the capacity is also given by def Cap1,2 (K, D) = inf |∇ϕ|2 dx : ϕ ∈ H01 (D) ∩ C(D), ϕ > 1 on K . D

But the space

H01 (D)

∩ C(D) is stable under the action of T :

ϕ ∈ H01 (D) ∩ C(D) ⇐⇒ T ◦ ϕ ∈ H01 (D) ∩ C(D). Consider the matrix A(x) = | det(DT (x))| ∗DT (x)( ∗DT (x))−1 . def

By assumption on T , the elements of the matrix A belong to L∞ (D) and αI ≤ A(x) ≤ α−1 I a.e. in D.

∃α > 0,

Deﬁne E(K) = {ϕ ∈ H01 (D) ∩ C(D) : ϕ > 1 on K}. For any ϕ ∈ E(T (K)), ϕ ◦ T ∈ E(K) and |∇ϕ|2 dx = A ∇(ϕ ◦ T ) · ∇(ϕ ◦ T ) dx ≥ α |∇(ϕ ◦ T )|2 dx. D

D

D

Let K be a compact subset such that Cap1,2 (K, D) = 0. For each ε > 0 there exists ϕ ∈ E(K) such that 2 |∇ϕ| dx ≤ ε ⇒ ∀ε, |∇(ϕ ◦ T )|2 dx ≤ ε/α or ϕ ◦ T ∈ E(T −1 (K)) D

D

⇒ ∀ε,

Cap1,2 (T −1 (K), D) ≤ ε/α

⇒ Cap1,2 (T −1 (K), D) = 0

and we can repeat the proof with T −1 in place of T . (b) E = G, G open. Let

Cap1,2 (G, D) = sup Cap1,2 (K, D) : ∀K ⊂ G, K compact = 0. Therefore capD (K) = 0, which implies that Cap1,2 (T (K), D) = 0 and hence

Cap1,2 (T (G), D) = sup Cap1,2 (K , D) : ∀K ⊂ T (G), K compact

= sup Cap1,2 (T (K), D) : ∀K ⊂ G, K compact = 0. (c) General case. Let def

Cap1,2 (E, D) = inf {capD (G) : ∀G ⊃ E, G open} .

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Chapter 8. Shape Continuity and Optimization

Therefore, for all ε > 0, there exists G ⊃ E open such that Cap1,2 (E, D) ≤ Cap1,2 (G, D) ≤ ε

⇒ Cap1,2 (T (E), D) ≤ Cap1,2 (T (G), D)

since T (E) ⊂ T (G) is open. By deﬁnition of Cap1,2 (G, D) we have ∀K ⊂ G,

Cap1,2 (K, D) ≤ ε

⇒ Cap1,2 (T (K), D) ≤ αε

⇒ Cap1,2 (T (G), D) = sup{Cap1,2 (T (K), D) : K ⊂ G} ≤ αε. Finally, for all ε > 0, Cap1,2 (T (E), D) ≤ αε and hence Cap1,2 (T (E), D) = 0.

7

Crack-Free Sets and Some Applications

In this section we show that the result for the second question raised in the introduction to this chapter is true for measurable sets in the following large family.

7.1

Deﬁnitions and Properties

Deﬁnition 7.1. (i) The interior boundary of a set Ω is deﬁned as def

∂i Ω = ∂Ω\∂Ω.

(7.1)

The exterior boundary of a set Ω is deﬁned as def

∂e Ω = ∂Ω\∂Ω.

(7.2)

(ii) A subset Ω of RN is said to be crack-free if ∂i Ω = ∅; its complement Ω is said to be crack-free if ∂e Ω = ∅. The intuitive terminology crack-free arises from the following identities: ∂i Ω = ∂Ω\∂Ω = int Ω\int Ω,

∂e Ω = ∂Ω\∂Ω = int Ω\int Ω.

(7.3)

A crack-free set Ω does not have internal cracks that would disappear after the cleaning operation with respect to the closure of the set2 . It is veriﬁed for sets that 2 Since ∂Ω = Ω ∩ Ω ⊂ Ω ∩ Ω = ∂Ω, the deﬁnition of a crack-free set Ω is equivalent to the property ∂Ω = ∂Ω. In 1994 A. Henrot [2] introduced the terminology Carath´ eodory set for such ¨ki, J. Sprekels, and D. Tiba [1], a set that was later adopted by D. Tiba [3], P. Neittaanma and A. Henrot and M. Pierre [4]. However, this terminology does not seem to be standard. For instance, in the literature on polynomial approximations in the complex plane C, a Carath´ eodory set is deﬁned as follows.

Deﬁnition 7.2 (O. Dovgoshey [1] or D. Gaier [1]). A bounded subset Ω of C is said to be a Carath´ eodory set if the boundary of Ω coincides with the boundary of the unbounded component of the complement of Ω. A Carath´ eodory domain is a Carath´ eodory set if, in addition, Ω is simply connected. In V. A. Martirosian and S. E. Mkrtchyan [1], Ω is further assumed to be measurable. This deﬁnition excludes not only interior cracks but also bounded holes inside the set Ω as can be seen from the example of the annulus Ω = {x ∈ R2 : 1 < |x| < 2} in R2 . So, it is more restrictive than ∂Ω = ∂Ω. In order to avoid any ambiguity, we choose to keep the intuitive terminology crack-free.

7. Crack-Free Sets and Some Applications

435

are locally the epigraph of a C 0 -function. Yet, it is also veriﬁed for sets that are not locally the epigraph of a C 0 -function. The following equivalent characterization of a crack-free set in terms of the oriented distance function was used as its deﬁnition in M. C. Delfour and J.´sio [43]. P. Zole Theorem 7.1. A subset Ω of RN is crack-free if and only if bΩ = bΩ . Its complement Ω is crack-free if and only if bΩ = bΩ (or, equivalently, bΩ = bint Ω ). Proof. If ∂Ω = ∂Ω, then |bΩ | = d∂Ω = d∂Ω = |bΩ |. Hence, dΩ + dΩ = |bΩ | = |bΩ | = dΩ + dΩ implies dΩ = dΩ and bΩ = bΩ . Conversely, if bΩ = bΩ , then d∂Ω = |bΩ | = |bΩ | = d∂Ω and ∂Ω = ∂Ω since both sets are closed. We also have the following equivalences: bΩ = bint Ω ⇐⇒ bΩ = bΩ ⇐⇒ Ω = Ω ⇐⇒ ∂Ω = ∂Ω ⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int Ω = int Ω

(7.4)

and bΩ = bint Ω ⇐⇒ bΩ = bΩ ⇐⇒ Ω = Ω ⇐⇒ ∂Ω = ∂Ω ⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int Ω = int Ω.

(7.5)

Remark 7.1. An open set Ω is crack-free if and only if Ω = int Ω. Indeed, by deﬁnition, Ω is crackfree if and only if Ω = Ω. But since Ω is open Ω = Ω and Ω = Ω = int Ω. We give the following general technical theorem and its corollary. ´sio [43]). Let K be a compact Theorem 7.2 (M. C. Delfour and J.-P. Zole subset of RN and let v ∈ W 1,p (RN ). Then v = 0 almost everywhere on K implies that v has a quasi-continuous representative v ∗ such that v ∗ |int K ∈ W01,p (int K). Corollary 1. Let Ω be a bounded set in RN and let v ∈ W 1,p (RN ). Then v = 0 almost everywhere on Ω implies that v has a quasi-continuous representative v ∗ such that v ∗ |int Ω ∈ W01,p (int Ω). Proof. Choose the quasi-continuous representative 1 def v dy v ∗ (x) = lim δ0 |B(x, δ)| B(x,δ)

(7.6)

of v and let {Gε } be the associated family of open subsets of (1, p)-capacity less than ε of Deﬁnition 6.1 (ii). (a) If there exists ε > 0 such that Gε ⊂ int K, then K ⊂ Gε and v ∗ is continuous in K. But, by assumption, v = 0 almost everywhere in the open set K. Hence, by deﬁnition of v ∗ , v ∗ = 0 everywhere in the open subset K ⊂ K, and, by continuity, v ∗ = 0 everywhere in K.

436

Chapter 8. Shape Continuity and Optimization

(b) If for all ε > 0, Gε ∩ K = ∅, then either x ∈ ∪ε>0 Gε ∩ K or x ∈ ∩ε>0 Gε ∩ ∂K, since K = int K ∪ ∂K = K ∪ ∂K. In the ﬁrst case, v ∗ (x) = 0 since x ∈ K. In the second case, Capp (∩ε>0 Gε ∩ ∂K) ≤ Capp (Gε ) < ε and ∩ε>0 Gε ∩ ∂K has zero (1, p)-capacity. From (a) and (b), v ∗ = 0 quasi-everywhere in K. Therefore by Theorem 5 in L. I. Hedberg and Th. H. Wolff [1], which generalizes Theorem 1.1 in L. I. Hedberg [2], v ∗ |int K ∈ W01,p (int K). Remark 7.2. In L. I. Hedberg [2, 3] a compact set K is said to be (1, p)-stable if v = 0 quasieverywhere on K implies that v|int K ∈ W01,p (int K). A suﬃcient condition (cf. Theorem 6.4 in L. I. Hedberg [2]) for the (1, p)-stability of K is that K be (1, p)-thick (1, p) quasi-everywhere on ∂K. In the proof of Theorem 7.2 we have shown that, under the weaker suﬃcient condition v = 0 almost everywhere on K, v = 0 quasi-everywhere on K. However, this condition would not be suﬃcient for the (2, p)-stability as shown in the example of L. I. Hedberg [3] (cf. Thm. 3.14, p. 100). Remark 7.3. There is also a result in dimension N = 2 in Theorem 7 of L. I. Hedberg and Th. H. Wolff [1]: for p, 2 ≤ p < ∞, and q, q −1 + p−1 = 1, K is (1, q)stable if and only if Capq (∂K ∩ e1,q (K)) = 0, where e1,q (K) = {x ∈ RN : K is (1, q)-thin at x}. It says that the part of ∂K where K is thin has zero (1, q)capacity. However, in the equivalence of parts (b) and (d) of Theorem 7, there seems to be a typo: the (1, q)-stability is deﬁned as follows: “K is (1, q)-stable if v = 0 q.e. on K implies that v ∈ W01,q (K).” It should probably read “v ∈ W01,q (int K).” We now turn to our speciﬁc problem. Given a measurable subset Ω of a bounded open holdall D ⊂ RN , deﬁne the closed subspace def W1,p (Ω; D) = ψ ∈ W01,p (D) : (1 − χΩ )ψ = 0 a.e. (7.7) of W01,p (D). This space provides a relaxation of the Dirichlet boundary value problem to domains that are only measurable. The next theorem will show that in fact W1,p (Ω; D) can be embedded in W01,p (int Ω). Furthermore, for crack-free measurable subsets Ω of D, W1,p (Ω; D) coincides with W01,p (int Ω). This will yield the continuity of the homogeneous Dirichlet problem with respect to the open domains Ω in the family L(D, r, O, λ) of subsets of D verifying a uniform fat segment property. Theorem 7.3. Let D be bounded open in RN . (i) For any measurable subset Ω ⊂ D such that int Ω = ∅, ∃ a quasi-continuous representative v ∗ 1,p v ∈ W (Ω; D) =⇒ 1,p such that v ∗ | int Ω ∈ W0 (int Ω) and W01,p (int Ω; D) ⊂ W1,p (Ω; D) ⊂ W01,p (int Ω; D).

(7.8)

7. Crack-Free Sets and Some Applications (ii) For any measurable subset Ω ⊂ D such that int Ω = ∅, ∃ a quasi-continuous representative v ∗ v ∈ W 1,p (Ω; D) =⇒ such that v ∗ |int Ω ∈ W01,p (int Ω) and Ω crack-free

437

(7.9)

and W01,p (int Ω; D) = W1,p (Ω; D) = W01,p (int Ω; D). Proof. (i) From Theorem 7.2 with K = Ω and v˜, the extension by zero of v from D to RN . Given ϕ ∈ W01,p (int Ω), the zero extension e0 ϕ ∈ W01,p (int Ω; D) and (1 − χint Ω )ϕ = 0 almost everywhere in D. This implies that (1 − χΩ )ϕ = 0 almost everywhere in D and ϕ ∈ W1,p (Ω; D). (ii) From (i) and the equivalences (7.4): int Ω = int Ω. Remark 7.4. The condition that v ∈ W1,p (Ω; D) is weaker than the capacitary condition of ´sio [12] and hence its conclusion that v ∗ ∈ W01,p (int Ω) D. Bucur and J.-P. Zole is also weaker. It gives information only about the function on ∂Ω rather than on the whole ∂Ω. The “crack-free” property of Deﬁnition 7.1 is purely geometric and even set theoretic. It is diﬀerent from and probably stronger than the capacitary ´sio [12]. Instead of forcing Ω to be condition of D. Bucur and J.-P. Zole “uniformly thick” at quasi all points of the boundary ∂Ω in order to preserve the zero Dirichlet condition on the whole boundary, it erases the “interior boundary” ∂Ω\∂Ω. Its advantage is that it does not require an a priori knowledge of the diﬀerential operator and does not necessitate the Maximum Principle or harmonic functions; its disadvantage is that it cannot handle problems where the trace of the function on ∂Ω\∂Ω is an important feature of the problem at hand. Yet, it should readily be extendable to Sobolev spaces on C 1,1 -submanifolds of codimension 1 to deal with the Laplace–Beltrami or shell equations.

7.2 7.2.1

Continuity and Optimization over L(D, r, O, λ) Continuity of the Classical Homogeneous Dirichlet Boundary Condition

In light of Theorem 4.1, it remains to ﬁnd families of sets with properties that would enable us to sharpen part (i) or that would verify the stability condition H01 (Ω; D) = H1 (Ω; D). In the second case, the family of subsets of a bounded holdall D that satisfy the uniform fat segment property meets all the requirements. Theorem 7.4. Let D be a bounded open nonempty subset in RN and Ω be open subsets of D. Denote by y(Ω) ∈ H01 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map Ω → y(Ω) : L(D, r, O, λ) → H01 (D)-strong

(7.10)

is continuous with respect to any of the topologies on L(D, r, O, λ) of Theorem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p (D), dΩ in C(D) and W 1,p (D), bΩ , C(D) in C(D) and W 1,p (D), χΩ in Lp (D), and χΩ in Lp (D).

438

Chapter 8. Shape Continuity and Optimization

Proof. Since all topologies are equivalent on L(D, r, O, λ), consider a domain Ω and a sequence {Ωn } such that dΩn → dΩ in W 1,p (D) for some p, 1 ≤ p < ∞. By Theorem 6.1 (ii) of Chapter 2, all sets in L(D, r, O, λ) are crack-free since int Ω = ∅,

Ω = int Ω = Ω

⇒ b Ω = bΩ .

By Theorem 7.3 (ii), H1 (Ω; D) = H01 (Ω; D). Finally, from part (ii) of Theorem 4.1, we get the continuity of the solutions of the classical Dirichlet problem. 7.2.2

Minimization/Maximization of the First Eigenvalue

Consider the ﬁrst eigenvalue λA (Ω) introduced in (2.1) A∇ϕ · ∇ϕ dx def A D λ (Ω) = inf1 |ϕ|2 dx 0=ϕ∈H0 (Ω;D) D

(7.11)

for the Laplace equation on the open Ω of D. It was shown in Theorem 2.2 that λA (Ω) is an upper semicontinuous function of Ω and that there is a solution to the maximization problem with respect to Ω. We now turn to the minimization of λA (Ω) with respect to Ω that requires the lower semicontinuity and hence the continuity of λA (Ω) with respect to Ω. To do that we choose the smaller family L(D, r, O, λ) of sets verifying a uniform fat segment property. Theorem 7.5. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). (i) The mapping dΩ → λA (Ω) : L(D, r, O, λ) → R ∪{+∞}

(7.12)

is continuous with respect to any of the topologies on L(D, r, O, λ) of Theorem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p (D), dΩ in C(D) and W 1,p (D), bΩ , C(D) in C(D) and W 1,p (D), χΩ in Lp (D), and χΩ in Lp (D). (ii) The minimization and maximization problems inf

λ(Ω) and

Ω∈L(D,r,O,λ)

sup

λ(Ω)

(7.13)

Ω∈L(D,r,O,λ)

have solutions in L(D, r, O, λ). Moreover, 0 < λA (D) ≤

inf Ω∈L(D,r,O,λ)

λ(Ω) ≤

sup

λ(Ω) ≤ β λI (O).

(7.14)

Ω∈L(D,r,O,λ)

Proof. (i) Continuity. Go back to the proof of Theorem 2.2. Given Ω ∈ L(D, r, O, λ), consider a sequence {Ωn } in L(D, r, O, λ) such that dΩn → dΩ in C(D). Let λA (Ωn ) and un ∈ H01 (Ωn ; D), un L2 (D) = 1, be the eigenvalue and an eigenvector associated with Ωn . By deﬁnition of L(D, r, O, λ), for all x ∈ ∂Ω, there exists Ax ∈ O(N) such that for all y ∈ B(x, r) ∩ Ω, y + Ax O ⊂ Ω. Fix x0 ∈ ∂Ω. There exist z ∈ B(x0 , r) ∩ Ω

7. Crack-Free Sets and Some Applications

439

and ε > 0 such that B(z, ε) ⊂ B(x0 , r) ∩ Ω. As a result B(z, ε) + Ax O ⊂ Ω and z + Ax O ⊂ Ω. By the compactivorous property, there exists N such that, for all n > N , z + Ax O ⊂ Ωn and ∀n > N,

λn = λA (Ωn ) ≤ λA (z + Ax O) ≤ βλI (z + Ax O) = βλI (O)

since the eigenvalue is independent of a translation and a rotation of the set. This means that the sequence {λn } of eigenvalues is bounded. By coercivity α∇un }2L2 (D) ≤ λn un 2L2 (D) = λn and {un } is bounded in H01 (D). There exist λ0 and u0 ∈ H1 (Ω; D) and subsequences, still indexed by n, such that λn → λ 0 ,

un L2 (D) = 1,

⇒ u0 L2 (D) = 1.

un u0 in H01 (D)-weak

But for all Ω ∈ L(D, r, O, λ) we have H1 (Ω; D) = H01 (Ω; D) and u0 ∈ H01 (Ω; D). For any ϕ ∈ D(Ω), K = supp ϕ ⊂ Ω is compact and there exists N (K) such that for all n ≥ N (K), supp ϕ ⊂ Ωn . Therefore for all n ≥ N (K) A∇un · ∇ϕ dx = λn un ϕ dx. un ∈ H01 (Ωn ; D), D

By letting n → ∞

D

∀ϕ ∈ D(Ω),

A∇u · ∇ϕ dx = λ0 D

u ϕ dx, D

and, by density of D(Ω) in H01 (Ω; D), the variational equation is veriﬁed in H01 (Ω; D). Hence, u0 ∈ H01 (Ω; D), uL2 (Ω) = 1, satisﬁes the variational equation A∇u0 · ∇ϕ dx = λ0 u ϕ dx. ∀ϕ ∈ H01 (Ω; D), D

D

Since λA (Ω) is the smallest eigenvalue with respect to H01 (Ω; D), λA (Ω) ≤ λ0 . To show that λ0 = λA (Ω), it remains to prove that λ0 ≤ λA (Ω). For each n A∇ϕn · ∇ϕn dx 1 A . ∀ϕn ∈ H0 (Ωn ; D), λ (Ωn ) ≤ D ϕn 2L2 (D) By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N such that supp ϕ ⊂ Ωn and ϕ ∈ D(Ωn ). Therefore for all n > N A∇ϕ · ∇ϕ dx A λ (Ωn ) ≤ D ϕ2L2 (D) and by going to the limit ∀ϕ ∈ D(Ω),

λ0 ≤

D

A∇ϕ · ∇ϕ dx . ϕ2L2 (D

440

Chapter 8. Shape Continuity and Optimization

By density this is true for all ϕ ∈ H01 (Ω; D), ϕ = 0, and λ0 ≤ λA (Ω). As a result λ0 = λA (Ω). Therefore the whole sequence {λA (Ωn )} converges to λA (Ω) and we have the continuity of the map dΩ → λA (Ω). The strong continuity in H01 (D) now follows from the convergence of un → u0 in L2 (D)-strong: α∇(un − u0 )2L2 (D) ≤ A∇(un − u0 ) · ∇(un − u0 ) dx D A∇un · ∇un dx + A∇u0 · ∇u0 dx − 2 A∇un · ∇u0 dx = D D D un · un dx + λ0 u0 · u0 dx − 2 A∇un · ∇u0 dx = λn D D D u0 · u0 dx − 2 A∇u0 · ∇u0 dx = 0. → 2λ0 D

D

(ii) This follows from the continuity and the compactness of L(D, r, O, λ) in all the mentioned topologies. The inequalities have been proven in part (i). Corollary 1. Under the assumptions of Theorem 7.5, the maximization and minimization problems (7.13) also have solutions in L(D, r, O, λ) under an equality or an inequality constraint of the form m(Ω) = α, m(Ω) ≤ α, or m(Ω) ≥ α provided that there exists Ω ∈ L(D, r, O, λ) such that m(Ω ) = α, m(Ω ) ≤ α, or m(Ω ) ≥ α. Proof. This follows from Theorem 13.1 and its Corollaries 1 and 2 of section 13 in Chapter 7, where it is shown that the convergence is strong not only in the Hausdorﬀ topologies but also for the associated characteristic functions in L1 (D), thus making the volume function continuous with respect to open domains in L(D, r, ω, λ). For the spaces of convex subsets recall from Theorem 8.6 of Chapter 6 and Theorem 10.2 of Chapter 7 that the compactness remains true in the W 1,1 (D)-topology.

8

Continuity under Capacity Constraints

We have established the continuity of the homogeneous Dirichlet problem, that is, the continuity of the function dΩ → u(Ω) : Cdc (D) → H01 (D)-strong for the family of domains verifying a uniform fat segment property. Domains with cracks lie between families of L(D, r, O, λ) type and Cdc (D). Topologies using the characteristic function χΩ are excluded since they do not see interior boundaries of zero measure. Families that contains sets with cracks must be chosen in such a way that the homogeneous Dirichlet boundary condition on the sequence of solutions u(Ωn ) in the converging sets Ωn is preserved by the solution u(Ω) on the boundary of the limit domain Ω in section 4. We have to be careful since the relative capacity of the

8. Continuity under Capacity Constraints

441

complement of the domains near the boundary must not vanish. In order to handle this point, we introduce the following concepts and terminology related to the local capacity of the complement near the boundary points. The following deﬁnition is due to J. Heinonen, T. Kilpelainen, and O. Martio [1, pp. 114–115]. Deﬁnition 8.1. (i) Given r > 0, a set E ⊂ RN , and a point x, the function def

capx,r (E) =

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

(8.1)

is called the capacity density function. (ii) We say that a set E is thick at a point x if

1

0

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

1/(p−1)

dr = ∞. r

(8.2)

This is called the Wiener criterion. (iii) We say that an open set Ω satisﬁes the Wiener density condition if ∀x ∈ ∂Ω, 0

1

Capp (Ω ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

1/(p−1)

dr = ∞. r

(8.3)

This means that the set Ω is thick at every boundary point x ∈ ∂Ω (that is, Ω has “suﬃcient” capacity around each boundary point). Remark 8.1. If Ω satisﬁes the (r, c)-capacity density condition, the complement is thick in any point of the boundary. The Wiener criterion can be strengthened by introducing the following condition (J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 127]). Deﬁnition 8.2. (i) Given r > 0, c > 0 , and an open set Ω, Ω is said to satisfy the (r, c)-capacity density condition if ∀x ∈ ∂Ω, ∀r , 0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

≥ c.

(8.4)

We also say that Ω is uniformly (r, c)-thick. (ii) Given r > 0, c > 0 , and an open set Ω, Ω is said to satisfy the strong (r, c)-capacity density condition if ∀x ∈ ∂Ω, ∀r , 0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) ≥ c. Capp (B(x, r ), B(x, 2r ))

We also say that Ω is strongly uniformly (r, c)-thick.

(8.5)

442

Chapter 8. Shape Continuity and Optimization

(iii) For r, 0 < r < 1, and c > 0 deﬁne the following family of open subsets of D: & Ω satisﬁes the strong def . (8.6) Oc,r (D) = Ω ⊂ D : (r, c)-capacity density condition

Example 8.1. Only a few capacities can be computed explicitly. The computation of the spherical condensers Capp (B(x, r), B(x, R)) and Capp (B(x, r), B(x, R)) leads to some simpliﬁcations in the denominator of the above expression (cf. J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 35]). For 0 < r < R < ∞ Capp (B(x, r), B(x, R)) = Capp (B(x, r), B(x, R)

(8.7)

and for p > 1 p−1 p−N 1−p |N − p| p−N N −1 , R p−1 − r p−1 ω p−1 Capp (B(x, r), B(x, R)) = 1−N R N −1 ω , log r

p = N, p = N,

where ω N −1 is the volume (Lebesgue measure) of the ball in RN −1 . In particular Capp (B(x, r), B(x, 2r)) = c1 (N, p) rN −p , N

Capp (B(x, r)) = Capp (B(x, r), R ) = c2 (N, p) r Capp ({x0 }, R ) = c3 (N, p) r N

N −p

1 < p, N −p

,

1 < p < N,

,

(8.8)

p < N,

for constants ci (N, p) that depend only on p and N . Remark 8.2. Deﬁnition 8.2 (i) uses a slightly diﬀerent deﬁnition of the previous capacity density function of Deﬁnition 8.1: def

capx,r (E) = instead of def

capx,r (E) =

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r)) Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

.

Note that Capp (B(x, r), B(x, 2r)) = Capp (B(x, r), B(x, 2r)) = c1 (N, p) rN −p ,

1 < p,

and Capp (E ∩ B(x, r), B(x, 2r)) ≤ Capp (E ∩ B(x, r), B(x, 2r)) imply that ∀r > 0, ∀x,

capx,r (E) ≥ capx,r (E)

8. Continuity under Capacity Constraints

443

and the strong (r, c)-capacity density condition implies the (r, c)-capacity density condition. But for an open subset Ω of the bounded open subset D of RN , the following two conditions are equivalent: ∀x ∈ ∂Ω, ∀ 0 < r < r, and ∀x ∈ ∂Ω, ∀0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) ≥c Capp (B(x, r ), B(x, 2r ))

Capp (Ω ∩ B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

≥ c,

(8.9)

(8.10)

and also from the Oc,r (D) family viewpoint the two deﬁnitions are equivalent. It is clear that (8.9) implies (8.10), but the converse is not obvious. Recall Theorem 4.1 and the notation un for the solution of (4.5) in H01 (Ωn ; D) and u ∈ H1 (Ω; D) for the weak limit in H01 (D) which satisﬁes (4.8). The following theorem gives the main continuity result. Theorem 8.1. Let D be a bounded open nonempty subset of RN of class C 2 for N ≥ 3 and Ω be an open subset of D. Assume that A is a matrix function that satisﬁes the conditions (4.4) and that the elements of A belong to C 1 (D). Denote by u(Ω) ∈ H01 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map Ω → u(Ω) : Oc,r (D) → H01 (D)-strong

(8.11)

is continuous with respect to the uniform topology C(D) on Cdc (D). Proof. When the dimension N of the space is equal to 1, H 1 (D) ⊂ C(D) and no approximation of f is necessary. When N ≥ 2, we ﬁrst prove the result for f ∈ H s (D), s > N/2 − 2 (since for D of class C 2 the corresponding solution will belong to H s+2 (D) ⊂ C(D)), and then, by approximation of f , we prove it for f ∈ H −1 (D). (i) First consider f ∈ H s (D), s > N/2−2, for N ≥ 2 (s = −1 for N = 1). The proof will make use of the following results from J. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 3, Thm. 3.70, p. 80]. Lemma 8.1. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisﬁed. Let v be an A-harmonic function; that is, v ∈ 1,p Wloc (Ω) is a weak solution of equation A∇v(Ω) · ∇ϕ dx = 0 (8.12) ∀ϕ ∈ D(Ω), D

in the open set Ω. Then there exists a continuous function u on Ω such that v(Ω) = u almost everywhere. Note that the continuous representative from Lemma 8.1 is in fact a quasicontinuous H01 (Ω) representative. Indeed, let v1 = v almost everywhere and let v1 be continuous on Ω. We want to show that v1 is a quasi-continuous representative of v. There exists a quasi-continuous representative v2 of v, which is equal to v almost

444

Chapter 8. Shape Continuity and Optimization

everywhere. So v1 is continuous, v2 is quasi-continuous, and v1 = v2 almost everywhere. Using J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.12], we get v1 = v2 quasi-everywhere. Always from J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 6.27, p. 122] we get the following lemma. Lemma 8.2. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisﬁed. Let Ω belong to Oc,r (D). If θ ∈ W 1,p (Ω) ∩ C(Ω) and if h is an A-harmonic function in Ω such that h − θ ∈ W01,p (Ω), then ∀x0 ∈ ∂Ω,

lim h(x) = θ(x0 ).

x→x0

Note that the fact that Ω belongs to Oc,r (D) involves the notion of thickness in any poi

Editor-in-Chief Ralph C. Smith, North Carolina State University

Editorial Board Athanasios C. Antoulas, Rice University Siva Banda, Air Force Research Laboratory Belinda A. Batten, Oregon State University John Betts, The Boeing Company (retired) Stephen L. Campbell, North Carolina State University Michel C. Delfour, University of Montreal Max D. Gunzburger, Florida State University J. William Helton, University of California, San Diego Arthur J. Krener, University of California, Davis Kirsten Morris, University of Waterloo Richard Murray, California Institute of Technology Ekkehard Sachs, University of Trier

Series Volumes Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second Edition Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical Approaches Speyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and Control Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design with MATLAB Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and Computation Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue- Based Approach ¸ Adaptive Control Tutorial Ioannou, Petros and Fidan, Barıs, Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming for Optimization of Dynamical Systems Huang, J., Nonlinear Output Regulation: Theory and Applications Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems Gunzburger, Max D., Perspectives in Flow Control and Optimization Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control Helton, J. William and James, Matthew R., Extending H1 Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives

Shapes and Geometries

Metrics, Analysis, Differential Calculus, and Optimization Second Edition

M. C. Delfour Université de Montréal Montréal, Québec Canada

J.-P. Zolésio

National Center for Scientific Research (CNRS) and National Institute for Research in Computer Science and Control (INRIA) Sophia Antipolis France

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2011 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. The research of the first author was supported by the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the National Sciences and Engineering Research Council of Canada; and the FQRNT program of the Ministère de l’Éducation du Québec. Library of Congress Cataloging-in-Publication Data Delfour, Michel C., 1943Shapes and geometries : metrics, analysis, differential calculus, and optimization / M. C. Delfour, J.-P. Zolésio. -- 2nd ed. p. cm. Includes bibliographical references and index. ISBN 978-0-898719-36-9 (hardcover : alk. paper) 1. Shape theory (Topology) I. Zolésio, J.-P. II. Title. QA612.7.D45 2011 514’.24--dc22 2010028846

is a registered trademark.

j

This book is dedicated to Alice, Jeanne, Jean, and Roger

Contents List of Figures Preface 1 Objectives and Scope of the Book . 2 Overview of the Second Edition . . 3 Intended Audience . . . . . . . . . 4 Acknowledgments . . . . . . . . . .

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xix . xix . xx . xxii . xxiii

1 Introduction: Examples, Background, and Perspectives 1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Geometry as a Variable . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of the Introductory Chapter . . . . . . . . . . . . . . 2 A Simple One-Dimensional Example . . . . . . . . . . . . . . . . . . 3 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Optimal Triangular Meshing . . . . . . . . . . . . . . . . . . . . . . . 6 Modeling Free Boundary Problems . . . . . . . . . . . . . . . . . . . 6.1 Free Interface between Two Materials . . . . . . . . . . . . . 6.2 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 7 Design of a Thermal Diﬀuser . . . . . . . . . . . . . . . . . . . . . . 7.1 Description of the Physical Problem . . . . . . . . . . . . . . 7.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 7.3 Reformulation of the Problem . . . . . . . . . . . . . . . . . . 7.4 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 7.5 Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Design of a Thermal Radiator . . . . . . . . . . . . . . . . . . . . . . 8.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . 8.2 Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . . 9 A Glimpse into Segmentation of Images . . . . . . . . . . . . . . . . 9.1 Automatic Image Processing . . . . . . . . . . . . . . . . . . 9.2 Image Smoothing/Filtering by Convolution and Edge Detectors 9.2.1 Construction of the Convolution of I . . . . . . . . 9.2.2 Space-Frequency Uncertainty Relationship . . . . . 9.2.3 Laplacian Detector . . . . . . . . . . . . . . . . . . . vii

1 1 1 3 3 4 6 7 10 11 12 13 13 14 16 16 17 18 18 20 21 21 22 23 23 25

viii

Contents 9.3

10

11

Objective Functions Deﬁned on the Whole Edge . . . . . . . 9.3.1 Eulerian Shape Semiderivative . . . . . . . . . . . . 9.3.2 From Local to Global Conditions on the Edge . . . 9.4 Snakes, Geodesic Active Contours, and Level Sets . . . . . . 9.4.1 Objective Functions Deﬁned on the Contours . . . . 9.4.2 Snakes and Geodesic Active Contours . . . . . . . . 9.4.3 Level Set Method . . . . . . . . . . . . . . . . . . . 9.4.4 Velocity Carried by the Normal . . . . . . . . . . . 9.4.5 Extension of the Level Set Equations . . . . . . . . 9.5 Objective Function Deﬁned on the Whole Image . . . . . . . 9.5.1 Tikhonov Regularization/Smoothing . . . . . . . . . 9.5.2 Objective Function of Mumford and Shah . . . . . . 9.5.3 Relaxation of the (N − 1)-Hausdorﬀ Measure . . . . 9.5.4 Relaxation to BV-, H s -, and SBV-Functions . . . . 9.5.5 Cracked Sets and Density Perimeter . . . . . . . . . Shapes and Geometries: Background and Perspectives . . . . . . . . 10.1 Parametrize Geometries by Functions or Functions by Geometries? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Shape Analysis in Mechanics and Mathematics . . . . . . . . 10.3 Characteristic Functions: Surface Measure and Geometric Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Distance Functions: Smoothness, Normal, and Curvatures . . 10.5 Shape Optimization: Compliance Analysis and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Shape Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Shape Calculus and Tangential Diﬀerential Calculus . . . . . 10.8 Shape Analysis in This Book . . . . . . . . . . . . . . . . . . Shapes and Geometries: Second Edition . . . . . . . . . . . . . . . . 11.1 Geometries Parametrized by Functions . . . . . . . . . . . . . 11.2 Functions Parametrized by Geometries . . . . . . . . . . . . . 11.3 Shape Continuity and Optimization . . . . . . . . . . . . . . 11.4 Derivatives, Shape and Tangential Diﬀerential Calculuses, and Derivatives under State Constraints . . . . . . . . . . . . . .

2 Classical Descriptions of Geometries and Their Properties 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation and Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Abelian Group Structures on Subsets of a Fixed Holdall D . 2.2.1 First Abelian Group Structure on (P(D), ) . . . . 2.2.2 Second Abelian Group Structure on (P(D), ) . . . 2.3 Connected Space, Path-Connected Space, and Geodesic Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Bouligand’s Contingent Cone, Dual Cone, and Normal Cone 2.5 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . .

26 26 27 28 28 28 29 30 31 32 32 32 33 33 35 36 36 39 41 41 43 44 46 46 47 48 50 52 53 55 55 56 56 56 57 58 58 59 60 60

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2.6

3

4 5

Sets 3.1 3.2

3.3 Sets Sets 5.1 5.2 5.3 5.4

6

Sets 6.1 6.2 6.3 6.4

6.5

2.5.2 The Space W0m,p (Ω) . . . . . . . . . . . . . . . . . . 61 2.5.3 Embedding of H01 (Ω) into H01 (D) . . . . . . . . . . 62 2.5.4 Projection Operator . . . . . . . . . . . . . . . . . . 63 Spaces of Continuous and Diﬀerentiable Functions . . . . . . 63 2.6.1 Continuous and C k Functions . . . . . . . . . . . . 63 2.6.2 H¨ older (C 0, ) and Lipschitz (C 0,1 ) Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6.3 Embedding Theorem . . . . . . . . . . . . . . . . . 65 2.6.4 Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to Path-Connected Domains via the Geodesic Distance 66 Locally Described by an Homeomorphism or a Diﬀeomorphism 67 Sets of Classes C k and C k, . . . . . . . . . . . . . . . . . . . 67 Boundary Integral, Canonical Density, and Hausdorﬀ Measures 70 3.2.1 Boundary Integral for Sets of Class C 1 . . . . . . . 70 3.2.2 Integral on Submanifolds . . . . . . . . . . . . . . . 71 3.2.3 Hausdorﬀ Measures . . . . . . . . . . . . . . . . . . 72 Fundamental Forms and Principal Curvatures . . . . . . . . . 73 Globally Described by the Level Sets of a Function . . . . . . . 75 Locally Described by the Epigraph of a Function . . . . . . . . 78 Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs and the Space H of Dominating Functions . . . . . . . . . . . 79 olderian/Lipschitzian Sets . . . . 87 Local C k, -Epigraphs and H¨ Local C k, -Epigraphs and Sets of Class C k, . . . . . . . . . . 89 Locally Lipschitzian Sets: Some Examples and Properties . . 92 5.4.1 Examples and Continuous Linear Extensions . . . . 92 5.4.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . 93 5.4.3 Boundary Measure and Integral for Lipschitzian Sets 94 5.4.4 Geodesic Distance in a Domain and in Its Boundary 97 5.4.5 Nonhomogeneous Neumann and Dirichlet Problems 100 Locally Described by a Geometric Property . . . . . . . . . . . 101 Deﬁnitions and Main Results . . . . . . . . . . . . . . . . . . 102 Equivalence of Geometric Segment and C 0 Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Equivalence of the Uniform Fat Segment and the Equi-C 0 Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . 109 Uniform Cone/Cusp Properties and H¨ olderian/Lipschitzian Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.1 Uniform Cone Property and Lipschitzian Sets . . . 114 6.4.2 Uniform Cusp Property and H¨ olderian Sets . . . . . 115 Hausdorﬀ Measure and Dimension of the Boundary . . . . . 116

3 Courant Metrics on Images of a Set 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Generic Constructions of Micheletti . . . . . . . . . . . . . 2.1 Space F(Θ) of Transformations of RN . . . . . . . 2.2 Diﬀeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

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123 123 124 124 136

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Contents Closed Subgroups G . . . . . . . . . . . . . . . . . . . . . . . Courant Metric on the Quotient Group F(Θ)/G . . . . . . . Assumptions for B k (RN , RN ), C k (RN , RN ), and C0k (RN , RN ) 2.5.1 Checking the Assumptions . . . . . . . . . . . . . . 2.5.2 Perturbations of the Identity and Tangent Space . . 2.6 Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) . . . . . . 2.6.1 Checking the Assumptions . . . . . . . . . . . . . . 2.6.2 Perturbations of the Identity and Tangent Space . . Generalization to All Homeomorphisms and C k -Diﬀeomorphisms . .

2.3 2.4 2.5

3

4 Transformations Generated by Velocities 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Metrics on Transformations Generated by Velocities . . . . . . . 2.1 Subgroup GΘ of Transformations Generated by Velocities 2.2 Complete Metrics on GΘ and Geodesics . . . . . . . . . . 2.3 Constructions of Azencott and Trouv´e . . . . . . . . . . . 3 Semiderivatives via Transformations Generated by Velocities . . 3.1 Shape Function . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gateaux and Hadamard Semiderivatives . . . . . . . . . . 3.3 Examples of Families of Transformations of Domains . . . 3.3.1 C ∞ -Domains . . . . . . . . . . . . . . . . . . . . 3.3.2 C k -Domains . . . . . . . . . . . . . . . . . . . . 3.3.3 Cartesian Graphs . . . . . . . . . . . . . . . . . 3.3.4 Polar Coordinates and Star-Shaped Domains . . 3.3.5 Level Sets . . . . . . . . . . . . . . . . . . . . . . 4 Unconstrained Families of Domains . . . . . . . . . . . . . . . . . 4.1 Equivalence between Velocities and Transformations . . . 4.2 Perturbations of the Identity . . . . . . . . . . . . . . . . 4.3 Equivalence for Special Families of Velocities . . . . . . . 5 Constrained Families of Domains . . . . . . . . . . . . . . . . . . 5.1 Equivalence between Velocities and Transformations . . . 5.2 Transformation of Condition (V2D ) into a Linear Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Continuity of Shape Functions along Velocity Flows . . . . . . . 5 Metrics via Characteristic Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Abelian Group Structure on Measurable Characteristic Functions 2.1 Group Structure on Xµ (RN ) . . . . . . . . . . . . . . . . 2.2 Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Complete Metric for Characteristic Functions in Lp -Topologies . . . . . . . . . . . . . . . . . . . . . . . . . 3 Lebesgue Measurable Characteristic Functions . . . . . . . . . . 3.1 Strong Topologies and C ∞ -Approximations . . . . . . . . 3.2 Weak Topologies and Microstructures . . . . . . . . . . . 3.3 Nice or Measure Theoretic Representative . . . . . . . . .

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5 6

7

xi 3.4 The Family of Convex Sets . . . . . . . . . . . . . . . . . . . 3.5 Sobolev Spaces for Measurable Domains . . . . . . . . . . . . Some Compliance Problems with Two Materials . . . . . . . . . . . 4.1 Transmission Problem and Compliance . . . . . . . . . . . . 4.2 The Original Problem of C´ea and Malanowski . . . . . . . . . 4.3 Relaxation and Homogenization . . . . . . . . . . . . . . . . Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . Caccioppoli or Finite Perimeter Sets . . . . . . . . . . . . . . . . . . 6.1 Finite Perimeter Sets . . . . . . . . . . . . . . . . . . . . . . . 6.2 Decomposition of the Integral along Level Sets . . . . . . . . 6.3 Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces . . . . . . . . . . . . . . . . . . . 6.4 Compactness and Uniform Cone Property . . . . . . . . . . . Existence for the Bernoulli Free Boundary Problem . . . . . . . . . . 7.1 An Example: Elementary Modeling of the Water Wave . . . 7.2 Existence for a Class of Free Boundary Problems . . . . . . . 7.3 Weak Solutions of Some Generic Free Boundary Problems . . 7.3.1 Problem without Constraint . . . . . . . . . . . . . 7.3.2 Constraint on the Measure of the Domain Ω . . . . 7.4 Weak Existence with Surface Tension . . . . . . . . . . . . .

6 Metrics via Distance Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniform Metric Topologies . . . . . . . . . . . . . . . . . . . . . . . 2.1 Family of Distance Functions Cd (D) . . . . . . . . . . . . . . 2.2 Pomp´eiu–Hausdorﬀ Metric on Cd (D) . . . . . . . . . . . . . . 2.3 Uniform Complementary Metric Topology and Cdc (D) . . . . c 2.4 Families Cdc (E; D) and Cd,loc (E; D) . . . . . . . . . . . . . . . 3 Projection, Skeleton, Crack, and Diﬀerentiability . . . . . . . . . . . 4 W 1,p -Metric Topology and Characteristic Functions . . . . . . . . . 4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 4.2 Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . . 5 Sets of Bounded and Locally Bounded Curvature . . . . . . . . . . . 5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Reach and Federer’s Sets of Positive Reach . . . . . . . . . . . . . . 6.1 Deﬁnitions and Main Properties . . . . . . . . . . . . . . . . 6.2 C k -Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 A Compact Family of Sets with Uniform Positive Reach . . . 7 Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Characterization of Convex Sets . . . . . . . . . . . . . . . . . . . . 8.1 Convex Sets and Properties of dA . . . . . . . . . . . . . . . . 8.2 Semiconvexity and BV Character of dA . . . . . . . . . . . . 8.3 Closed Convex Hull of A and Fenchel Transform of dA . . . . 8.4 Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and c (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cd,loc

223 224 228 228 235 239 240 244 245 251 252 254 258 258 260 262 262 264 265 267 267 268 268 269 275 278 279 292 292 296 299 301 303 303 310 315 316 318 318 320 322 323

xii

Contents 9

Compactness Theorems for Sets of Bounded Curvature . . . . . . . . 324 9.1 Global Conditions in D . . . . . . . . . . . . . . . . . . . . . 325 9.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 327

7 Metrics via Oriented Distance Functions 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Family of Oriented Distance Functions Cb (D) . . . . . . 2.2 Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . 3 Projection, Skeleton, Crack, and Diﬀerentiability . . . . . . . . . . . 4 W 1,p (D)-Metric Topology and the Family Cb0 (D) . . . . . . . . . . . 4.1 Motivations and Main Properties . . . . . . . . . . . . . . . . 4.2 Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . . 5 Boundary of Bounded and Locally Bounded Curvature . . . . . . . . 5.1 Examples and Limit of Tubular Norms as h Goes to Zero . . 6 Approximation by Dilated Sets/Tubular Neighborhoods . . . . . . . 7 Federer’s Sets of Positive Reach . . . . . . . . . . . . . . . . . . . . . 7.1 Approximation by Dilated Sets/Tubular Neighborhoods . . . 7.2 Boundaries with Positive Reach . . . . . . . . . . . . . . . . . 8 Boundary Smoothness and Smoothness of bA . . . . . . . . . . . . . 9 Sobolev or W m,p Domains . . . . . . . . . . . . . . . . . . . . . . . . 10 Characterization of Convex and Semiconvex Sets . . . . . . . . . . . 10.1 Convex Sets and Convexity of bA . . . . . . . . . . . . . . . . 10.2 Families of Convex Sets Cb (D), Cb (E; D), and Cb,loc (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 BV Character of bA and Semiconvex Sets . . . . . . . . . . . 11 Compactness and Sets of Bounded Curvature . . . . . . . . . . . . . 11.1 Global Conditions on D . . . . . . . . . . . . . . . . . . . . . 11.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . . 12 Finite Density Perimeter and Compactness . . . . . . . . . . . . . . 13 Compactness and Uniform Fat Segment Property . . . . . . . . . . . 13.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Equivalent Conditions on the Local Graph Functions . . . . . 14 Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 De Giorgi Perimeter of Caccioppoli Sets . . . . . . . . . . . . 14.2 Finite Density Perimeter . . . . . . . . . . . . . . . . . . . . . 15 The Families of Cracked Sets . . . . . . . . . . . . . . . . . . . . . . 16 A Variation of the Image Segmentation Problem of Mumford and Shah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 16.2 Cracked Sets without the Perimeter . . . . . . . . . . . . . . 16.2.1 Technical Lemmas . . . . . . . . . . . . . . . . . . . 16.2.2 Another Compactness Theorem . . . . . . . . . . . 16.2.3 Proof of Theorem 16.1 . . . . . . . . . . . . . . . . . 16.3 Existence of a Cracked Set with Minimum Density Perimeter

335 335 337 337 339 344 349 349 352 354 355 358 361 361 363 365 373 375 375 379 380 381 382 382 385 387 387 391 393 393 394 394 400 400 401 401 402 402 405

Contents

xiii 16.4

Uniform Bound or Penalization Term in the Objective Function on the Density Perimeter . . . . . . . . . . . . . . . 407

8 Shape Continuity and Optimization 1 Introduction and Generic Examples . . . . . . . . . . . . . . . . . . . 1.1 First Generic Example . . . . . . . . . . . . . . . . . . . . . . 1.2 Second Generic Example . . . . . . . . . . . . . . . . . . . . . 1.3 Third Generic Example . . . . . . . . . . . . . . . . . . . . . 1.4 Fourth Generic Example . . . . . . . . . . . . . . . . . . . . . 2 Upper Semicontinuity and Maximization of the First Eigenvalue . . 3 Continuity of the Transmission Problem . . . . . . . . . . . . . . . . 4 Continuity of the Homogeneous Dirichlet Boundary Value Problem . 4.1 Classical, Relaxed, and Overrelaxed Problems . . . . . . . . . 4.2 Classical Dirichlet Boundary Value Problem . . . . . . . . . . 4.3 Overrelaxed Dirichlet Boundary Value Problem . . . . . . . . 4.3.1 Approximation by Transmission Problems . . . . . . 4.3.2 Continuity with Respect to X(D) in the Lp (D)-Topology . . . . . . . . . . . . . . . . . . . . 4.4 Relaxed Dirichlet Boundary Value Problem . . . . . . . . . . 5 Continuity of the Homogeneous Neumann Boundary Value Problem 6 Elements of Capacity Theory . . . . . . . . . . . . . . . . . . . . . . 6.1 Deﬁnition and Basic Properties . . . . . . . . . . . . . . . . . 6.2 Quasi-continuous Representative and H 1 -Functions . . . . . . 6.3 Transport of Sets of Zero Capacity . . . . . . . . . . . . . . . 7 Crack-Free Sets and Some Applications . . . . . . . . . . . . . . . . 7.1 Deﬁnitions and Properties . . . . . . . . . . . . . . . . . . . . 7.2 Continuity and Optimization over L(D, r, O, λ) . . . . . . . . 7.2.1 Continuity of the Classical Homogeneous Dirichlet Boundary Condition . . . . . . . . . . . . . . . . . . 7.2.2 Minimization/Maximization of the First Eigenvalue . . . . . . . . . . . . . . . . . . . . . . . 8 Continuity under Capacity Constraints . . . . . . . . . . . . . . . . . 9 Compact Families Oc,r (D) and Lc,r (O, D) . . . . . . . . . . . . . . . 9.1 Compact Family Oc,r (D) . . . . . . . . . . . . . . . . . . . . 9.2 Compact Family Lc,r (O, D) and Thick Set Property . . . . . 9.3 Maximizing the Eigenvalue λA (Ω) . . . . . . . . . . . . . . . 9.4 State Constrained Minimization Problems . . . . . . . . . . . 9.5 Examples with a Constraint on the Gradient . . . . . . . . .

409 409 411 411 411 412 412 417 418 418 421 423 423

9 Shape and Tangential Diﬀerential Calculuses 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Diﬀerentiation in Topological Vector Spaces 2.1 Deﬁnitions of Semiderivatives and Derivatives . 2.2 Derivatives in Normed Vector Spaces . . . . . . 2.3 Locally Lipschitz Functions . . . . . . . . . . . 2.4 Chain Rule for Semiderivatives . . . . . . . . .

457 457 458 458 461 465 465

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424 425 426 429 429 431 432 434 434 437 437 438 440 447 447 450 452 453 454

xiv

Contents

3

4

5

6

2.5 Semiderivatives of Convex Functions . . . . . . . . . . . 2.6 Hadamard Semiderivative and Velocity Method . . . . . First-Order Shape Semiderivatives and Derivatives . . . . . . . 3.1 Eulerian and Hadamard Semiderivatives . . . . . . . . . 3.2 Hadamard Semidiﬀerentiability and Courant Metric Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Perturbations of the Identity and Gateaux and Fr´echet Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Shape Gradient and Structure Theorem . . . . . . . . . Elements of Shape Calculus . . . . . . . . . . . . . . . . . . . . 4.1 Basic Formula for Domain Integrals . . . . . . . . . . . 4.2 Basic Formula for Boundary Integrals . . . . . . . . . . 4.3 Examples of Shape Derivatives . . . . . . . . . . . . . . 4.3.1 Volume of Ω and Surface Area of Γ . . . . . . 4.3.2 H 1 (Ω)-Norm . . . . . . . . . . . . . . . . . . . 4.3.3 Normal Derivative . . . . . . . . . . . . . . . . Elements of Tangential Calculus . . . . . . . . . . . . . . . . . 5.1 Intrinsic Deﬁnition of the Tangential Gradient . . . . . 5.2 First-Order Derivatives . . . . . . . . . . . . . . . . . . 5.3 Second-Order Derivatives . . . . . . . . . . . . . . . . . 5.4 A Few Useful Formulae and the Chain Rule . . . . . . . 5.5 The Stokes and Green Formulae . . . . . . . . . . . . . 5.6 Relation between Tangential and Covariant Derivatives 5.7 Back to the Example of Section 4.3.3 . . . . . . . . . . . Second-Order Semiderivative and Shape Hessian . . . . . . . . 6.1 Second-Order Derivative of the Domain Integral . . . . 6.2 Basic Formula for Domain Integrals . . . . . . . . . . . 6.3 Nonautonomous Case . . . . . . . . . . . . . . . . . . . 6.4 Autonomous Case . . . . . . . . . . . . . . . . . . . . . 6.5 Decomposition of d2 J(Ω; V (0), W (0)) . . . . . . . . . .

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. . . .

467 469 471 471

. . . 476 . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

10 Shape Gradients under a State Equation Constraint 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Min Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 An Illustrative Example and a Shape Variational Principle 2.2 Function Space Parametrization . . . . . . . . . . . . . . . 2.3 Diﬀerentiability of a Minimum with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Application of the Theorem . . . . . . . . . . . . . . . . . . 2.5 Domain and Boundary Integral Expressions of the Shape Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Transport of H0k (Ω) by W k,∞ -Transformations of RN . . . 4.2 Laplacian and Bi-Laplacian . . . . . . . . . . . . . . . . . . 4.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . .

476 479 482 482 484 486 486 487 488 491 492 495 496 497 498 498 501 501 502 504 505 510 515

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519 519 521 521 522

. 523 . 526 . . . . . .

530 532 535 536 537 546

Contents 5

6

xv Saddle 5.1 5.2 5.3 5.4

Point Formulation and Function Space Parametrization . . An Illustrative Example . . . . . . . . . . . . . . . . . . . . Saddle Point Formulation . . . . . . . . . . . . . . . . . . . Function Space Parametrization . . . . . . . . . . . . . . . Diﬀerentiability of a Saddle Point with Respect to a Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Application of the Theorem . . . . . . . . . . . . . . . . . . 5.6 Domain and Boundary Expressions for the Shape Gradient Multipliers and Function Space Embedding . . . . . . . . . . . . . 6.1 The Nonhomogeneous Dirichlet Problem . . . . . . . . . . . 6.2 A Saddle Point Formulation of the State Equation . . . . . 6.3 Saddle Point Expression of the Objective Function . . . . . 6.4 Veriﬁcation of the Assumptions of Theorem 5.1 . . . . . . .

. . . .

551 551 552 553

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555 559 561 562 562 563 564 566

Elements of Bibliography

571

Index of Notation

615

Index

619

List of Figures 1.1 1.2 1.3 1.4 1.5 1.6

Graph of J(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Column of height one and cross section area A under the load . Triangulation and basis function associated with node Mi . . . . . Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . . Heat spreading scheme for high-power solid-state devices. . . . . (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω; . . . . . . . . . . . . . . . . . . . . . (C) Surface D generating Ω. 1.7 Volume Ω and its cross section. . . . . . . . . . . . . . . . . . . . and its generating surface A. . . . . . . . . . . . . . . 1.8 Volume Ω 1.9 Image I of objects and their segmentation in the frame D. . . . . 1.10 Image I containing black curves or cracks in the frame D. . . . . 1.11 Example of a two-dimensional strongly cracked set. . . . . . . . . 1.12 Example of a surface with facets associated with a ball. . . . . .

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4 5 8 11 14

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15 19 20 22 22 35 37 68 79 91 92

2.8

Diﬀeomorphism gx from U (x) to B. . . . . . . . . . . . . . . . . . . Local epigraph representation (N = 2). . . . . . . . . . . . . . . . . . Domain Ω0 and its image T (Ω0 ) spiraling around the origin. . . . . . Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin. . . . Examples of arbitrary and axially symmetrical O around the direction d = Ax (0, eN ). . . . . . . . . . . . . . . . . . . . . . . . . . The cone x + Ax C(λ, ω) in the direction Ax eN . . . . . . . . . . . . . Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α , h(θ) = θ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. . .

4.1

Transport of Ω by the velocity ﬁeld V . . . . . . . . . . . . . . . . . . 171

2.1 2.2 2.3 2.4 2.5 2.6 2.7

5.1 5.2 5.3 5.4 5.5 5.6

. . . . . . . . . . . . . Smiling sun Ω and expressionless sun Ω. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 . . . . . . . . . . . . . . Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . The function f (x, y) = 56 (1 − |x| − |y|)6 . . . . . . . . . . . . . Optimal distribution and isotherms with k1 = 2 (black) and k2 (white) for the problem of section 4.1. . . . . . . . . . . . . . . Optimal distribution and isotherms with k1 = 2 (black) and k2 (white) for the problem of C´ea and Malanowski. . . . . . . . . xvii

. . . . . . . . . . . . =1 . . . =1 . . .

109 114 118 119

220 227 228 234 235 239

xviii

List of Figures

5.7

The staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1 6.2 6.3 6.4 6.5

Skeletons Sk (A), Sk (A), and Sk (∂A) = Sk (A) ∪ Sk (A). Nonuniqueness of the exterior normal. . . . . . . . . . . . . Vertical stripes of Example 4.1. . . . . . . . . . . . . . . . . ∇dA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . Set of critical points of A. . . . . . . . . . . . . . . . . . . .

. . . . .

280 286 293 301 318

7.1 7.2

∇bA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . . W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R2 with uniformly bounded density perimeter to a set with empty interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example of a two-dimensional strongly cracked set. . . . . . . . . . . The two-dimensional strongly cracked set of Figure 7.3 in an open frame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The two open components Ω1 and Ω2 of the open domain Ω for N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

356

7.3 7.4 7.5

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387 396 400 406

Preface

1

Objectives and Scope of the Book

The objective of this book is to give a comprehensive presentation of mathematical constructions and tools that can be used to study problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. In that context, a good analytical framework and good modeling techniques must be able to handle the occurrence of singular behaviors whenever they are compatible with the mechanics or the physics of the problems at hand. In some optimization problems, the natural intuitive notion of a geometric domain undergoes mutations into relaxed entities such as microstructures. So the objects under consideration need not be smooth open domains, or even sets, as long as they still makes sense mathematically. This book covers the basic mathematical ideas, constructions, and methods that come from diﬀerent ﬁelds of mathematical activities and areas of applications that have often evolved in parallel directions. The scope of research is frighteningly broad because it touches on areas that include classical geometry, modern partial differential equations, geometric measure theory, topological groups, and constrained optimization, with applications to classical mechanics of continuous media such as ﬂuid mechanics, elasticity theory, fracture theory, modern theories of optimal design, optimal location and shape of geometric objects, free and moving boundary problems, and image processing. Innovative modeling or new issues raised in some applications force a new look at the fundamentals of well-established mathematical areas such as geometry, to relax basic notions of volume, perimeter, and curvature or boundary value problems, and to ﬁnd suitable relaxations of solutions. In that spirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notion of volume to the one of measure on an equivalence class of measurable sets in 1907. He was followed in that endeavor in the early 1950s by the celebrated work of E. De Giorgi, who used the relaxed notion of perimeter deﬁned on the class of Caccioppoli sets to solve Plateau’s problem of minimal surfaces. The material that is pertinent to the study of geometric objects and the entities and functions that are deﬁned on them would necessitate an encyclopedic investment to bring together the basic theories and their ﬁelds of applications. This objective is obviously beyond the scope of a single book and two authors. The xix

xx

Preface

coverage of this book is more modest. Yet, it contains most of the important fundamentals at this stage of evolution of this expanding ﬁeld. Even if shape analysis and optimization have undergone considerable and important developments on the theoretical and numerical fronts, there are still cultural barriers between areas of applications and between theories. The whole ﬁeld is extremely active, and the best is yet to come with fundamental structures and tools beginning to emerge. It is hoped that this book will help to build new bridges and stimulate cross-fertilization of ideas and methods.

2

Overview of the Second Edition

The second edition is almost a new book. All chapters from the ﬁrst edition have been updated and, in most cases, considerably enriched with new material. Many chapters or parts of chapters have been completely rewritten following the developments in the ﬁeld over the past 10 years. The book went from 9 to 10 chapters with a more elaborate sectioning of each chapter in order to produce a much more detailed table of contents. This makes it easier to ﬁnd speciﬁc material. A series of illustrative generic examples has been added right at the beginning of the introductory Chapter 1 to motivate the reader and illustrate the basic dilemma: parametrize geometries by functions or functions by geometries? This is followed by the big picture: a section on background and perspectives and a more detailed presentation of the second edition. The former Chapter 2 has been split into Chapter 2 on the classical descriptions and properties of domains and sets and a new Chapter 3, where the important material on Courant metrics and the generic constructions of A. M. Micheletti have been reorganized and expanded. Basic deﬁnitions and material have been added and regrouped at the beginning of Chapter 2: Abelian group structure on subsets of a set, connected and path-connected spaces, function spaces, tangent and dual cones, and geodesic distance. The coverage of domains that verify some segment property and have a local epigraph representation has been considerably expanded, and Lipschitzian (graph) domains are now dealt with as a special case. The new Chapter 3 on domains and submanifolds that are the image of a ﬁxed set considerably expands the material of the ﬁrst edition by bringing up the general assumptions behind the generic constructions of A. M. Micheletti that lead to the Courant metrics on the quotient space of families of transformations by subgroups of isometries such as identities, rotations, translations, or ﬂips. The general results apply to a broad range of groups of transformations of the Euclidean space and to arbitrary closed subgroups. New complete metrics on the whole spaces of homeomorphisms and C k -diﬀeomorphisms are also introduced to extend classical results for transformations of compact manifolds to general unbounded closed sets and open sets that are crack-free. This material is central in classical mechanics and physics and in modern applications such as imaging and detection. The former Chapter 7 on transformations versus ﬂows of velocities has been moved right after the Courant metrics as Chapter 4 and considerably expanded. It now specializes the results of Chapter 3 to spaces of transformations that are

2. Overview of the Second Edition

xxi

generated by the ﬂow of a velocity ﬁeld over a generic time interval. One important motivation is to introduce a notion of semiderivatives as well as a tractable criterion for continuity with respect to Courant metrics. Another motivation for the velocity point of view is the general framework of R. Azencott and A. Trouv´e starting in 1994 with applications in imaging. They construct complete metrics in relation with geodesic paths in spaces of diﬀeomorphisms generated by a velocity ﬁeld. The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and 5 on distance and oriented distance functions have become Chapters 5, 6, and 7. Those chapters have been renamed Metrics Generated by . . . in order to emphasize one of the main thrusts of the book: the construction of complete metrics on shapes and geometries.1 Those chapters emphasize the function analytic description of sets and domains: construction of metric topologies and characterization of compact families of sets or submanifolds in the Euclidean space. In that context, we are now dealing with equivalence classes of sets that may or may not have an invariant open or closed representative in the class. For instance, they include Lebesgue measurable sets and Federer’s sets of positive reach. Many of the classical properties of sets can be recovered from the smoothness or function analytic properties of those functions. The former Chapter 6 on optimization of shape functions has been completely rewritten and expanded as Chapter 8 on shape continuity and optimization. With meaningful metric topologies, we can now speak of continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc., compact families of sets, and existence of optimal geometries. The chapter concentrates on continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the objective function is deﬁned as an inﬁmum over a family of functions over a ﬁxed domain or set such as the eigenvalue problems. We ﬁrst characterize the continuity of the transmission problem and the upper semicontinuity of the ﬁrst eigenvalue of the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of deﬁnition since they require diﬀerent constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial diﬀerential equations. An introduction is also given to the concepts and results from capacity theory from which very general families of sets stable with respect to boundary conditions can be constructed. Note that some material has been moved from one chapter to another. For instance, section 7 on the continuity of the Dirichlet boundary problem in the former Chapter 3 has been merged with the content of the former Chapter 4 in the new Chapter 8. The former Chapters 8 and 9 have become Chapters 9 and 10. They are devoted to a modern version of the shape calculus, an introduction to the tangential diﬀerential calculus, and the shape derivatives under a state equation constraint. In Chapters 3, 5, 6, and 7, we have constructed complete metric spaces of geometries. Those spaces are nonlinear and nonconvex. However, several of them have a group 1 This is in line with current trends in the literature such as in the work of the 2009 Abel Prize ´moli and G. winner M. Gromov [1] and its applications in imaging by G. Sapiro [1] and F. Me Sapiro [1] to identify objects up to an isometry.

xxii

Preface

structure and, in some cases, it is possible to construct C 1 -paths in the group from velocity ﬁelds. This leads to the notion of Eulerian semiderivative that is somehow the analogue of a derivative on a smooth manifold. In fact, two types of semiderivatives are of interest: the weaker Gateaux style semiderivative and the stronger Hadamard style semiderivative. In the latter case, the classical chain rule is still available even for nondiﬀerentiable functions. In order to prepare the ground for shape derivatives, an enriched self-contained review of the pertinent material on semiderivatives and derivatives in topological vector spaces is provided. The important Chapter 10 concentrates on two generic examples often encountered in shape optimization. The ﬁrst one is associated with the so-called compliance problems, where the shape functional is itself the minimum of a domain-dependent energy functional. The special feature of such functionals is that the adjoint state coincides with the state. This obviously leads to considerable simpliﬁcations in the analysis. In that case, it will be shown that theorems on the diﬀerentiability of the minimum of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the minimizer is not unique. The second one will deal with shape functionals that can be expressed as the saddle point of some appropriate Lagrangian. As in the ﬁrst example, theorems on the diﬀerentiability of the saddle point of a functional with respect to a real parameter readily give explicit expressions of the Eulerian semiderivative even when the solution of the saddle point equations is not unique. Avoiding the diﬀerentiation of the state equation with respect to the domain is particularly advantageous in shape problems.

3

Intended Audience

The targeted audience is applied mathematicians and advanced engineers and scientists, but the book is also suitable for a broader audience of mathematicians as a relatively well-structured initiation to shape analysis and calculus techniques. Some of the chapters are fairly self-contained and of independent interest. They can be used as lecture notes for a mini-course. The material at the beginning of each chapter is accessible to a broad audience, while the latter sections may sometimes require more mathematical maturity. Thus the book can be used as a graduate text as well as a reference book. It complements existing books that emphasize speciﬁc mechanical or engineering applications or numerical methods. It can be considered ´sio [9], Introduction a companion to the book of J. Sokolowski and J.-P. Zole to Shape Optimization, published in 1992. Earlier versions of parts of this book have been used as lecture notes in graduate courses at the Universit´e de Montr´eal in 1986–1987, 1993–1994, 1995–1996, and 1997–1998 and at international meetings, workshops, or schools: S´eminaire de Math´ematiques Sup´erieures on Shape Optimization and Free Boundaries (Montr´eal, Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis (K´enitra, Morocco, December 1993), course of the COMETT MATARI European Program on Shape Optimization and Mutational Equations (Sophia-Antipolis, France, September 27 to October 1, 1993), CRM Summer School on Boundaries,

4. Acknowledgments

xxiii

Interfaces and Transitions (Banﬀ, Canada, August 6–18, 1995), and CIME course on Optimal Design (Troia, Portugal, June 1998).

4

Acknowledgments

The ﬁrst author is pleased to acknowledge the support of the Canada Council, which initiated the work presented in this book through a Killam Fellowship; the constant support of the National Sciences and Engineering Research Council of Canada; ´ and the FQRNT program of the Minist`ere de l’Education du Qu´ebec. Many thanks also to Louise Letendre and Andr´e Montpetit of the Centre de Recherches Math´ematiques, who provided their technical support, experience, and talent over the extended period of gestation of this book. Michel Delfour Jean-Paul Zol´esio August 13, 2009

Chapter 1

Introduction: Examples, Background, and Perspectives 1 1.1

Orientation Geometry as a Variable

The central object of this book1 is the geometry as a variable. As in the theory of functions of real variables, we need a diﬀerential calculus, spaces of geometries, evolution equations, and other familiar concepts in analysis when the variable is no longer a scalar, a vector, or a function, but is a geometric domain. This is motivated by many important problems in science and engineering that involve the geometry as a modeling, design, or control variable. In general the geometric objects we shall consider will not be parametrized or structured. Yet we are not starting from scratch, and several building blocks are already available from many ﬁelds: geometric measure theory, physics of continuous media, free boundary problems, the parametrization of geometries by functions, the set derivative as the inverse of the integral, the parametrization of functions by geometries, the Pomp´eiu–Hausdorﬀ metric, and so on. As is often the case in mathematics, spaces of geometries and notions of derivatives with respect to the geometry are built from well-established elements of functional analysis and diﬀerential calculus. There are many ways to structure families of geometries. For instance, a domain can be made variable by considering 1 The numbering of equations, theorems, lemmas, corollaries, deﬁnitions, examples, and remarks is by chapter. When a reference to another chapter is necessary it is always followed by the words in Chapter and the number of the chapter. For instance, “equation (2.18) in Chapter 9.” The text of theorems, lemmas, and corollaries is slanted; the text of deﬁnitions, examples, and remarks is normal shape and ended by a square . This makes it possible to aesthetically emphasize certain words especially in deﬁnitions. The bibliography is by author in alphabetical order. For each author or group of coauthors, there is a numbering in square brackets starting with [1]. A ´ [3] and a reference to an reference to an item by a single author is of the form J. Dieudonne item with several coauthors S. Agmon, A. Douglis, and L. Nirenberg [2]. Boxed formulae or statements are used in some chapters for two distinct purposes. First, they emphasize certain important deﬁnitions, results, or identities; second, in long proofs of some theorems, lemmas, or corollaries, they isolate key intermediary results which will be necessary to more easily follow the subsequent steps of the proof.

1

2

Chapter 1. Introduction: Examples, Background, and Perspectives

the images of a ﬁxed domain by a family of diﬀeomorphisms that belong to some function space over a ﬁxed domain. This naturally occurs in physics and mechanics, where the deformations of a continuous body or medium are smooth, or in the numerical analysis of optimal design problems when working on a ﬁxed grid. This construction naturally leads to a group structure induced by the composition of the diﬀeomorphisms. The underlying spaces are no longer topological vector spaces but groups that can be endowed with a nice complete metric space structure by introducing the Courant metric. The practitioner might or might not want to use the underlying mathematical structure associated with his or her constructions, but it is there and it contains information that might guide the theory and inﬂuence the choice of the numerical methods used in the solution of the problem at hand. The parametrization of a ﬁxed domain by a ﬁxed family of diﬀeomorphisms obviously limits the family of variable domains. The topology of the images is similar to the topology of the ﬁxed domain. Singularities that were not already present there cannot be created in the images. Other constructions make it possible to considerably enlarge the family of variable geometries and possibly open the doors to pathological geometries that are no longer open sets with a nice boundary. Instead of parametrizing the domains by functions or diﬀeomorphisms, certain families of functions can be parametrized by sets. A single function completely speciﬁes a set or at least an equivalence class of sets. This includes the distance functions and the characteristic function, but also the support function from convex analysis. Perhaps the best known example of that construction is the Pomp´eiu–Hausdorﬀ metric topology. This is a very weak topology that does not preserve the volume of a set. When the volume, the perimeter, or the curvatures are important, such functions must be able to yield relaxed deﬁnitions of volume, perimeter, or curvatures. The characteristic function that preserves the volume has many applications. It played a fundamental role in the integration theory of Henri Lebesgue at the beginning of the 20th century. It was also used in the 1950s by E. De Giorgi to deﬁne a relaxed notion of perimeter in the theory of minimal surfaces. Another technique that has been used successfully in free or moving boundary problems, such as motion by mean curvature, shock waves, or detonation theory, is the use of level sets of a function to describe a free or moving boundary. Such functions are often the solution of a system of partial diﬀerential equations. This is another way to build new tools from functional analysis. The choice of families of function parametrized sets or of families of set parametrized functions, or other appropriate constructions, is obviously problem dependent, much like the choice of function spaces of solutions in the theory of partial diﬀerential equations or optimization problems. This is one aspect of the geometry as a variable. Another aspect is to build the equivalent of a diﬀerential calculus and the computational and analytical tools that are essential in the characterization and computation of geometries. Again, we are not starting from scratch and many building blocks are already available, but many questions and issues remain open. This book aims at covering a small but fundamental part of that program. We had to make diﬃcult choices and refer the reader to appropriate books and references for background material such as geometric measure theory and specialized topics such as homogenization theory and microstructures which are available in excellent

2. A Simple One-Dimensional Example

3

books in English. It was unfortunately not possible to include references to the considerable literature on numerical methods, free and moving boundary problems, and optimization.

1.2

Outline of the Introductory Chapter

We ﬁrst give a series of generic examples where the shape or the geometry is the modeling, control, or optimization variable. They will be used in the subsequent chapters to illustrate the many ways such problems can be formulated. The ﬁrst example is the celebrated problem of the optimal shape of a column formulated by Lagrange in 1770 to prevent buckling. The extremization of the eigenvalues has also received considerable attention in the engineering literature. The free interface between two regions with diﬀerent physical or mechanical properties is another generic problem that can lead in some cases to a mixing or a microstructure. Two typical problems arising from applications to condition the thermal environment of satellites are described in sections 7 and 8. The ﬁrst one is the design of a thermal diﬀuser of minimal weight subject to an inequality constraint on the output thermal power ﬂux. The second one is the design of a thermal radiator to eﬀectively radiate large amounts of thermal power to space. The geometry is a volume of revolution around an axis that is completely speciﬁed by its height and the function which speciﬁes its lateral boundary. Finally, we give a glimpse at image segmentation, which is an example of shape/geometric identiﬁcation problems. Many chapters of this book are of direct interest to imaging sciences. Section 10 presents some background and perspectives. A fundamental issue is to ﬁnd tractable and preferably analytical representations of a geometry as a variable that are compatible with the problems at hand. The generic examples suggest two types of representations: the ones where the geometry is parametrized by functions and the ones where a family of functions is parametrized by the geometry. As is always the case, the choice is very much problem dependent. In the ﬁrst case, the topology of the variable sets is ﬁxed; in the second case the families of sets are much larger and topological changes are included. The book presents the two points of view. Finally, section 11 sketches the material in the second edition of the book.

2

A Simple One-Dimensional Example

A general feature of minimization problems with respect to a shape or a geometry subject to a state equation constraint is that they are generally not convex and that, when they have a solution, it is generally not unique. This is illustrated in ´a [2]: minimize the objective function the following simple example from J. Ce a def 2 |ya (x) − 1| dx, J(a) = 0

where a ≥ 0 and ya is the solution of the boundary value problem (state equation) d2 ya def (x) = −2 in Ωa = (0, a), dx2

dya (0) = 0, dx

ya (a) = 0.

(2.1)

4

Chapter 1. Introduction: Examples, Background, and Perspectives

Here the one-dimensional geometric domain Ωa = ]0, a[ is the minimizing variable. We recognize the classical structure of a control problem, except that the minimizing variable is no longer under the integral sign but in the limits of the integral sign. One consequence of this diﬀerence is that even the simplest problems will usually not be convex or convexiﬁables. They will require a special analysis. In this example it is easy to check that the solution of the state equation is ya (x) = a2 − x2

and J(a) =

8 5 4 3 a − a + a. 15 3

The graph of J, shown in Figure 1.1, is not the graph of a convex function. Its global minimum in a0 = 0, local maximum in a1 , and local minimum in a2 , 3 3 1 1 def def a1 = 1 − √ , a2 = 1+ √ , 4 4 3 3 are all diﬀerent.

a1

a2

Figure 1.1. Graph of J(a). To avoid a trivial solution, a strictly positive lower bound must be put on a. A unique minimizing solution is obtained for a ≥ a1 where the gradient of J is zero. For 0 < a < a2 , the minimum will occur at the preset lower or upper bound on a.

3

Buckling of Columns

The next example illustrates the fact that even simple problems can be nondiﬀerentiable with respect to the geometry. This is generic of all eigenvalue problems when the eigenvalue is not simple. One of the early optimal design problems was formulated by J. L. Lagrange [1] in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by the Danish mathematician and astronomer T. Clausen [1] in 1849. It consists in ﬁnding the best proﬁle of a vertical column of ﬁxed volume to prevent buckling.

3. Buckling of Columns

5

It turns out that this problem is in fact a hidden maximization of an eigenvalue. Many incorrect solutions had been published until 1992. This problem and other problems related to columns have been revisited in a series of papers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, but a complete theoretical and numerical solution for the buckling of a column was given only in 1992 by S. J. Cox and M. L. Overton [1]. The diﬃculty was that the eigenvalue is not simple and hence not diﬀerentiable with respect to the geometry. Consider a normalized column of unit height and unit volume (see Figure 1.2). Denote by the magnitude of the normalized axial load and by u the resulting transverse displacement. Assume that the potential energy is the sum of the bending and elongation energies 1 1 EI |u |2 dx − |u |2 dx, 0

0

where I is the second moment of area of the column’s cross section and E is its Young’s modulus. For suﬃciently small load the minimum of this potential energy with respect to all admissible u is zero. Euler’s buckling load λ of the column is the largest for which this minimum is zero. This is equivalent to ﬁnding the following minimum: 1 EI |u |2 dx def 0 λ = inf , (3.1) 1 |2 dx 0=u∈V |u 0 where V = H02 (0, 1) corresponds to the clamped case, but other types of boundary conditions can be contemplated. This is an eigenvalue problem with a special Rayleigh quotient. Assume that E is constant and that the second moment of area I(x) of the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

1

x

normalized load

cross section area A(x)

0

Figure 1.2. Column of height one and cross section area A under the load .

6

Chapter 1. Introduction: Examples, Background, and Perspectives

cross-sectional area A(x),

I(x) = c A(x)

1

A(x) dx = 1.

and 0

Normalizing λ by cE and taking into account the engineering constraints ∃0 < A0 < A1 , ∀x ∈ [0, 1],

0 < A0 ≤ A(x) ≤ A1 ,

we ﬁnally get

1

A |u |2 dx , 1 |2 dx 0=u∈V A∈A |u 0 1 def 2 A(x) dx = 1 . A = A ∈ L (0, 1) : A0 ≤ A ≤ A1 and sup λ(A),

def

λ(A) =

inf

0

(3.2) (3.3)

0

4

Eigenvalue Problems

Let D be a bounded open Lipschitzian domain in RN and A ∈ L∞ (D; L(RN , RN )) be a matrix function deﬁned on D such that ∗

A=A

and αI ≤ A ≤ βI

(4.1)

for some coercivity and continuity constants 0 < α ≤ β and ∗A is the transpose of A. Consider the minimization or the maximization of the ﬁrst eigenvalue sup λA (Ω) A∇ϕ · ∇ϕ dx Ω∈A(D) def A Ω inf 1 , (4.2) λ (Ω) = A |ϕ|2 dx 0=ϕ∈H0 (Ω) inf λ (Ω) Ω Ω∈A(D) where A(D) is a family of admissible open subsets of D (cf., for instance, sections 2, 7, and 9 of Chapter 8). In the vectorial case, consider the following linear elasticity problem: ﬁnd U ∈ H01 (Ω)3 such that Cε(U )·· ε(W ) dx = F · W dx (4.3) ∀W ∈ H01 (Ω)3 , Ω

Ω

for some distributed loading F ∈ L2 (Ω)3 and a constitutive law C which is a bilinear symmetric transformation of def def

σij τij Sym3 = τ ∈ L(R3 ; R3 ) : ∗ τ = τ , σ·· τ = 1≤i,j≤3

(L(R3 ; R3 ) is the space of all linear transformations of R3 or 3 × 3-matrices) under the following assumption. Assumption 4.1. The constitutive law is a transformation C ∈ Sym3 for which there exists a constant α > 0 such that Cτ ·· τ ≥ α τ ·· τ for all τ ∈ Sym3 .

5. Optimal Triangular Meshing

7

For instance, for the Lam´e constants µ > 0 and λ ≥ 0, the special constitutive law Cτ = 2µ τ + λ tr τ I satisﬁes Assumption 4.1 with α = 2µ. The associated bilinear form is def Cε(U )·· ε(W ) dx, aΩ (U, W ) = Ω

where U is a vector function, D(U ) is the Jacobian matrix of U , and def

ε(U ) =

1 (D(U ) + ∗D(U )) 2

is the strain tensor. The ﬁrst eigenvalue is the minimum of the Rayleigh quotient aΩ (U, U ) 1 3 (Ω) , U = 0 . : ∀U ∈ H λ(Ω) = inf 0 |U |2 dx Ω A typical problem is to ﬁnd the sensitivity of the ﬁrst eigenvalue with respect to the shape of the domain Ω. In 1907, J. Hadamard [1] used displacements along the normal to the boundary Γ of a C ∞ -domain to compute the derivative of the ﬁrst eigenvalue of the clamped plate. As in the case of the column, this problem is not diﬀerentiable with respect to the geometry when the eigenvalue is not simple.

5

Optimal Triangular Meshing

The shape calculus that will be developed in Chapters 9 and 10 for problems governed by partial diﬀerential equations (the continuous model ) will be readily applicable to their discrete model as in the ﬁnite element discretization of elliptic boundary value problems. However, some care has to be exerted in the choice of the formula for the gradient, since the solution of a ﬁnite element discretization problem is usually less smooth than the solution of its continuous counterpart. Most shape objective functionals will have two basic formulas for their shape gradient: a boundary expression and a volume expression. The boundary expression is always nicer and more compact but can be applied only when the solution of the underlying partial diﬀerential equation is smooth and in most cases smoother than the ﬁnite element solution. This leads to serious computational errors. The right formula to use is the less attractive volume expression that requires only the same smoothness as the ﬁnite element solution. Numerous computational experiments conﬁrm that fact (cf., for instance, E. J. Haug and J. S. Arora [1] or E. J. Haug, K. K. Choi, and V. Komkov [1]). With the volume expression, the gradient of the objective function with respect to internal and boundary nodes can be readily obtained by plugging in the right velocity ﬁeld. A large class of linear elliptic boundary value problems can be expressed as the minimum of a quadratic function over some Hilbert space. For instance, let Ω be a bounded open domain in RN with a smooth boundary Γ. The solution u of the boundary value problem −∆u = f in Ω,

u = 0 on Γ

8

Chapter 1. Introduction: Examples, Background, and Perspectives

is the minimizing element in the Sobolev space H01 (Ω) of the energy functional def E(v, Ω) = |∇v|2 − 2f v dx, Ω def |∇u|2 dx. J(Ω) = inf1 E(v, Ω) = E(u, Ω) = − v∈H0 (Ω)

Ω

The elements of this problem are a Hilbert space V , a continuous symmetrical coercive bilinear form on V , and a continuous linear form on V . With this notation ∃u ∈ V,

E(u) = inf E(v), v∈V

def

E(v) = a(v, v) − 2 (v)

and u is the unique solution of the variational equation ∃u ∈ U, ∀v ∈ V,

a(u, v) = (v).

In the ﬁnite element approximation of the solution u, a ﬁnite-dimensional subspace Vh of V is used for some small mesh parameter h. The solution of the approximate problem is given by ∃uh ∈ Vh ,

E(uh ) = inf E(vh ), vh ∈Vh

⇒ ∃uh ∈ Uh , ∀vh ∈ Vh ,

a(uh , vh ) = (vh ).

It is easy to show that the error can be expressed as follows: a(u − uh , u − uh ) = u − uh 2V = 2 [E(uh ) − E(u)] . Assume that Ω is a polygonal domain in RN . In the ﬁnite element method, the ¯ domain is partitioned into a set τh of small triangles by introducing nodes in Ω def

M = {Mi ∈ Ω : 1 ≤ i ≤ p}

def

def

∂M = {Mi ∈ ∂Ω : p + 1 ≤ i ≤ p + q}

and M = M ∪ ∂M

for some integers p ≥ N + 1 and q ≥ 1 (see Figure 1.3). Therefore the triangularization τh = τh (M ), the solution space Vh = Vh (M ), and the solution uh = uh (M ) are functions of the positions of the nodes of the set M . Assuming that the total 0

0

0 Mi

1 0

0 0

Figure 1.3. Triangulation and basis function associated with node Mi .

5. Optimal Triangular Meshing

9

number of nodes is ﬁxed, consider the following optimal triangularization problem: def

inf j(M ), M

j(M ) = E(uh (τh (M )), Ω) =

inf

vh ∈Vh (M )

E(vh , Ω),

|∇(u − uh )|2 dx = 2 E(uh , Ω(τh (M ))) − E(u, Ω) Ω

= 2 J(Ω(τh (M ))) − J(Ω) , def |∇uh |2 dx. J(Ω(τh (M ))) = inf E(vh , Ω(τh (M ))) = E(uh , Ω(τh (M ))) = − u −

uh 2V

=

vh ∈Vh

Ω

The objective is to compute the partial derivative of j(M ) with respect to the th component (Mi ) of the node Mi : ∂j (M ). ∂(Mi ) This partial derivative can be computed by using the velocity method for the special ´sio [3]) velocity ﬁeld (cf. M. C. Delfour, G. Payre, and J.-P. Zole Vi (x) = bMi (x) e , where bMi ∈ Vh is the (piecewise P 1 ) basis function associated with the node Mi : bMi (Mj ) = δij for all i, j. In that method each point X of the plane is moved according to the solution of the vector diﬀerential equation dx (t) = V (x(t)), dt

x(0) = X.

def

This yields a transformation X → Tt (X) = x(t; X) : R2 → R2 of the plane, and it is natural to introduce the following notion of semiderivative: def

dJ(Ω; V ) = lim

t0

J(Tt (Ω)) − J(Ω) . t

For t ≥ 0 small, the velocity ﬁeld must be chosen in such a way that triangles are moved onto triangles and the point Mi is moved in the direction e : Mi → Mit = Mi + t e . This is achieved by choosing the following velocity ﬁeld: Vi (t, x) = bMit (x) e , where bMit is the piecewise P 1 basis function associated with node Mit : bMit (Mj ) = δij for all i, j. This yields the family of transformations Tt (x) = x + t bMi (x) e which moves the node Mi to Mi + t e and hence ∂j (M ) = dJ(Ω; Vi ). ∂(Mi )

10

Chapter 1. Introduction: Examples, Background, and Perspectives Going back to our original example, introduce the shape functional def 2 |∇u| dx, E(Ω, v) = |∇v|2 − 2 f v dx. J(Ω) = inf1 E(Ω, v) = − v∈H0 (Ω)

Ω

Ω

In Chapter 9, we shall show that we have the following boundary and volume expressions for the derivative of J(Ω):

dJ(Ω; V ) =

2 ∂u dJ(Ω; V ) = − V · n dΓ, Γ ∂n A (0)∇u · ∇u − 2 [div V (0)f + ∇f · V (0)] u dx,

Ω

A (0) = div V (0) I − ∗DV (0) − DV (0). For a P 1 -approximation def

¯ : v|K ∈ P 1 (K), ∀K ∈ τh Vh = v ∈ C 0 (Ω) and the trace of the normal derivative on Γ is not deﬁned. Thus, it is necessary to use the volume expression. For the velocity ﬁeld Vi DVi = e ∗∇bMi , div DVi = e · ∇bMi , A (0) = e · ∇bMi I − e ∗∇bMi − ∇bMi ∗ e . Since ∂j (M ) = dJ(Ω; Vi ), ∂(Mi ) we ﬁnally obtain the formula for the derivative of the function j(M ) with respect to node Mi in the direction e : ∂j (M ) = [ e · ∇bMi I − e ∗∇bMi − ∇bMi ∗ e ] ∇; uh · ∇uh ∂(Mi ) Ω − 2 [ e · ∇bMi f + ∇f · e bMi ] uh dx. Since the support of bMi consists of the triangles having Mi as a vertex, the gradient with respect to the nodes can be constructed piece by piece by visiting each node.

6

Modeling Free Boundary Problems

The ﬁrst step towards the solution of a shape optimization is the mathematical modeling of the problem. Physical phenomena are often modeled on relatively smooth or nice geometries. Adding an objective functional to the model will usually push the system towards rougher geometries or even microstructures. For instance, in the optimal design of plates the optimization of the proﬁle of a plate led to highly oscillating proﬁles that looked like a comb with abrupt variations ranging from zero

6. Modeling Free Boundary Problems

11

to maximum thickness. The phenomenon began to be understood in 1975 with the paper of N. Olhoff [1] for circular plates with the introduction of the mechanical notion of stiﬀeners. The optimal plate was a virtual plate, a microstructure, that is a homogenized geometry. Another example is the Plateau problem of minimal surfaces that experimentally exhibits surfaces with singularities. In both cases, it is mathematically natural to replace the geometry by a characteristic function, a function that is equal to 1 on the set and 0 outside the set. Instead of optimizing over a restricted family of geometries, the problem is relaxed to the optimization over a set of measurable characteristic functions that contains a much larger family of geometries, including the ones with boundary singularities and/or an arbitrary number of holes.

6.1

Free Interface between Two Materials

´a and K. Malanowski [1] Consider the optimal design problem studied by J. Ce in 1970, where the optimization variable is the distribution of two materials with diﬀerent physical characteristics within a ﬁxed domain D. It cannot a priori be assumed that the two regions are separated by a smooth interface and that each region is connected. This problem will be covered in more details in section 4 of Chapter 5. Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assume for the moment that the domain D is partitioned into two subdomains Ω1 and Ω2 separated by a smooth interface ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 1.4. Domain Ω1 (resp., Ω2 ) is made up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y be the solution of the transmission problem − k1 y = f in Ω1 y = 0 on ∂D

and − k2 y = f in Ω2 , ∂y ∂y + k2 = 0 on Ω1 ∩ Ω2 , and k1 ∂n1 ∂n2

(6.1)

where n1 (resp., n2 ) is the unit outward normal to Ω1 (resp., Ω2 ) and f is a given function in L2 (D). Assume that k1 > k2 . The objective is to maximize the equivalent of the compliance f y dx (6.2) J(Ω1 ) = − D

Ω1

Ω2

Figure 1.4. Fixed domain D and its partition into Ω1 and Ω2 .

12

Chapter 1. Introduction: Examples, Background, and Perspectives

over all domains Ω1 in D subject to the following constraint on the volume of material k1 which occupies the part Ω1 of D: m(Ω1 ) ≤ α,

0 < α < m(D)

(6.3)

for some constant α. If χ denotes the characteristic function of the domain Ω1 , / Ω1 , χ(x) = 1 if x ∈ Ω1 and 0 if x ∈ the compliance J(χ) = J(Ω1 ) can be expressed as the inﬁmum over the Sobolev space H01 (D) of an energy functional deﬁned on the ﬁxed set D: J(χ) = def

min

ϕ∈H01 (D)

E(χ, ϕ),

(k1 χ + k2 (1 − χ)) |∇ϕ|2 − 2 χf ϕ dx.

E(χ, ϕ) =

(6.4) (6.5)

D

J(χ) can be minimized or maximized over some appropriate family of characteristic functions or with respect to their relaxation to functions between 0 and 1 that would correspond to microstructures. As in the eigenvalue problem, the objective function is an inﬁmum, but here the inﬁmum is over a space that does not depend on the function χ that speciﬁes the geometric domain. This will be handled by the special techniques of Chapter 10 for the diﬀerentiation of the minimum of a functional.

6.2

Minimal Surfaces

The celebrated Plateau’s problem, named after the Belgian physicist and professor J. A. F. Plateau [1] (1801–1883), who did experimental observations on the geometry of soap ﬁlms around 1873, also provides a nice example where the geometry is a variable. It consists in ﬁnding the surface of least area among those bounded by a given curve. One of the diﬃculties in studying the minimal surface problem is the description of such surfaces in the usual language of diﬀerential geometry. For instance, the set of possible singularities is not known. Measure theoretic methods such as k-currents (k-dim surfaces) were used by E. R. Reifenberg [1, 2, 3, 4] around 1960, H. Federer and W. H. Fleming [1] in 1960 (normals and integral currents), F. J. Almgren, Jr. [1] in 1965 (varifolds), and H. Federer [5] in 1969. In the early 1950s, E. De Giorgi [1, 2, 3] and R. Caccioppoli [1] considered a hypersurface in the N -dimensional Euclidean space RN as the boundary of a set. In order to obtain a boundary measure, they restricted their attention to sets whose characteristic function is of bounded variation. Their key property is an associated natural notion of perimeter that extends the classical surface measure of the boundary of a smooth set to the larger family of Caccioppoli sets named after the celebrated Neapolitan mathematician Renato Caccioppoli.2 2 In 1992 his tormented personality was remembered in a ﬁlm directed by Mario Martone, The Death of a Neapolitan Mathematician (Morte di un matematico napoletano).

7. Design of a Thermal Diﬀuser

13

Caccioppoli sets occur in many shape optimization problems (or free boundary problems), where a surface tension is present on the (free) boundary, such as in the free interface water/soil in a dam (C. Baiocchi, V. Comincioli, E. Magenes, and G. A. Pozzi [1]) in 1973 and in the free boundary of a water wave (M. Souli ´sio [1, 2, 3, 4, 5]) in 1988. More details will be given in Chapter 5. and J.-P. Zole

7

Design of a Thermal Diﬀuser

Shape optimization problems are everywhere in engineering, physics, and medicine. We choose two illustrative examples that were proposed by the Canadian Space Program in the 1980s. The ﬁrst one is the design of a thermal diﬀuser to condition the thermal environment of electronic devices in communication satellites; the second one is the design of a thermal radiator that will be described in the next section. There are more and more design and control problems coming from medicine. For instance, the design of endoprotheses such as valves, stents, and coils in blood vessels or left ventricular assistance devices (cardiac pumps) in interventional cardiology helps to improve the health of patients and minimize the consequences and costs of therapeutical interventions by going to mini-invasive procedures.

7.1

Description of the Physical Problem

This problem arises in connection with the use of high-power solid-state devices (HPSSD) in communication satellites (cf. M. C. Delfour, G. Payre, and J.´sio [1]). An HPSSD dissipates a large amount of thermal power (typ. P. Zole > 50 W) over a relatively small mounting surface (typ. 1.25 cm2 ). Yet, its junction temperature is required to be kept moderately low (typ. 110◦ C). The thermal resistance from the junction to the mounting surface is known for any particular HPSSD (typ. 1◦ C/W), so that the mounting surface is required to be kept at a lower temperature than the junction (typ. 60◦ C). In a space application the thermal power must ultimately be dissipated to the environment by the mechanism of radiation. However, to radiate large amounts of thermal power at moderately low temperatures, correspondingly large radiating areas are required. Thus we have the requirement to eﬃciently spread the high thermal power ﬂux (TPF) at the HPSSD source (typ. 40 W/cm2 ) to a low TPF at the radiator (typ. 0.04 W/cm2 ) so that the source temperature is maintained at an acceptably low level (typ. < 60◦ C) at the mounting surface. The eﬃcient spreading task is best accomplished using heatpipes, but the snag in the scheme is that heatpipes can accept only a limited maximum TPF from a source (typ. max 4 W/cm2 ). Hence we are led to the requirement for a thermal diﬀuser. This device is inserted between the HPSSD and the heatpipes and reduces the TPF at the source (typ. > 40 W/cm2 ) to a level acceptable to the heatpipes (typ. > 4 W/cm2 ). The heatpipes then suﬃciently spread the heat over large space radiators, reducing the TPF from a level at the diﬀuser (typ. 4 W/cm2 ) to that at the radiator (typ. 0.04 W/cm2 ). This scheme of heat spreading is depicted in Figure 1.5. It is the design of the thermal diﬀuser which is the problem at hand. We may assume that the HPSSD presents a uniform thermal power ﬂux to the diﬀuser

14

Chapter 1. Introduction: Examples, Background, and Perspectives schematic drawing not to scale

radiator to space heatpipes

heatpipe saddle

thermal diﬀuser

high-power solid-state diﬀuser

Figure 1.5. Heat spreading scheme for high-power solid-state devices. at the HPSSD/diﬀuser interface. Heatpipes are essentially isothermalizing devices, and we may assume that the diﬀuser/heatpipe saddle interface is indeed isothermal. Any other surfaces of the diﬀuser may be treated as adiabatic.

7.2

Statement of the Problem

Assume that the thermal diﬀuser is a volume Ω symmetrical about the z-axis (cf. Figure 1.6 (A)) whose boundary surface is made up of three regular pieces: the mounting surface Σ1 (a disk perpendicular to the z-axis with center in (r, z) = (0, 0)), the lateral adiabatic surface Σ2 , and the interface Σ3 between the diffuser and the heatpipe saddle (a disk perpendicular to the z-axis with center in (r, z) = (0, L)). The temperature distribution over this volume Ω is the solution of the stationary heat equation k∆T = 0 (∆T , the Laplacian of T ) with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω): k

∂T = qin on Σ1 , ∂n

k

∂T = 0 on Σ2 , ∂n

T = T3 (constant) on Σ3 ,

(7.1)

where n always denotes the outward unit normal to the boundary surface Σ and ∂T /∂n is the normal derivative to the boundary surface Σ, ∂T = ∇T · n (∇T = the gradient of T ). ∂n

(7.2)

The parameters appearing in (7.1) are k = thermal conductivity (typ. 1.8W/cm×◦ C), qin = uniform inward thermal power ﬂux at the source (positive constant).

7. Design of a Thermal Diﬀuser

15

The radius R0 of the mounting surface Σ1 is ﬁxed so that the boundary surface Σ1 is already given in the design problem. For practical considerations, we assume that the diﬀuser is solid without interior hollows or cutouts. The class of shapes for the diﬀuser is characterized by the design parameter L > 0 and the positive function R(z), 0 < z ≤ L, with R(0) = R0 > 0. They are volumes of revolution Ω about the z-axis generated by the surface A between the z-axis and the function R(z) (cf. Figure 1.6 (B)), that is, def

Ω = (x, y, z) : 0 < z < L, x2 + y 2 < R(z)2 .

(7.3)

So the shape of Ω is completely speciﬁed by the length L > 0 and the function R(z) > 0 on the interval [0, L]. z

z Σ3

L

ζ

L A R(z)

S3

1 S4

D

Σ2 0

0

Σ1

S2 y

0

0

y

R0

(A)

0

0

(B)

1 S1

ρ (C)

Figure 1.6. (A) Volume Ω and its boundary Σ; (B) Surface A generating Ω; (C) Surface D generating Ω. Assuming that the diﬀuser is made up of a homogeneous material of uniform density (no hollow) the design objective is to minimize the volume def

J(Ω) =

L

R(z)2 dz

dx = π Ω

(7.4)

0

subject to a uniform constraint on the outward thermal power ﬂux at the interface Σ3 between the diﬀuser and the heatpipe saddle: sup −k

p∈Σ3

∂T ∂T (p) ≤ qout or k + qout ≥ 0 on Σ3 , ∂z ∂n

(7.5)

where qout is a speciﬁed positive constant. It is readily seen that the minimization problem (7.4) subject to the constraint (7.5) (where T is the solution of the heat equation with the boundary conditions (7.1)) is independent of the ﬁxed temperature T3 on the boundary Σ3 . In other words the optimal shape Ω, if it exists, is independent of T3 . As a result, from now on we set T3 equal to 0.

16

7.3

Chapter 1. Introduction: Examples, Background, and Perspectives

Reformulation of the Problem

In a shape optimization problem the formulation is important from both the theoretical and the numerical viewpoints. In particular condition (7.5) is diﬃcult to numerically handle since it involves the pointwise evaluation of the normal derivative on the piece of boundary Σ3 . This problem can be reformulated as the minimization of T on Σ3 , where T is now the solution of a variational inequality. Consider the following minimization problem over the subspace of functions that are positive or zero on Σ3 : def

(7.6) V + (Ω) = v ∈ H 1 (Ω) : v|Σ3 ≥ 0 , 1 inf |∇v|2 dx − qin v dΣ + qout v dΣ. (7.7) v∈V + (Ω) Ω 2 Σ1 Σ3 H 1 (Ω) is the usual Sobolev space on the domain Ω, and the inequality on Σ3 has to be interpreted quasi-everywhere in the capacity sense. Leaving aside those technicalities, the minimizing solution of (7.7) is characterized by ∂T ∂T = qin on Σ1 , k = 0 on Σ2 , −k ∆T = 0 in Ω, k ∂n ∂n ∂T ∂T + qout ≥ 0, T k + qout = 0 on Σ3 . T ≥ 0, k ∂n ∂n

(7.8)

The former constraint (7.5) is veriﬁed and replaced by the new constraint T = 0 on Σ3 .

(7.9)

If there exists a nonempty domain Ω of the form (7.3) such that T = 0 on Σ3 , the problem is feasible. In this formulation the pointwise constraint on the normal derivative of the temperature on Σ3 has been replaced by a pointwise constraint on the less demanding trace of the temperature on Σ3 . Yet, we now have to solve a variational inequality instead of a variational equation for the temperature T .

7.4

Scaling of the Problem

In the above formulations the shape parameter L and the shape function R are not independent of each other since the function R is deﬁned on the interval [0, L]. This motivates the following changes of variables and the introduction of the dimensionless temperature y: x y z , y → ξ2 = , z → ζ = , 0 ≤ ζ ≤ 1, x → ξ1 = R0 R0 L R(Lζ) ˜ = L , R(ζ) ˜ L = , R0 R0 k y(ξ1 , ξ2 , ζ) = T (R0 ξ1 , R0 ξ2 , Lζ), Lqin def ˜ 2 . D = (ξ1 , ξ2 , ζ) : 0 < ζ < 1, ξ 2 + ξ 2 < R(z) 1

2

7. Design of a Thermal Diﬀuser

17

˜ now appears as a coeﬃcient in the partial diﬀerential equation The parameter L 2 ∂ y ∂2y ∂2y 2 ˜ L + 2 + 2 = 0 in D 2 ∂ξ1 ∂ξ2 ∂ζ with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D: ∂y = 1 on S1 , ∂νA

∂y = 0 on S2 , ∂νA

y = 0 on S3 ,

(7.10)

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface S, ∂y ∂y ∂y ∂y 2 ˜ + ν3 . = L ν1 + ν2 ∂νA ∂ξ1 ∂ξ2 ∂ζ Finally, the optimal design problem depends only on the ratio q = qout /qin through the constraint ∂y qout + ≥ 0 on S3 . ∂νA qin ˜ > 0 and the function R ˜ > 0 now deﬁned The design variables are the parameter L on the ﬁxed interval [0, 1].

7.5

Design Problem

The fact that this speciﬁc design problem can be reduced to ﬁnding a parameter and a function gives the false or unfounded impression that it can now be solved by standard mathematical programming and numerical methods. Early work on such problems revealed a diﬀerent reality, such as oscillating boundaries and convergence towards nonphysical designs. Clearly, the geometry refused to be handled by standard methods without a better understanding of the underlying physics and its inception in the modeling of the geometric variable. At the theoretical level, the existence of solution requires a concept of continuity with respect to the geometry of the solution of either the heat equation with an inequality constraint on the TPF or the variational inequality with an equality constraint on the temperature. The other element is the lower semicontinuity of the objective functional that is not too problematic for the volume functional as long as the chosen topology on the geometry preserves the continuity of the volume functional. For instance, the classical Hausdorﬀ metric topology does not preserve the volume. In the context of ﬂuid mechanics (cf., for instance, O. Pironneau [1]), it means that a drag minimizing sequence of sets with constant volume may converge to a set with twice the volume (cf. Example 4.1 in Chapter 6). A wine making industry exploiting the convergence in the Hausdorﬀ metric topology could yield miraculous proﬁts. Other serious issues are, for instance, the lack of diﬀerentiability of the solution of a variational inequality at the continuous level that will inadvertently aﬀect the diﬀerentiability or the evolution of a gradient method at the discrete level. We shall

18

Chapter 1. Introduction: Examples, Background, and Perspectives

see that there is not only one topology for shapes but a whole range that selectively preserve some but not all of the geometrical features. Again the right choice is problem dependent, much like the choice of the right Sobolev space in the theory of partial diﬀerential equations.

8

Design of a Thermal Radiator

Current trends indicate that future communications satellites and spacecrafts will grow ever larger, consume ever more electrical power, and dissipate larger amounts of thermal energy. Various techniques and devices can be deployed to condition the thermal environment for payload boxes within a spacecraft, but it is desirable to employ those which oﬀer good performance for low cost, low weight, and high reliability. A thermal radiator (or radiating ﬁn) which accepts a given TPF from a payload box and radiates it directly to space can oﬀer good performance and high reliability at low cost. However, without careful design, such a radiator can be unnecessarily bulky and heavy. It is the mass-optimized design of the thermal radiator ´which is the problem at hand (cf. M. C. Delfour, G. Payre, and J.-P. Zole sio [2]). We may assume that the payload box presents a uniform TPF (typ. 0.1 to 1.0 W/cm2 ) into the radiator at the box/radiator interface. The radiating surface is a second surface mirror which consists of a sheet of glass whose inner surface has silver coating. We may assume that the TPF out of the radiator/space interface is governed by the T 4 radiation law, although we must account also for a constant TPF (typ. 0.01 W/cm2 ) into this interface from the sun. Any other surfaces of the radiator may be treated as adiabatic. Two constraints restrict freedom in the design of the thermal radiator: (i) the maximum temperature at the box/radiator interface is not to exceed some constant (typ. 50◦ ); and (ii) no part of the radiator is to be thinner than some constant (typ. 1 mm).

8.1

Statement of the Problem

Assume that the radiator is a volume Ω symmetrical about the z-axis (cf. Figure 1.7) whose boundary surface is made up of three regular pieces: the contact surface Σ1 (a disk perpendicular to the z-axis with center at the point (r, z) = (0, 0)), the lateral adiabatic surface Σ2 , and the radiating surface Σ3 (a disk perpendicular to the z-axis with center at (r, z) = (0, L)). More precisely

Σ1 = (x, y, z) : z = 0 and x2 + y 2 ≤ R02 ,

Σ2 = (x, y, z) : x2 + y 2 = R(z)2 , 0 ≤ z ≤ L , (8.1)

2 2 2 Σ3 = (x, y, z) : z = L and x + y ≤ R(L) , where the radius R0 > 0 (typ. 10 cm), the length L > 0, and the function R : [0, L] → R,

R(0) = R0 ,

are given (R, the ﬁeld of real numbers).

R(z) > 0, 0 ≤ z ≤ L,

(8.2)

8. Design of a Thermal Radiator

19

z

z

Σ3

L Σ2 y

0

0

R0 Σ1

r

Figure 1.7. Volume Ω and its cross section. The temperature distribution (in Kelvin degrees) over the volume Ω is the solution of the stationary heat equation ∆T = 0

(the Laplacian of T )

(8.3)

with the following boundary conditions on the surface Σ = Σ1 ∪ Σ2 ∪ Σ3 (the boundary of Ω): k

∂T = qin on Σ1 , ∂n

k

∂T = 0 on Σ2 , ∂n

k

∂T = −σεT |T |3 + qs on Σ3 , ∂n

(8.4)

where n denotes the outward normal to the boundary surface Σ, ∂T /∂n is the normal derivative on the boundary surface Σ, and ∂T = ∇T · n (∇T = the gradient of T ). ∂n The parameters appearing in (8.1)–(8.4) are k = thermal conductivity (typ. 1.8 W/cm×◦ C), qin = uniform inward thermal power ﬂux at the source (typ. 0.1 to 1.0 W/cm2 ), σ = Boltzmann’s constant (5.67 × 10−8 W/m2 K4 ), ε = surface emissivity (typ. 0.8), qs = solar inward thermal power ﬂux (0.01 W/cm2 ). The optimal design problem consists in minimizing the volume L def R(z)2 dz J(Ω) = π

(8.5)

0

over all length L > 0 and shape function R subject to the constraint T (x, y, z) ≤ Tf

(typ. 50◦ C),

∀(x, y, z) ∈ Σ1 .

In this analysis we shall drop the requirement (ii) in the introduction.

(8.6)

20

8.2

Chapter 1. Introduction: Examples, Background, and Perspectives

Scaling of the Problem

As in the case of the diﬀuser, it is convenient to introduce the following dimensionless coordinates and temperature y (see Figure 1.8): x → ξ1 =

x , R0

y → ξ2 =

y , R0

z → ζ =

z , L

0 ≤ ζ ≤ 1,

R(Lζ) ˜ = L , R(ζ) ˜ L = , R0 R0 1/3 σεR0 T (R0 ξ1 , R0 ξ2 , Lζ), y(ξ1 , ξ2 , ζ) = k def ˜ 2 . D = (ξ1 , ξ2 , ζ) : 0 < ζ < 1, ξ 2 + ξ 2 < R(z) 1

2

˜ now appears as a coeﬃcient in the partial diﬀerential equation The parameter L 2 ∂ y ∂2y ∂ 2y 2 ˜ L + = 0 in D + ∂ξ12 ∂ξ22 ∂ζ 2 with the following boundary conditions on the boundary S = S1 ∪ S2 ∪ S3 of D: 1/3 qin ∂y σεR0 ˜ on S1 , R0 =L ∂νA k k ∂y =0 ∂νA

(8.7)

on S2 ,

∂y 3 ˜ ˜ + Ly|y| =L ∂νA

σεR0 k

1/3

qin qs R0 k qin

on S3 ,

where ν denotes the outward normal to the boundary surface S and ∂y/∂νA is the conormal derivative to the boundary surface Σ, ∂y ˜ 2 ν1 ∂y + ν2 ∂y + ν3 ∂y . =L ∂νA ∂ξ1 ∂ξ2 ∂ζ

ζ

ζ

˜3 Σ

1

˜2 Σ

S3

1 A

r 1 ˜1 Σ

0

S2 r

0

1 S1

and its generating surface A. Figure 1.8. Volume Ω

9. A Glimpse into Segmentation of Images

9

21

A Glimpse into Segmentation of Images

The study of the problem of linguistic or visual perceptions was initiated by several pioneering authors, such as H. Blum [1] in 1967, D. Marr and E. Hildreth [1] in 1980, and D. Marr [1] in 1982. It involves specialists of psychology, artiﬁcial intelligence, and experimentalists such as D. H. Hubel and T. N. Wiesel [1] in 1962 and F. W. Campbell and J. G. Robson [1] in 1968. In the ﬁrst part of this section, we revisit the pioneering work of D. Marr and E. Hildreth [1] in 1980 on the smoothing of the image by convolution with a suﬃciently diﬀerentiable normalized function as a function of the scaling parameter. We extend the space-frequency uncertainty principle to N -dimensional images. It is the analogue of the Heisenberg uncertainty principle of quantum mechanics. We revisit the Laplacian ﬁlter and we generalize the linearity assumption of D. Marr and E. Hildreth [1] from linear to curved contours. In the second part, we show how shape analysis methods and the shape and tangential calculuses can be applied to objective functionals deﬁned on the whole contour of an image. We anticipate Chapters 9 and 10 on shape derivatives by the velocity method and show how they can be applied to snakes, active geodesic contours, and level sets (cf., for instance, the book of S. Osher and N. Paragios [1]). It shows that the Eulerian shape semiderivative is the basic ingredient behind those representations, including the case of the oriented distance function. In all cases, the evolution equation for the continuous gradient descent method is shown to have the same structure. For more material along the same lines, the reader is referred to M. Dehaes [1] and M. Dehaes and M. C. Delfour [1].

9.1

Automatic Image Processing

The ﬁrst level of image processing is the detection of the contours or the boundaries of the objects in the image. For an ideal image I : D → R deﬁned in an open two-dimensional frame D with values in an interval of greys continuously ranging from white to black (see Figure 1.9), the edges of an object correspond to the loci of discontinuity of the image I (cf. D. Marr and E. Hildreth [1]), also called “step edges” by D. Marr [1]. As can be seen from Figure 1.9, the loci of discontinuity may only reveal part of an object hidden by another one and a subsequent and diﬀerent level of processing is required. A more diﬃcult case is the detection of black curves or cracks in a white frame (cf. Figure 1.10), where the function I becomes a measure supported by the curves rather than a function. In practice, the frame D of the image is divided into periodically spaced cells P (square, hexagon, diamond) with a quantized value or pixel from 256 grey levels. For small squares a piecewise linear continuous interpolation or higher degree C 1 -interpolation can be used to remove the discontinuities at the intercell boundaries. In addition, observations and measurements introduce noise or perturbations and the interpolated image I needs to be further smoothed or ﬁltered. When the characteristics of the noise are known, an appropriate ﬁlter can do the job (see, for instance, the use of low pass ﬁlters in A. Rosenfeld and M. Thurston [1], A. P. Witkin [1], and A. L. Yuille and T. Poggio [1]).

22

Chapter 1. Introduction: Examples, Background, and Perspectives

frame D levels of grey I

open set Ω

Figure 1.9. Image I of objects and their segmentation in the frame D.

frame D

Figure 1.10. Image I containing black curves or cracks in the frame D. Given an ideal grey level image I deﬁned in a ﬁxed bounded open frame D we want to identify the edges or boundaries of the objects contained in the image as shown in Figure 1.9. The edge identiﬁcation or segmentation problem is a lowlevel processing of a real image which should also involve higher-level processings or tasks (cf. M. Kass, M. Witkin, and D. Terzopoulos [1]). Intuitively the edges coincide with the loci of discontinuities of the image function I. At such points the norm |∇I| of the gradient ∇I is inﬁnite. Roughly speaking the segmentation of an ideal image I consists in ﬁnding the loci of discontinuity of the function I. The idea is now to ﬁrst smooth I by convolution with a suﬃciently diﬀerentiable function. This operation will be followed and/or combined with the use of an edge detector as the zero crossings of the Laplacian. In the literature the term ﬁlter often applies to both the ﬁlter and the detector. In this section we shall make the distinction between the two operations.

9.2

Image Smoothing/Filtering by Convolution and Edge Detectors

In this section we revisit the work of D. Marr and E. Hildreth [1] in 1980 on the smoothing of the image I by convolution with a suﬃciently diﬀerentiable normalized function ρ as a function of the scaling parameter ε > 0. In the second part of this

9. A Glimpse into Segmentation of Images

23

section we revisit the Laplacian ﬁlter and we generalize the linearity assumption of D. Marr and E. Hildreth [1] from linear to curved contours. 9.2.1

Construction of the Convolution of I

Let ρ : RN → R, N ≥ 1, be a suﬃciently smooth function such that ρ(x) dx = 1. ρ≥0 and RN

Associate with the image I, ρ, and a scaling parameter ε > 0 the normalized convolution x−y 1 def Iε (x) = (I ∗ ρε )(x) = N dy, (9.1) I(y) ρ ε ε RN where for x ∈ RN def

ρε (x) =

1 x ρ εN ε

and RN

ρε (x) dx = 1.

(9.2)

The function ρε plays the role of a probability density and ρε (x) dx of a probability measure. Under appropriate conditions Iε converges to the original image I as ε → 0. For a suﬃciently small ε > 0 the loci of discontinuity of the function I are transformed into loci of strong variation of the gradient of the convolution Iε . When ρ has compact support, the convolution acts locally around each point in a neighborhood whose size is of order ε. A popular choice for ρ is the Gaussian with integral normalized to one in RN deﬁned by 2 1 1 def e− 2 |x| and GN (x) dx = 1 GN (x) = √ ( 2π)N RN and the normalized Gaussian of variance ε GN ε (x) =

1 x 2 1 1 √ e− 2 | ε | N ε ( 2π)N

and RN

GN ε (x) dx = 1.

(9.3)

As noted in L. Alvarez, P.-L. Lions, and J.-M. Morel [1] in dimension N = 2, the function u(t) = G2√2t ∗ I is the solution of the parabolic equation ∂u (t, x) = ∆u(t, x), ∂t 9.2.2

u(0, x) = I(x).

Space-Frequency Uncertainty Relationship

N It is interesting to compute the Fourier transform F(GN ε ) of Gε to make explicit the N relationship between the mean square deviations of Gε and its Fourier transform N F(GN ε ). For ω ∈ R , deﬁne the Fourier transform of a function f by 1 def f (x) e−i ω·x . (9.4) F(f )(ω) = √ ( 2π)N RN

24

Chapter 1. Introduction: Examples, Background, and Perspectives

In applications the integral of the square of f often corresponds to an energy. We shall refer to the L2 -norm of f as the energy norm and the function f 2 (x) as the energy density. The Fourier transform (9.4) of the normalized Gaussian (9.3) is 2 1 1 1 def −i ω·x √ )(ω) = GN dx = √ e− 2 |εω| . (9.5) F(GN ε ε (x) e N N ( 2π) ( 2π) RN D. Marr and E. Hildreth [1] notice that there is an uncertainty relationship between the mean square deviation with respect to the energy density f (x)2 and the mean square deviation with respect to the energy density F(f )(ω)2 of its Fourier transform in dimension 1 using a result√of R. Bracewell [1, pp. 160–161], where he uses the constant 1/(2π) instead of 1/ 2π in the deﬁnition of the Fourier transform that yields 1/(4π) instead of 1/2 as a lower bound to the product of the two mean square deviations. This uncertainty relationship generalizes to dimension N . First deﬁne the notions of centro¨ıd and variance with respect to the energy density f (x)2 . Given a function f ∈ L1 (RN ) ∩ L2 (RN ) and x ∈ RN , deﬁne the centro¨ıd as 2 N x f (x) dx x = R f (x)2 dx RN def

(9.6)

and the variance as def

∆x2 = x − x2 =

|x|2 f (x)2 dx − |x|2 . 2 dx f (x) RN

RN

Theorem 9.1 (Uncertainty relationship). Given N ≥ 1 and f ∈ W 1,1 (RN ) ∩ W 1,2 (RN ) such that −ix f ∈ L1 (RN ) ∩ L2 (RN ), ∆x ∆ω ≥

N . 2

(9.7)

For f = GN ε 1 x 2 1 1 N ε2 √ , e− 2 | ε | ⇒ ∆x2 = N ε ( 2π)N 2 2 1 1 N fˆ(ω) = F(GN e− 2 |εω| ⇒ ∆ω2 = ε )(ω) = √ 2 ε2 ( 2π)N

f (x) = GN ε (x) =

⇒ ∆x∆ω =

N 2

N for GN ε and F(Gε ).

(9.8) (9.9) (9.10)

So the normalized Gaussian ﬁlter is indeed an optimal ﬁlter since it achieves the lower bound in all dimensions as stated by D. Marr and E. Hildreth [1] in dimension 1 in 1980. In the context of quantum mechanics, this relationship is the analogue of the Heisenberg uncertainty principle.

9. A Glimpse into Segmentation of Images 9.2.3

25

Laplacian Detector

One way to detect the edges of a regular object is to start from the convolutionsmoothed image Iε . Given a direction v, |v| = 1, and a point x ∈ R2 an edge point will correspond to a local minimum or maximum of the directional derivative def f (t) = ∇Iε (x + tv) · v with respect to t. Denote by tˆ such a point. Then a necessary condition for an extremal is D2 Iε (x + tˆv)v · v = 0. The point x ˆ = x + tˆv is a zero crossing following the terminology of D. Marr and E. Hildreth [1] of the second-order directional derivative in the direction v. Thus we are looking for the pairs (ˆ x, vˆ) verifying the necessary condition D2 Iε (ˆ x)ˆ v · vˆ = 0

(9.11)

and, more precisely, lines or curves C such that ∀x ∈ C, ∃v(x), |v(x)| = 1,

D2 Iε (x)v(x) · v(x) = 0.

(9.12)

This condition is necessary in order that, in each point x ∈ C, there exists a direction v(x) such that ∇Iε (x) · v is extremal. In order to limit the search to points x rather than to pairs (x, v), the Laplacian detector was introduced in D. Marr and E. Hildreth [1] under two assumptions: the linear variation of the intensity along the edges C of the object and the condition of zero crossing of the second-order derivative in the direction normal to C. Assumption 9.1 (Linear variation condition). The intensity Iε in a neighborhood of lines parallel to C is locally linear (aﬃne). Assumption 9.2. The zero-crossing condition is veriﬁed in all points of C in the direction of the normal n at the point, that is, D2 Iε n · n = 0 on C.

(9.13)

Under those two assumptions and for a line C, it is easy to show that the points of C verify the necessary condition ∆Iε = 0 on C.

(9.14)

Such conditions can be investigated for edges C that are the boundary Γ of a smooth domain Ω ⊂ RN of class C 2 by using the tangential calculus developed in section 5.2 of Chapter 9 and M. C. Delfour [7, 8] for objects in RN , N ≥ 1, and not just in dimension N = 2. Indeed we want to ﬁnd points x ∈ Γ such that the function ∇Iε (x)·n(x) is an extremal of the function f (t) = ∇Iε (x+t ∇bΩ (x))·∇bΩ (x) in t = 0. This yields the local necessary condition D2 Iε ∇bΩ · ∇bΩ = D2 Iε n · n = 0 on Γ of Assumption 9.2. As for Assumption 9.1, its analogue is the following.

26

Chapter 1. Introduction: Examples, Background, and Perspectives

Assumption 9.3. The restriction to the curve C of the gradient of the intensity Iε is a constant vector c, that is, ∇Iε = c,

c is a constant vector on C.

(9.15)

Indeed the Laplacian of Iε on C can be decomposed as follows (cf. section 5.2 of Chapter 9): ∆Iε = div Γ ∇Iε + D2 Iε n · n, where div Γ ∇Iε is the tangential divergence of ∇Iε that can be deﬁned as follows: div Γ ∇Iε = div (∇Iε ◦ pC )|C and pC is the projection onto C. Hence, under Assumption 9.3, div Γ (∇Iε ) = 0 on C since div Γ (∇Iε ) = div (∇Iε ◦ pΓ )|Γ = div (c)|Γ = 0, and, in view of (9.13) of Assumption 9.2, D2 Iε n · n = 0 on C. Therefore ∆Iε = div Γ (∇Iε ) + D2 Iε n · n = 0 on C, and we obtain the necessary condition ∆Iε = 0 on C.

9.3

(9.16)

Objective Functions Deﬁned on the Whole Edge

With the pioneering work of M. Kass, M. Witkin, and D. Terzopoulos [1] in 1988 we go from a local necessary condition at a point of the edge to a global necessary condition by introducing objective functionals deﬁned on the entire edge of an object. Here many computations and analytical studies can be simpliﬁed by adopting the point of view that a closed curve in the plane is the boundary of a set and using the whole machinery developed for shape and geometric analysis and the tangential and shape calculuses in Chapter 9. 9.3.1

Eulerian Shape Semiderivative

In this section we brieﬂy summarize the main elements of the velocity method. Deﬁnition 9.1. Given D, ∅ = D ⊂ RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D is the holdall or the universe. A shape functional is a map J : A → E from an admissible family A in P(D) with values in a topological vector space E (usually R).

9. A Glimpse into Segmentation of Images

27

Given a velocity ﬁeld V : [0, τ ] × RN → RN (the notation V (t)(x) = V (t, x) will often be used), consider the transformations def

T (t, X) = x(t, X),

t ≥ 0, X ∈ RN ,

(9.17)

where x(t, X) = x(t) is deﬁned as the ﬂow of the diﬀerential equation dx (t) = V (t, x(t)), t ≥ 0, dt

x(0, X) = X

(9.18)

(here the notation x → Tt (x) = T (t, x) : RN → RN will be used). The Eulerian shape semiderivative of J in Ω in the direction V is deﬁned as def

dJ(Ω; V ) = lim

t0

J(Ωt ) − J(Ω) t

(9.19)

(when the limit exists in E), where Ωt = Tt (Ω) = {Tt (x) : x ∈ Ω}. Under appropriate assumptions on the family {V (t)}, the transformations {Tt } are homeomorphisms that transport the boundary Γ of Ω onto the boundary Γt of Ωt and the interior Ω onto the interior of Ωt . 9.3.2

From Local to Global Conditions on the Edge

For simplicity, drop the subscript ε of the convolution-smoothed image Iε and assume that the edge of the object is the boundary Γ of an open domain Ω of class C 2 . Following V. Caselles, R. Kimmel, and G. Sapiro [1], it is important to choose objective functionals that are intrinsically deﬁned and do not depend on an arbitrary parametrization of the boundary. For instance, given a frame D = ]0, a[ × ]0, b[ and a smoothed image I : D → R, ﬁnd an extremum of the objective functional ∂I def dΓ, (9.20) E(Ω) = Γ ∂n where the integrand is the normal derivative of I. Using the velocity method the Eulerian shape directional semiderivative is given by the expression (cf. Chapter 9) ∂I ∂ ∂I + V · n dΓ, (9.21) dE(Ω; V ) = H ∂n ∂n ∂n Γ where H = ∆bΩ is the mean curvature and n = ∇bΩ is the outward unit normal. Proceeding in a formal way a necessary condition would be ∂I ∂ ∂I + = 0 on Γ H ∂n ∂n ∂n ⇒ ∆bΩ ∇I · ∇bΩ + ∇(∇I · ∇bΩ ) · ∇bΩ = 0 ⇒ ∆bΩ ∇I · ∇bΩ + D2 I∇bΩ · ∇bΩ + D2 bΩ ∇I · ∇bΩ = 0 ⇒ D2I n · n + H

∂I = 0 on Γ. ∂n

(9.22)

28

Chapter 1. Introduction: Examples, Background, and Perspectives

This global condition is to be compared with the local condition (9.13). It can also be expressed in terms of the Laplacian and the tangential Laplacian of I as ∆I − ∆Γ I = 0 on Γ,

(9.23)

which can be compared with the local condition (9.16), ∆I = 0. This arises from the decomposition of the Laplacian of I with respect to Γ using the following identity for a smooth vector function U : div U = div Γ U + DU n · n,

(9.24)

where the tangential divergence of U is deﬁned as div Γ U = div (U ◦ pΓ )|Γ and pΓ is the projection onto Γ. Applying this to U = ∇I and recalling the deﬁnition of the tangential Laplacian ∆I = div Γ ∇I + D 2 I n · n = ∆Γ I + H

∂I + D2 I n · n, ∂n

where the tangential gradient ∇Γ I and the Laplace–Beltrami operator ∆Γ I are deﬁned as ∇Γ I = ∇(I ◦ pΓ )|Γ

9.4 9.4.1

and

∆Γ I = div Γ (∇Γ I).

Snakes, Geodesic Active Contours, and Level Sets Objective Functions Deﬁned on the Contours

In the literature the objective or energy functional is generally made up of two terms: one (image energy) that depends on the image and one (internal energy) that speciﬁes the smoothness of Γ. A general form of objective functional is def E(Ω) = g(I) dΓ, g(I) is a function of I. (9.25) Γ

The directional semiderivative with respect to a velocity ﬁeld V is given by ∂ Hg(I) + dE(Ω; V ) = g(I) n · V dΓ. (9.26) ∂n Γ This gradient will make the snakes move and will activate the contours. 9.4.2

Snakes and Geodesic Active Contours

If a gradient descent method is used to minimize (9.25) starting from an initial curve C 0 = C, the iterative process is equivalent to following the evolution C t (boundary of the smooth domain Ωt ) of the closed curve C given by the equation ∂C t = − [Ht gI + (∇gI · nt )] nt on C t , ∂t

(9.27)

9. A Glimpse into Segmentation of Images

29

where Ht is the mean curvature, nt is the unit exterior normal, and the right-hand side of the equation is formally the “derivative” of (9.25) given by (9.26). Equation (9.27) is referred to as the geodesic ﬂow. For gI = 1 it is the motion by mean curvature ∂C t = −Ht nt on C t . ∂t 9.4.3

(9.28)

Level Set Method

The idea to represent the contours C t by the zero-level set of a function ϕt (x) = ϕ(t, x) for a function ϕ : [0, τ ] × RN → R by setting

C t = x ∈ RN : ϕ(t, x) = 0 = ϕ−1 t {0} and to replace (9.27) by an equation for ϕ is due to S. Osher and J. A. Sethian [1] in 1988. This approach seems to have been simultaneously introduced in image ´, T. Coll, and F. Dibos [1] in 1993 under processing by V. Caselles, F. Catte the name “geometric partial diﬀerential equations” with, in addition to the mean curvature term, a “transport” term and by R. Malladi, J. A. Sethian, and B. C. Vemuri [1] in 1995 under the name “level set approach” combined with the notion of “extension velocity.” Let (t, x) → ϕ(t, x) : [0, τ ] × RN → R be a smooth function and Ω be a subset N of R of boundary Γ = Ω ∩ Ω such that

int Ω = x ∈ RN : ϕ(0, x) < 0 and Γ = x ∈ RN : ϕ(0, x) = 0 . (9.29) Let V : [0, τ ]×RN → RN be a suﬃciently smooth velocity ﬁeld so that the transformations {Tt } are diﬀeomorphisms. Moreover, assume that the images Ωt = Tt (Ω) verify the following properties: for all t ∈ [0, τ ]

int Ωt = x ∈ RN : ϕ(t, x) < 0 and Γt = x ∈ RN : ϕ(t, x) = 0 . (9.30) Assuming that the function ϕt (x) = ϕ(t, x) is at least of class C 1 and that ∇ϕt = 0 on ϕ−1 t {0}, the total derivative with respect to t of ϕ(t, Tt (x)) for x ∈ Γ yields d ∂ ϕ(t, Tt (x)) + ∇ϕ(t, Tt (x)) · Tt (x) = 0. ∂t dt

(9.31)

By substituting the velocity ﬁeld in (9.18), we get ∂ ϕ(t, Tt (x)) + ∇ϕ(t, Tt (x)) · V (t, Tt (x)) = 0, ∀ Tt (x) ∈ Γt ∂t ∂ ϕt + ∇ϕt · V (t) = 0 on Γt , t ∈ [0, τ ]. ⇒ ∂t

(9.32)

This last equation, the level set evolution equation, is veriﬁed only on the boundaries or fronts Γt , 0 ≤ t ≤ τ . Can we ﬁnd a representative ϕ in the equivalence class

−1 [ϕ]Ω,V = ϕ : ϕ−1 t {0} = Γt and ϕt {< 0} = int Ωt , ∀t ∈ [0, τ ]

30

Chapter 1. Introduction: Examples, Background, and Perspectives

of functions ϕ that verify conditions (9.29) and (9.30) in order to extend (9.32) from Γt to the whole RN or at least almost everywhere in RN ? Another possibility is to consider the larger equivalence class

[ϕ]Γ,V = ϕ : ϕ−1 t {0} = Γt , ∀t ∈ [0, τ ] . 9.4.4

Velocity Carried by the Normal

In D. Adalsteinsson and J. A. Sethian [1], J. Gomes and O. Faugeras [1], and R. Malladi, J. A. Sethian, and B. C. Vemuri [1], the front moves under the eﬀect of a velocity ﬁeld carried by the normal with a scalar velocity that depends on the curvatures of the level set or the front. So we are led to consider velocity ﬁelds V of the form V (t)|Γt = v(t) nt ,

x ∈ Γt ,

(9.33)

for a scalar velocity (t, x) → v(t)(x) : [0, τ ] × RN → R. By using the expression ∇ϕt /|∇ϕt | of the exterior normal nt as a function of ∇ϕt , (9.32) now becomes ∂ ϕt + v(t) |∇ϕt | = 0 on Γt . ∂t

(9.34)

For instance, consider the example of the minimization of the total length of the curve C of (9.27) for the metric g(I) dC as introduced by V. Caselles, R. Kimmel, and G. Sapiro [1]. By using the computation (9.26) of the shape semiderivative with respect to the velocity ﬁeld V of the objective functional (9.25), a natural direction of descent is given by

v(t)nt ,

∂ v(t) = − Ht g(I) + g(I) ∂nt

on Γt .

(9.35)

The normal nt and the mean curvature Ht can be expressed as a function of ∇ϕt : ∇ϕt ∇ϕt ∇ϕt and Ht = div Γt nt = div Γt = div nt = . (9.36) |∇ϕt | |∇ϕt | |∇ϕt | Γt By substituting in (9.34), we get the following evolution equation: ∂ ∇ϕt ∂ϕt g(I) ∇ϕt · − Ht g(I) + = 0 on Γt ∂t ∂nt |∇ϕt |

(9.37)

and ∂ϕt − Ht g(I) + ∂ g(I) |∇ϕt | = 0 ∂t ∂nt ϕ = ϕ0 0

on Γt , on Γ0 .

(9.38)

9. A Glimpse into Segmentation of Images

31

By substituting expressions (9.36) for nt and Ht in terms of ∇ϕt , we ﬁnally get ∇ϕt ∂ϕt − div ∇ϕt g(I) + ∇g(I) · |∇ϕt | = 0 ∂t |∇ϕt | |∇ϕt | ϕ = ϕ0 0

on Γt ,

(9.39)

on Γ0 .

The negative sign arises from the fact that we have chosen the outward rather than the inward normal. The main references to the existence and uniqueness theorems related to (9.39) can be found in Y. G. Chen, Y. Giga, and S. Goto [1]. 9.4.5

Extension of the Level Set Equations

Equation (9.39) on the fronts Γt is not convenient from both the theoretical and the numerical viewpoints. So it would be desirable to be able to extend (9.39) in a small tubular neighborhood of thickness h def

Uh (Γt ) = x ∈ RN : dΓt (x) < h

(9.40)

of the front Γt for a small h > 0 (dΓt , the distance function to Γt ). In theory, the velocity ﬁeld associated with Γt is given by ∇ϕt ∇ϕt ∇ϕt (9.41) V (t) = − div g(I) + ∇g(I) · on Γt . |∇ϕt | |∇ϕt | |∇ϕt | Given the identities (9.36), the expressions of nt and Ht extend to a neighborhood of Γt and possibly to RN if ∇ϕt is suﬃciently smooth and ∇ϕt = 0 on Γt . Equation (9.39) can be extended to RN at the price of violating the assumptions (9.29)–(9.30), either by loss of smoothness of ϕt or by allowing its gradient to be zero: ∇ϕt ∂ϕt − div ∇ϕt g(I) + ∇g(I) · |∇ϕt | = 0 ∂t |∇ϕt | |∇ϕt | ϕ = ϕ0 0

in RN ,

(9.42)

in RN .

´, T. Coll, and F. Dibos [1] In fact, the starting point of V. Caselles, F. Catte was the following geometric partial diﬀerential equation: ∂ϕt − div ∇ϕt g(I) + ν g(I) |∇ϕt | = 0 in RN , ∂t |∇ϕt | (9.43) ϕ = ϕ0 in RN 0

for a constant ν > 0 and the function g(I) = 1/(1+|∇I|2 ). They prove the existence, in dimension 2, of a viscosity solution unique for initial data ϕ0 ∈ C([0, 1] × [0, 1]) ∩ W 1,∞ ([0, 1] × [0, 1]) and g ∈ W 1,∞ (R2 ).

32

9.5 9.5.1

Chapter 1. Introduction: Examples, Background, and Perspectives

Objective Function Deﬁned on the Whole Image Tikhonov Regularization/Smoothing

The convolution is only one approach to smoothing an image I ∈ L2 (D) deﬁned in a frame D. Another way is to use the Tikhonov regularization: given ε > 0, ﬁnd a function Iε ∈ H 1 (D) that minimizes the objective functional def 1 ε|∇ϕ|2 + |ϕ − I|2 dx, Eε (ϕ) = 2 D where the value of ε can be suitably adjusted to get the desired degree of smoothness. The minimizing function Iε ∈ H 1 (D) is solution of the following variational equation: ∀ϕ ∈ H 1 (D),

ε ∇Iε · ∇ϕ + (Iε − I) ϕ dx = 0.

(9.44)

D

The price to pay now is that the computation is made on the whole image. This can be partly compensated by combining in a single operation or level of processing the smoothing of the image with the detection of the edges. To do that D. Mumford and J. Shah [1] in 1985 and [2] in 1989 introduced a new objective functional that received a lot of attention from the mathematical and engineering communities. 9.5.2

Objective Function of Mumford and Shah

Given a grey level ideal image in a ﬁxed bounded open frame D we are looking for an open subset Ω of D such that its boundary reveals the edges of the two-dimensional objects contained in the image of Figure 1.9. Deﬁnition 9.2. Let D be a bounded open subset of RN with Lipschitzian boundary. (i) An image in the frame D is speciﬁed by a function I ∈ L2 (D). (ii) We say that {Ωj }j∈J is an open partition of D if {Ωj }j∈J is a family of disjoint connected open subsets of D such that mN (∪j∈J Ωj ) = mN (D)

and

mN (∂ ∪j∈J Ωj ) = 0,

where mN is the N -dimensional Lebesgue measure. Denote by P(D) the family of all such open partitions of D. The idea behind the formulation of D. Mumford and J. Shah [1] in 1985 is to do a Tikhonov regularization Iε,j ∈ H 1 (Ωj ) of I on each element Ωj of the partition and then minimize the sum of the local minimizations over Ωj over all open

9. A Glimpse into Segmentation of Images

33

partitions of the image D: ﬁnd an open partition P = {Ωj }j∈J in P(D) solution of the minimization problem inf inf1 ε|∇ϕj |2 + |ϕj − I|2 dx (9.45) P ∈P(D)

j∈J

ϕj ∈H (Ωj )

Ωj

for some ﬁxed constant ε > 0. Observe that without the condition mN (∪j∈J Ωj ) = mN (D) the empty set would be a solution of the problem. The question of existence requires a more speciﬁc family of open partitions or a penalization term that controls the “length” of the interfaces in some sense: inf inf1 ε|∇ϕj |2 + |ϕj − I|2 dx + c HN −1 (∂ ∪j∈J Ωj ) (9.46) P ∈P(D)

j∈J

ϕj ∈H (Ωj )

Ωj

for some c > 0, where HN −1 is the Hausdorﬀ (N − 1)-dimensional measure. Equivalently, we are looking for Ω = ∪j∈J Ωj open in D such that χΩ = χD a.e. in D (χΩ and χD , characteristic functions) solution of the following minimization problem: inf inf1 ε|∇ϕ|2 + |ϕ − I|2 dx + c HN −1 (∂Ω). (9.47) Ω open ⊂D ϕ∈H (Ω) χΩ =χD a.e.

Ω

In general, the Hausdorﬀ measure is not a lower semicontinuous functional. So either the (N − 1)-Hausdorﬀ measure HN −1 is relaxed to a lower semicontinuous notion of perimeter or the minimization problem is reformulated with respect to a more suitable family of open domains and/or a space of functions larger than H 1 (Ω). Another way of looking at the problem would be to minimize the number J of connected subsets of the open partition, but this seems more diﬃcult to formalize. 9.5.3

Relaxation of the (N − 1)-Hausdorﬀ Measure

The choice of a relaxation of the (N − 1)-Hausdorﬀ measure HN −1 is critical. Here the ﬁnite perimeter of Caccioppoli (cf. section 6 in Chapter 5) reduces to the perimeter of D since the characteristic function of Ω is almost everywhere equal to the characteristic function of D. However, the relaxation of HN −1 (∂Ω) to the ´sio [8] (N − 1)-dimensional upper Minkowski content by D. Bucur and J.-P. Zole is much more interesting and in view of its associated compactness theorem (cf. section 12 in Chapter 7) it yields a unique solution to the problem. 9.5.4

Relaxation to BV-, H s -, and SBV-Functions

The other avenue to explore is to reformulate the minimization problem with respect to a space of functions larger than H 1 (Ω). One possibility is to use functions with possible jumps, such as functions of bounded variations. In dimension 1, the BV-functions can be decomposed into an

34

Chapter 1. Introduction: Examples, Background, and Perspectives

absolutely continuous function in W 1,1 (0, 1) plus a jump part at a countable number of points of discontinuity. Such functions have their analogue in dimension N . With this in mind one could consider the penalized objective function def JBV (ϕ) = |ϕ − I|2 dx + εϕ2BV(D) , (9.48) D

which is similar to the Tikhonov regularization when rewritten in the form def JH 1 (ϕ) = |ϕ − I|2 dx + εϕ2H 1 (D) = (1 + ε)|ϕ − I|2 dx + ε|∇ϕ|2 dx, D

D

where we square the penalization term to make it diﬀerentiable. Unfortunately, the square of the BV-norm is not diﬀerentiable. Going back to Figure 1.9, assume that the triangle T has an intensity of 1, the circle C has an intensity of 2/3, and the remaining part of the square S has an intensity of 1/3. The remainder of the image is set equal to 0. Then the image functional is precisely 2 1 I(x) = χT (x) + χC + χS\(T ∪C) . 3 3 We know from Theorem 6.9 in section 6.3 of Chapter 5 that the characteristic function of a Lipschitzian set is a BV-function. Therefore, if we minimize the objective functional (9.48) over all ϕ ∈ BV(D), we get exactly ϕˆ = I. If we insist on formulating the minimization problem over a Hilbert space as for the Tikhonov regularization, the norm of the penalization term can be chosen in the space H s (D) for some s ∈ (0, 1]: def JH s (ϕ) = |ϕ − I|2 dx + ε ϕ2H s (D) . D

For 0 < s < 1/2, the norm is given by the expression def

ϕH s (D) =

dx D

dy D

|ϕ(y) − ϕ(x)|2 |y − x|N +2s

and the relaxation is very close to the one in BV(D) since BV(D)∩L∞ (D) ⊂ H s (D), 0 < s < 1/2 (cf. Theorem 6.9 (ii) in Chapter 5). However, as seducing as those relaxations can be, it is not clear that the objectives of the original formulation are preserved. Take the BV-formulation. It is implicitly assumed that there are clean jumps across the interfaces. This is true in dimension 1, but not in dimension strictly greater than 1 as explained in L. Ambrosio [1], where he points out that the distributional gradient of a BV-function which is a vector measure has three parts: an absolutely continuous part, a jump part, and a nasty Cantor part. The space BV(D) contains pathological functions of Cantor–Vitali type that are continuous with an approximate diﬀerential 0 almost everywhere, and this class of functions is dense in L2 (D), making the inﬁmum of

9. A Glimpse into Segmentation of Images

35

JBV over BV(D) equal to zero without giving information about the segmentation. To get around this diﬃculty, he introduces the smaller space SBV(D) of functions whose distributional gradient does not have a Cantor part. He considers the mathematical framework for the minimization of the objective functional def |ϕ − I|2 + ε|∇ϕ|2 dx + c HN −1 (K) J(ϕ, K) = D\K

with respect to all closed sets K ⊂ Ω and ϕ ∈ H 1 (Ω\K), where I ∈ L∞ (D). This problem makes sense and has a solution in SBV(D) when K is replaced by the set Sϕ of points where the function ϕ ∈ SBV(D) has a jump, that is, def

|ϕ − I|2 + ε|∇ϕ|2 dx + c HN −1 (Sϕ ),

J(ϕ) =

D

that turns out to be well-deﬁned on SBV(D). 9.5.5

Cracked Sets and Density Perimeter

The alternate avenue to explore is to reformulate the minimization problem with respect to a more suitable family of open domains (cf. section 15 of Chapter 7). An additional reason to do that would be to remove the term on the length of the interface. When a penalization on the length of the segmentation is included, long slender objects are not “seen” by the numerical algorithms since they have too large a perimeter. In order to retain a segmentation of piecewise H 1 -functions ´sio [38] introduced the without a perimeter term, M. C. Delfour and J.-P. Zole family of cracked sets (see Figure 1.11) that yields a compactness theorem in the W 1,p -topology associated with the oriented distance function (cf. Chapter 7).

Figure 1.11. Example of a two-dimensional strongly cracked set. The originality of this approach is that it does not require a penalization term on the length of the segmentation and that, within the set of solutions, there exists one with minimum density perimeter as deﬁned by D. Bucur and ´sio [8]. J.-P. Zole

36

Chapter 1. Introduction: Examples, Background, and Perspectives

Theorem 9.2. Let D be a bounded open subset of RN and α > 0 and h > 0 be real numbers.3 Consider the families Γ = ∅ and ∀x ∈ Γ, ∃d, |d| = 1, def , F(D, h, α) = Ω ⊂ D : dΓ (x + td) ≥ α such that inf (9.49) 0 0 there exists a compact subset K of Ω such that, for all x ∈ Ω ∩ K, |∂ α f (x)| ≤ ε. Denote by C0k (Ω) the space of all such functions. Clearly C0k (Ω) ⊂ C k (Ω) ⊂ Bk (Ω) ⊂ C k (Ω). Endowed with the norm (2.16), C0k (Ω), C k (Ω), and Bk (Ω) are Banach spaces. Finally % % % def def def C k (Ω), C0∞ (Ω) = C0k (Ω), and B(Ω) = Bk (Ω). C ∞ (Ω) = k≥0

k≥0

k≥0

function f : Ω → R is uniformly continuous if for each ε > 0 there exists δ > 0 such that for all x and y in Ω such that |x − y| < δ we have |f (x) − f (y)| < ε. 3A

2. Notation and Deﬁnitions

65

When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by C0k (Ω)m or C0k (Ω, Rm ), C k (Ω)m or C k (Ω, Rm ), B k (Ω)m or Bk (Ω, Rm ), C k (Ω)m or C k (Ω, Rm ), etc. We quote the following classical compactness theorem. Theorem 2.4 (Ascoli–Arzel`a theorem). Let Ω be a bounded open subset of RN . A subset K of C(Ω) is precompact in C(Ω) provided the following two conditions hold: (i) there exists a constant M such that for all f ∈ K and x ∈ Ω, |f (x)| ≤ M ; (ii) for every ε > 0 there exists δ > 0 such that if f ∈ K, x, y in Ω, and |x−y| < δ, then |f (x) − f (y)| < ε. 2.6.2

H¨ older (C 0, ) and Lipschitz (C 0,1 ) Continuous Functions

Given λ, 0 < λ ≤ 1, a function f is (0, λ)-H¨ older continuous in Ω if ∃c > 0, ∀x, y ∈ Ω,

|f (y) − f (x)| ≤ c |x − y|λ .

When λ = 1, we also say that f is Lipschitz or Lipschitz continuous. Similarly for k ≥ 1, f is (k, λ)-H¨ older continuous in Ω if |∂ α f (y) − ∂ α f (x)| ≤ c |x − y|λ .

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω,

Denote by C k,λ (Ω) the space of all (k.λ)-H¨older continuous functions on Ω. Deﬁne for k ≥ 0 the subspaces4 & ∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω def (2.20) C k,λ (Ω) = f ∈ C k (Ω) : α |∂ f (y) − ∂ α f (x)| ≤ c |x − y|λ of C k (Ω). By deﬁnition for each α, 0 ≤ |α| ≤ k, ∂ α f has a unique, bounded, continuous extension to Ω. Endowed with the norm |∂ α f (y) − ∂ α f (x)| def f C k,λ (Ω) = max f C k (Ω) , max sup (2.21) |x − y|λ 0≤|α|≤k x,y∈Ω x=y

C k,λ (Ω) is a Banach space. Finally denote by C0k,λ (Ω) the space C k,λ (Ω) ∩ C0k (Ω). 2.6.3

Embedding Theorem

In general C k+1 (Ω) ⊂ C k,1 (Ω), but the inclusion is true for a large class of domains, including convex domains.5 We ﬁrst quote the following embedding theorem. 4 The notation C k,λ (Ω) should not be confused with the notation C k,λ (Ω) for (k, λ)-H¨ older continuous functions in Ω without the uniform boundedness assumption in Ω. In particular, C k,λ (RN ) is contained in but not equal to C k,λ (RN ). 5 For instance the convexity can be relaxed to the more general condition: there exists M > 0 such that for all x and y in Ω, there exists a path γx,y in Ω (that is, a C 1 injective map from the

interval [0, 1] into Ω with γ(0) = x, γ(1) = y, and

1 0

γ (t) dt ≤ M |x − y|).

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Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 2.5 (R. A. Adams [1]). Let k ≥ 0 be an integer and 0 < ν < λ ≤ 1 be real numbers. Then the following embeddings exist: C k+1 (Ω) → C k (Ω),

(2.22)

(Ω) → C (Ω),

(2.23)

C C

k,λ

k,λ

k

(Ω) → C

k,ν

(Ω).

(2.24)

If Ω is bounded, then the embeddings (2.23) and (2.24) are compact. If Ω is convex, we have the further embeddings C k+1 (Ω) → C k,1 (Ω), C

k+1

(Ω) → C

k,ν

(Ω).

(2.25) (2.26)

If Ω is convex and bounded, then embeddings (2.22) and (2.26) are compact. As a consequence of the second part of the theorem, the deﬁnition of C k,λ (Ω) simpliﬁes when Ω is convex: def

C k,λ (Ω) = f ∈ Bk (Ω) : ∀α, |α| = k, ∃c > 0, ∀x, y ∈ Ω, |∂ α f (y) − ∂ α f (x)| ≤ c |x − y|λ ,

(2.27)

and its norm is equivalent to the norm def

|∂ α f (y) − ∂ α f (x)| . |x − y|λ x,y∈Ω

f C k,λ (Ω) = f C k (Ω) + max sup |α|=k

(2.28)

x=y

When f is a vector function from Ω to Rm , the corresponding spaces will be denoted by C k,λ (Ω)m or C k,λ (Ω, Rm ). 2.6.4

Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to Path-Connected Domains via the Geodesic Distance

By Rademacher’s theorem (cf., for instance, L. C. Evans and R. F. Gariepy [1]) k+1,∞ C k,1 (Ω) ⊂ Wloc (Ω)

and, when Ω is convex, C k,1 (Ω) = W k+1,∞ (Ω). This last convexity assumption can be relaxed as follows. From Rademacher’s ´zis [1, p. 154]) theorem and the inequality (cf. H. Bre ∀u ∈ W 1,∞ (Ω) and ∀x, y ∈ Ω, we have the following theorem.

|u(x) − u(y)| ≤ ∇uL∞ (Ω) distΩ (x, y)

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

67

Theorem 2.6. Let Ω be a bounded, path-connected, open subset of RN such that ∃c, ∀x, y ∈ Ω,

distΩ (x, y) ≤ c |x − y|.

(2.29)

Then we have W 1,∞ (Ω) = C 0,1 (Ω) algebraically and topologically, and there exists a constant c such that ∀u ∈ W 1,∞ (Ω) and ∀x, y ∈ Ω,

|u(x) − u(y)| ≤ c ∇uL∞ (Ω) |x − y|.

Hence for all integers k ≥ 0, W k+1,∞ (Ω) = C k,1 (Ω) algebraically and topologically. This is true for Lipschitzian domains (D. Gilbarg and N. S. Trudinger [1]). Corollary 1. Let Ω be a bounded, open, path-connected, and locally Lipschitzian6 subset of RN . Then property (2.29) and Theorem 2.6 are veriﬁed. Proof. By Theorem 5.8 later in section 5.4 of this chapter.

3

Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

Open domains in the Euclidean space RN are classically described by introducing at each point of their boundary a local diﬀeomorphism (that is, deﬁned in a neighborhood of the point) that locally ﬂattens the boundary. The smoothness of the boundary is determined by the smoothness of the local diﬀeomorphism. After introducing the main deﬁnition in section 3.1, we brieﬂy recall the deﬁnition of an (embedded) submanifold of codimension greater than or equal to 1 in section 3.2 in order to make sense of the associated boundary integral and, more generally, the integral over smooth submanifolds of arbitrary dimension in RN . The deﬁnition of the integral can be generalized by introducing the Hausdorﬀ measures that extend the integration theory from smooth submanifolds to arbitrary Hausdorﬀ measurable subsets of RN , thus making the writing of the integral completely independent of the choice of the local diﬀeomorphisms. Section 3.3 completes the section by deﬁning the fundamental forms and curvatures for a smooth submanifold of RN of codimension 1.

3.1

Sets of Classes C k and C k,

Let {e1 , . . . , eN } be the standard unit orthonormal basis in RN . We use the notation ζ = (ζ , ζN ) for a point ζ = (ζ1 , . . . , ζN ) in RN , where ζ = (ζ1 , . . . , ζN −1 ). Denote by B the open unit ball in RN and deﬁne the sets def

B0 = {ζ ∈ B : ζN = 0} , def

B+ = {ζ ∈ B : ζN > 0} ,

def

B− = {ζ ∈ B : ζN < 0} .

(3.1) (3.2)

The main elements entering Deﬁnition 3.1 are illustrated in Figure 2.1 for N = 2. 6 Cf. Deﬁnition 5.2 of section 5 and Theorem 5.3, which says that a locally Lipschitzian domain is equi-Lipschitzian when ∂Ω is bounded.

68

Chapter 2. Classical Descriptions of Geometries and Their Properties ζN gx B+ ζ

set Ω B0 ∂Ω

x U (x)

hx = gx−1

hx (x) = (0, 0) B−

ball B

Figure 2.1. Diﬀeomorphism gx from U (x) to B. Deﬁnition 3.1. Let Ω be a subset of RN such that ∂Ω = ∅, 0 ≤ k be an integer or +∞, and 0 ≤ ≤ 1 be a real number. (i)

- Ω is said to be locally of class C k at x ∈ ∂Ω if there exist (a) a neighborhood U (x) of x, and (b) a bijective map gx : U (x) → B with the following properties: def gx ∈ C k U (x); B , hx = gx−1 ∈ C k B; U (x) , int Ω ∩ U (x) = hx (B+ ), def Γx = ∂Ω ∩ U (x) = hx (B0 ), B0 = gx (Γx ).

(3.3) (3.4) (3.5)

- Given 0 < < 1, Ω is said to be locally (k, )-H¨ olderian at x ∈∂Ω if ∈ C k, U (x), B with conditions (a) and (b) are satisﬁed with a map g x −1 k, B, U (x) . inverse hx = gx ∈ C - Ω is said to be locally k-Lipschitzian at x ∈ ∂Ω if conditions (a) and−1(b) k,1 are satisﬁed with a map g U (x), B with inverse hx = gx ∈ ∈ C x C k,1 B, U (x) . (ii)

- Ω is said to be locally of class C k if, for each x ∈ ∂Ω, Ω is locally of class C k at x. - Given 0 < < 1, Ω is said to be locally (k, )-H¨ olderian if, for each x ∈ ∂Ω, Ω is locally (k, )-H¨olderian at x. - Ω is said to be locally k-Lipschitzian if, for each x ∈ ∂Ω, Ω is locally k-Lipschitzian at x.

Remark 3.1. By deﬁnition int Ω = ∅ and int Ω = ∅. The above deﬁnitions are usually given for an open set Ω called a domain. This terminology naturally arises in partial diﬀerential equations, where this open set is indeed the domain in which the solution

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

69

of the partial diﬀerential equation is deﬁned.7 At this point it is not necessary to assume that the set Ω is open. The family of diﬀeomorphisms {gx : x ∈ ∂Ω} characterizes the equivalence class of sets [Ω] = {A ⊂ RN : int A = int Ω and ∂A = ∂Ω} with the same interior and boundary. The sets int Ω and ∂Ω are invariants for the equivalence class [Ω] of sets of class C k, . The notation Γ for ∂Ω and the standard terminology domain for the unique (open) set int Ω associated with the class [Ω] will be used. In what follows, a C k, -mapping gx with C k, -inverse will be called a C k, -diﬀeomorphism. Consider the solution y = y(Ω) of the Dirichlet problem on the bounded open domain Ω: −∆y = f in Ω,

y = g on ∂Ω.

(3.6)

When Ω is of class C ∞ , the solution y(Ω) can be sought in any Sobolev space H m (Ω), m ≥ 1, by choosing suﬃciently smooth data f and g and appropriate compatibility conditions. However, when Ω is only of class C k , 1 ≤ k < ∞, m cannot be made arbitrary large by choosing smoother data f and g and appropriate compatibility conditions. The reader is referred to R. Dautray and J.-L. Lions [1, Chap. VII, sect. 3, pp. 1271–1304] for classical smoothness results of solutions to elliptic problems in domains of class C k . It is important to understand that we shall consider shape problems involving the solution of a boundary value problem over domains with minimal smoothness. Since we know that, at least for elliptic problems, the smoothness of the solution depends not only on the smoothness of the data but also on the smoothness of the domain. This issue will be of paramount importance to make sure that the shape problems are well posed. Remark 3.2. We shall see in section 5 that domains that are locally the epigraph of a C k, , k ≥ 1 (resp., Lipschitzian), function are of class C k, (resp., C 0,1 ), but domains which are of class C 0,1 are not necessarily locally the epigraph of a Lipschitzian function (cf. Examples 5.1 and 5.2). Nevertheless globally C 0,1 -mappings with a C 0,1 inverse are important since they transport Lp -functions onto Lp -functions and W 1,p -functions ˇas [1, Lems. 3.1 and 3.2, pp. 65–66]). onto W 1,p -functions (cf. J. Nec For sets of class C 1 , the unit exterior normal to the boundary Γ = ∂Ω can be characterized through the Jacobian matrices of gx and hx . By deﬁnition of B0 , {e1 , . . . , eN −1 } ⊂ B0 and the tangent space Ty Γ, Γ = ∂Ω, at y to Γx is the vector space spanned by the N − 1 vectors {Dhx (ζ , 0)ei : 1 ≤ i ≤ N − 1} ,

(ζ , 0) = gx (y) ∈ B0 ,

(3.7)

7 Classically for k ≥ 1 a bounded open domain Ω in RN is said to be of class C k if its boundary is a C k -submanifold of RN of codimension 1 and Ω is located on one side of its boundary Γ = ∂Ω (cf. S. Agmon [1, Def. 9.2, p. 178]).

70

Chapter 2. Classical Descriptions of Geometries and Their Properties

where Dhx (ζ) is the Jacobian matrix of hx at the point ζ: def

(Dhx )m = ∂m (hx ) . So from (3.7) a normal vector ﬁeld to Γx at y ∈ Γx is given by mx (y) = − ∗(Dhx )−1 (ζ , 0) eN = − ∗Dgx (y) eN ,

hx (ζ , 0) = y,

(3.8)

since −mx (y) · Dhx (ζ , 0) ei = eN · ei = δiN ,

1 ≤ i ≤ N.

Thus the outward unit normal ﬁeld n(y) at y ∈ Γx is given by (Dhx )−1 (ζ , 0) eN , | ∗(Dhx )−1 (ζ , 0) eN | ∗ (Dhx )−1 (h−1 x (y)) eN . n(y) = − ∗ −1 | (Dhx ) (h−1 x (y)) eN |

∀hx (ζ , 0) = y ∈ Γx , ∀y ∈ Γx ,

n(y) = −

∗

(3.9)

It can be veriﬁed that n is uniquely deﬁned on Γ by checking that for y ∈ Γx ∩ Γx , n is uniquely deﬁned by (3.9).

3.2 3.2.1

Boundary Integral, Canonical Density, and Hausdorﬀ Measures Boundary Integral for Sets of Class C 1

The family of neighborhoods U (x) associated with all the points x of Γ is an open cover of Γ. If Γ = ∂Ω is assumed to be compact, then there exists a ﬁnite open subcover: that is, there exists a ﬁnite sequence of points {xj : 1 ≤ j ≤ m} of Γ such that Γ ⊂ U1 ∪ · · · ∪ Um , where Uj = U (xj ). For simplicity, index all the previous symbols by j instead of xj . The boundary integration on Γ is obtained by using a partition of unity {rj : 1 ≤ j ≤ m} for the family of open neighborhoods {Uj : 1 ≤ j ≤ m} of Γ: r ∈ D(Uj ), 0 ≤ rj (x) ≤ 1, j m (3.10) rj (x) = 1 in a neighborhood U of Γ, j=1

such that U ⊂ ∪m j=1 Uj , where D(Uj ) is the set of all inﬁnitely continuously diﬀerentiable functions with compact support in Uj . If f ∈ C(Γ), then (f rj ) ◦ hj ∈ C(B0 ),

1 ≤ j ≤ m.

Deﬁne the boundary integral of f rj on Γj as def f rj dΓ = (f rj ) ◦ hj (ζ , 0) ωj (ζ ) dζ , Γj

B0

Γj = U (xj ) ∩ Γ,

(3.11)

(3.12)

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

71

where ωj = ωxj and ωx is the density term ωx (ζ ) = |mx (hx (ζ , 0))| | det Dhx (ζ , 0)|, ∗

mx (y) = − (Dhx )

−1

(h−1 x (y)) eN

= − Dgx (y) eN .

From this deﬁne the boundary integral of f on Γ as m def f dΓ = f rj dΓ. Γ

j=1

(3.13)

∗

(3.14)

Γj

The expression on the right-hand side of (3.12) results from the parametrization of the boundary Γj by B0 through the diﬀeomorphism hj . 3.2.2

Integral on Submanifolds

In diﬀerential geometry there is a general procedure to deﬁne the canonical density for a d-dimensional submanifold V in RN parametrized by a C k -mapping, k ≥ 1 (cf., for instance, M. Berger and B. Gostiaux [1, Def. 2.1.1, p. 48 (56) and Prop. 6.62, p. 214 (239)]). Deﬁnition 3.2. Let ∅ = S ⊂ RN , let k ≥ 1 and 1 ≤ d ≤ N be integers, and let 0 ≤ ≤ 1 be a real number. (i) Given x ∈ S, S is said to be locally a C k - (resp., C k, -) submanifold of dimension d at x in RN if there exists an open subset U (x) of RN containing k k, x and a C - (resp., C -) diﬀeomorphism gx from U (x) onto its open image gx U (x) , such that gx U (x) ∩ S = gx U (x) ∩ Rd , where Rd = (x1 , . . . , xd , 0, . . . , 0) ∈ RN : ∀(x1 , . . . , xd ) ∈ Rd . (ii) S is said to be a C k - (resp., C k, -) submanifold of dimension d in RN if, for each x ∈ S, it is locally a C k - (resp., C k, -) submanifold of dimension d at x in RN . For k ≥ 1 the canonical density ωx on the submanifold S at a point y ∈ U (x) ∩ S is given by ' ωx = | det B|, where the d × d matrix B is given by (B)ij = (Dhx ei ) · (Dhx ej ), 1 ≤ i, j ≤ d,

hx = gx−1 on gx (U (x)).

In the special case d = N − 1, it is easy to verify that ∗ C C ∗C c ∗ , B = ∗C C, Dhx Dhx = ∗ ∗ cC cc

72

Chapter 2. Classical Descriptions of Geometries and Their Properties

where C is the (N × (N − 1))-matrix and c is the N -vector deﬁned by Cij = {Dhx }ij ,

1 ≤ i ≤ N, 1 ≤ j ≤ N − 1,

c = Dhx eN .

Denote by M (A) the matrix of cofactors associated with a matrix A: M (A)ij is equal to the determinant of the matrix obtained after deleting the ith row and the jth column times (−1)i+j . Then M (A) = (det A) ∗ A−1 , M ( ∗ A) = ∗ M (A), and for two invertible matrices A1 and A2 , M (A1 A2 ) = M (A1 )M (A2 ). As a result det B = M ( ∗Dhx Dhx )N N = eN · M ( ∗Dhx Dhx )eN , where M ( ∗Dhx Dhx )N N is the N N -cofactor of the matrix ∗Dhx Dhx . Then M ( ∗Dhx Dhx )N N = eN · M ( ∗Dhx Dhx )eN = eN · M ( ∗Dhx ) M (Dhx )eN = |M (Dhx )eN |2 . In view of the previous considerations and (3.13) √ det B = |M (Dhx )eN | = | det Dhx | | ∗(Dhx )−1 eN | = ωx . 3.2.3

Hausdorﬀ Measures

Deﬁnition 3.2 gives the classical construction of a d-dimensional surface measure on the boundary of a C 1 -domain. In 1918, F. Hausdorff [1] introduced a ddimensional measure in RN which gives the same surface measure for smooth submanifolds but is deﬁned on all measurable subsets of RN . When d = N , it is equal to the Lebesgue measure. To complete the discussion we quote the deﬁnition from F. Morgan [1, p. 8] or L. C. Evans and R. F. Gariepy [1, p. 60]. Deﬁnition 3.3. For any subset S of RN , deﬁne the diameter of S as def

diam (S) = sup{|x − y| : x, y ∈ S}. Let α(d) denote the Lebesgue measure of the unit ball in Rd . The d-dimensional Hausdorﬀ measure Hd (A) of a subset A of RN is deﬁned by the following process. For δ small, cover A eﬃciently by countably many sets Sj with diam (Sj ) ≤ δ, add up all the terms d

α(d) (diam (Sj )/2) , and take the limit as δ → 0: def

Hd (A) = lim

δ0

inf

A⊂∪Sj diam (Sj )≤δ

j

α(d)

d diam (Sj ) , 2

where the inﬁmum is taken over all countable covers {Sj } of A whose members have diameter at most δ.

3. Sets Locally Described by an Homeomorphism or a Diﬀeomorphism

73

For 0 ≤ d < ∞, Hd is a Borel regular measure, but not a Radon measure for d < N since it is not necessarily ﬁnite on each compact subset of RN . The Hausdorﬀ dimension of a set A ⊂ RN is deﬁned as def

Hdim (A) = inf{0 ≤ s < ∞ : Hs (A) = 0}.

(3.15)

By deﬁnition Hdim (A) ≤ N and ∀k > Hdim (A),

Hk (A) = 0.

If a submanifold S of dimension d, 1 ≤ d < N , of Deﬁnitions 3.2 and 3.3 is characterized by a single C 1 -diﬀeomorphism g, that is, h = g −1 ,

g(S) = Rd and S = h(Rd ), then for any Lebesgue-measurable set E ⊂ Rd ω dx = Hd (h(E)). E

This is a generalization to submanifolds of codimension greater than 1 of formula (3.12). The deﬁnition of the Hausdorﬀ measure and the Hausdorﬀ dimension extend from integers d to reals s, 0 ≤ s ≤ ∞, by modifying Deﬁnition 3.3 as follows. Deﬁnition 3.4. For any real s, 0 ≤ s ≤ ∞, the s-dimensional Hausdorﬀ measure Hs (A) of a subset A of RN is deﬁned by the following process. For δ small, cover A eﬃciently by countably many sets Sj with diam (Sj ) ≤ δ, and add all the terms α(s) (diam (Sj )/2)d , where def

α(s) =

π s/2 , Γ(s/2 + 1)

Take the limit as δ → 0: def

Hs (A) = lim

δ0

inf

def

∞

Γ(t) =

A⊂∪Sj diam (Sj )≤δ

e−x xt−1 .

(3.16)

0

j

α(s)

d diam (Sj ) , 2

where the inﬁmum is taken over all countable covers {Sj } of A whose members have diameter at most δ. The Hausdorﬀ dimension is deﬁned by the same formula (3.15).

3.3

Fundamental Forms and Principal Curvatures

Consider a set Ω locally of class C 2 in RN . Its boundary Γ = ∂Ω is an (N − 1)dimensional submanifold of RN of class C 2 . At each point x ∈ Γ there is a C 2 diﬀeomorphism gx from a neighborhood U (x) of x onto B. Denoting its inverse by

74

Chapter 2. Classical Descriptions of Geometries and Their Properties

hx = gx−1 , the covariant basis at a point y ∈ U (x) ∩ Γ is deﬁned as def

aα (y) =

∂hx (ζ , 0), ∂ζα

α = 1, . . . , N − 1,

hx (ζ , 0) = y,

and aN is chosen as the inward unit normal : def

aN (y) =

∗

−1 (Dhx (h−1 eN x (y))) . −1 | ∗(Dhx (hx (y)))−1 eN |

The standard convention that the Greek indices range from 1 to N − 1 and the Roman indices from 1 to N will be followed together with Einstein’s rule of summation over repeated indices. The associated contravariant basis {ai } = {ai (y)} is deﬁned from the covariant one {ai } = {ai (y)} as ai · aj = δij , where δij is the Kronecker index function. The ﬁrst, second, and third fundamental forms a, b, and c are deﬁned as def

aαβ = aα · aβ ,

def

bαβ = −aα · aN,β ,

def

cαβ = bλα bλβ ,

where aN,β =

∂aN , ∂ζβ

bλα = aλ · aµ bµα .

The above deﬁnitions extend to sets of class C 1,1 for which hx ∈ C 1,1 (B) and hence hx ∈ C 1,1 (B0 ). So by Rademacher’s theorem in dimension N − 1 (cf., for instance, L. C. Evans and R. F. Gariepy [1]), hx ∈ W 2,∞ (B0 ) and the deﬁnitions of the second and third fundamental forms still make sense HN −1 almost everywhere on Γ. The eigenvalues of bαβ are the (N − 1) principal curvatures κi , 1 ≤ i ≤ N − 1, of the submanifold Γ. The mean curvature H and the Gauss curvature K are deﬁned as N −1 N −1 ( 1 def def H = κα and K = κα . N − 1 α=1 α=1 The choice of the inner normal for aN is necessary to make the principal curvatures of the sphere (boundary of the ball) positive. The factor 1/(N − 1) is used to make the mean curvature of the unit sphere equal to 1 in all dimensions. The reader should keep in mind the long-standing diﬀerences in usage between geometry and partial diﬀerential equations, where the outer unit normal is used in the integration-byparts formulae for the Euclidean space RN . For integration by parts on submanifolds of RN the sum of the principal curvatures will naturally occur rather than the mean curvature. It will be convenient to redeﬁne H as the sum of the principal curvatures and introduce the notation H for the classical mean curvature: N −1 1 κα , H = κα . H = N − 1 α=1 α=1 def

N −1

def

(3.17)

4. Sets Globally Described by the Level Sets of a Function

4

75

Sets Globally Described by the Level Sets of a Function

From the deﬁnition of a set of class C k, , k ≥ 1 and 0 ≤ ≤ 1, the set Ω can also be locally described by the level sets of the C k, -function def

fx (y) = gx (y) · eN ,

(4.1)

since by deﬁnition int Ω ∩ U (x) = {y ∈ U (x) : fx (y) > 0}, ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0}. The boundary ∂Ω is the zero level set of fx and the gradient ∇fx (y) = ∗Dgx (y)eN = 0 is normal to that level set. Thus the exterior normal to Ω is given by n(y) = −

∗ ∗ ∇fx (y) Dgx (y)eN (Dhx (gx (y)))−1 eN =− ∗ =− ∗ . |∇fx (y)| | Dgx (y)eN | | (Dhx (gx (y)))−1 eN |

From this local construction of the functions {fx : x ∈ ∂Ω}, a global function on RN can be constructed to characterize Ω. Theorem 4.1. Given k ≥ 1, 0 ≤ ≤ 1, and a set Ω in RN of class C k, with compact boundary, there exists a Lipschitz continuous f : RN → R such that int Ω = {y ∈ RN : f (y) > 0}

and

∂Ω = {y ∈ RN : f (y) = 0}

(4.2)

and a neighborhood W of ∂Ω such that f ∈ C k, (W ) and ∇f = 0 on W and n = −

∇f , |∇f |

(4.3)

where n is the outward unit normal to Ω on ∂Ω. Proof. Construction of the function f . Fix x ∈ ∂Ω and consider the function fx deﬁned by (4.1). By continuity of ∇fx , let V (x) ⊂ U (x) be a neighborhood of x such that 1 ∀y ∈ V (x), |∇fx (y) − ∇fx (x)| ≤ |∇fx (x)|. 2 Furthermore, let W (x) be a bounded open neighborhood of x such that W (x) ⊂ V (x). For ∂Ω compact there exists a ﬁnite subcover {Wj = W (xj ) : 1 ≤ j ≤ m} of ∂Ω for a ﬁnite sequence {xj : 1 ≤ j ≤ m} ⊂ ∂Ω. Index all the previous symbols by j instead of xj and deﬁne W = ∪m j=1 Wj . Let {rj : 1 ≤ j ≤ m} be a partition of unity for {Vj = V (xj )} such that r ∈ D(Vj ), 0 ≤ rj (x) ≤ 1, 1 ≤ j ≤ m, j m (4.4) rj (x) = 1 in W , j=1

76

Chapter 2. Classical Descriptions of Geometries and Their Properties

where, by deﬁnition, W is an open neighborhood of ∂Ω such that W ⊂ V = ∪m j=1 Vj and D(Vj ) is the set of all inﬁnitely continuously diﬀerentiable functions with compact support in Vj . Deﬁne the function f : RN → R: def

f (y) = dW ∪Ωc − dW ∪Ω +

m j=1

rj (y)

fj (y) , |∇fj (xj )|

where dA (x) = inf{|y − x| : y ∈ A} is the distance function from a point x to a nonempty set A of RN . The function dA is Lipschitz continuous with Lipschitz constant equal to 1. Since for all j, fj ∈ C k, (Vj ), rj ∈ D(Vj ), and W j ⊂ Vj , rj fj ∈ C k, (Vj ) with compact support in Vj . Therefore since k ≥ 1, rj fj is Lipschitz continuous in RN . So, by deﬁnition, f is Lipschitz continuous on RN as the ﬁnite sum of Lipschitz continuous functions on RN . Properties (4.2). Introduce the index set def

J(y) = {j : 1 ≤ j ≤ m, rj (y) > 0} for y ∈ V . For all y ∈ W , J(y) = ∅, dW ∪Ωc = 0 = dW ∪Ω , and f (y) =

rj (y)

j∈J(y)

fj (y) . |∇fj (xj )|

(4.5)

For y ∈ ∂Ω = W ∩ ∂Ω, fj (y) = 0 for all j ∈ J(y) and hence f (y) = 0; for y ∈ int Ω ∩ W , fj (y) > 0 and rj (y) > 0 for all j ∈ J(y) and hence f (y) > 0; for y ∈ int Ωc ∩ W , fj (y) < 0 and)rj (y) > 0 for all j ∈ J(y) and hence f (y) < 0. For y ∈ Ω\W , fj ≥ 0, rj ≥ 0, m rj fj ≥ 0, dW ∪Ωc > 0, dW ∪Ω = 0, and f > 0; )mj=1 c for y ∈ Ω \W , fj ≤ 0, rj ≥ 0, j=1 rj fj ≤ 0, dW ∪Ωc = 0, dW ∪Ω > 0, and f < 0. So we have proven that ∂Ω ⊂ f −1 (0), int Ω ⊂ {f > 0}, and int Ωc ⊂ {f < 0}. Hence f has the properties (4.2). Properties (4.3). Recall that on W the function f is given by expression (4.5). It belongs to C k, (W ), and, a fortiori, to C 1 (W ), as the ﬁnite sum of C k, -functions on W since k ≥ 1. The gradient of f in W is given by ∇f (y) =

∇rj (y)

j∈J(y)

fj (y) ∇fj (y) + rj (y) . |∇fj (xj )| |∇fj (xj )|

It remains to show that it is nonzero on ∂Ω. On ∂Ω ∩ Vj , fj = 0 and ∀j ∈ J(y),

∇fj (y) = −n(y). |∇fj (y)|

Therefore in W , ∇f can be rewritten in the form |∇fj (y)| rj (y) ∇f (y) = −n(y) |∇fj (xj )| j∈J(y) and |∇fj (y)| rj (y) 1 − ∇f (y) + n(y) = n(y) . |∇fj (xj )| j∈J(y)

4. Sets Globally Described by the Level Sets of a Function Finally, by construction of W ,

|∇f (y)|≥ |n(y)| − |n(y)| ≥1−

j∈J(y)

rj (y)

j∈J(y)

77

|∇fj (y)| rj (y) 1 − |∇fj (xj )|

1 1 = 2 2

and ∇f = 0 in W . This proves properties (4.3). This theorem has a converse. Theorem 4.2. Associate with a continuous function f : RN → R the set def

Ω = {y ∈ RN : f (y) > 0}.

(4.6)

f −1 (0) = {y ∈ RN : f (y) = 0} = ∅

(4.7)

Assume that def

and that there exists a neighborhood V of f −1 (0) such that f ∈ C k, (V ) for some k ≥ 1 and 0 ≤ ≤ 1 and that ∇f = 0 in f −1 (0). Then Ω is a set of class C k, , int Ω = Ω

and

∂Ω = f −1 (0).

(4.8)

Proof. By continuity of f , Ω is open, int Ω = Ω, Ω is closed, Ω ⊂ {y ∈ RN : f (y) ≥ 0},

Ω = Ω = {y ∈ RN : f (y) ≤ 0},

and ∂Ω ⊂ f −1 (0). Conversely for each x ∈ ∂Ω, deﬁne the function def

g(t) = f (x + t∇f (x)). There exists δ > 0 such that for all |t| < δ, x + t∇f (x) ∈ V and the function g is C 1 in (−δ, δ). Hence since g(0) = 0 and g (t) = ∇f (x + t∇f (x)) · ∇f (x)

t

∇f (x + s∇f (x)) · ∇f (x) ds.

f (x + t∇f (x)) = 0

By continuity of ∇f in V and the fact that ∇f (x) = 0, there exists δ , 0 < δ < δ, such that ∀s, 0 ≤ |s| ≤ δ ,

∇f (x + s∇f (x)) · ∇f (x) ≥

1 |∇f (x)|2 > 0. 2

Hence for all t, 0 < t ≤ δ , f (x + t∇f (x)) > 0. So any point in f −1 (0) can be approximated by a sequence {xn = x + tn ∇f (x) : n ≥ 1, 0 < tn ≤ δ } in Ω, tn → 0, and f −1 (0) ⊂ Ω. Similarly, using a sequence of negative tn ’s, f −1 (0) ⊂ Ω and hence f −1 (0) ⊂ ∂Ω. This proves (4.8).

78

Chapter 2. Classical Descriptions of Geometries and Their Properties

Fix x ∈ ∂Ω = f −1 (0). Since ∇f (x) = 0, deﬁne the unit vector eN (x) = ∇f (x)/|∇f (x)|. Associate with eN (x) unit vectors e1 (x), . . . , eN −1 (x), which form an orthonormal basis in RN with eN (x). Deﬁne the map gx : V → RN as f (y) def N −1 gx (y) = {(y − x) · eα (x)}α=1 , ⇒ gx ∈ C k, (V ; RN ). |∇f (x)| The transpose of the Jacobian matrix of gx is given by ∗

Dgx (y) = (e1 (x), . . . , eN −1 (x), ∇f (y)/|∇f (x)|)

and Dgx (x) = I, the identity matrix in the {ei (x)} reference system. By the inverse mapping theorem gx has a C k, inverse hx in some neighborhood U (x) of x in V . Therefore, Ω ∩ U (x) = {y ∈ U (x) : f (x) > 0} = int Ω ∩ U (x), {y ∈ U (x) : f (x) = 0} = ∂Ω ∩ U (x), and the set Ω is of class C k, . We complete this section with the important theorem of Sard. ´ [3, Vol. III, sect. 16.23, p. 167]). Let X and Y Theorem 4.3 (J. Dieudonne be two diﬀerential manifolds, f : X → Y be a C ∞ -mapping, and E be the set of critical points of f . Then f (E) is negligible in Y , and Y − f (E) is dense in Y . Combining this theorem with Theorem 4.1, this means that, for almost all t in the range of a C ∞ -function f : RN → R, the set

y ∈ RN : f (x) > t is of class C ∞ in the sense of Deﬁnition 3.1. The following theorem extends and completes Sard’s theorem. Theorem 4.4 (H. Federer [5, Thm. 3.4.3, p. 316]). If m > ν ≥ 0 and k ≥ 1 are integers, A is an open subset of Rm , B ⊂ A, Y is a normed vector space, and f : A → Y is a map of class k,

dim Im Df (x) ≤ ν for x ∈ B,

then Hν+(m−ν)/k (f (B)) = 0.

5

Sets Locally Described by the Epigraph of a Function

After diﬀeomorphisms and level sets, the local epigraph description provides a third point of view and another way to characterize the smoothness of a set or a domain. For instance, Lipschitzian domains, which are characterized by the property that their boundary is locally the epigraph of a Lipschitzian function, play a central role in the theory of Sobolev spaces and partial diﬀerential equations. They can equivalently be characterized by the geometric uniform cone property. We shall see that sets that are locally the epigraph of C k, -functions are equivalent to sets that are locally of class C k, for k ≥ 1.

5. Sets Locally Described by the Epigraph of a Function

5.1

79

Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs and the Space H of Dominating Functions

Let eN in RN be a unit reference vector and introduce the following notation: H = {eN }⊥ , def

∀ζ ∈ RN ,

ζ = PH (ζ), def

def

ζN = eN · ζ

(5.1)

for the associated reference hyperplane H orthogonal to eN , the orthogonal projection PH onto H, and the normal component ζN of ζ. The vector ζ is equal to ζ + ζN eN and will often be denoted by (ζ , ζN ). In practice the vector eN is chosen as eN = (0, . . . , 0, 1), but the actual form of the unit vector eN is not important. A direction in RN is speciﬁed by a unit vector d ∈ RN , |d| = 1. Alternatively, it can be speciﬁed by AeN for some matrix A in the orthogonal subgroup of N × N matrices O(N) = {A : ∗A A = A ∗A = I} , def

(5.2)

where ∗A is the transposed matrix of A. Conversely, for any unit vector d in RN , |d| = 1, there exists8 A ∈ O(N) such that d = AeN . For all x ∈ RN and A ∈ O(N), it is easy to check that |Ax| = |x| = |A−1 x| for all x ∈ RN . The main elements of Deﬁnition 5.1 are illustrated in Figure 2.2 for N = 2. ζN graph of ax

set Ω Ax eN ∂Ω

ζ x

0 Vx ⊂ H = {eN }⊥ U(x)

Figure 2.2. Local epigraph representation (N = 2). Deﬁnition 5.1. Let eN be a unit vector in RN , H be the hyperplane {eN }⊥ , and Ω be a subset of RN with nonempty boundary ∂Ω. (i) Ω is said to be locally a C 0 epigraph if for each x ∈ ∂Ω there exist (a) an open neighborhood U(x) of x; (b) a matrix Ax ∈ O(N); the orthonormal basis with respect to d by adding N −1 unit vectors d1 , d2 , . . . , dN −1 and construct the matrix whose columns are the vectors A = [d1 d2 . . . dN −1 d] for which AeN = d. 8 Complete

80

Chapter 2. Classical Descriptions of Geometries and Their Properties (c) a bounded open neighborhood Vx of 0 in H such that U(x) ⊂ {y ∈ RN : PH (A−1 x (y − x)) ∈ Vx };

(5.3)

(d) and a function ax ∈ C 0 (Vx ) such that ax (0) = 0 and & ζ ∈ Vx , U(x) ∩ ∂Ω = U(x) ∩ x + Ax (ζ + ζN eN ) : ζN = ax (ζ ) & ζ ∈ Vx U(x) ∩ int Ω = U(x) ∩ x + Ax (ζ + ζN eN ) : . ζN > ax (ζ )

(5.4)

(5.5)

(ii) Ω is said to be a C 0 epigraph if it is locally a C 0 epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

(iii) Ω is said to be an equi-C 0 epigraph if it is a C 0 epigraph and the family of functions {ax : x ∈ ∂Ω} is uniformly bounded and equicontinuous: ∃c > 0 such that ∀x ∈ ∂Ω, ∀ξ ∈ V, ∀ε > 0, ∃δ > 0 such that ∀x ∈ ∂Ω, ∀y, ∀z ∈ V such that |z − y| < δ,

|ax (ξ )| ≤ c, (5.6) |ax (z) − ax (y)| < ε.

Remark 5.1. Conditions (5.3), (5.4), and (5.5) are equivalent to the following three conditions: & ζ ∈ Vx U(x) ∩ ∂Ω ⊂ x + Ax (ζ + ζN eN ) : , (5.7) ζN = ax (ζ ) & ζ ∈ Vx , (5.8) U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : ζN > ax (ζ ) & ζ ∈ V x U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : . (5.9) ζN < ax (ζ ) In particular, Ω is locally a C 0 epigraph if and only if Ω is locally a C 0 epigraph. As a result, conditions (5.3), (5.4), and (5.5) are also equivalent to conditions (5.3), (5.4), and the following condition: & ζ ∈ Vx (5.10) U(x) ∩ int Ω ⊂ x + Ax (ζ + ζN eN ) : ζN < ax (ζ ) in place of condition (5.5).

5. Sets Locally Described by the Epigraph of a Function

81

Remark 5.2. Note that from (5.3) PH (A−1 x (U(x) − x)) ⊂ Vx and that this yields the condition PH (U ) ⊂ V in part (iii) of the deﬁnition. Remark 5.3. It is always possible to redeﬁne U(x) − x or Vx to be open balls or open hypercubes centered in 0. For instance, in the case of balls, for Vx = BH (0, ρ) ⊂ Vx , choose the associated neighborhood U (x) = U(x) ∩ {y ∈ RN : PH (A−1 x (y − x)) ∈ BH (0, ρ)}. Similarly, for U (x) = B(x, r) ⊂ U(x) choose the associated neighborhood Vx = −1 PH (A−1 x (B(x, r)−x) = BH (0, r) ⊂ Vx , since, from (5.3), Vx ⊃ PH (Ax (U(x)−x)) ⊃ −1 PH (Ax B(x, r) − x)) = BH (0, r). In both cases properties (5.3) to (5.5) are veriﬁed for the new neighborhoods. Remark 5.4. Sets Ω ⊂ RN that are locally C 0 epigraphs have a boundary ∂Ω with zero N -dimensional Lebesgue measure. The three cases considered in Deﬁnition 5.1 diﬀer only when the boundary ∂Ω is unbounded. But we ﬁrst introduce some notation. Notation 5.1. (i) Under condition (5.3), a point y in U(x) ⊂ RN is represented by (ζ , ζN ) ∈ Vx ×R, where ζ = PH (A−1 x (y − x)) ∈ Vx ⊂ H, def

ζN = A−1 x (y − x) · eN (x). def

We can also identify H with RN −1 = {(z , 0) ∈ RN : z ∈ RN −1 }. (ii) The graph, epigraph, and hypograph of ax : Vx → R will be denoted as follows: def

A0x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ζN = ax (ζ )} , def

A+ x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ∀ζN > ax (ζ )} , def

A− x = {x + Ax (ζ + ζN eN ) : ∀ζ ∈ Vx , ∀ζN < ax (ζ )} . def

The equi-C 0 epigraph case is important since we shall see in Theorem 5.2 that the three cases of Deﬁnition 5.1 are equivalent when the boundary ∂Ω is compact. We now show that, for an arbitrary boundary, the equicontinuity of the graph functions {ax : x ∈ ∂Ω} can be expressed in terms of the continuity in 0 of a dominating function h. For this purpose, deﬁne the space of dominating functions as follows: def

H = {h : [0, ∞[ → [0, ∞[ : h(0) = 0 and h is continuous in 0} .

(5.11)

By continuity in 0, h is locally bounded in 0: ∀c > 0, ∃ρ > 0 such that ∀θ ∈ [0, ρ],

|h(θ)| ≤ c.

The only part of the function h that will really matter is in a bounded neighborhood of zero. Its extension outside to [0, ∞[ can be relatively arbitrary.

82

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 5.1. Let Ω be a C 0 -epigraph. The following conditions are equivalent: (i) Ω is an equi-C 0 -epigraph. (ii) There exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ζ , ξ ∈ BH (0, ρ) such that |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |). (5.12)

(iii) There exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ξ ∈ BH (0, ρ),

ax (ξ ) ≤ h(|ξ |).

(5.13)

Inequalities (5.12) and (5.13) for h ∈ H are also veriﬁed with lim sup h that also belongs to H. Proof. Since the ax ’s are uniformly continuous, inequalities (5.12) and (5.13) with h ∈ H imply the same inequalities with lim sup h that also belongs to H by taking the pointwise lim sup of both sides of the inequalities. (ii) ⇒ (i). For h ∈ H, there exists ρ1 > 0 such that 0 < ρ1 ≤ ρ/2 and |h(θ)| ≤ 1 for all θ ∈ [0, ρ1 ]. By construction BH (0, ρ1 ) ⊂ V and from inequality (5.12) ∀ζ , ξ ∈ BH (0, ρ1 ), |ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |), ∀ ξ ∈ BH (0, ρ1 ), |ax (ξ )| ≤ h(|ξ |) ≤ 1 and the family {ax : BH (0, ρ1 ) → R : x ∈ ∂Ω} is equicontinuous and uniformly bounded. (i) ⇒ (ii). By Deﬁnition 5.1 (iii), there exist a neighborhood V of 0 in H and a neighborhood U of 0 in RN such that for all x ∈ ∂Ω Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U,

and the family {ax ∈ C 0 (V ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous. Choose r > 0 such that B(0, 3r) ⊂ U . From property (5.3) in Deﬁnition 5.1

⇒ BH (0, 3r) ⊂ V. U(x) ⊂ y ∈ RN : PH A−1 x (y − x) ∈ V Deﬁne the modulus of continuity ∀θ ∈ [0, 3r],

h(θ) = sup sup

sup

ζ ∈H x∈∂Ω ξ ∈V |ζ |=1 ξ +θζ ∈V

|ax (θζ + ξ ) − ax (ξ )|.

(5.14)

The function h(θ), as the sup of the family θ → |ax (θζ + ξ ) − ax (ξ )| of continuous functions with respect to (ζ , ξ ), is lower semicontinuous with respect to θ and bounded in [0, 3r] since the ax ’s are uniformly bounded. By construction h(0) = 0 and by equicontinuity of the ax , for all ε > 0, there exists δ > 0 such that for all θ ∈ [0, 3r], |θ| < δ, ∀x ∈ ∂Ω, ∀ζ ∈ H, |ζ | = 1,

|ax (θζ + ξ ) − ax (ξ )| < ε

⇒ |h(θ)| < ε,

5. Sets Locally Described by the Epigraph of a Function

83

where h is continuous in 0. In view of Remark 5.3, choose ρ = 3r and the new neighborhoods V = BH (0, ρ) and U = {ξ ∈ B(0, ρ) : PH ξ ∈ V } = B(0, ρ) and extend h by the constant h(ρ) to [ρ, ∞[ . Therefore h ∈ H and, by deﬁnition of h, for all x ∈ ∂Ω and for all ξ and ζ in BH (0, ρ) such that |ξ − ζ | < ρ ξ −ζ |ax (ξ ) − ax (ζ )| = ax ζ + |ξ − ζ | − a (ζ ) x ≤ h(|ξ − ζ |) |ξ − ζ | ⇒ ∀x ∈ ∂Ω, ∀ξ , ζ ∈ BH (0, ρ), |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |).

(ii) ⇒ (iii) Choose ζ = 0 in inequality (5.12): ∀ξ ∈ BH (0, ρ) such that |ξ | < ρ,

ax (ξ ) ≤ |ax (ξ )| ≤ h(|ξ |).

(iii) ⇒ (ii). In view of Remark 5.3, choose the new neighborhoods V = BH (0, ρ) and U = U ∩ {y ∈ RN : PH (y) ∈ BH (0, ρ)} and the restriction of the functions {ax : V → R : ∀x ∈ ∂Ω} to V . Choose r > 0 such that B(0, 4r) ⊂ U (hence 4r ≤ ρ) and consider the region O = {ζ + ζN eN : ζ ∈ BH (0, 2r) and h(|ζ |) < ζN < r} . def

For any x ∈ ∂Ω and ζ ∈ O, consider the point yζ = x + Ax (ζ + ζN eN ). It is readily ' 2 seen that |yζ − x| = |ζ| < (2r) + r 2 < 3r and that yζ ∈ U(x). In addition y ζ ∈ A+ x . From (5.13) ax (ζ ) ≤ h(|ζ |) for all ζ ∈ V = BH (0, ρ) and hence ax (ζ ) ≤ + h(|ζ |) < ζN for all ζ ∈ O (2r < ρ). As a result yζ ∈ U(x) ∩ Ax = U(x) ∩ int Ω ⊂ int Ω and ∀x ∈ ∂Ω,

x + Ax O ⊂ int Ω.

(5.15)

Given ζ and ξ in BH (0, 2r), consider the two points yζ = x + Ax (ζ + ax (ζ )eN ) ∈ A0x = U(x) ∩ ∂Ω, def

yξ = x + Ax (ξ + ax (ξ )eN ) ∈ A0x = U(x) ∩ ∂Ω. def

Since yξ and yζ belong to U(x) ∩ ∂Ω, yξ does not belong to yζ + Ax O ⊂ int Ω or / Ax O. Hence equivalently yξ − yζ ∈ ξ − ζ + (ax (ξ ) − ax (ζ ))eN = A−1 / O. x (yξ − yζ ) ∈ For |ξ − ζ | < 2r, this is equivalent to saying that one of the following inequalities is veriﬁed: h(|ξ − ζ |) ≥ Ax eN · (yξ − yζ ) = ax (ξ ) − ax (ζ ) or r ≤ Ax eN · (yξ − yζ ) = ax (ξ ) − ax (ζ ) by deﬁnition of ax and the fact that ξ − ζ ∈ BH (0, 2r) ⊂ BH (0, ρ) = V . But the second inequality contradicts the continuity of ax . Hence the ﬁrst inequality holds and, by interchanging the roles of ξ and ζ , we get inequality (5.12) with a new ρ equal to 2r.

84

Chapter 2. Classical Descriptions of Geometries and Their Properties

Theorem 5.2. When the boundary ∂Ω is compact, the three properties of Deﬁnition 5.1 are equivalent: locally C 0 epigraph

⇐⇒

C 0 epigraph

⇐⇒

equi-C 0 epigraph.

Under such conditions, there exist neighborhoods V of 0 in H and U of 0 in RN such that PH U ⊂ V and the neighborhoods Vx of 0 in H and the neighborhoods U(x) of x in RN can be chosen of the form Vx = V

and

∃Ax ∈ O(N) such that U(x) = x + Ax U.

(5.16)

Moreover, the family of functions {ax : V → R : x ∈ ∂Ω} can be chosen uniformly bounded and equicontinuous with respect to x ∈ ∂Ω. In addition, there exist ρ > 0 such that BH (0, ρ) ⊂ V and a function h ∈ H such that ∀x ∈ ∂Ω, ∀ξ , ζ ∈ BH (0, ρ), |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |). (5.17)

Proof. It is suﬃcient to show that a locally C 0 epigraph is an equi-C 0 epigraph. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx ) ⊂ U(x). Since ∂Ω is compact, m there exist a ﬁnite sequence {xi }m i=1 of points of ∂Ω and a ﬁnite subcover {Bi }i=1 , Bi = B(xi , Ri ), Ri = rxi . (i) We ﬁrst claim that ∃R > 0, ∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi . We proceed by contradiction. If this is not true, then for each n ≥ 1, ∃xn ∈ ∂Ω, ∀i, 1 ≤ i ≤ m, B xn , 1/n ⊂ Bi , / Bi . ⇒ ∀i, 1 ≤ i ≤ m, ∃yin ∈ B xn , 1/n such that yin ∈ Since ∂Ω is compact, there exist a subsequence of {xn } and x ∈ ∂Ω such that xnk → x as k goes to inﬁnity. Hence for each i, 1 ≤ i ≤ m, |yink − x| ≤ |yink − xnk | + |xnk − x| ≤

1 + |xnk − x| → 0 nk

and yink → x as k goes to inﬁnity. For each i, the set Bi is closed. Thus x ∈ Bi and m * / Bi . yink ∈ Bi → x ∈ Bi ⇒ ∃ x ∈ ∂Ω such that x ∈ i=1

But this contradicts the fact that {Bi }m i=1 is an open cover of ∂Ω. This property is obviously also satisﬁed for all neighborhoods N of 0 such that ∅ = N ⊂ B(0, R). (ii) Construction of the neighborhoods Vx and U (x) and the maps Ax and ax . From part (i) for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi = B(xi , Ri ) ⊂ U(xi ) ⇒

PH (A−1 xi (B(x, R)

− xi )) ⊂ PH (A−1 xi (U(xi ) − xi )) ⊂ Vxi

⇒ PH (A−1 xi (B(x − xi , R))) ⊂ Vxi .

5. Sets Locally Described by the Epigraph of a Function

85

Since x ∈ ∂Ω ∩ U(xi ), it is of the form x = xi + Axi (ξi + axi (ξi )eN ),

ξi = PH (A−1 xi (x − xi ))

−1 −1 ⇒ PH (A−1 xi (B(x − xi , R))) = PH (Axi (x − xi )) + PH (Axi (B(0, R)))

= ξi + BH (0, R)

⇒ ξi + BH (0, R) ⊂ Vxi .

(5.18)

Choose the neighborhoods def

def

U = B(0, R) and V = BH (0, R). Associate with x ∈ ∂Ω the following new neighborhoods and functions: Vx = V, def

Ax = Axi , def

U(x) = x + Ax U, def

ζ → ax (ζ ) = axi (ζ + ξi ) − axi (ξi ) : V → R . def

It is readily seen from identity (5.18) that ax is well-deﬁned, ax (0) = 0, and ax ∈ C 0 (V ). Since the family {axi : 1 ≤ i ≤ m} is ﬁnite, ax C 0 (V ) ≤ 2 axi C 0 (Vx

i

)

≤ 2 max axi C 0 (Vx ) < ∞ 1≤i≤m

i

and the family {ax ; x ∈ ∂Ω} is uniformly bounded. Since each axi is uniformly continuous, for all ε > 0 there exists δi > 0 such that ∀ζ1 , ζ2 ∈ Vxi such that |ζ1 − ζ2 | < δi ,

|axi (ζ1 ) − axi (ζ2 )| < ε.

And since the family {axi : 1 ≤ i ≤ m} is ﬁnite pick δ = min1≤i≤m δi to get the following: for all ε > 0, there exists δ > 0 such that for all i ∈ {1, . . . , m}, ∀ζ1 , ζ2 ∈ Vxi such that |ζ1 − ζ2 | < δ,

|axi (ζ1 ) − axi (ζ2 )| < ε.

Therefore, for all ε > 0, there exists δ > 0 such that ∀x ∈ ∂Ω, ∀ζ1 , ζ2 ∈ V such that |ζ1 − ζ2 | < δ,

|ax (ζ1 ) − ax (ζ2 )| < ε

(5.19)

and the family {ax ; x ∈ ∂Ω} is uniformly equicontinuous. By construction, properties (a) and (b) of Deﬁnition 5.1 are veriﬁed for U (x) and Ax . By construction, properties (a) and (b) are veriﬁed for U (x) and Ax . Always by construction in x ∈ ∂Ω U (x) = x + Ax B(0, R) = x + B(0, R) = B(x, R), PH ((Ax )−1 (U (x) − x)) = PH (B(0, R)) = BH (0, R) = V = Vx

⇒ U (x) ⊂ y ∈ RN : PH ((Ax )−1 (y − x)) ∈ Vx

86

Chapter 2. Classical Descriptions of Geometries and Their Properties

and condition (c) of Deﬁnition 5.1 is veriﬁed. For condition (d) of Deﬁnition 5.1 in x ∈ ∂Ω, let xi be the point associated with x such that B(x, R) ⊂ U(xi ). Since U (x) = B(x, R), U (x) ∩ ∂Ω ⊂ U(xi ) ∩ ∂Ω = {xi + Axi (ζ + axi (ζ )eN ) : ∀ζ ∈ Vxi } ∩ ∂Ω and for all y ∈ U (x) ∩ ∂Ω there exists ξ ∈ Vxi such that y = xi + Axi (ξ + axi (ξ )eN ) = x + Axi (ξ + axi (ξ )eN ) − (x − xi ). But x also has a representation with respect to xi x = xi + Axi (ξi + axi (ξi )eN ),

⇒ y = x + Axi (ξ −

ξi

ξi ∈ Vxi

+ (axi (ξ ) − axi (ξi ))eN )

that can be rewritten as y = x + Axi (ξ − ξi + (axi ((ξ − ξi ) + ξi ) − axi (ξi ))eN ) ⇒ ζ = PH (A−1 xi (y − x)) = ξ − ξi ∈ V def

−1 (U (x) − x)) ⊂ Vx = V . Finally, since by (c) PH (A−1 xi (U (x) − x)) = PH ((Ax ) there exists ζ ∈ V such that

y = x + Axi (ζ + (axi (ζ + ξi ) − axi (ξi ))eN ) = x + Axi (ζ + ax (ζ )eN ) and property (5.4) is veriﬁed. The constructions and the proof of property (5.5) are similar: U (x) ∩ int Ω ⊂ U(xi ) ∩ int Ω = U(xi ) ∩ xi + Axi (ζ + ζN eN ) : ∀ζ ∈ Vxi , ∀ζN > axi (ζ ) and for all y ∈ U (x)∩int Ω there exists (ξ , ξN ) ∈ Vxi ×R such that ξN > axi (ξ ) and y = xi + Axi (ξ + ξN eN ) = x + Axi (ξ + ξN eN ) − (x − xi ) = x + Axi (ξ − ξi + (ξN − axi (ξi ))eN ) ⇒ ζ = PH (A−1 xi (y − x)) = ξ − ξi ∈ V, def

ζN = (A−1 xi (y − x)) · eN = ξN − axi (ξi ) def

−1 (U (x) − x)) ⊂ Vx = V . Finally, since by (c) PH (A−1 xi (U (x) − x)) = PH ((Ax ) there exists (ζ , ζN ) ∈ V × R such that

y = x + Axi (ζ + ax (ζ )eN ), ζN = ξN − axi (ξi ) = ξN − axi (ξ ) + axi (ξ ) − axi (ξi ) = ξN − axi (ξ ) + axi (ζ + ξi ) − axi (ξi ) = ξN − axi (ξ ) + ax (ζ ) > ax (ζ )

since ξN > axi (ξ ) and property (5.5) in (d) is veriﬁed. (iii) The other properties and (5.17) follow from Theorem 5.1.

5. Sets Locally Described by the Epigraph of a Function

5.2

87

Local C k, -Epigraphs and H¨ olderian/Lipschitzian Sets

We extend the three deﬁnitions of C 0 epigraphs to C k, epigraphs. C 0,1 is the important family of Lipschitzian sets and C 0, , 0 < < 1, of H¨ olderian sets. Deﬁnition 5.2. Let Ω be a subset of RN such that ∂Ω = ∅. Let k ≥ 0, 0 ≤ ≤ 1. (i) Ω is said to be locally a C k, epigraph if for each x ∈ ∂Ω there exist (a) an open neighborhood U(x) of x; (b) a matrix Ax ∈ O(N); (c) a bounded open neighborhood Vx of 0 in H such that U(x) ⊂ {y ∈ RN : PH (A−1 x (y − x)) ∈ Vx };

(5.20)

(d) and a function ax ∈ C k, (Vx ) such that ax (0) = 0 and U(x) ∩ ∂Ω = U(x) ∩

x + Ax (ζ + ζN eN ) :

U(x) ∩ int Ω = U(x) ∩

&

ζ ∈ Vx

,

ζN = ax (ζ )

x + Ax (ζ + ζN eN ) :

(5.21)

&

ζ ∈ Vx ζN > ax (ζ )

,

(5.22)

where ζ = (ζ1 , . . . , ζN −1 ) ∈ RN −1 . We shall say that Ω is locally Lipschitzian in the C 0,1 case and locally H¨ olderian of index in the C 0, , 0 < < 1, case. (ii) Ω is said to be a C k, epigraph if it is locally a C k, epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x)−x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

def

Vx = V

def

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

olderian of index We shall say that Ω is Lipschitzian in the C 0,1 case and H¨ in the C 0, , 0 < < 1, case. (iii) Given (k, ) = (0, 0), Ω is said to be an equi-C k. epigraph if it is a C k, epigraph and ∃c > 0, ∀x ∈ ∂Ω,

ax C k, (V ) ≤ c,

(5.23)

where f C k, (V ) is the norm on the space C k, (V ) as deﬁned in (2.21): def

f C k, (V ) = f C k (V ) + max

sup

0≤|α|≤k x,y∈V x=y

def

C k,0 (V ) = C k (V )

and

|∂ α f (y) − ∂ α f (x)| , |y − x| def

if 0 < ≤ 1,

f C k,0 (V ) = f C k (V ) .

88

Chapter 2. Classical Descriptions of Geometries and Their Properties Ω is said to be an equi-C 0,0 epigraph if it is an equi-C 0 epigraph in the sense of Deﬁnition 5.1 (iii). We shall say that Ω is equi-Lipschitzian in the C 0,1 case and equi-H¨ olderian of index in the C 0, , 0 < < 1, case.

Remark 5.5. In Deﬁnition 5.1 (iii) for (k, ) = (0, 0) it follows from condition (5.23) that the functions {ax : x ∈ ∂Ω} are uniformly equicontinuous on V . In the case (k, ) = (0, 0) this is no longer true. So we have to include it by using Deﬁnition 5.1 (iii). Theorem 5.2 readily extends to the C k, case. Theorem 5.3. When the boundary ∂Ω is compact, the three types of sets of Deﬁnition 5.2 are equi-C k, epigraphs in the sense of Deﬁnitions 5.2 (iii) and 5.1 (iii). Proof. The proof is the same as the one of Theorem 5.2 for the C 0 case. The equiC k, property is obtained by choosing in the proof of Theorem 5.2 the common constant as c = max1≤i≤m cxi , where cxi is the constant associated with axi on Vxi , axi C k (Vxi ) ≤ cxi . Sets Ω that are locally a C k, epigraph are Lebesgue measurable in RN , their “volume” is locally ﬁnite, and their boundary has zero “volume.” Theorem 5.4. Let k ≥ 0, let 0 ≤ ≤ 1, and let Ω be locally a C k, epigraph. (i) The complement Ω is also locally a C k, epigraph, and int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω, int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω. Moreover, for all x ∈ ∂Ω m U(x) ∩ ∂Ω = 0

⇒ m ∂Ω = 0,

(5.24)

(5.25)

where m is the N -dimensional Lebesgue measure. (ii) If Ω = ∅ is open, then Ω = int Ω. Proof. (i) The fact that the complement of Ω is locally a C k, epigraph follows from Remark 5.1. To complete the proof, it is suﬃcient to establish the ﬁrst two properties of the ﬁrst line of (5.24). The second line is the ﬁrst line applied to Ω. Since U(x) is a neighborhood of x, there exists r > 0 such that B(x, 3r) ⊂ U(x). Since ax ∈ C 0 (Vx ), ax (0) = 0, and Vx is a neighborhood of 0, there exists ρ, 0 < ρ < r, such that BH (0, ρ) ⊂ Vx and for all ζ , |ζ | < ρ, |ax (ζ )| ≤ r. Given ζ ∈ BH (0, ρ), the point yζ = x + A(ζ + ax (ζ )eN ) ∈ ∂Ω ∩ U(x). Given a real α, |α| < r, consider the point zα = x + A(ζ + (ax (ζ ) + α)eN ). By construction |zα − x| = |A(ζ + (ax (ζ ) + α)eN )| = |ζ + (ax (ζ ) + α)eN | < 3r and zα ∈ U(x). Hence the segment (yζ , yζ + rAx eN ) ⊂ A+ ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅. Also the segment (yζ , yζ − rAx eN ) ⊂ A− ∩ U(x) = int Ω ∩ U(x) and int Ω = ∅.

5. Sets Locally Described by the Epigraph of a Function

89

Finally, associate with yζ ∈ ∂Ω and the sequence αn = r/n the points zαn ∈ int Ω. As n goes to inﬁnity, zn → yζ and ∂Ω ⊂ int Ω and hence Ω = int Ω. From the following Lemma 5.1 we get the third identity in the ﬁrst line of (5.24). (ii) From part (i) ∂(int Ω) = ∂Ω and by Lemma 5.1 (ii) int Ω = Ω. Since Ω is open, Ω = int Ω = Ω and int Ω = Ω = Ω. Lemma 5.1. ⇐⇒

(i) ∂(int Ω) = ∂Ω (ii) ∂(int Ω) = ∂Ω

int Ω = Ω.

⇐⇒

int Ω = Ω.

Proof. It is suﬃcient to prove (i). (⇒) By deﬁnition of the closure, Ω = int Ω ∪ ∂Ω = int Ω ∪ ∂(int Ω) = int Ω. Conversely (⇐) By deﬁnition ∂Ω = Ω ∩ Ω, Ω ∩ Ω = int Ω ∩ Ω = int Ω ∩ (int Ω) = int Ω ∩ (int Ω) = ∂(int Ω).

5.3

Local C k, -Epigraphs and Sets of Class C k,

Sets or domains which are locally the epigraph of a C k, -function, k ≥ 0, 0 ≤ ≤ 1 (resp., locally Lipschitzian), are sets of class C k, (resp., C 0,1 ). However, we shall see from Examples 5.1 and 5.2 that a domain of class C 0,1 is generally not locally the epigraph of a Lipschitzian function. Theorem 5.5. Let , 0 ≤ ≤ 1, be a real number. (i) If Ω is locally a C k, epigraph for k ≥ 0, then it is locally of class C k, . (ii) If Ω is locally of class C k, for k ≥ 1, then it is locally a C k, epigraph. Proof. (i) For each point x ∈ ∂Ω, let U(x), Vx , and ax be the associated neighborhoods and the C k, -function. Deﬁne the C k, -mappings for ξ = (ξ , ξN ) ∈ Vx × R hx (ξ) = x + Ax (ξ + [ξN + ax (ξ )] eN ), def

−1 −1 gx (y) = (PH A−1 x (y − x), eN · Ax (y − x) − ax (PH Ax (y − x))). def

It is easy to verify that hx is the inverse of gx , hx (gx (y)) = y in U(x) and gx (hx (ξ)) = ξ in gx (U(x)), −1 and that since for y ∈ U(x) ∩ ∂Ω, eN · A−1 x (y − x) = ax (PH Ax (y − x)

gx (U(x) ∩ ∂Ω) ∈ H = {(ξ , ξN ) ∈ RN : ξN = 0}, −1 and that since for y ∈ U(x) ∩ int Ω, eN · A−1 x (y − x) > ax (PH Ax (y − x)

gx (U(x) ∩ int Ω) ⊂ {(ξ , ξN ) ∈ RN : ξN > 0}. The set Ω is of class C k, .

(5.26) (5.27)

90

Chapter 2. Classical Descriptions of Geometries and Their Properties

(ii) In the discussion preceding Theorem 4.1, we showed that a set of class C k, is locally the level set of a C k, -function. From the deﬁnition of a set of class C k, , k ≥ 1 and 0 ≤ ≤ 1, the set Ω can be locally described by the level sets of the C k, -function def

fx (y) = gx (y) · eN , since by deﬁnition int Ω ∩ U (x) = {y ∈ U (x) : fx (y) > 0}, ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0}. The boundary ∂Ω is the zero level set of fx and the gradient ∇fx (y) = ∗Dgx (y)eN = 0 is normal to that level set. The exterior normal to Ω is given by n(y) = −

∗ ∗ Dgx (y)eN (Dhx (gx (y)))−1 eN ∇fx (y) =− ∗ =− ∗ . |∇fx (y)| | Dgx (y)eN | | (Dhx (gx (y)))−1 eN |

Since fx is C 1 and ∇fx (x) = 0, choose a smaller neighborhood of x, still denoted by U (x), such that ∀y ∈ U (x),

∇fx (y) · ∇fx (x) > 0.

To construct the graph function around the point x ∈ Γ = ∂Ω, choose Ax ∈ O(N) such that Ax eN = −n(x) and H = {eN }⊥ . Consider the C k, -function, k ≥ 1, λ(ζ , ζN ) = fx (x + Ax (ζ + ζN eN )) . def

By construction, λ(0, 0) = 0 and 0 = ∇λ(0, 0) · (0, ξN ) = ∇fx (x) · (0, ξN ) = ξN |∇fx (x)|

⇒ ξN = 0.

Therefore, by the implicit function theorem, there exist a neighborhood Vx ⊂ B0 of (0, 0) and a function ax ∈ C k. (Vx ) such that λ(0, 0) = 0 and ∀ζ ∈ Vx ,

λ(ζ , ax (ζ )) = fx (x + Ax (ζ + ax (ζ )eN ) = 0.

By construction, x + Ax (ζ + ax (ζ )eN ) ∈ Γ ∩ U (x). Choose U(x) = U (x) ∩ {x + Ax (ζ + ζN eN ) : ζ ∈ Vx and ζN ∈ R}. def

By construction, Vx and U(x) satisfy condition (5.3). Moreover, from (3.4)–(3.5) int Ω ∩ U (x) = hx (B+ ) and ∂Ω ∩ U (x) = hx (B0 ) int Ω ∩ U (x) = {y ∈ U (x) : f (y) > 0}, x ⇒ ∂Ω ∩ U (x) = {y ∈ U (x) : fx (y) = 0},

5. Sets Locally Described by the Epigraph of a Function

91

and since U(x) ⊂ U (x) we get conditions (5.4) and (5.5): int Ω ∩ U(x) = {y ∈ U(x) : fx (y) > 0} = A+ ∩ U(x), ∂Ω ∩ U(x) = {y ∈ U(x) : fx (y) = 0} = A0 ∩ U(x). Recall that, by construction of U(x), for all (ζ , ζN ) ∈ A−1 x (U(x) − {x}), ζN > ax (ζ ) ⇐⇒ λ(ζ , ζN ) = fx (x + Ax (ζ + ζN eN )) > 0, since ∇fx (x) ∂λ (ζ , ζN ) = ∇fx (x + Ax (ζ + ζN en )) · Ax eN = ∇fx (x + Ax ζ) · >0 ∂ζN |∇fx (x)| −1 in the neighborhood A−1 x (U(x) − {x}) ⊂ Ax (U (x) − {x}) of (0, 0).

Example 5.1 (R. Adams, N. Aronszajn, and K. T. Smith [1]). Consider the open convex (Lipschitzian) set Ω0 = {ρ eiθ : 0 < ρ < 1, 0 < θ < π/2} and its image Ω = T (Ω0 ) by the C 0,1 -diﬀeomorphism (see Figure 2.3) T (ρ eiθ ) = ρ ei(θ−log ρ) ,

T −1 (ρ eiθ ) = ρ ei(θ+log ρ) .

It is readily seen that, as ρ goes to zero, the image of the two pieces of the boundary of Ω0 corresponding to θ = 0 and θ = π/2 begin to spiral around the origin. As a result Ω is not locally the epigraph of a function at the origin.

1

1

0

0 0 (a) quarter disk Ω0

1

0

1

(b) T (Ω0 ) spirals around 0

Figure 2.3. Domain Ω0 and its image T (Ω0 ) spiraling around the origin. Example 5.2. This second example can be found in F. Murat and J. Simon [1], where it is attributed to M. Zerner. Consider the Lipschitzian function λ deﬁned on [0, 1] as follows: λ(0) = 0, and on each interval [1/3n+1 , 1/3n ] 2 1 1 ≤ s ≤ n+1 , 2 s − , n+1 n+1 3 3 3 def λ(s) = 1 2 1 −2 s − , ≤ s ≤ n, n n+1 3 3 3

92

Chapter 2. Classical Descriptions of Geometries and Their Properties

where n ranges over all integers n ≥ 0. Associate with λ and a real δ > 0 the set def

Ω = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, |x2 − λ(x1 )| < δ x1 } (see Figure 2.4). The set Ω is the image of the triangle def

Ω0 = {(x1 , x2 ) ∈ R2 : 0 < x1 < 1, |x2 | < δ x1 } through the C 0,1 -homeomorphism def

T (x1 , x2 ) = (x1 , x2 + λ(x1 )),

T −1 (y1 , y2 ) = (y1 , y2 − λ(y1 )).

Since the triangle Ω0 is Lipschitzian, its image is a set of class C 0,1 . But Ω is not locally Lipschitzian in (0, 0) since Ω zigzags like a lightning bolt as it gets closer to the origin. Thus, however small the neighborhood around (0, 0), a direction eN (0, 0) cannot be found to make the domain locally the epigraph of a function.

1

1

0

0 0

1 (a) triangle Ω0

0

1 (b) T (Ω0 ) zigzags to 0

Figure 2.4. Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin.

5.4

Locally Lipschitzian Sets: Some Examples and Properties

The important subfamily of sets that are locally a Lipschitzian epigraph (that is, locally a C 0,1 epigraph) enjoys most of the properties of smooth domains. Convex sets and domains that are locally the epigraph of a C 1 - or smoother function are locally Lipschitzian. In a bounded path-connected Lipschitzian set, the geodesic distance between any two points is uniformly bounded. Their boundary has zero volume and locally ﬁnite boundary measure (cf. Theorem 5.7 below). Sobolev spaces deﬁned on Ω have linear extension to RN . 5.4.1

Examples and Continuous Linear Extensions

According to this deﬁnition the whole space RN and the closed unit ball with its center or a small crack removed are not locally Lipschitzian since in the ﬁrst case

5. Sets Locally Described by the Epigraph of a Function

93

the boundary is empty and in the second case the conditions cannot be satisﬁed at the center or along the crack, which is a part of the boundary. Similarly the set ∞ *

def

Ω =

Ωn ,

def

Ωn =

n=1

1 1 y ∈ RN : y − n < n+2 2 2

is not locally Lipschitzian since the conditions of Deﬁnition 5.2 (i) are not satisﬁed in 0 ∈ ∂Ω. However, the set def

Ω =

∞ * n=1

Ωn ,

def

Ωn =

y ∈ R : |y − n| < N

1

2n+2

is locally Lipschitzian, but not Lipschitzian or equi-Lipschitzian in the sense of Deﬁnition 5.2. One of the important properties of a bounded (open) Lipschitzian domain Ω is the existence of a continuous linear extension of functions of the Sobolev space H k (Ω) from Ω to functions in H k (RN ) deﬁned on RN , that is, (5.28) E : H k (Ω) → H k (RN ) and ∀ϕ ∈ H k (Ω), (Eϕ) Ω = ϕ (cf. the Calder´ on extension theorem in R. A. Adams [1, p. 83, Thm. 4.32, p. 91] and ˇas [1, Thm. 3.10, p. 80]). This property is also important in the existence of J. Nec optimal domains when it is uniform for a given family. For instance, this will occur for the family of domains satisfying the uniform cone property of section 6.4.1. 5.4.2

Convex Sets

Theorem 5.6. Any convex subset Ω of RN such that Ω = RN and int Ω = ∅ is locally Lipschitzian, and for each x ∈ ∂Ω the neighborhood Vx and the function ax can be chosen convex. Proof. (i) Construction of the function ax . Start with the unit vector eN and the hyperplane H orthogonal to eN . Pick any point x0 in the interior of Ω and choose ε > 0 such that B(x0 , 2ε) ⊂ int Ω. Since Ω = RN , ∂Ω = ∅ and we can associate with each x ∈ ∂Ω the direction d(x) = (x0 − x)/|x0 − x| and the matrix Ax ∈ O(N) such that d(x) = Ax eN . Note that the hyperplane Ax H is orthogonal to the vector / Ω and the minimum d(x) = Ax eN . By convexity, the point x− 0 = x − (x0 − x) ∈ distance δ from x− to Ω is such that 0 < δ < |x − x |. As a result there exists ρ, 0 0 0 < ρ < ε, such that for all ζ ∈ H, |ζ | < ρ, the line Lζ = {x + Ax (ζ + ζN eN ) : |ζN | ≤ |x − x0 |} def

from the point x0 + Ax ζ to x− 0 + Ax ζ in the direction Ax eN contains a point of Ω and a point of its complement. Hence there exists ζˆN , |ζˆN | ≤ |x − x0 |, such that yˆ = x + Ax (ζ + ζˆN eN ) ∈ ∂Ω ∩ Lζ minimizes ζN = Ax eN · (y − x) over all y ∈ Ω ∩ Lζ . If yˆ1 and yˆ2 are two distinct minimizing points such that ζˆ1N = ζˆ2N ,

94

Chapter 2. Classical Descriptions of Geometries and Their Properties

then yˆ1 = x + Ax (ζ + ζˆ1N eN ) = x + Ax (ζ + ζˆ2N eN ) = yˆ2 . As a result, for all ζ ∈ H, |ζ | < ρ, the function ax (ζ ) =

def

inf

Ax eN · (y − x)

y∈Ω∩Lζ

is well-deﬁned and ﬁnite and there exists a unique ζˆN , |ζˆN | < |x − x0 |, such that ax (ζ ) = ζˆN and yˆ = x + Ax (ζ + ζˆN eN ) is the unique minimizer. Choose the neighborhoods def

U(x) =

y∈R : N

&

|PH A−1 x (y − x)| < ρ |A−1 x (y − x) · eN | < |x − x0 |

,

def

Vx = {z ∈ H : |z| < ρ} .

Then condition (5.20) of Deﬁnition 5.2 is satisﬁed and it is easy to verify that the function ax satisﬁes conditions (5.21) and (5.22). (ii) Convexity of the function ax . By construction the neighborhoods Vx and U(x) are convex. The set U(x) ∩ Ω is convex. Then recall from (5.21)–(5.22) that U(x) ∩ {x + Ax (ζ + ζN eN ) : ζ ∈ Vx , ζN ≥ ax (ζ )} = U(x) ∩ Ω. Thus the set on the left-hand side is convex. In particular, ∀y 1 , y 2 ∈ Ω ∩ U(x), ∀α ∈ [0, 1],

αy 1 + (1 − α)y 2 ∈ Ω ∩ U(x).

In particular, for any two y 1 and y 2 in ∂Ω ∩ U(x),

j y j = x + Ax (ζ j + ζN eN ),

j ζN = ax (ζ j ), j = 1, 2,

and y α = αy 1 + (1 − α)y 2 = x + Ax [αζ 1 + (1 − α)ζ 2 ] ∈ Ω ∩ U(x) 1 2 + (1 − α)ζN ≥ ax αζ 1 + (1 − α)ζ 2 ⇒ αζN ⇒ αax (ζ 1 ) + (1 − α)ax (ζ 2 ) ≥ ax αζ 1 + (1 − α)ζ 2 and ax is convex. (iii) The function ax is Lipschitzian. It is suﬃcient to show that ax is bounded in Vx . Then we conclude from I. Ekeland and R. Temam [1, Lem. 3.1, p. 11] that the function is continuous and convex and hence Lipschitz continuous in Vx . By construction the minimizing point yˆ belongs to U(x), which is bounded. 5.4.3

Boundary Measure and Integral for Lipschitzian Sets

The boundary measure on ∂Ω, which was deﬁned in section 3.2 from the local C 1 -diﬀeomorphism, can also be deﬁned from the local graph representation of the boundary and extended to the Lipschitzian case. Assuming that ∂Ω is compact, there is a ﬁnite subcover of open neighborhoods U(x) for ∂Ω that can be represented by a ﬁnite family of Lipschitzian graphs. Let

5. Sets Locally Described by the Epigraph of a Function

95

def

{Uj }m j=1 , Uj = U(xj ), be a ﬁnite open cover of ∂Ω corresponding to some ﬁnite m sequence {xj }m j=1 of points of ∂Ω. Denote by eN , H, {Vj }j=1 , Vj = Vxj , and {aj }, aj = axj , the associated elements of Deﬁnition 5.2. Introduce the notation Γ = ∂Ω,

Γj = Γ ∩ Uj , 1 ≤ j ≤ m,

(5.29)

and the map hj = hxj and its inverse gj = gxj as deﬁned in (5.26)–(5.27): hx (ξ) = x + Ax (ξ + [ξN + ax (ξ )] eN ), def

−1 −1 gx (y) = (PH A−1 x (y − x), eN · Ax (y − x) − ax (PH Ax (y − x))). def

By Rademacher’s theorem the Lipschitzian function ax is diﬀerentiable almost everywhere in Vx and belongs to W 1,∞ Vx (cf., for instance, L. C. Evans and R. F. Gariepy [1]). Since hj is deﬁned from the Lipschitzian function aj , Dhj is deﬁned almost everywhere in Vx ×R, but also HN −1 almost everywhere on Vx ×{0}. Thus the canonical density for C 1 -domains still makes sense and is given by the same formula (3.13): ωj (ζ ) = ωxj (ζ ) = | det Dhj (ζ , 0)| | ∗(Dhj )−1 (ζ , 0) eN |. def

(5.30)

It is easy to verify that for almost all ξ ∈ Vj Dhj (ξ , ξN ) = Aj

... 0 .. . .. , det Dh (ξ , ξ ) = 1, .. j N . . .. . 0 . . . ∂N −1 aj (ξ ) 1

1

0 .. .

0 .. . 0 ∂1 aj (ξ )∂2 aj (ξ )

...

where ∂i aj is the partial derivative of aj with respect to the ith component of ζ = (ζ1 , . . . , ζN −1 ) ∈ Vj . The matrix Dhj is invertible and its coeﬃcients belong to L∞ and ∗(Aj )−1 = Aj . By direct computation

1

0 ∗ (Dhj )−1 (ξ , ξN ) = Aj ... . ..

0 .. .

...

−∂1 aj (ξ )

0 .. .

.

0 .. .

...

0

, −∂N −1 aj (ξ ) 1

= |(Aj )−1 ∗(Dhj )−1 (ζ , 0) eN | =

1 + |∇aj (ζ )|2 .

0

..

0

−∂2 aj (ξ ) .. .

(5.31)

(5.32)

and ﬁnally ωj (ζ ) = | ∗(Dhj )−1 (ζ , 0) eN |

96

Chapter 2. Classical Descriptions of Geometries and Their Properties

Let {r1 , . . . , rm } be a partition of unity for the Uj ’s, that is, r ∈ D(Uj ), 0 ≤ rj (x) ≤ 1, j m rj (x) = 1 in a neighborhood U of Γ

(5.33)

j=1

0 such that U ⊂ ∪m j=1 Uj . For any function f in C (Γ) the functions fj and fj ◦ hj deﬁned as

fj (y) = f (y)rj (y), fj ◦ hj (ζ , 0) = fj (x + Aj (ξ + aj (ξ ) eN ),

y ∈ Γj , ζ ∈ Vj ,

(5.34)

respectively, belong to C 0 (Γj ) and C 0 (Vj ). The integral of f on Γ is then deﬁned as

def

f dΓ = Γ

m j=1

fj dΓj , Γj

def

fj (hj (ζ , 0)) ωj (ζ ) dζ .

fj dΓ = Γj

(5.35)

Vj

Since ωj ∈ L∞ (Vj ), 1 ≤ j ≤ m, this integral is also well-deﬁned for all f in L1 (Γ); that is, the function fj ◦ hj ωj belongs to L1 (Vj ) for all j, 1 ≤ j ≤ m. As in section 2, the tangent plane to Γj at y = hj (ζ , 0) is deﬁned by Dhj (ζ , 0) Aj H, where Aj H is the hyperplane orthogonal to Aj eN . An outward normal ﬁeld to Γj is given by ∂1 aj (ζ ) .. . mx (ζ) = − ∗(Dhj )−1 (ζ , ζN )eN = Aj ∂N −1 aj (ζ ) −1 and the unit outward normal to Γj at y = hj (ζ , 0) ∈ Γj is given by ∂1 aj (ζ ) .. 1 . Aj n(y) = n(hj (ζ , 0)) = ' . ∂N −1 aj (ζ ) 1 + |∇aj (ζ )|2 −1

(5.36)

The matrix Aj can be removed by redeﬁning the functions ax on the space Aj H orthogonal to Aj eN . The formulae also hold for C k, -domains, k ≥ 1. We have seen that sets that are locally a C k, epigraph, k ≥ 0, have nice properties. For instance, their boundary has zero “volume.” However, they generally have no locally ﬁnite “surface measure” (cf. M. C. Delfour, N. Doyon, and J.´sio [1, 2] for speciﬁc examples) as we shall see in section 6.5. Fortunately, P. Zole Lipschitzian sets have locally ﬁnite “surface measure.”

5. Sets Locally Described by the Epigraph of a Function Theorem 5.7. Let Ω ⊂ RN be locally Lipschitzian. Then ' HN −1 U(x) ∩ ∂Ω ≤ 1 + c2x mN −1 Vx ,

97

(5.37)

where mN −1 and HN −1 are the respective (N − 1)-dimensional Lebesgue measure and (N − 1)-dimensional Hausdorﬀ measure, and cx > 0 is the Lipschitz constant of ax in Vx . If, in addition, ∂Ω is compact, then HN −1 (∂Ω) < ∞.

(5.38)

Proof. From Theorem 5.4 and from formula (5.32),

1/2 1/2 dζ ≤ 1 + c2x mN −1 Vx . 1 + |∇ax (ζ )|2 HN −1 U(x) ∩ ∂Ω = V (x)

When the boundary of Ω is compact, then we can ﬁnd a ﬁnite subcover of open neighborhoods {U(xj )}m j=1 . From the previous estimate, HN −1 (∂Ω) is bounded by a ﬁnite sum of bounded terms. 5.4.4

Geodesic Distance in a Domain and in Its Boundary

We can now justify property (2.29) in Theorem 2.6. Theorem 5.8. Let Ω ⊂ RN be bounded, open, and locally Lipschitzian. (i) If Ω is path-connected, then ∃cΩ , ∀x, y ∈ Ω,

distΩ (x, y) ≤ cΩ |x − y|.

(5.39)

dist∂Ω (x, y) ≤ c∂Ω |x − y|.

(5.40)

(ii) If ∂Ω is path-connected, then ∃c∂Ω , ∀x, y ∈ ∂Ω,

Remark 5.6. The boundary of a path-connected, bounded, open, and locally Lipschitzian set is generally not path-connected. As an example consider the set {x ∈ R2 : 1 < |x| < 2} whose boundary is made up of the two disjoint circles of radii 1 and 2. Remark 5.7. An open subset of RN with compact path-connected boundary is generally not pathconnected as can be seen from a domain in R2 made up of two tangent squares. Proof. For each x ∈ ∂Ω, there exists rx > 0 such that B(x, rx ) ⊂ U(x). Since ∂Ω is compact, there exist a ﬁnite sequence {xi }m i=1 of points of ∂Ω and a ﬁnite , B = B(x , R ), R = r . Moreover, from part (i) of the proof subcover {Bi }m i i i i x i i=1 of Theorem 5.2, ∃R > 0, ∀x ∈ ∂Ω, ∃i, 1 ≤ i ≤ m, such that B(x, R) ⊂ Bi .

98

Chapter 2. Classical Descriptions of Geometries and Their Properties

Always from part (i) of the proof of Theorem 5.2, the neighborhoods U and V can be chosen as def

U = B(0, R)

and

def

V = BH (0, R),

and to each point x ∈ ∂Ω we can associate the following new neighborhoods and functions: Vx = V, def

Ax = Axi , def

U(x) = x + Ax U, def

ζ → ax (ζ ) = axi (ζ + ξi ) − axi (ξi ) : V → R . def

Since V = BH (0, R) is path-connected and for each ζ ∈ V the map ζN → x + Ax [ζ + ζN eN ] is continuous, Lζ = B(x, R) ∩ {x + Ax [ζ + ζN eN ] : ζN ∈ R} def

is a (path-connected) line segment. This is nice, but we need to further reduce the size of the neighborhoods U (x) and Vx to make the piece of boundary ∂Ω ∩ U (x) and hence Ω∩U (x) path-connected. Since the family of functions {ax (ζ ) : x ∈ ∂Ω} is uniformly bounded and equicontinuous, there exists c > 0 such that ∀x ∈ ∂Ω,

∀ζ1 , ζ2 ∈ V,

|ax (ζ2 ) − ax (ζ1 )| ≤ c |ζ2 − ζ1 |.

Choose ρ, 0 < ρ ≤ R/(4c + 4). Then ∀ζ ∈ BH (0, ρ), |ax (ζ )| ≤ R/4 ⇒ |Ax (ζ + ax (ζ )eN )| < R ⇒ {x + Ax [ζ + ax (ζ )eN ] : ζ ∈ BH (0, ρ)} ⊂ ∂Ω ∩ B(x, R).

∀x ∈ ∂Ω,

Since BH (0, ρ) is path-connected and the map ζ → x + Ax [ζ + ax (ζ )eN ] is continuous, the image of BH (0, ρ) is path-connected. Redeﬁne V as BH (0, ρ) and U as B(0, R) ∩ {y ∈ RN : PH (y) ∈ BH (0, ρ)} and take the restrictions of the functions ax to BH (0, ρ). From this construction, for all x ∈ ∂Ω the new sets U(x ) ∩ ∂Ω and U(x ) ∩ int Ω are path-connected. Using the same notation for the ﬁnite sequence xi ∈ ∂Ω, let {U (xi ) : 1 ≤ i ≤ k} be the new ﬁnite subcovering of ∂Ω. For each x ∈ int Ω there exists 0 < rx < R/2 such that B(x, rx ) ⊂ int Ω. Consider the new set ΩR = Ω\ ∪ki=1 U (xi ). Since ΩR is compact, there exists a ﬁnite open covering of ΩR by the sets {B(xi , ri ) : k + 1 ≤ i ≤ m}, xi ∈ ΩR , and ri = rxi . By construction B(xi , ri ) ⊂ int Ω. (i) We ﬁrst show that distΩ (x, y) is uniformly bounded for any two points of Ω. Since Ω is covered by a ﬁnite number of balls, it is suﬃcient to show that the geodesic distance between any two points x and y in each ball is bounded. For k + 1 ≤ i ≤ m and x, y ∈ B(xi , ri ), distΩ (x, y) = |x − y| < R. For 1 ≤ i ≤ k, and x, y ∈ U (xi ), there is a graph representation x = xi + Ai (ζ + ζN eN ) and y = xi + Ai (ξ + ξN eN ). Choose as ﬁrst path the line from x = xi + Ai (ζ + ζN eN ) to xi + Ai (ζ + ai (ζ )eN ), follow the boundary along the second path {λζ +(1−λ)ξ , ai (λζ +(1−λ)ξ ) : λ ∈ [0, 1]} from xi +Ai (ζ +ai (ζ )eN )

5. Sets Locally Described by the Epigraph of a Function

99

to xi +Ai (ξ +ai (ξ )eN ), and ﬁnally as third path the line from xi +Ai (ξ +ai (ξ )eN ) to y = xi + Ai (ξ + ξN eN ). Its length is bounded by 1' |ζ − ξ |2 + |∇ai (λζ + (1 − λ)ξ ) · (ζ − ξ )|2 dλ + 2R, 2R + 0

√ which is bounded by 2R + 1 + c2 2ρ + 2R. Finally we proceed by contradiction. By deﬁnition of the geodesic distance distΩ (x, y)/|x − y| ≥ 1. If (5.39) is not true, there exist sequences {xn } and {yn } in Ω such that distΩ (xn , yn )/|xn − yn | goes to +∞, and hence, by boundedness of distΩ (xn , yn ), |xn − yn | goes to zero. Since Ω is bounded there exist subsequences, still denoted by {xn } and {yn }, and a point x ∈ Ω such that xn → x and yn → x. If x ∈ int Ω, there exists r > 0 such that B(x, r ) ⊂ int Ω and N such that for all n > N , xn , yn ∈ B(x, r ), where distΩ (xn , yn ) = |xn − yn | → 0 and we get a contradiction. If x ∈ ∂Ω, PH (B(x, ρ) − x) = BH (0, ρ) = V . There exists N such that, for all n > N , xn , yn ∈ B(0, ρ) and there exist (ζn , ζnN ), and (ξn , ξnN ), ζn , ξn ∈ BH (0, ρ), ξnN ≥ ax (ξn ), and ζnN ≥ ax (ζn ) such that xn = x + Ax [ξn + ξnN eN ] and yn = x + Ax [ζn + ζnN eN ]. The path {p(λ); 0 ≤ λ ≤ 1} deﬁned as λζn + (1 − λ)ξn p(λ) = x + Ax ax (λζn + (1 − λ)ξn ) + λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )] stays in B(x, R) ∩ Ω. Indeed λζn + (1 − λ)ξn ∈ BH (0, ρ), eN · A−1 x (p(λ) − x) = ax (λζn + (1 − λ)ξn ) + λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )] ≥ ax (λζn + (1 − λ)ξn ) , and p(λ) ∈ B(x, R). This last property follows from the estimate ' |p(λ) − x| ≤ |λζn + (1 − λ)ξn |2 + |ax (λζn + (1 − λ)ξn ) |2 + |λ [ζnN − ax (ζn )] + (1 − λ) [ξnN − ax (ξn )]| ' ≤ 1 + c2 |λζn + (1 − λ)ξn | + λ |ζnN − ax (ζn )| + (1 − λ) |ξnN − ax (ξn )| ' ' ≤ 1 + c2 ρ + λ 2ρ + (1 − λ) 2ρ = (2 + 1 + c2 ) ρ < R. Since the path lies in B(x, R) ∩ Ω, its length can be estimated by computing ζn − ξn p (λ) = Ax ∇ax (λζn + (1 − λ)ξn ) · (ζn − ξn ) + [ζnN − ξnN ] + ax (ξn ) − ax (ζn ) ⇒ |p (λ)| ≤ |ζn − ξn | + 2c|ζn − ξn | ≤ (1 + 2c) |yn − xn | 1 ⇒ distΩ (xn , yn ) ≤ |p (λ)| dλ ≤ (1 + 2c) |yn − xn | ≤ (1 + 2c) 2ρ < R 0

and we again get a contradiction. (ii) The case where ∂Ω is path-connected follows by the same constructions and proof as in part (i).

100 5.4.5

Chapter 2. Classical Descriptions of Geometries and Their Properties Nonhomogeneous Neumann and Dirichlet Problems

With the help of the previous deﬁnition of boundary measure in section 5.4.3, we can now make sense of the nonhomogeneous Neumann and Dirichlet problems (for the Laplace equation) in Lipschitzian domains. Assuming that the functions aj belong to W 1,∞ (Vj ), it can be shown that the classical Stokes divergence theorem holds for such domains. Given a bounded smooth domain D in RN and a Lipschitzian domain Ω in D, then ∀ ϕ ∈ C 1 (D, RN ), div ϕ

dx = ϕ

· n dΓ, (5.41) Ω

Γ

where the outward unit normal ﬁeld n is deﬁned by (5.36) for almost all ∈ Γj (with y = hx (ζ , 0), ζ ∈ Vj ) as (∂1 aj (ζ ), . . . , ∂N −1 aj (ζ ), −1) ' . 1 + |∇aj (ζ )|2

∗

n(y) = Aj

The trace of a function g in W 1,1 (D) on Γ is deﬁned through a W 1,∞ (D)N -extension N of the normal n as def gϕ dΓ = div (gϕN ) dx (5.42) ∀ϕ ∈ D(RN ), Γ

Ω

(cf. S. Agmon, A. Douglis, and L. Nirenberg [1, 2]). The right-hand side is well-deﬁned in the usual sense so that by the Stokes divergence theorem we obtain the trace g Γ deﬁned on Γ. This trace is uniquely deﬁned, denoted by γΓ g, and (5.43) γΓ ∈ L W 1,1 (Ω), L1 (Γ) . Let Ω be a Lipschitzian domain in D. Given f ∈ L2 (D) and g ∈ H 1 (D), consider the following problem: ∃y ∈ H 1 (Ω) such that ∀ϕ ∈ H 1 (Ω), ∇y · ∇ϕ dx = f ϕ dx + g ϕ dΓ. Ω

If the condition

Ω

(5.44)

Γ

f dx + Ω

g dΓ = 0

(5.45)

Γ

is satisﬁed, then by the Lax–Milgram theorem, problem (5.44)–(5.45) has a unique solution in H 1 (Ω)/ R: any two solutions of problem (5.44) can diﬀer only by a constant. For instance, we can associate with the solution y = y(Ω) in H 1 (Ω)/ R of (5.44) the following objective function: def def 2 J(Ω) = c Ω, y(Ω) , c(Ω, ϕ) = (|∇ϕ(x)| − G) dx (5.46) Ω

for some G in L2 (D).

6. Sets Locally Described by a Geometric Property

101

For the nonhomogeneous Dirichlet problem let Ω be a Lipschitzian domain in D and consider the weak form of the problem ∃y ∈ H 1 (Ω) such that ∀ϕ ∈ H 2 (Ω) ∩ H01 (Ω), ∂ϕ y ϕ dx = − g f ϕ dx − dΓ + ∂n Ω Γ Ω

(5.47)

for data (f, g) in L2 (D) × H 1 (D). If this problem has a solution y, then by Green’s theorem ∂ϕ ∂ϕ dΓ = − g dΓ + ∇y · ∇ϕ dx − y f ϕ dx (5.48) ∂n Ω Γ Γ ∂n Ω and −y = f ∈ L2 (Ω),

1

y|Γ = g ∈ H 2 (Γ).

Here, for instance, we can associate with y the objective function def def 1 |ϕ − G|2 dx, G ∈ L2 (D). J(Ω) = c Ω, y(Ω) , c(Ω, ϕ) = 2 Ω

6

(5.49)

(5.50)

Sets Locally Described by a Geometric Property

A large class of sets Ω can be characterized by geometric segment properties. The basic segment property is equivalent to the property that the set is locally a C 0 epigraph (cf. section 6.2). For instance, it is suﬃcient to get the density of C k (Ω) in the Sobolev space W m,p (Ω) for any m ≥ 1 and k ≥ m. In this section, we establish the equivalence of the segment, the uniform segment, and the more recent9 uniform fat segment properties with the respective locally C 0 , the C 0 , and the equi-C 0 epigraph properties (cf. sections 6.2, 6.3, and 6.4). Under the uniform fat segment property the local functions all have the same modulus of continuity speciﬁed by the continuity of the dominating function at the origin (cf. section 6.3). For sets with a compact boundary the three segment properties are equivalent (cf. section 6.1). However, for sets with an unbounded boundary the uniform segment property is generally too meager to make the local epigraphs uniformly bounded and equicontinuous. Section 6.4 specializes the uniform fat segment property to the uniform cusp and cone properties, which imply that ∂Ω is, respectively, an equi-H¨olderian and equi-Lipschitzian epigraph. Section 6.5 discusses the existence of a locally ﬁnite boundary measure. We have already seen in section 5.4.3 that Lipschitzian sets have a locally ﬁnite boundary or surface (that is Hausdorﬀ HN −1 ) measure. So we can make sense of boundary conditions associated with a partial diﬀerential equation on the domain. In general, this is not true for H¨ olderian domains. To illustrate that point, we construct examples of H¨ olderian sets for which the Hausdorﬀ dimension of the boundary is strictly greater than N − 1 and hence HN −1 (∂Ω) = +∞. 9 This was introduced starting with the uniform cusp property in M. C. Delfour and J.-P. Zo´sio [37] in 2001 and further reﬁned and generalized in the sequence of papers by M. C. Delfour, le ´sio [1, 2] in 2005 and M. C. Delfour and J.-P. Zole ´sio [43] in 2007. N. Doyon, and J.-P. Zole

102

6.1

Chapter 2. Classical Descriptions of Geometries and Their Properties

Deﬁnitions and Main Results

In this section we use the notation introduced at the beginning of section 5: eN is a unit vector in RN and H = {eN }⊥ . The open segment between two distinct points x and y of RN will be denoted by def

(x, y) = {x + t(y − x) : ∀t, 0 < t < 1}. Deﬁnition 6.1. Let Ω be a subset of RN such that ∂Ω = ∅. (i) Ω is said to satisfy the segment property if ∀x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃Ax ∈ O(N), such that

∀y ∈ B(x, r) ∩ Ω,

(y, y + λAx eN ) ⊂ int Ω.

(ii) Ω is said to satisfy the uniform segment property if ∃r > 0, ∃λ > 0 such that ∀x ∈ ∂Ω, ∃Ax ∈ O(N), such that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λAx eN ) ⊂ int Ω.

(iii) Ω is said to satisfy the uniform fat segment property if there exist r > 0, λ > 0, and an open region O of RN containing the segment (0, λeN ) and not 0 such that for all x ∈ ∂Ω, ∃Ax ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω,

y + Ax O ⊂ int Ω.

(6.1)

Of course, when Ω is an open domain, Ω = int Ω, and we are back to the standard deﬁnition. Yet, as in Deﬁnition 3.1, those deﬁnitions involve only ∂Ω, Ω, and int Ω and the properties remain true for all sets in the class

[Ω]b = A ⊂ RN : A = Ω and ∂A = ∂Ω . A set having the segment property must have an (N −1)-dimensional boundary and cannot simultaneously lie on both sides of any given part of its boundary. In fact, a domain satisfying the segment property is locally a C 0 epigraph in the sense of Deﬁnition 5.1, as we shall see in section 6.2. We ﬁrst give a few general properties. Theorem 6.1. Let Ω be a subset of RN such that ∂Ω = ∅. (i) If ∀x ∈ ∂Ω, ∃λ > 0, ∃Ax ∈ O(N),

(x, x + λAx eN ) ∈ int Ω,

then int Ω = ∅ and int Ω = Ω

and

∂(int Ω) = ∂Ω.

(6.2)

6. Sets Locally Described by a Geometric Property

103

(ii) Ω satisﬁes the segment property if and only if Ω satisﬁes the segment property. In particular int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω, int Ω = ∅, int Ω = Ω, and ∂(int Ω) = ∂Ω. Proof. (i) Pick any point x ∈ ∂Ω. By deﬁnition, for all µ ∈ ]0, 1[ , xµ = x+µ λ AeN ∈ int Ω and int Ω = ∅. As a result there exists a sequence µn > 0 → 0 such that int Ω xµn → x

⇒ ∂Ω ⊂ int Ω ⊂ Ω

⇒ Ω ⊂ int Ω ⊂ Ω.

The second identity follows from Lemma 5.1 (i). (ii) Assume that Ω satisﬁes the segment property. For any x ∈ ∂Ω, there exist r > 0, λ > 0, and A ∈ O(N) such that for all y ∈ B(x, r) ∩ Ω, (y, y + λAeN ) ⊂ int Ω. We claim that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λ(−A)eN ) ⊂ int Ω.

Indeed, if there exist z ∈ B(x, r) ∩ Ω and α ∈ (0, λ) such that y − αAeN ∈ Ω, then y = (y − αAeN ) + αAeN ∈ int Ω, which leads to a contradiction. In the other direction we interchange Ω and Ω and repeat the proof. Finally, the two sets of properties follow from part (i) applied to Ω and Ω. We summarize the equivalences between the epigraph and segment properties that will be detailed in Theorems 6.5, 6.6, and 6.7 with the associated constructions and additional motivation. Theorem 6.2. Let Ω be a subset of RN such that ∂Ω = ∅. (i) (a) Ω is locally a C 0 epigraph if and only if Ω has the segment property; (b) Ω is a C 0 epigraph if and only if Ω has the uniform segment property; (c) Ω is an equi-C 0 epigraph if and only if Ω has the uniform fat segment property. (ii) If, in addition, ∂Ω is compact, the six properties are equivalent and there exists a dominating function h ∈ H that satisﬁes the properties of Theorems 5.1 and 5.2. Proof. (i) From Theorems 6.5, 6.6, and 6.7. (ii) When ∂Ω is compact, from part (i) and the equivalences in Theorem 5.2. We quote the following density results from R. A. Adams [1, Thm. 3.18, p. 54]. Theorem 6.3. If Ω has the segment property, then the set

f |int Ω : ∀f ∈ C0∞ (RN ) of restrictions of functions of C0∞ (RN ) to int Ω is dense in W m,p (int Ω) for 1 ≤ p < ∞ and m ≥ 1. In particular, C k (int Ω) is dense in W m,p (int Ω) for any m ≥ 1 and k ≥ m.

104

6.2

Chapter 2. Classical Descriptions of Geometries and Their Properties

Equivalence of Geometric Segment and C 0 Epigraph Properties

As for sets that are locally a C 0 epigraph in Deﬁnition 5.1, the three cases of Deﬁnition 6.1 diﬀer only when ∂Ω is unbounded. We ﬁrst deal with the ﬁrst two and will come back to the third one in section 6.3. Theorem 6.4. Let ∂Ω = ∅ be compact. Then the segment property and the uniform segment property of Deﬁnition 6.1 are equivalent. Proof. It is suﬃcient to show that the segment property implies the uniform segment property. Since ∂Ω is compact there exists a ﬁnite open subcover {Bi }m i=1 , Bi = of points of ∂Ω. From part (i) of B(xi , rxi ), of ∂Ω for some ﬁnite sequence {xi }m i=1 the proof of Theorem 5.2 ∃r > 0, ∀x ∈ ∂Ω,

∃i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi .

Deﬁne λ = min1≤i≤m λi > 0. Therefore for each x ∈ ∂Ω, there exists i, 1 ≤ i ≤ m, such that B(x, r) ⊂ Bi and hence, by choosing Ax = Axi ∈ O(N), we get ∀y ∈ Ω ∩ Bi ,

(y, y + λAx eN ) ⊂ (y, y + λi Axi eN ) ⊂ int Ω.

By restricting the above property to B(x, r) ⊂ Bi , ∀y ∈ Ω ∩ B(x, r),

(y, y + λAx eN ) ⊂ (y, y + λi Ax eN ) ⊂ int Ω.

Since r and λ are independent of x ∈ ∂Ω, Ω has the uniform segment property. Theorem 6.5. Let Ω be a subset of RN such that ∂Ω = ∅. (i) If Ω satisﬁes the segment property, then Ω is locally a C 0 epigraph. For x ∈ ∂Ω, let B(x, rx ), λx , and Ax be the associated open ball, the height, and the rotation matrix. Then there exists ρx , def

0 < ρx ≤ rxλ = min {rx , λx /2} ,

(6.3)

which is the largest radius such that

BH (0, ρx ) ⊂ PH (A−1 x (y − x)) : ∀y ∈ B(x, rxλ ) ∩ ∂Ω . The neighborhoods of Deﬁnition 5.1 can be chosen as def

Vx = BH (0, ρx ) and

def U(x) = B(x, rxλ ) ∩ y ∈ RN : PH (A−1 x (y − x)) ∈ Vx ,

(6.4)

where10 BH (0, ρx ) denotes the open ball of radius ρx in the hyperplane H. For each ζ ∈ Vx , there exists a unique yζ ∈ ∂Ω∩U(x) such that PH A−1 x (yζ −x) = ζ and the function ζ → ax (ζ ) = (yζ − x) · (Ax eN ) : Vx → R def

10 Note

that U(x) − {x} = Ax [B(0, rxλ ) ∩ {ζ : PH ζ ∈ Vx }].

6. Sets Locally Described by a Geometric Property

105

is well-deﬁned, bounded, ∀ζ ∈ Vx ,

|ax (ζ )| < rxλ ,

(6.5)

and uniformly continuous in Vx , that is, ax ∈ C 0 (Vx ). (ii) Conversely, if Ω is locally a C 0 epigraph, both Ω and Ω satisfy the segment property. Proof. (i) By assumption for each x ∈ ∂Ω, ∃r > 0, ∃λ > 0, ∃A ∈ O(N) such that ∀y ∈ B(x, r) ∩ Ω,

(y, y + λAeN ) ⊂ int Ω.

Consider the set def

W = PH A−1 (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω ,

rλ = min{r, λ/2}.

Therefore ∀ζ ∈ W,

∃y ∈ B(x, r) ∩ ∂Ω,

PH A−1 (y − x) = ζ .

If there exist y1 , y2 ∈ B(x, rλ )∩∂Ω such that PH A−1 (y1 −x) = PH A−1 (y2 −x) = ζ , then y2 ∈ (y1 , y1 + AeN · (y2 − y1 ) AeN ) ⊂ (y1 , y1 + λAeN ) ∈ int Ω since |y2 − y1 | < 2rλ < λ and we get a contradiction. Therefore, the map ζ → a(ζ ) = AeN · (y − x) = eN · A−1 (y − x) : W → R def

is well-deﬁned and a(0) = 0. (W is open and 0 ∈ W ). By deﬁnition, 0 ∈ W ⊂ BH (0, rλ ). Given any point ξ ∈ W , the segment property is satisﬁed at the point yξ = x + A(ξ + a(ξ )eN ) ∈ ∂Ω ∩ B(x, rλ ), def

and (yξ , yξ + λAeN ) ⊂ int Ω and (yξ , yξ − λAeN ) ⊂ int Ω by Theorem 6.1 (ii): (yξ , yξ + λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ B(x, rλ ), (yξ , yξ − λAeN ) ∩ B(x, rλ ) ⊂ int Ω ∩ ∩B(x, rλ ). Therefore, there exists a neighborhood BH (ξ , R) of ξ such that BH (ξ , R) ⊂ BH (0, rλ ) and for all ζ ∈ BH (ξ , R), Lζ ∩ int Ω ∩ B(x, rλ ) = ∅, where Lζ = x + A(ζ + R eN ) is the line through x + Aζ in the direction AeN . If ξ is not an interior point of W , there exists a sequence {ζn } ⊂ BH (ξ , R), ζn → ξ , such that for all n, (Lζn ∩ B(x, rλ )) ∩ ∂Ω = ∅. Therefore, either Lζn ∩ B(x, rλ ) ⊂ int Ω or Lζn ∩ B(x, rλ ) ⊂ int Ω. Since for points ζn ∈ BH (ξ , R) we already know that Lζn ∩ B(x, rλ ) ∩ int Ω = ∅, then Lζn ∩ B(x, rλ ) ⊂ int Ω. Choose def 1 α = a(ξ ) − rλ2 − |ξ |2 2

106

Chapter 2. Classical Descriptions of Geometries and Their Properties

and the sequence zn = x + A(ζn + αeN ) → z = yξ − def

def

1 2

a(ξ ) +

rλ2 − |ξ |2 AeN .

We get the following estimate: |z − x|

N − α NΩ,i (ε) ≤

m i=1

NΩ,i (ε) ≤ m

1 εN −α

(rλ + ε)

⇒ NΩ (ε)εβ ≤ εβ−N +α m (rλ + ε) ⇒ ∀β > N − α,

N −1

N −1

α √ N − 1 + ε1−α c

α √ N − 1 + ε1−α c

Hβ (∂Ω) = 0.

By deﬁnition, the Hausdorﬀ dimension of ∂Ω is less than or equal to N − α.

118

Chapter 2. Classical Descriptions of Geometries and Their Properties

It is possible to construct examples of sets verifying the uniform cusp property for which the Hausdorﬀ dimension of the boundary is strictly greater than N − 1 and hence HN −1 (∂Ω) = +∞. Example 6.1. This following two-dimensional example of an open domain with compact boundary satisfying the uniform cusp condition for the function h(θ) = θ α , 0 < α < 1, can easily be generalized to an N -dimensional example. Consider the open domain def

Ω = {(x, y) : −1 < x ≤ 0 and 0 < y < 2} ∩ {(x, y) : 0 < x < 1 and f (x) < y < 2} ∩ {(x, y) : 1 ≤ x < 2 and 0 < y < 2} in R2 (see Figure 2.7), where f : [0, 1] → R is deﬁned as follows: def

f (x) = dC (x)α ,

0 ≤ x ≤ 1,

and C is the Cantor set on the interval [0, 1] and dC (x) is the distance function from the point x to the set C (dC is uniformly Lipschitzian of constant 1). This function is equal to 0 on C. Any point in [0, 1]\C belongs to one of the intervals of length 3−k , k ≥ 1, which has been deleted from [0, 1] in the sequential construction of the Cantor set. Therefore the distance function dC (x) is equal to the distance function to the two end points of that interval. In view of this special structure it can be shown that ∀x, y ∈ [0, 1],

|dC (y)α − dC (x)α | ≤ |y − x|α .

Denote by Γ the piece of the boundary ∂Ω speciﬁed by the function f = dC . On Γ the uniform cusp condition is veriﬁed with ρ = 1/6, λ = (1/6)α , and h(θ) = θ α (see

2 y

1

domain Ω

C(λ, h, ρ, e2 )

f (x) 0

−1

x 0

1

2

Figure 2.7. Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α , h(θ) = θα .

6. Sets Locally Described by a Geometric Property

0

1/9

2/9

1/3

119

2/3

7/9

8/9

1

Figure 2.8. f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. Figure 2.8). Clearly the number NΩ (ε) of hypercubes of dimension N and side ε required to cover ∂Ω is greater than the number NΓ (ε) of hypercubes of dimension N and side ε required to cover Γ. The construction of the Cantor set is done by sequentially deleting intervals. At step k = 0 the interval (1/3, 2/3) of width 3−1 is removed. At step k a total of 2k intervals of width 3−(k+1) are removed. Thus if we pick ε = 3−(k+1) , the interval [0, 1] can be covered with exactly 3(k+1) intervals. Here we are interested in ﬁnding a lower bound to the total number of squares of side ε necessary to cover Γ. For this purpose we keep only the 2k intervals removed at step k. Vertically it takes α −1 −(k+1) α 2−1 3−(k+1) 2 3 ≥ − 1. 3−(k+1) 3−(k+1) Then we have for β ≥ 0 0 NΩ (ε) ≥ NΓ (ε)≥ 2k

2−1 3−(k+1) 3−(k+1)

1

α −1

k ≥ 2k−α 3(k+1)(1−α) − 2k = 3(1−α) 2 2−α 3(1−α) − 2k and NΩ (ε)(3

−k 1+β

)

−k(1+β)

≥3

≥ 3

−(α+β)

3

(1−α)

k 2

2

k 2

2

−α (1−α)

3

−α (1−α)

3

−

−2

2 3(1+β)

k

k .

The second term goes to zero as k goes to inﬁnity. The ﬁrst term goes to inﬁnity as k goes to inﬁnity if 3−(α+β) 2 > 1, that is, 0 < α+β < ln 2/ ln 3. Under this condition, H1+β (∂Ω) = H1+β (Γ) = +∞ for all 0 < α < ln 2/ ln 3 and all 0 ≤ β < ln 2/ ln 3 − α.

120

Chapter 2. Classical Descriptions of Geometries and Their Properties

Therefore given 0 < α < ln 2/ ln 3 ∀β, 0 ≤ β + α < ln 2/ ln 3,

H1+β (∂Ω) = +∞

and the Hausdorﬀ dimension of ∂Ω is strictly greater than 1. Given 0 < α < 1, it is possible to construct an optimal example of a set verifying the uniform cusp property for which the Hausdorﬀ dimension of the boundary is exactly N − α and hence HN −1 (∂Ω) = +∞. Example 6.2 (Optimal example of a set that veriﬁes the uniform cusp property with h(θ) = |θ|α , 0 < α < 1, and whose boundary has Hausdorﬀ dimension exactly equal to N − α.). For that purpose, we need a generalization of the Cantor set. Denote by C1 the Cantor set. Recall that each x, 0 ≤ x ≤ 1, can be written uniquely (if we make a certain convention) as ∞ aj (3, x) x= , 3j j=1 where aj (3, x) can be regarded as the jth digit of x written in basis 3. From this the Cantor set is characterized as follows: x ∈ C1 ⇐⇒ ∀j, aj (3, x) = 1. Similarly for an arbitrary integer k ≥ 1, each x ∈ [0, 1] can be uniquely written in the form ∞ aj (2k + 1, x) x= (2k + 1)j j=1 and we can deﬁne the set Ck as x ∈ Ck ⇐⇒ ∀j, aj (2k + 1, x) = k. In a certain sense, if k1 > k2 , Ck1 contains more points than Ck2 . We now use these sets to construct the family of set Dk as follows: x ∈ D1 ⇐⇒ 2x ∈ C1 and for k > 1, x ∈ Dk ⇐⇒ 2k+1 (x − 2k ) ∈ Ck . 2 Note that if k1 = k2 , Dk1 Dk2 = ∅ since the Dk ’s contain only points from the def interval [1 − 2k−1 , 1 − 2k ]. Consider now the set D = ∪∞ k=1 Dk and go back to def

Example 6.1 with the function f replaced by the function f (x) = dD (x)α . Again it can be shown that for all x, y ∈ [0, 1], |dD (y)α − dD (x)α | ≤ |y − x|α . Note that on the interval [1 − 2k−1 , 1 − 2k ] we have dD (x)α = dDk (x)α . Denote by Γ the piece of boundary ∂Ω speciﬁed by the function f = dD and by Γk the part of boundary ∂Ω speciﬁed by the function f = dD (= dDk ) on the interval [1 − 2k−1 , 1 − 2k ]. Once again on Γ the uniform cusp property is veriﬁed

6. Sets Locally Described by a Geometric Property

121

with ρ = 1/6, λ = (1/6)α , and h(θ) = θα . Clearly the number NΩ (ε) of hypercubes of dimension N and side ε required to cover ∂Ω is greater than the number NΓk (ε) of hypercubes of dimension N and side ε required to cover Γk . The construction of the set Ck is also done sequentially by deleting intervals. At step j = 0 the interval ]k/(2k + 1), (k + 1)/(2k + 1)[ of width (2k + 1)−1 is removed. At step j a total of 2j intervals of width (2j + 1)−(j+1) are removed. If we consider the intervals that remain at step j, a total of 2j+1 nonempty disjoint intervals of width (k/(2k + 1))j+1 remain in the set Ck . Each of these intervals contains a gap of length (k/(2k + 1))j+1 1/(2k + 1) created at step j + 1. If we construct the set Dk in the same way, at step j a total of 2j nonempty disjoint intervals of width (k/(2k + 1))j+1 1/2k remain in the set Dk . Each of these intervals contains a gap of length (k/(2k + 1))j+1 1/(2k (2k + 1)). Pick j+1 1 k ε= k 2 2k + 1 and look for a lower bound on the number of squares of side ε necessary to cover Γk . For this purpose, consider only the 2j+1 nonempty disjoint intervals remaining at step j. As they each contain a gap of length j+1 1 k 2k + 1 2k (2k + 1) vertically it takes 1α 0 j+1 j+1 k 1 2k + 1 k 2 2k + 1 2k+1 (2k + 1) k 1α 0 j+1 j+1 1 2k + 1 k k 2 −1 ≥ 2k + 1 2k+1 (2k + 1) k ε-cubes. Then we have for β ≥ 0 00 1α 1 j+1 j+1 k 1 2k + 1 j+1 k NΩ (ε) ≥ NΓ (ε) ≥ 2 2 −1 2k + 1 2k+1 (2k + 1) k j+1 2(2k + 1)kα 2k ≥ − 2j+1 k(2k + 1)α 2α(k+1) (2k + 1)α j+1 2k(1−α)−α − 2j+1 = 2(2k + 1)1−α k α−1 (2k + 1)α and hence ε1+β NΩ (ε) 0 j+1 11+β k(1−α)−α 1 k 1−α α−1 j+1 2 j+1 ≥ k 2(2k + 1) − 2 2k 2k + 1 (2k + 1)α 0 0 α+β 1j+1 j+1 11+β 2k(1−α)−α 1 k k j+1 2 −2 . ≥ 2k + 1 2k 2k + 1 2k(1+β) (2k + 1)α

122

Chapter 2. Classical Descriptions of Geometries and Their Properties

The second term goes to zero as j goes to inﬁnity. The ﬁrst term goes to inﬁnity as j goes to inﬁnity if ∀k,

k 2k + 1

α+β 2 > 1,

that is ∀k,

0 0 such that for all x such that |x| > ρ0 , |g(x)| < ε. But f also belongs to C00 (RN , RN ) and

126

Chapter 3. Courant Metrics on Images of a Set

there exists ρ ≥ 2ρ0 such that for all x such that |x| > ρ, |f (x)| < ρ0 . In particular, |x + f (x)| ≥ |x| − |f (x)| > ρ − ρ0 ≥ ρ0 and hence |g(x + f (x))| < ε. In conclusion for all ε > 0, there exists ρ > 0 such that for all x such that |x| > ρ, |g(x + f (x))| < ε and g ◦ (I + f ) ∈ C00 (RN , RN ). (4) Θ = C 0,1 (RN , RN ). This is the space of bounded Lipschitz continuous functions on RN that is contained in C 0 (RN , RN ). For f, g ∈ C 0,1 (RN , RN ), f ◦ (I + g)C 0 = f C 0 < ∞. Since g is Lipschitz with Lipschitz constant c(g), I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is also Lipschitz with constant c(f ) (1 + c(g)). Proof of Theorem 2.1. I ∈ F(Θ) for f = 0. If I + f ∈ F(Θ), (I + f )−1 is bijective, (I + f )−1 − I ∈ Θ, (F −1 )−1 = F = I + f , and (F −1 )−1 − I = f ∈ Θ. Hence (I + f )−1 ∈ F(Θ) and I = (I + f ) ◦ (I + f )−1 = (I + f )−1 + f ◦ (I + f )−1 ⇒ (I + f )−1 − I = −f ◦ (I + f )−1 . The last property to check is the composition (I + f ) ◦ (I + g). It is bijective as the composition of two bijective transformations. Moreover, by linearity of Θ and Assumption 2.1 (I + f ) ◦ (I + g) − I = g + f ◦ (I + g) ∈ Θ.

(2.4)

From (2.4) and the invertibility of the composition [(I + f ) ◦ (I + g)]−1 − I = −[g + f ◦ (I + g)] ◦ (I + g)−1 ◦ (I + f )−1 ∈ Θ by using Assumption 2.1 twice. Conversely, if (2.4) is true for all f ∈ Θ and I + g ∈ F(Θ), by linearity of Θ, Assumption 2.1 is veriﬁed. Associate with F ∈ F(Θ) the following function of I and F : def

d(I, F ) =

inf

n

F =F1 ◦···◦Fn Fi ∈F (Θ) i=1

Fi − IΘ + Fi−1 − IΘ ,

(2.5)

where the inﬁmum is taken over all ﬁnite factorizations of F in F(Θ) of the form F = F1 ◦ · · · ◦ Fn ,

Fi ∈ F(Θ).

In particular d(I, F ) = d(I, F −1 ). Extend this function to all F and G in F(Θ): d(F, G) = d(I, G ◦ F −1 ). def

By deﬁnition, d is right-invariant since for all F , G, and H in F(Θ) d(F, G) = d(F ◦ H, G ◦ H).

(2.6)

2. Generic Constructions of Micheletti

127

To show that d is a metric5 on F(Θ) necessitates a second assumption. Assumption 2.2. (Θ, · ) is a normed vector space of mappings from RN into RN and for each r > 0 there exists a continuous function c0 (r) such that ∀{fi }ni=1 ⊂ Θ such that

n

fi < α < r;

(2.7)

(I + f1 ) ◦ · · · ◦ (I + fn ) − I < α c0 (r).

(2.8)

i=1

then

Theorem 2.2. Under Assumptions 2.1 and 2.2, d is a right-invariant metric on F(Θ). Example 2.2. (1), (2), (3) Since the norm on the three spaces B0 (RN , RN ), C 0 (RN , RN ), and C00 (RN , RN ) is the same and Assumption 2.2 involves only the norm, it is suﬃcient to check it for B 0 (RN , RN ). Consider {fi }ni=1 ⊂ B0 (RN , RN ) such that n fi C 0 < α < r. (2.9) i=1

Using the convention (I + fn ) ◦ (I + fn ) = (I + fn ) in the summation (I + f1 ) ◦ · · · ◦ (I + fn ) − I = (I + fn ) − I +

n−1

(I + fi ) ◦ · · · ◦ (I + fn ) − (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

= fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α < r

⇒ c0 (r) = r.

i=1

(4) Θ = C 0,1 (RN , RN ). Consider a sequence {fi }ni=1 ⊂ C 0,1 (RN , RN ) such that n

max {fi C 0 , c(fi )} < α < r.

(2.10)

i=1 5A

function d : X × X → R is said to be a metric on X if (cf. J. Dugundji [1])

(i) d(F.G) ≥ 0, for all F, G, (ii) d(F, G) = 0 ⇐⇒ F = G, (iii) d(F, G) = d(G, F ), for all F, G, (iv) d(F, H) ≤ d(F, G) + d(G, H), for all F, G, H. A metric d on a group (F , ◦) is said to be right-invariant if for all H ∈ F , d(F ◦H, G◦H) = d(F, G).

128

Chapter 3. Courant Metrics on Images of a Set

From the previous example (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α.

i=1

In addition, (I + f1 ) ◦ · · · ◦ (I + fn ) − I = fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ c((I + f1 ) ◦ · · · ◦ (I + fn ) − I) ≤ c(fn ) +

n−1

c(fi ) (1 + c(fi+1 )) . . . (1 + c(f1 ))

i=1

≤ c(fn ) + ≤

n

n−1

)i+1 c(fi ) e

i=1

)n

c(fi ) e[

i=1

k=1

c(fi )]

c(fk )

< α eα

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0,1 < α max{eα , 1} < α er and we can choose c0 (r) = er . Proof of Theorem 2.2. Properties (i) and (iii) are veriﬁed by deﬁnition. For the triangle inequality (iv), F ◦ H −1 = (F ◦ G−1 ) ◦ (G ◦ H −1 ) is a ﬁnite factorization of F ◦ H −1 . Consider arbitrary ﬁnite factorizations of F ◦ G−1 and G ◦ H −1 : F ◦ G−1 = (I + α1 ) ◦ · · · ◦ (I + αm ), G◦H

−1

= (I + β1 ) ◦ · · · ◦ (I + βn ),

I + αi ∈ F(Θ), I + βj ∈ F(Θ).

This yields a new ﬁnite factorization of F ◦ H −1 . By deﬁnition of d, d(I, F ◦ H −1 ) ≤

m

αi + αi ◦ (I + αi )−1 +

i=1

n

βj + βj ◦ (I + βj )−1 ,

i=j

and by taking inﬁma over each factorization, d(F, H) = d(I, F ◦ H −1 ) ≤ d(I, F ◦ G−1 ) + d(I, G ◦ H −1 ) = d(F, G) + d(G, H) using the symmetry property (iii). To verify (ii) it is suﬃcient to show that F = I ⇐⇒ d(I, F ) = 0. Clearly I ◦ I is a factorization of F = I and 0 ≤ d(I, F ) ≤ 0. Conversely, if d(I, F ) = 0, for ε, 0 < ε < 1, there exists a ﬁnite factorization F = F1 ◦ · · · ◦ Fn such that n

Fi − I + Fi−1 − I < ε < 1.

i=1

By Assumption 2.2, F − I + F −1 − I < ε c0 (1), which implies that F − I = 0 and F −1 − I = 0. This completes the proof.

2. Generic Constructions of Micheletti

129

An important aspect of the previous construction was to build in the triangle inequality to make d a (right-invariant) metric. However, there are other ways to introduce a topology on F(Θ). The interest of this speciﬁc choice will become apparent later on. For instance, the original Courant metric of A. M. Micheletti [1] in 1972 was constructed from the following slightly diﬀerent right-invariant metric on F(Θ): given H ∈ F(Θ), consider ﬁnite factorizations H1 ◦ · · · ◦ Hm , Hi ∈ F(Θ), 1 ≤ i ≤ m, of H and ﬁnite factorizations G1 ◦ · · · ◦ Gn , Gj ∈ F(Θ), 1 ≤ j ≤ n, of H −1 , and deﬁne m n def d1 (I, H) = inf Hi − I + inf Gi − I, (2.11) H=H1 ◦···◦Hm i=1 m≥1

H −1 =G1 ◦···◦Gn i=1 n≥1

where the inﬁma are taken over all ﬁnite factorizations of H and H −1 in F(Θ) and ∀F, G ∈ F(Θ),

d1 (G, F ) = d1 (I, F ◦ G−1 ). def

(2.12)

Another example is the topology generated by the right-invariant semimetric 6 ∀H ∈ F(Θ),

d0 (I, H) = H − I + H −1 − I, def

∀F, G ∈ F(Θ),

d0 (G, F ) = d0 (I, F ◦ G−1 ). def

(2.13)

Given r > 0 and F ∈ F(Θ), let Br (F ) = {G ∈ F(Θ) : d0 (F, G) < r}. Let τ0 be the weakest topology on F(Θ) such that the family {Br (F ) : 0 ≤ r < ∞} is the base for τ0 . By deﬁnition d1 (F, G) ≤ d(F, G) ≤ d0 (F, G) and the injections (F, d1 ) → (F, d) → (F, d0 ) are continuous. Theorem 2.3. Under Assumptions 2.1 and 2.2, the topologies generated by d1 , d, and d0 are equivalent. Proof. Since d0 , d, and d1 are right-invariant on the group F(Θ), it is suﬃcient to prove the equivalence for d0 and d1 around I. For all ε > 0 with δ = ε ∀H ∈ F(Θ), d0 (I, H) < δ,

d1 (I, H) < ε,

since d1 (I, H) ≤ d0 (I, H). Conversely, by Assumption 2.2, for any δ, 0 < δ < 1, d1 (I, H) < δ < 1

⇒ d0 (I − H) = H − I + H −1 − I < 2δ c(1).

So for any ε > 0, pick δ = min{1, ε/c(1)}/2. Hence ∀ε > 0, ∃δ > 0,

∀H ∈ F(Θ), d1 (I, H) < δ,

d0 (I, H) < ε.

This concludes the proof of the equivalences. 6 Given a space X, a function d : X × X → R is said to be a semimetric if (i) d(F.G) ≥ 0, for all F, G,

(ii) d(F, G) = 0 ⇐⇒ F = G, (iii) d(F, G) = d(G, F ), for all F, G. This notion goes back to Fr´echet and Menger.

130

Chapter 3. Courant Metrics on Images of a Set F. Murat and J. Simon [1] in 1976 considered as candidates for Θ the spaces W k+1,c (RN , RN ) def = f ∈ W k,∞ (RN , RN ) : ∀ 0 ≤ |α| ≤ k + 1, ∂ α f ∈ C(RN , RN )

and W k+1,∞ (RN , RN ), k ≥ 0. The ﬁrst space W k+1,c (RN , RN ) is equivalent to the space C k+1 (RN , RN ), k ≥ 0, algebraically and topologically. The second space W k+1,∞ (RN , RN ) coincides with C k,1 (RN , RN ), k ≥ 0. For the spaces C k+1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, they obtained the following pseudo triangle inequality for the semimetric d0 : d0 (F1 ,F3 ) ≤d0 (F1 , F2 ) + d0 (F2 , F3 ) + d0 (F1 , F2 ) d0 (F2 , F3 ) P [d0 (F1 , F2 ) + d0 (F2 , F3 )] ,

(2.14) ∀F1 , F2 , F3 ,

where P : R+ → R+ is a continuous increasing function. With that additional property, they called d0 a pseudodistance 7 and showed that they could construct a metric from it. Lemma 2.1. Assume that the semimetric d0 satisﬁes the pseudo triangle inequality (2.14). Then, for all α, 0 < α < 1, there exists a constant ηα > 0 such that the (α) function d0 : F(Θ) × F(Θ) → R+ deﬁned as def

(α)

d0 (F1 , F2 ) = inf{d0 (F1 , F2 ), ηα }α is a metric on E. The Micheletti construction with the metric d1 was given in 2001 for the spaces C k (RN , RN ) and C k,1 (RN , RN ), k ≥ 0, in M. C. Delfour and J.-P. Zo´sio [37] so that pseudodistances can be completely bypassed. The metrics d and le d1 constructed from the semimetric d0 under the general Assumptions 2.1 and 2.2 are both more general and interesting since they are of geodesic type by the use of inﬁma over ﬁnite factorizations (I + θ1 ) ◦ · · · ◦ (I + θn ) of I + θ when the semimetric d0 is interpreted as an energy n

d0 (I + θi , I) =

i=1

n

θi + − θi ◦ (I + θi )−1

(2.15)

i=1

7 This terminology is not standard. In the literature a pseudometric or pseudodistance is usually reserved for a function d : X × X → R satisfying

(i) d(F.G) ≥ 0,

for all F, G,

(ii’) d(F, G) = 0 ⇐= F = G, (iii) d(F, G) = d(G, F ),

for all F, G.

(iv) d(F, H) ≤ d(F, G) + d(G, H),

for all F, G, H.

Condition (ii) for a metric is weakened to condition (ii’) for a pseudodistance.

2. Generic Constructions of Micheletti

131

over families of piecewise constant trajectories T : [0, 1] → F(Θ) starting from I at time 0 to I + θ at time 1 in the group F(Θ) by assigning times ti , 0 = t0 < t1 < · · · < tn < tn+1 = 1, at each jump: I, 0 ≤ t < t1 , (I + θ1 ), t 1 < t < t2 , t 2 < t < t3 , (I + θ2 ) ◦ (I + θ1 ), def ... T (t) = (2.16) (I + θi ) ◦ · · · ◦ (I + θ1 ), ti < t < ti+1 , ... (I + θn ) ◦ · · · ◦ (I + θ1 ), tn < t ≤ 1. The jump in the group F(Θ) at time ti is given by − −1 [T (t)]ti = T (t+ −I i ) ◦ T (ti ) def

= [(I + θi ) ◦ · · · ◦ (I + θ1 )] ◦ [(I + θi−1 ) ◦ · · · ◦ (I + θ1 )]−1 − I = θi as an element of Θ that could be viewed as the tangent space to F(Θ) ⊂ I + Θ. The choice of d in a geodesic context is more interesting than the choice of d1 that would involve a double inﬁma in the energy (2.15). At this stage, in view of the equivalence Theorem 2.3, working with the metric d or d1 is completely equivalent. Assumption 2.3. (Θ, · ) is a normed vector space of mappings from RN into RN and there exists a continuous function c1 such that for all I + f ∈ F(Θ) and g ∈ Θ g ◦ (I + f ) ≤ g c1 (f ).

(2.17)

The next assumption is a kind of uniform continuity. Assumption 2.4. (Θ, · ) is a normed vector space of mappings from RN into RN . For each g ∈ Θ, for all ε > 0, there exists δ > 0 such that ∀γ ∈ Θ such that γ < δ,

g ◦ (I + γ) − g < ε.

(2.18)

Theorem 2.4. Under Assumptions 2.1 to 2.4, F(Θ) is a topological (metric) group. Example 2.3 (Checking Assumption 2.3). (1), (2), (3) Since the norm on the three spaces B 0 (RN , RN ), C 0 (RN , RN ), and C00 (RN , RN ) is the same and Assumption 2.2 involves only the norm, it is suﬃcient to check it for B 0 (RN , RN ). Consider g ∈ B0 (RN , RN ) and I +f ∈ F(B 0 (RN , RN )). Since I + f is a bijection g ◦ (I + f )C 0 ≤ gC 0 and the constant function c1 (r) = 1 satisﬁes Assumption 2.3.

(2.19)

132

Chapter 3. Courant Metrics on Images of a Set

(4) Θ = C 0,1 (RN , RN ). Consider g ∈ C 0,1 (RN , RN ) and I + f ∈ F(C 0,1 (RN , RN )). From (1) g ◦ (I + f )C 0 ≤ gC 0 and since all the functions are Lipschitz c(g ◦ (I + f )) ≤ c(g) (1 + c(f )) ⇒ g ◦ (I + f )C 0,1 ≤ gC 0,1 (1 + c(f )) ≤ gC 0,1 (1 + f C 0,1 ) and the continuous function c1 (r) = 1 + r satisﬁes Assumption 2.3. Assumption 2.4 is a little trickier. It involves not only norms but also a “uniform continuity” of the function g that is not veriﬁed for g ∈ B0 (RN , RN ) and for the Lipschitz part c(g) of the norm of C 0,1 (RN , RN ). Example 2.4 (Checking Assumption 2.4). (1) Θ = B0 (RN , RN ). In that space there exists some g that is not uniformly continuous. For that g there exists ε > 0 such that for all n > 0, there exists xn , yn , |yn − xn | < 1/n such that |g(yn ) − g(xn )| ≥ ε. Deﬁne the function γn (x) = yn − xn . Therefore, there exists ε > 0 such that for all n with γn C 0 < 1/n, g ◦ (I + γn ) − gC 0 ≥ |g(xn + γn (xn )) − g(xn )| = |g(xn + yn − xn ) − g(xn )| ≥ ε and Assumption 2.4 is not veriﬁed. However, B0 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : B 0 (RN , RN ) → R is continuous. (2) and (3) Θ = C 0 (RN , RN ). By the uniform continuity of g ∈ C 0 (RN , RN ), for all ε > 0 there exists δ > 0 such that ∀x, y ∈ RN ,

|g(y) − g(x)| < ε/2.

For γC 0 < δ ∀x ∈ RN , ∀x ∈ RN ,

|(I + γ)(x) − x| = |γ(x)| ≤ γC 0 < δ,

|g(x + γ(x)) − g(x)| < ε/2

⇒ g ◦ (I + γ) − gC 0 ≤ ε/2 < ε

and Assumption 2.4 is veriﬁed. In addition, C 0 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : C 0 (RN , RN ) → R is continuous. (4) Θ = C 0,1 (RN , RN ). Since g ∈ C 0,1 (RN , RN ) ∀x, y ∈ RN ,

|g(y) − g(x)| < c(g) |y − x|.

Therefore for def

γC 0,1 = max{γC 0 , c(γ)} < δ, γC 0 < δ and we get g ◦ (I + γ) − gC 0 ≤ c(g) γC 0 < c(g) δ.

2. Generic Constructions of Micheletti

133

Unfortunately, for the part c(g ◦ (I + γ)) of the C 0,1 -norm, for all x = y ∈ RN |g(y + γ(y)) − g(y) − (g(x + γ(x)) − g(x))| ≤ c(g) + c(g) c(I + γ), |y − x| c(g ◦ (I + γ) − g) ≤ c(g) [1 + c(γ)] < c(g) [1 + δ] and we cannot satisfy Assumption 2.4. However, C 0,1 (RN , RN ) ⊂ C 0 (RN , RN ) and for all x ∈ RN , the mapping f → f (x) : C 0,1 (RN , RN ) → R is continuous. Remark 2.1. Since the spaces C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) have the same norm and Assumptions 2.2 and 2.3 involve only the norms, they will always be veriﬁed for C0k (RN , RN ) and C k (RN , RN ) if they are veriﬁed for B k (RN , RN ), k ≥ 0. Solely Assumption 2.1 will require special attention. The fact that Assumption 2.4 is not always veriﬁed will not prevent us from proving that the space (F(Θ), d) is complete. The completeness will be proved in Theorem 2.6 under Assumptions 2.1 to 2.3, where Assumption 2.4 will be replaced by the condition Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous. The following theorem will be proved in Theorems 2.11, 2.12, and 2.13 of section 2.5 and Theorems 2.15 and 2.16 of section 2.6. Theorem 2.5. Let k ≥ 0 be an integer. (i) C0k (RN , RN ) and C k (RN , RN ) satisfy Assumptions 2.1 to 2.4, C0k (RN , RN ) ⊂ C k (RN , RN ) ⊂ C 0 (RN , RN ), and, for all x ∈ RN , the mapping f → f (x) : C k (RN , RN ) → RN is continuous. (ii) B k (RN , RN ) and C k,1 (RN , RN ) verify Assumptions 2.1 to 2.3, C k,1 (RN , RN ) ⊂ B k (RN , RN ) ⊂ C 0 (RN , RN ), and, for all x ∈ RN , the mapping f → f (x) : B k (RN , RN ) → RN is continuous. Proof of Theorem 2.4. From J. L. Kelley [1], it is suﬃcient to prove that ∀F ∈ F(Θ), ∀H ∈ F(Θ) such that d(I, H) → 0,

d(I, F −1 ◦ H ◦ F ) → 0.

Rewrite F −1 ◦ H ◦ F as follows: F −1 ◦ H ◦ F − I = F −1 ◦ [I + H − I] ◦ F − F −1 ◦ F

= F −1 ◦ (I + γ) − F −1 ◦ F

= (F −1 − I) ◦ (I + γ) − (F −1 − I) ◦ F + γ ◦ F, γ = H − I and g = (F −1 − I) ◦ (I + γ) − (F −1 − I). def

def

By Assumption 2.3, γ ◦ F ≤ γ c1 (F − I),

g ◦ F ≤ g c1 (F − I);

134

Chapter 3. Courant Metrics on Images of a Set

by Assumption 2.4, for all ε > 0, there exists δ, 0 < δ < min{1, ε/(4c0 (1) c1 (F − I))}, such that γ = H − I < δ c0 (1)

⇒

g =(F −1 − I) ◦ (I + γ) − (F −1 − I) < ε/(4 c1 (F − I)).

Combining the above properties ∀H such that d(I, H) < δ, ⇒ F

−1

H − I < δ c0 (1)

◦ H ◦ F − I ≤ γ c1 (F − I) + ε/2 ≤ δ c0 (1) c1 (F − I) + ε/4 ≤ ε/2.

By Assumption 2.2, for all H such that d(I, H) < δ < 1, we have H − I < δ c0 (1) and H −1 − I < δ c0 (1). Finally, for all ε > 0, there exists δ > 0 such that d(I, H) < δ implies F −1 ◦ H ◦ F − I ≤ ε/2. By a similar argument, for all ε > 0, there exists δ > 0 such that d(I, H) < δ implies F −1 ◦ H −1 ◦ F − I ≤ ε/2. Therefore, for all ε > 0 there exists δ0 = min{δ, δ } > 0 such that for all H, d(I, H) < δ0 , d(I, F −1 ◦ H ◦ F ) ≤ F −1 ◦ H ◦ F − I + F −1 ◦ H −1 ◦ F − I < ε and F(Θ) is a topological group for the metric d. Theorem 2.6. (i) Let Assumptions 2.1 to 2.3 be veriﬁed for a Banach space Θ contained in C 0 (RN , RN ) such that, for all x ∈ RN , the mapping f → f (x) : Θ → RN is continuous. Then (F(Θ), d) is a complete metric space. (ii) Let Assumptions 2.1 to 2.4 be veriﬁed for a Banach space Θ. Then F(Θ) is a complete topological (metric) group. Remark 2.2. F(C 0,1 (RN , RN )) is a complete metric space since the assumptions of Theorem 2.6 are veriﬁed, but it is not a topological group since Assumption 2.4 is not veriﬁed. Proof of Theorem 2.6. (i) Let {Fn } be a Cauchy sequence in F(Θ). (a) Boundedness of {Fn − I} and {Fn−1 − I} in Θ. For each ε, 0 < ε < 1, there exists N such that, for all m, n > N , d(Fm , Fn ) < ε. By the triangle inequality |d(I, Fm ) − d(I, Fn )| ≤ d(Fm , Fn ) < ε, {d(I, Fn } is Cauchy, and, a posteriori, is bounded by some constant L. For each n, n ) such that there exists a factorization Fn = (I + f1n ) ◦ · · · ◦ (I + fν(n)

ν(n)

0≤

fin + fin ◦ (I + f1n )−1 − d(I, Fn ) < ε < 1

i=1

ν(n)

⇒

i=1

fin + fin ◦ (I + fin )−1 < L + ε < L + 1.

2. Generic Constructions of Micheletti

135

By Assumption 2.2 Fn − I + Fn−1 − I < 2(L + ε) c0 (L + 1) and the sequences {Fn − I} and {Fn−1 − I} are bounded in Θ. (b) Convergence of {Fn − I} and {Fn−1 − I} in Θ. For any m, n −1 Fn − Fm = (Fn ◦ Fm − I) ◦ Fm

and by Assumption 2.3 −1 −1 Fm − Fn = (Fn ◦ Fm − I) ◦ Fm ≤ Fn ◦ Fm − I c1 (Fm − I) −1 ⇒ Fm − Fn ≤ c+ 1 Fn ◦ Fm − I,

def

c+ 1 = sup c1 (Fm − I) < ∞ m

since {Fm − I} is bounded. For all ε, 0 < ε < c0 (1) c+ 1 , there exists N such that ∀n, m, > N,

−1 d(I, Fn ◦ Fm ) = d(Fm , Fn ) < ε/(c0 (1) c+ 1 ) < 1.

By Assumption 2.2 + −1 Fn ◦ Fm − I < c0 (1)ε/(c0 (1) c+ 1 ) = ε/c1

⇒ ∀n, m, > N,

−1 Fm − I − (Fn − I) ≤ c+ 1 Fn ◦ Fm − I < ε

and {Fn − I} is a Cauchy sequence in Θ that converges to some F − I ∈ Θ since Θ is a Banach space. Similarly, we get that {Fn−1 − I} is a Cauchy sequence in Θ that converges to some G − I ∈ Θ. (c) F is bijective, G = F −1 , and F ∈ F(Θ). For all n, Fn , Fn−1 , F , and G are continuous since Θ ⊂ C 0 (RN , RN ). By continuity of f → f (x) : Θ → RN , f → x + f (x) : Θ → RN is continuous and [Fn−1 (x) − x| + x → [G(x) − x] + x. By the same argument for Fn , x = Fn (Fn−1 (x)) → F (G(x)) = (F ◦ G)(x) and for all x ∈ RN , (F ◦ G)(x) = x. Similarly, starting from Fn−1 ◦ Fn = I, we get (G ◦ F )(x) = x. Hence, G ◦ F = I = F ◦ G. So F is bijective, F − I, F −1 − I ∈ Θ, and F ∈ F(Θ). (d) Convergence of Fn → F in F(Θ). To check that Fn → F in F(Θ) recall that by Assumption 2.3 d(Fn , F ) = d(I, F ◦ Fn−1 ) ≤I − F ◦ Fn−1 + I − Fn ◦ F −1 = (Fn − F ) ◦ Fn−1 + (F − Fn ) ◦ F −1 ≤ c1 (Fn−1 − I)Fn − F + c1 (F −1 − I)F − Fn ≤ max c1 (Fn−1 − I) + c1 (F −1 − I) F − Fn , n

which goes to zero as Fn − I goes to F − I in Θ. (ii) The proof (ii) diﬀers from the proof of (i) only in part (c). Consider G ◦ F − I = G ◦ F − G ◦ Fn + (G − Fn−1 ) ◦ Fn .

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Chapter 3. Courant Metrics on Images of a Set

Since {Fn − I} is bounded in Θ and Fn−1 → G, the term (G − Fn−1 ) ◦ Fn goes to 0 as n → ∞ by Assumption 2.3. The second terms can be rewritten as G ◦ F − G ◦ Fn = F − Fn + (G − I) ◦ F − (G − I) ◦ Fn = F − Fn + g ◦ F − g ◦ Fn , where g = G − I ∈ Θ. The term F − Fn goes to zero. As for the second terms

g ◦ F − g ◦ Fn = g ◦ [I + (F − Fn ) ◦ Fn−1 ] − g ◦ Fn . Let gn = g ◦ [I + (F − Fn ) ◦ Fn−1 ] − g and let γn = (F − Fn ) ◦ Fn−1 . By boundedness of {Fn − I} and {Fn−1 − I} in Θ and Assumption 2.3, γn = (F − Fn ) ◦ Fn−1 ≤ F − Fn c1 (Fn−1 − I) ≤ c+ 1 F − Fn , gn ◦ Fn ≤ gn c1 (Fn − I) ≤ c+ 1 gn , −1 c+ 1 = sup{c1 (Fn − I), c1 (Fn − I)} < ∞. def

n

By Assumption 2.4, for all ε > 0 there exists δ > 0 such that γn < δ c+ 1

⇒ gn = g ◦ (I + γn ) − g < ε/c+ 1.

By combining for all ε > 0, there exists δ > 0 such that F − Fn < δ

+ ⇒ γn ≤ c+ 1 F − Fn < c1 δ + ⇒ gn ◦ Fn < c+ 1 ε/c1 = ε ⇒ F − Fn < δ ⇒ g ◦ F − g ◦ Fn < ε.

As a result G ◦ F − I is bounded by an expression that goes to zero as n → ∞ and G ◦ F = I. By an analogous argument, we can prove that F ◦ G − I = 0, that is, F −1 = G. So F is bijective, F − I, F −1 − I ∈ Θ. Hence F ∈ F(Θ).

2.2

Diﬀeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

Consider as other candidates for Θ the spaces k N N ∞ k N N B(RN , RN ) = ∩∞ k=0 B (R , R ) = ∩k=0 C (R , R ),

(2.20)

k N N C0∞ (RN , RN ) = ∩∞ k=0 C0 (R , R )

(2.21)

def

def

and the resulting groups F(B(RN , RN ))

I + f : f ∈ B(RN , RN ), (I + f ) bijective and (I + f )−1 − I ∈ B(RN , RN ) and F(C0∞ (RN , RN ))

I + f : f ∈ C0∞ (RN , RN ), (I + f ) bijective and (I + f )−1 − I ∈ C0∞ (RN , RN )

2. Generic Constructions of Micheletti

137

that satisfy Assumption 2.1 but not the other assumptions since B(RN , RN ) and C0∞ (RN , RN ) are Fr´echet but not Banach spaces. To get around this, observe that k N N ∞ k N N F(B(RN , RN )) = ∩∞ k=0 F(B (R , R )) = ∩k=0 F(C (R , R )), k N N F(C0∞ (RN , RN )) = ∩∞ k=0 F(C0 (R , R )),

where (F(C k (RN , RN )), dk ) and (F(C0k (RN , RN )), dk ) are topological (metric) groups with the same metrics {dk } and the following monotony property: dk (F, G) ≤ dk+1 (F, G) resulting from the monotony of the norms f C k = max f C i ≤ 0≤i≤k

max f C i = f C k+1 .

0≤i≤k+1

Theorem 2.7. The spaces F(B(RN , RN )) and F(C0∞ (RN , RN )) are complete topological (metric) groups for the distance def

d∞ (F, G) =

∞ 1 dk (F, G) , 2k 1 + dk (F, G)

(2.22)

k=0

where dk is the metric associated with C k (RN , RN ) and C0k (RN , RN ).8 This theorem is a consequence of the following general lemma. Lemma 2.2. Assume that (Fk , dk ), k ≥ 0, is a family of groups each complete with respect to their metric dk and that ∀k ≥ 0,

Fk+1 ⊂ Fk

and

dk (F, G) ≤ dk+1 (F, G), ∀F, G ∈ ∩∞ k=0 Fk .

(2.23)

Then the intersection F∞ = ∩∞ k=0 Fk is a group that is complete with respect to the metric d∞ deﬁned in (2.22). If, in addition, each (Fk , dk ) is a topological (metric) group, then F∞ is a topological (metric) group. Proof. By deﬁnition, F∞ is clearly a group. (i) d∞ is a metric on F∞ . d∞ (F, G) = 0 implies that dk (F, G) = 0 for all k and F = G and conversely. Also d∞ (F, G) = d∞ (G, F ). The function x/(1 + x) is monotone strictly increasing with x ≥ 0. Therefore, dk (F, H) ≤ dk (F, G)+dk (G, H) implies dk (F, H) dk (F, G) dk (G, H) ≤ + 1 + dk (F, H) 1 + dk (F, G) + dk (G, H) 1 + dk (F, G) + dk (G, H) dk (G, H) dk (F, G) + , ≤ 1 + dk (F, G) 1 + dk (G, H) ∞ ∞ ∞ 1 dk (F, H) 1 dk (F, G) 1 dk (G, H) ≤ + 2k 1 + dk (F, H) 2k 1 + dk (F, G) 2k 1 + dk (G, H) k=0

k=0

k=0

and d∞ (F, H) ≤ d∞ (F, G) + d∞ (G, H). 8 In view of (2.20), it is important to note that F (B(RN , RN )) is a topological group by using the family (F (C k (RN , RN )), dk ) in Lemma 2.2 and not the family (F (B k (RN , RN )), dk ).

138

Chapter 3. Courant Metrics on Images of a Set

(ii) (F∞ .d∞ ) is complete. Let {Fn } be a Cauchy sequence in F∞ ; that is, for all ε, 0 < ε < 1/2, there exists N0 such that for n, m > N0 , d∞ (Fn , Fm ) < ε. In particular ∞

1 dk (Fn , Fm ) d0 (Fn , Fm ) ≤ N1 , d∞ (Fn , Fm ) < ε/22 . Then ∞

1 dk (Fn , Fm ) 1 1 d1 (Fn , Fm ) ≤ < 2 ε ⇒ d1 (Fn , Fm ) < ε/(1 − ε) < 2ε 2 1 + d1 (Fn , Fm ) 2k 1 + dk (Fn , Fm ) 2 k=0

and {Fn } is a Cauchy sequence in F1 that converges to some G1 in F1 . By continuous injection of F1 into F0 , G1 = F ∈ F0 ∩ F1 . Proceeding in that way, we get that there exists F ∈ F∞ = ∩∞ k=0 Fk such that Fn → F in Fk for all k ≥ 0. Therefore, F∞ is complete.

2.3

Closed Subgroups G

Given an arbitrary nonempty subset Ω0 of RN , consider the family def

X (Ω0 ) = F (Ω0 ) ⊂ RN : ∀F ∈ F(Θ)

(2.24)

of images of Ω0 by the elements of F(Θ). By introducing the subgroup def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 } ,

(2.25)

[F ] → F (Ω0 ) : F(Θ)/G(Ω0 ) → X (Ω0 )

(2.26)

we obtain a bijection

between the set of images X (Ω0 ) and the quotient space F(Θ)/G(Ω0 ). Using this bijection, the topological structure of X (Ω0 ) will be identiﬁed with the topological structure of the quotient space. Lemma 2.3. Let Assumptions 2.1 to 2.3 hold with Θ ⊂ C 0 (RN , RN ) and, for all x ∈ RN , let the mapping f → f (x) : Θ → RN be continuous.9 If the nonempty subset Ω0 of RN is closed or if Ω0 = int Ω0 ,10 the family def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

(2.27)

is a closed subgroup of F(Θ). 9 This 10 Such

means that F(Θ) ⊂ I + Θ ⊂ Hom(RN , RN ). a set will be referred to as a crack-free set in Deﬁnition 7.1 (ii) of Chapter 8. Indeed,

by deﬁnition Ω is crack-free if Ω = Ω. If, in addition, Ω is open, then Ω = Ω = Ω and Ω = Ω = int Ω.

2. Generic Constructions of Micheletti

139

Proof. G(Ω0 ) is clearly a subgroup of F(Θ). It is suﬃcient to show that for any sequence {Fn } in G(Ω0 ) that converges to F in F(Θ), then F ∈ G(Ω0 ). The other properties are straightforward. If {Fn } converges to F in F(Θ), then by Assumption 2.3 we have Fn ◦ F −1 − I + F ◦ Fn−1 − I → 0. From F ◦ Fn−1 − I → 0, for each x ∈ Ω0 the sequence {F (Fn−1 (x))} ⊂ F (Ω0 ) converges to x ∈ Ω0 and hence Ω0 ⊂ F (Ω0 ). From the convergence Fn ◦F −1 −I → 0 for each x ∈ F (Ω0 ), F −1 (x) ∈ Ω0 and the sequence {Fn (F −1 (x))} ⊂ Ω0 converges to x ∈ F (Ω0 ) and hence F (Ω0 ) ⊂ Ω0 . Therefore, Ω0 ⊂ F (Ω0 ) ⊂ Ω0 and Ω0 = F (Ω0 ). Since F is a homeomorphism, for all A, F (A) = F (A) and F (A) = F (A). As a result int F (A) = F (int A). In view of the identity Ω0 = F (Ω0 ), F (Ω0 ) = F (Ω0 ) = Ω0

and

int Ω0 = int F (Ω0 ) = F (int Ω0 ).

From this, if Ω0 is closed, Ω0 = Ω0 and F (Ω0 ) = Ω0 . If Ω0 = int Ω0 , we also get F (Ω0 ) = Ω0 . In both cases F ∈ G(Ω0 ). Remark 2.3. A. M. Micheletti [1] assumes that Ω0 is a bounded connected open domain of class C 3 in order to make all the images F (Ω0 ) bounded connected open domains of class C 3 with Θ = C03 (RN , RN ). Open domains of class C k , k ≥ 1, are locally C k epigraphs and domains that are locally C 0 epigraphs are crack-free11 by Theorem 5.4 (ii) of Chapter 2. Remark 2.4. F. Murat and J. Simon [1] were mainly interested in families of images of locally Lipschitzian (epigraph) domains. Recall that F(C 1 (RN , RN )) transports locally Lipschitzian (epigraph) domains onto locally Lipschitzian (epigraph) domains (cf. A. Bendali [1] and A. Djadane [1]), but that F(C 0,1 (RN , RN )) does not, as shown in Examples 5.1 and 5.2 of section 5.3 in Chapter 2. Of special interest are the closed submanifolds of RN of codimension greater than or equal to 1. For instance, we can choose Ω0 = S n , 2 ≤ n < N , the n-sphere of radius 1 centered in 0. We get a family of closed curves in R3 for n = 2 and of closed surfaces in R3 for n = 3. Similarly, by choosing Ω0 = B1n (0), the closed unit ball of dimension n, we get a family of curves for n = 1 and a family of surfaces for n = 2. The choice of [0, 1]n yields a family of surfaces with corners for n = 2. Explicit expressions can be obtained for normals, curvatures, and density measures as in Chapter 2. Example 2.5 (Closed surfaces in R3 ). For Ω0 = S 2 in R3 , f ∈ C 1 (R3 , R3 ), and F = I + f , the tangent ﬁeld at the point ξ ∈ S 2 is Tξ S 2 = {ξ}⊥ and the unit normal is ξ. At a point x = F (ξ) of the image 11 Cf.

Deﬁnition 7.1 (ii) of Chapter 8.

140

Chapter 3. Courant Metrics on Images of a Set

F (S 2 ), the tangent space and the unit normal are given by TF (ξ) F (S 2 ) = DF (ξ){ξ}⊥ and n(F (x)) =

∗

DF (ξ)−1 ξ . | ∗DF (ξ)−1 ξ|

Example 2.6 (Curves in RN ). Let eN be a unit vector in RN , let H = {eN }⊥ , and let def

Ω0 = {ζN eN : ζN ∈ [0, 1]} ⊂ RN . For f ∈ C 1 (RN , RN ), and F = I + f , the tangent ﬁeld at the point ξ ∈ Ω0 is Tξ Ω0 = R eN and the normal space (Tξ Ω0 )⊥ is H. At a point x = F (ξ) of the curve F (Ω0 ), the tangent and normal spaces are given by TF (ξ) F (Ω0 ) = R DF (ξ)eN

TF (ξ) F (Ω0 )⊥ = {DF (ξ)eN }⊥ .

and

In practice, one might want to use other subgroups, such as all the translations by some vector a ∈ RN : def

Fa (x) = a + x, ⇒

Fa−1 (x)

x ∈ RN ,

− a + x = F−a (x)

⇒ Fa − I = a ∈ Θ ⇒

Fa−1

− I = −a ∈ Θ

(2.28) (2.29)

for all Θ that contain the constant functions such as Bk (RN , RN ), C k (RN , RN ), and C k,1 (RN , RN ), but not C0k (RN , RN ). Therefore def

Gτ = Fa : ∀a ∈ RN

(2.30)

is a subgroup of F(Θ). In that case, it is also a closed subgroup of F(Θ): Fa ◦ Fb (x) = a + b + x = Fa+b (x),

F0 = I

and (Gτ , ◦) is isomorphic to the one-dimensional closed abelian group (RN , +). This means that all the topologies on Gτ are equivalent and (Gτ , ◦) is closed in F(Θ).

2.4

Courant Metric on the Quotient Group F (Θ)/G

When F(Θ) is a topological right-invariant metric group and G is a closed subgroup of F(Θ), the quotient metric def

dG (F ◦ G, H ◦ G) =

˜ inf d(F ◦ G, H ◦ G)

˜ G,G∈G

(2.31)

induces a complete metric topology on the quotient group F(Θ)/G.12 The quotient metric dG(Ω0 ) with Θ = C0k (RN , RN ) was referred to as the Courant metric by 12 Cf.,

for instance, D. Montgomery and L. Zippin [1 sect. 1.22, p. 34, sect. 1.23, p. 36])

(i) If G is a topological group whose open sets at e have a countable basis, then G is metrizable and, moreover, there exists a metric which is right-invariant. (ii) If G is a closed subgroup of a metric group F, then F/G is metrizable.

2. Generic Constructions of Micheletti

141

A. M. Micheletti [1]. In that case F(Θ) is a topological (metric) group. However, we have seen that for Θ = C 0,1 (RN , RN ) and B0 (RN , RN ), F(Θ) is not a topological group. Yet, the completeness of (F (Θ), d) and the right-invariance of d are suﬃcient to recover the result for an arbitrary closed subgroup G of F. It is important to have the weakest possible assumptions on G to secure the widest range of applications. Theorem 2.8. Assume that (F, d) is a group with right-invariant metric d that is complete for the topology generated by d. For any closed subgroup G of F, the function dG : F × F → R, def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G), G∈G

(2.32)

is a right-invariant metric on F(Θ)/G and the space (F/G, dG ) is complete. The topology induced by dG coincides with the quotient topology of F/G. Proof. (i) F/G is clearly a group with unit element G under the composition law def

[F ◦ G] ◦ [H ◦ G] = [F ◦ H] ◦ G. (ii) (dG is a metric). Since d is right-invariant, the deﬁnition (2.31) is equivalent to def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G). G∈G

(2.33)

Since d is symmetrical, so is dG . For the triangle inequality let g1 , g2 ∈ G: dG (F ◦ G, H ◦ G) ≤ d(F, H ◦ g1 ◦ g2 ) ≤ d(F, G ◦ g2 ) + d(G ◦ g2 , H ◦ g1 ◦ g2 ) ≤ d(F, G ◦ g2 ) + d(G, H ◦ g1 ) by right-invariance. The triangle inequality for dG follows by taking the inﬁma over g1 , g2 ∈ G. Since F/G is a group, it remains to show that dG (I ◦ G, H ◦ G) = 0 if and only if H ∈ G. Clearly dG (I ◦G, I ◦G) ≤ d(I, I) = 0. Conversely, let dG (I ◦G, H ◦G) = 0. For all n > 0, there exists gn ∈ G such that d(gn−1 , H) = d(I, H ◦ gn ) < 1/n. Therefore gn−1 → H in F(Θ). But since G is closed limn→∞ gn−1 ∈ G and H ∈ G. In particular, the canonical mapping F → π(F ) = F ◦ G : F(Θ) → F(Θ)/G is continuous since dG (I ◦ G, F ◦ G) ≤ d(I, F ) for all F ∈ F(Θ). (iii) (Completeness). Consider a Cauchy sequence {Fn ◦ G} in F/G. It is suﬃcient to show that there exists a subsequence which converges to some F ◦ G ∈ F(Θ)/G. Construct the subsequence {Fν ◦ G} such that dG (Fν ◦ G, Fν+1 ◦ G) < 1/(2ν ). Then there exists a sequence {Hν } in F such that (i) Hν ∈ Fν ◦ G and (ii) d(Hν , Hν+1 ) < 1/(2ν ). We proceed by induction. By deﬁnition of dG , if dG (F1 ◦ G, F2 ◦G) < 1/2, there exist H1 ∈ F1 ◦G and H2 ∈ F2 ◦G such that d(H1 , H2 ) < 1/2. Similarly, if Hν ∈ Fν ◦ G, it follows that there exists Hν+1 with the required property. Because dG (Fν ◦ G, Fν+1 ◦ G) < 1/2ν , there exist G1 and G2 in G such that d(Fν ◦ G1 , Fν+1 ◦ G2 ) < 1/2ν . If G3 is such that Fν ◦ G1 = Hν ◦ G3 , then we can

142

Chapter 3. Courant Metrics on Images of a Set

choose Hν+1 = Fν+1 ◦ G2 ◦ G−1 3 since d(Hν , Fν+1 ◦ G2 ◦ G−1 3 ) = d(Hν ◦ G3 , Fν+1 ◦ G2 ) = d(Fν ◦ G1 , Fν+1 ◦ G2 ). It is easy to check that {Hν } is a Cauchy sequence in F. Indeed ∀i < j,

d(Hi , Hj ) ≤

j−1

d(Hn , Hn+1 ) ≤

n=i

1 1 1 + · · · + j ≤ i−1 . 2i 2 2

By completeness of F, {Hν } converges to some H in F. From (ii) the canonical map π : F → F/G is continuous. So the sequence {π(Hν ) = Fν ◦ G} converges to π(H) = H ◦ G in F/G. R (iv) Associate with G the equivalence relation F ∼ H if F ◦ H −1 ∈ G. By deﬁnition, the quotient topology on F/R is the ﬁnest topology T on F/R for which the canonical map F → π(F ) : F → F/R is continuous. But, by deﬁnition, F/R = F/G and, from part (ii), π is also continuous for the right-invariant quotient metric. Therefore, the identity map i : (F/R, T ) → (F/G, dG ) is continuous. The topology T in [I] is deﬁned by the fundamental system of neighborhoods {π(F ) : d(F, I) < ε}. But dG (π(F ), π(I)) ≤ d(F, I) < ε and i({π(F ) : d(F, I) < ε}) ⊂ {π(F ) : dG ([F ], [I]) < ε} that is a neighborhood of [I] in the (F/G, dG ) topology. Therefore i−1 is continuous and the two topologies coincide. We now have all the ingredients to conclude. Theorem 2.9. Let Θ be any of the spaces Bk (RN , RN ), C k (RN , RN ), C0k (RN , RN ), C k,1 (RN , RN ), k ≥ 0, or C ∞ (RN , RN ), C0∞ (RN , RN ). (i) The group (F(Θ), d) is a complete right-invariant metric space. For Θ equal to C k (RN , RN ) or C0k (RN , RN ), 0 ≤ k ≤ ∞, (F(Θ), d) is also a topological group. (ii) For any closed subgroup G of F(Θ), the function dG : F(Θ) × F(Θ) → R, def

dG (F ◦ G, H ◦ G) = inf d(F, H ◦ G), G∈G

(2.34)

is a right-invariant metric on F(Θ)/G and the space (F(Θ)/G, dG ) is complete. The topology induced by dG coincides with the quotient topology of F(Θ)/G. (iii) Let Ω0 , ∅ = Ω0 ⊂ RN , be closed or let Ω0 = int Ω0 . Then def

G(Ω0 ) = {F ∈ F(Θ) : F (Ω0 ) = Ω0 }

(2.35)

is a closed subgroup of F(Θ). Finally, it is interesting to recall that A. M. Micheletti [1] used the quotient metric to prove the following theorem.

2. Generic Constructions of Micheletti

143

Theorem 2.10. Fix k = 3, the Courant metric of F(C03 (RN , RN ))/G(Ω0 ), and a bounded open connected domain Ω0 of class C 3 in RN . Let X (Ω0 ) be the family of images of Ω0 by elements of F(C03 (RN , RN ))/G(Ω0 ). The subset of all bounded open domains Ω in X (Ω0 ) such that the spectrum of the Laplace operator −∆ on Ω with homogeneous Dirichlet conditions on the boundary ∂Ω does not have all its eigenvalues simple is of the ﬁrst category.13 This theorem says that, up to an arbitrarily small perturbation of the domain, the eigenvalues of the Laplace operator of a C 3 -domain can be made simple.

2.5

Assumptions for Bk (RN , RN ), C k (RN , RN ), and C0k (RN , RN )

2.5.1

Checking the Assumptions

By deﬁnition, for all integers k ≥ 0 C0k (RN , RN )) ⊂ C k (RN , RN ) ⊂ Bk (RN , RN ) ⊂ B0 (RN , RN ) ⊂ C 0 (RN , RN ) ∀x ∈ RN ,

f → f (x) : B0 (RN , RN ) → RN

with continuous injections between the spaces and continuity of the evaluation map. The spaces are all Banach spaces endowed with the same norm:14 def

f Ck =

max ∂ α f C .

0≤|α|≤k

As a consequence, the evaluation map f → f (x) is continuous for all the spaces. The following results are quoted directly from A. M. Micheletti [1] and apply to all spaces Θ ⊂ C k (RN , RN ). Denote by S(x)[y1 ][y2 ], . . . , [yk ] at the point x ∈ RN a k-linear form with arguments y1 , . . . , yk . Lemma 2.4. Assume that F and G are two mappings from RN to RN such that F is k-times diﬀerentiable in an open neighborhood of x and G is k-times diﬀerentiable in an open neighborhood of F (x). Then the kth derivative of G ◦ F in x is the sum of a ﬁnite number of k-linear applications on RN of the form . / (h1 , h2 , . . . , hk ) → G() (F (x)) F (λ1 ) (x)[h1 ][h1 ] . . . [hλ1 ] / . (2.36) . . . F (λ ) (x)[hk−λ +1 ] . . . [hk ] , where = 1, . . . , k and λ1 + · · · + λ = k. 13 A set B is said to be nowhere dense if its closure has no interior or, alternatively, if B is dense. A set is said to be of the ﬁrst category if it is the countable union of nowhere dense sets (cf. J. Dugundji [1, Def. 10.4, p. 250]). 14 Here

RN is endowed with the norm |x| =

)N

i=1

x2i

1/2

.

144

Chapter 3. Courant Metrics on Images of a Set

Proof. We proceed by induction on k. The result is trivially true for k = 1. Assuming that it is true for k − 1, we prove that it is also true for k. This is an obvious consequence of the observation that for a mapping from RN into L((RN )k−1 , RN ) of the form x → G() (F (x))[F (λ1 ) (x)] . . . [F (λ ) (x)], where = 1, . . . , k − 1 and λ1 + · · · + λ = k − 1, the diﬀerential in x is G(+1) (F (x))[F (1) ][F (λ1 ) (x)] . . . [F (λ ) (x)] + G() (F (x))[F (λ1 +1) (x)] . . . [F (λ ) (x)] + · · · + G() (F (x))[F (λ1 ) (x)] . . . [F (λ +1) (x)], (2.37) and the result is true for k. Lemma 2.5. Given g and f in C k (RN , RN ), let ψ = g ◦ (I + f ). Then for each x ∈ RN |ψ(x)| = |g(x + f (x))|, |ψ

(1)

(x)| ≤ |g (1) (x + f (x))| [1 + |f (1) (x)|],

|ψ (i) (x)| ≤ |g(1) (x + f (x))| |f (i) (x)| +

i

|g (j) (x + f (x))| aj (|f (1) (x)|, . . . , |f (i−1) (x)|)

j=2

for i = 2, . . . , k, where aj is a polynomial with positive coeﬃcients. Proof. This is an obvious consequence of Lemma 2.4. Theorem 2.11. Assumptions 2.1 and 2.3 are veriﬁed for the spaces C0k (RN , RN ), C k (RN , RN ), and B k (RN , RN ), k ≥ 0. Proof. (i) Assumption 2.3. From Lemma 2.5 it readily follows that f ◦ (I + g) ∈ Bk (RN , RN ) for all f ∈ B k (RN , RN ) and I + g in F(B k (RN , RN )). Moreover, ψC = f C , ψ

(1)

C ≤ f (1) C (1 + g (1) C ),

ψ (i) C ≤ f (1) C g (i) C +

i

f (j) C aj (g (1) C , . . . , g (i−1) C )

j=2

for i = 2, . . . , k, where aj is a polynomial. Hence ψC k = max ψ i C ≤ f C k c1 (gC k ) i=0,...,k

for some polynomial function c1 that depends only on k.

2. Generic Constructions of Micheletti

145

(ii) Assumption 2.1. The case of Bk (RN , RN ) is a consequence of (i). For C (RN , RN ) and each i, ψ(i) is uniformly continuous as the ﬁnite sum and product of compositions of uniformly continuous functions by Lemma 2.4. As for C0k (RN , RN ), for all I + g ∈ F(C0k (RN , RN )) and f ∈ C0k (RN , RN ), h = f ◦ (I + g) ∈ C k (RN , RN ) from the previous case. It remains to show that h ∈ C0k (RN , RN ). This amounts to showing that |h(x)| and |h(i) (x)| go to zero as |x| → ∞, i = 1, . . . , k. We again proceed by induction on k. Since |g(x)| → 0 as |x| → ∞, |x + g(x)| ≥ |x| − |g(x)| → ∞ as |x| → ∞ and |h(x)| = |f (x + g(x))| → 0 as |x| → ∞ and Assumption 2.1 is true for k = 0. Always, from Lemma 2.5 and the identity h(x) = f (x + g(x)), k

|h(1) (x)| ≤ |f (1) (x)| [1 + |g (1) (x)|] → 0 as |x| → 0 and Assumption 2.1 is true for k = 1. For k ≥ 2 and 2 ≤ i ≤ k, again, from Lemma 2.5 |h(i) (x)| ≤ |f (1) (x + g(x))| g (i) C +

i

|f (j) (x + g(x))| aj (g (1) C , . . . , g (i−1) C )

j=2

for i = 2, . . . , k. Since for each j, |f (j) (x)| → 0 as |x| → ∞, by the same argument, |f (j) (x + g(x))| → 0 as |x| → ∞ and hence the whole right-hand side goes to 0. Theorem 2.12. Assumption 2.2 is veriﬁed for the spaces C0k (RN , RN ), C k (RN , RN ), and Bk (RN , RN ), k ≥ 0. It is a consequence of the following lemma of A. M. Micheletti [1]. Lemma 2.6. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0 with the following property: if the sequence f1 , . . . , fn in C r (RN , RN ) is such that n fi C r < α, 0 < α < s, (2.38) i=1

then for the map F = (I + fn ) ◦ · · · ◦ (I + f1 ), F − IC r ≤ α c(r, s).

(2.39)

Proof. We again proceed by induction on r. For simplicity deﬁne def

Fi = (I + fi ) ◦ · · · ◦ (I + f1 ),

Fn = F.

By the deﬁnition of F we have F − I = f1 + f2 ◦ (I + f1 ) + · · · + fn ◦ (I + fn−1 ) ◦ · · · ◦ (I + f1 ) = f1 + f2 ◦ F1 + · · · + fn ◦ Fn−1 .

(2.40)

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Chapter 3. Courant Metrics on Images of a Set

)n Then F − IC 0 ≤ i=1 fi C 0 ≤ α and c(0, s) = 1 for all integers s > 0. From (2.40) and Lemma 2.5 we further get sup |(F − I)(1) (x)| ≤ f1 C 1 + f2 C 1 [1 + f1 C 1 ] x

≤

n i=1

fi C 1

n (

+ · · · + fn C 1 [1 + fn−1 C 1 ] . . . [1 + f1 C 1 ] (2.41) 0 n 1 )n (1 + fi C 1 ) ≤ fi C 1 e( i=1 fi C 1 ) ≤ α eα ≤ α es

i=1

i=1

and F − IC 1 ≤ α eα . Hence c(1, s) = es for all integers s > 0. We now show that if the result is true for r − 1, it is true for r. We have to evaluate |(Fn − I)(r) (x)|. It is obvious that Fn − I = (I + fn ) ◦ Fn−1 − I = Fn−1 − I + fn ◦ Fn−1 and hence (Fn − I)(r) (x) = (Fn−1 − I)(r) (x) + (fn ◦ Fn−1 )(r) (x).

(2.42)

For r ≥ 2 from Lemma 2.5 we have |(fn ◦ Fn−1 )(r) (x)| ≤ |fn(1) (Fn−1 (x))| |(Fn−1 − I)(r) (x)| r |fn(j) (Fn−1 (x))| aj (|(Fn−1 − I)(1) (x)|, . . . , |(Fn−1 − I)(r−1) (x)|). + j=2

By the induction assumption |(Fn−1 − I)(i) (x)| ≤ (Fn−1 − I)C r−1 ≤ α c(r − 1, s), i = 1, . . . , r − 1, and the fact that aj is a polynomial dependent on r for j = 2, . . . , r, there exists a constant L(r, s) such that |(fn ◦ Fn−1 )(r) (x)| r |fn(j) (Fn−1 (x))| ≤ |fn(1) (Fn−1 (x))| |(Fn−1 − I)(r) (x)| + L(r, s) j=2

(2.43)

≤ fn C r |(Fn−1 − I)(r) (x)| + (r − 1)L(r, s)fn C r . Deﬁne M (r, s) = (r − 1)L(r, s). From (2.42) and (2.43), we get |(Fn − I)(r) (x)| ≤ [1 + fn C r ] |(Fn−1 − I)(r) (x)| + M (r, s)fn C r . After repeating this procedure n − 1 times we have |(Fn − I)(r) (x)| ≤ [1 + fn C r ] . . . [1 + f2 C r ]f1 C r + [1 + fn C r ] . . . [1 + f2 C r ]M (r, s)f2 C r + · · · + M (r, s)fn C r ≤ max{M (r, s), 1} eα α. Henceforth c(r, s) = max {M (r, s), 1}es . Theorem 2.13. Assumption 2.4 is veriﬁed for C k (RN , RN ) and C0k (RN , RN ). It is a consequence of the following slightly modiﬁed versions of lemmas of A. M. Micheletti [1].

2. Generic Constructions of Micheletti

147

Lemma 2.7. Given f ∈ C k (RN , RN ) and γ ∈ C 0 (RN , RN ), for all ε > 0, there exists δ > 0 such that ∀γ, γC 0 < δ,

sup f (i) ◦ (I + γ) − f (i) C 0 < ε.

(2.44)

0≤i≤k

Proof. Since f and its derivatives f (i) all belong to C 0 (RN , RN ), they are uniformly continuous functions. Hence, for each ε > 0 there exists δ > 0 such that ∀x, y ∈ RN , |x − y| < δ,

sup |f (i) (y) − f (i) (x)| < ε. 0≤i≤k

In particular for γ ∈ C 0 (RN , RN ) such that γC 0 < δ |x + γ(x) − x| = |γ(x)| ≤ γC 0 < δ

⇒ sup |f (i) (x + γ(x)) − f (i) (x)| < ε 0≤i≤k

and we get the result. Lemma 2.8. For each g ∈ C k (RN , RN ), for all ε > 0, there exists δ > 0 such that ∀γ ∈ C k (RN , RN ) such that γC k < δ,

g ◦ (I + γ) − gC k < ε.

(2.45)

Proof. Again proceed by induction on k. For k = 0 the result is a consequence of Lemma 2.7. Then check that if it is true for k − 1 it is true for k. From Lemma 2.4 the kth derivative of g ◦(I +γ)−g is the sum of a ﬁnite number of k-linear mappings of the form g (r) (x + γ(x))[(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] − g (r) (x)[I (λ1 ) (x)] . . . [I (λr ) (x)] = g (r) (x + γ(x)) − g (r) (x) [(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] + g (r) (x)[(I + γ)(λ1 ) (x)] . . . [(I + γ)(λr ) (x)] − g (r) (x)[I (λ1 ) (x)] . . . [I (λr ) (x)], where r = 1, . . . , k and λ1 + · · · + λr = k. Since |(I + γ)(λ1 ) (x)| |(I + γ)(λ2 ) (x)| . . . |(I + γ)(λr ) (x)| ≤ (γC k + 1)r , the norm of the mapping can be bounded above by the following expression: sup |g (r) (x + γ(x)) − g (r) (x)| (γC k + 1)r + γC k p(γC k , gC k ), x

(2.46)

where p is a polynomial. From this upper bound and Lemma 2.7 the result of the lemma is true for k. 2.5.2

Perturbations of the Identity and Tangent Space

By construction F(Θ) ⊂ I + Θ

148

Chapter 3. Courant Metrics on Images of a Set

and the tangent space in each point of the aﬃne space I + Θ is Θ. In general, I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there is a suﬃciently small ball B around 0 in Θ such that the so-called perturbations of the identity, I + B, are contained in F(Θ). In particular, for θ ∈ B, the family of transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C 1 -path in F(Θ) and the tangent space to each point of F(Θ) can be shown to be exactly Θ. Theorem 2.14. Let Θ be equal to C0k (RN , RN ), C k (RN , RN ), or B k (RN , RN ), k ≥ 1. (i) The map f → I + f : B(0, 1) ⊂ Θ → F(Θ) is continuous. (ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ. Remark 2.5. The generic framework of Micheletti is similar to an inﬁnite-dimensional Riemannian manifold. Proof. (i) For k ≥ 1 and f suﬃciently small, F is bijective and has a unique inverse. Given y ∈ RN , consider the mapping def

S(x) = y − f (x),

x ∈ RN .

Then for any x1 and x2 , S(x2 ) − S(x1 ) = −[f (x2 ) − f (x1 )] ⇒ |S(x2 ) − S(x1 )| ≤ |f (x2 ) − f (x1 )| ≤ f (1) C 0 |x2 − x1 |. For f (1) C 1 < 1, S is a contraction and, for each y, there exists a unique x such that y − f (x) = S(x) = x, [I + f ](x) = y. Therefore I + f is bijective. But in order to show that F ∈ F(B k (RN , RN )) we also need to prove that [I + f ]−1 ∈ B k (RN , RN )). The Jacobian matrix of F (x) is equal to I + f (1) (x) and ∀x ∈ RN ,

|F (1) (x) − I| ≤ F (1) − IC 0 = f (1) C 0 .

If there exists x ∈ RN such that F (1) (x) is not invertible, then for some 0 = y ∈ RN , F (1) (x)y = 0 and |y| = |F (1) (x)y − y| ≤ |F (1) (x) − I||y| ≤ f (1) C 0 |y|

⇒ 1 ≤ f (1) C 0 < 1,

which is a contradiction. Therefore, for all x ∈ RN , F (1) (x) is invertible. As a result the conditions of the implicit function theorem are met and from L. Schwartz [3, Vol. 1, Thm. 29, p. 294, Thm. 31, p. 299] we have that F −1 = [I + f ]−1 ∈ C k (RN , RN ). It remains to prove that F −1 = [I+f ]−1 ∈ Bk (RN , RN ) and the continuity. We again proceed by induction on k. Set y = F (x). By deﬁnition g(y) = F −1 (y) − y = x − F (x) = −f (x) and gC 0 = f C0 . Always, from Lemma 2.5 and the identity g(y) = −f (x) = −f (F −1 (y)) = −f (y + g(y)), |g (1) (y)| ≤ |f (1) (x)| [1 + |g(1) (y)|]

⇒ |g (1) (y)| [1 − |f (1) (x)|] ≤ |f (1) (x)|.

2. Generic Constructions of Micheletti

149

Since |f (1) (x)| ≤ f C0 < 1 0 ≤ |g (1) (y)| ≤

|f (1) (x)| 1 − f C0

⇒ g(1) C 0 ≤

|f (1) (x)| f (1) C0 ≤ . 1 − f C0 1 − f C0

(2.47)

For k ≥ 2, again from Lemma 2.5, we have )k 0 ≤ |g

(k)

j=2

(y)| ≤

)k j=2

≤

|f (j) (x)| aj (|g (1) (y)|, . . . , |g (k−1) (y)|) 1 − |f (1) (x)| |f

(j)

(x)| aj (|g (1) (y)|, . . . , |g (k−1) (y)|) 1 − f (1) C 0

(2.48) .

So there exists a constant ck such that F −1 − ICk ≤ ck F − IC k = ck f C k . Therefore, for k ≥ 1, ∀f ∈ Bk (RN , RN ) such that f (1) C 0 < 1,

F = I + f ∈ F(B k (RN , RN )).

Note that the condition is on f (1) C 0 and not f C k , but by deﬁnition f (1) C 0 ≤ f C 1 ≤ f C k and the condition f C k < 1 yields the same result. In particular, the map f → I + f : B(0, 1) ⊂ Bk (RN , RN ) → F ( B k (RN , RN )) is continuous. All this can now be specialized to the spaces C0k (RN , RN ) and C k (RN , RN ) by using the identity (I + f )−1 − I = −f ◦ (I + f )−1 . (ii) Since F(Θ) is contained in the aﬃne space I + Θ, any tangent element to F(Θ) will be contained in Θ. From part (i) for k ≥ 1 and for all θ ∈ Θ there exists τ > 0 such that for all t, 0 ≤ t ≤ τ , I + tΘ ∈ F(Θ). For any F ∈ F(Θ), t → (I + tθ) ◦ F : [0, τ ] → F(Θ) is a continuous path in F(Θ) and the limit of (I + tθ) − I [(I + tθ) ◦ F ] ◦ F −1 − I = =θ t t will be an element of the tangent space TF F(Θ) in F to F(Θ). Therefore for all F ∈ F(Θ), TF F(Θ) = Θ.

2.6 2.6.1

Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) Checking the Assumptions

By deﬁnition, for all integers k ≥ 0 C

k,1

N

def

(RN , R ) =

∀α, 0 ≤ |α| ≤ k, ∃c > 0, ∀x, y ∈ Ω f ∈ C (RN , R ) : α |∂ f (y) − ∂ α f (x)| ≤ c |x − y| k

N

&

150

Chapter 3. Courant Metrics on Images of a Set

endowed with the same norm:15 def f C k,1 = max max ∂ α f C , ck (f ) = max {f C k , ck (f )} ,

(2.49)

0≤|α|≤k

def

|f (y) − f (x)| and ∀k ≥ 1, |y − x| y=x

c(f ) = sup

def

ck (f ) =

c(∂ α f ),

(2.50)

|α|=k

and the convention c0 (f ) = c(f ). C0k,1 (RN , RN ) is the space of functions f ∈ C k,1 (RN , RN ) such that f and all its derivatives go to zero at inﬁnity. They are all Banach spaces. By deﬁnition C0k,1 (RN , RN ) ⊂ C k,1 (RN , RN ) ⊂ B0 (RN , RN ) ⊂ C 0 (RN , RN ), ∀x ∈ RN ,

f → f (x) : B 0 (RN , RN ) → RN

with continuous injections between the spaces and continuity of the evaluation map. As a consequence, the evaluation map f → f (x) is continuous for all the spaces. We deal only with the case C k,1 (RN , RN ). The case C0k,1 (RN , RN ) is obtained from the case C k,1 (RN , RN ) as in the previous section. Theorem 2.15. Assumptions 2.1 and 2.3 are veriﬁed for the spaces C0k,1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0. Proof. The composition and multiplication of bounded and Lipschitz functions is bounded and Lipschitz. For f ∈ C 0,1 (RN , RN ) and I + g ∈ F(C 0,1 (RN , RN )), f ◦ (I + g)C 0 = f C 0 < ∞. Since g is Lipschitz with Lipschitz constant c(g), I + g is also Lipschitz with Lipschitz constant 1 + c(g) and the composition is also Lipschitz with constant c(f ◦ (I + g)) ≤ c(f ) (1 + c(g)). Hence f ◦ (I + g)C 0,1 ≤ f C 0,1 (1 + c(g)) ≤ f C 0,1 (1 + gC 0,1 ) and the continuous function c1 (r) = 1 + r satisﬁes Assumption 2.3. For k = 1, D(f ◦ (I + g)) = Df ◦ (I + g) (I + Dg) and D(f ◦ (I + g))C ≤ Df C (1 + DgC ), c(D(f ◦ (I + g))) ≤ c(Df ) (1 + c(g)) (1 + DgC ) + Df C (1 + c(Dg)) ⇒ D(f ◦ (I + g))C 0,1 ≤ Df C 0,1 (1 + DgC 0,1 ) (1 + c(g)) ≤ Df C 0,1 (1 + DgC 0,1 ) (1 + gC 0,1 ) ⇒ f ◦ (I + g)C 1,1 ≤ f C 1,1 (1 + DgC 0,1 ) (1 + gC 0,1 ) ≤ f C 1,1 (1 + gC 1,1 )2 . The general case is obtained by induction over k. Theorem 2.16. Assumption 2.2 is veriﬁed for C0k,1 (RN , RN ) and C k,1 (RN , RN ), k ≥ 0.

)N 2 1/2 15 N Here R

is endowed with the norm |x| =

i=1

xi

.

2. Generic Constructions of Micheletti

151

Proof. Consider a sequence {fi }ni=1 ⊂ C 0,1 (RN , RN ) such that n

max{fi C 0 , c(fi )} < α < r.

(2.51)

i=1

From Theorem 2.12 (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0 ≤

n

fi C 0 < α.

i=1

In addition, (I + f1 ) ◦ · · · ◦ (I + fn ) − I = fn +

n−1

fi ◦ (I + fi+1 ) ◦ · · · ◦ (I + fn )

i=1

⇒ c((I + f1 ) ◦ · · · ◦ (I + fn ) − I) ≤ c(fn ) + ≤ c(fn ) + ≤

n

n−1 i=1 n−1

c(fi ) (1 + c(fi+1 )) . . . (1 + c(f1 )) )i+1 c(fi ) e

k=1

c(fk )

i=1

)n

c(fi ) e[

i=1

c(fi )]

< α eα

i=1

⇒ (I + f1 ) ◦ · · · ◦ (I + fn ) − IC 0,1 < α max{eα , 1} < α er and we can choose c0 (r) = er for k = 0. We use the same technique for k ≥ 1 but with heavier computations. 2.6.2

Perturbations of the Identity and Tangent Space

By construction F(Θ) ⊂ I + Θ and the tangent space in each point of the aﬃne space I + Θ is Θ. In general, I + Θ is not a subset of F(Θ), but, for some spaces of transformations Θ, there is a suﬃciently small ball B around 0 in Θ such that the so-called perturbations of the identity, I + B, are contained in F(Θ). In particular, for θ ∈ B, the family of transformations Tt = I + tθ ∈ F(Θ), 0 ≤ t ≤ 1, is a C 1 -path in F(Θ) and the tangent space to each point of F(Θ) can be shown to be exactly Θ. Theorem 2.17. Let Θ be equal to C0k,1 (RN , RN ) or C k,1 (RN , RN ), k ≥ 0. (i) The map f → I + f : B(0, 1) ⊂ Θ → F(Θ) is continuous. (ii) For all F ∈ F(Θ), the tangent space TF F(Θ) to F(Θ) is Θ. Remark 2.6. The generic framework of Micheletti is similar to an inﬁnite-dimensional Riemannian manifold.

152

Chapter 3. Courant Metrics on Images of a Set

Proof. (i) Consider perturbations of the identity of the form F = I + f , f ∈ C 0,1 (RN , RN ). Consider for any y ∈ RN the map def

x → S(x) = y − f (x) : RN → RN . For all x1 and x2 S(x2 ) − S(x1 ) = −[f (x2 ) − f (x1 )], |S(x2 ) − S(x1 )| = |f (x2 ) − f (x1 )| ≤ c(f ) |x2 − x1 |, and f is contracting for c(f ) < 1. Hence for each y ∈ RN there exists a unique x ∈ RN such that ⇒ y − f (x) = x

S(x) = x

⇒ F (x) = y def

and F is bijective. It remains to prove that g = F −1 − I ∈ C 0,1 (RN , RN ) to conclude that I + f ∈ F(C 0,1 (RN , RN )). For any y ∈ RN and x = F −1 (y) F −1 (y) − y = x − F (x), gC 0 = F −1 − IC 0 = sup |F −1 (y) − y| = sup |(I − F )(F −1 (y))| y∈RN

y∈RN

= sup |(I − F )(x)| = F − IC 0 < ∞ x∈RN

since F : RN → RN is bijective and gC 0 = f C 0 . For all y1 and y2 and xi = F −1 (yi ), |g(y2 ) − g(y1 )| = |F −1 (y2 ) − y2 − (F −1 (y1 ) − y1 )| ≤ |x2 − F (x2 ) − (x1 − F (x1 )| = |f (x2 ) − f (x1 )| ≤ c(f )|x2 − x1 | ≤ c(f )|F −1 (y2 ) − F −1 (y1 )| = c(f )|g(y2 ) − g(y1 ) + (y2 − y1 )| ⇒ |g(y2 ) − g(y1 )| ≤

c(f ) |y2 − y1 | 1 − c(f )

⇒ c(g) ≤

c(f ) < ∞. 1 − c(f )

Therefore ∀f ∈ C 0,1 (RN , RN ) such that c(f ) < 1,

F = I + f ∈ F(C 0,1 (RN , RN )).

Moreover since c(f ) ≤ f C 0,1 , the condition f C 0,1 < 1 yields c(f ) < 1 and I +f ∈ F(C 0,1 (RN , RN )). Furthermore, the map f → I + f : B(0, 1) ⊂ C 0,1 (RN , RN ) → F(C 0,1 (RN , RN )) is continuous. We use the same technique for k ≥ 1 plus the arguments of Theorem 2.14 (i). (ii) Same proof as Theorem 2.14 (ii).

3. Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

3

153

Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

With a variation of the generic constructions associated with the Banach space Θ of section 2, it is possible to construct a metric on the whole space of homeomorphisms Hom(RN , RN ) or on the whole space Diﬀ k (RN , RN ) of C k -diﬀeomorphisms of RN , k ≥ 1, or of an open subset D of RN . First, recall that, for an open subset D of RN , the topology of the vector space C k (D, RN ) can be speciﬁed by the family of seminorms def

∀K compact ⊂ D,

nK (F ) = F C k (K) .

(3.1)

This topology is equivalent to the one generated by the monotone increasing subfamily {Ki }i≥1 of compact sets 1 def Ki = x ∈ D : dD (x) ≥ and |x| ≤ i , i ≥ 1, (3.2) i where dD (x) = inf y∈D |y − x|. This leads to the construction of the metric def

δk (F, G) =

∞ 1 nKi (F − G) i 1 + n (F − G) 2 Ki i=1

(3.3)

that generates the same topology as the initial family of seminorms and makes C k (D, RN ) a Fr´echet space with metric δk . Given an open subset D of RN , consider the group Homk (D) def

= F ∈ C k (D, RN ) : F : D → D is bijective and F −1 ∈ C k (D, RN )

(3.4)

of transformations of D. For k = 0, Hom0 (D) = Hom(D) and for k ≥ 1, Homk (D) = Diﬀ k (D), where the condition F −1 ∈ C k (D, RN ) is redundant. One possible choice of a topology on Homk (D) is the topology induced by C k (D, RN ). This is the so-called weak topology on Homk (D) (cf., for instance, M. Hirsch [1, Chap. 2]). However, Homk (D) is not necessarily complete for that topology. In order to get completeness, we adapt the constructions of A. M. Micheletti [1]. Associate with F ∈ Homk (D) and each compact subset K ⊂ D the following function of I and F : def

qK (I, F ) =

inf

n

F =F1 ◦···◦Fn F ∈Homk (D) =1

nK (I, F ) + nK (I, F−1 ),

(3.5)

where the inﬁmum is taken over all ﬁnite factorizations F = F1 ◦ · · · ◦ Fn , Fi ∈ Homk (D), of F in Homk (D). By construction, qK (I, F −1 ) = qK (I, F ). Extend this deﬁnition to all pairs F and G in Homk (D): qK (F, G) = qK (I, G ◦ F −1 ). def

(3.6)

154

Chapter 3. Courant Metrics on Images of a Set

The function qK is a pseudometric 16 and the family {qK : K compact ⊂ D} deﬁnes a topology on Homk (D). This topology is metrizable for the metric def

dk (F, G) =

∞ 1 qKi (F, G) , 2i 1 + qKi (F, G) i=1

(3.7)

constructed from the monotone increasing subfamily {Ki }i≥1 of compact sets deﬁned in (3.2). By deﬁnition, dk (F, G) = dk (G, F ) and dk is symmetrical. It is right-invariant since for all F , G, and H in Homk (D)) ∀F, G, and H ∈ Homk (D), ∀F ∈ Homk (D),

dk (F, G) = dk (F ◦ H, G ◦ H), dk (I, F ) = dk (I, F −1 ).

Theorem 3.1. Homk (D), k ≥ 0, is a group under composition, dk is a rightinvariant metric on Homk (D), and (Homk (D), dk ) is a complete metric space. We need Lemmas 2.4 and 2.5 and the analogues of Lemma 2.6 and Theorem 2.11 for the spaces C k (K, RN ), K a compact subset of D, whose elements are bounded and uniformly continuous. Lemma 3.1. Given integers r ≥ 0 and s > 0 there exists a constant c(r, s) > 0 with the following property: for any compact K ⊂ D and a sequence F1 , . . . , Fn in C r (K, RN ) is such that n

F − IC r (K) < α,

0 < α < s;

(3.8)

=1

then for the map F = Fn ◦ · · · ◦ F1 , F − IC r (K) ≤ α c(r, s).

(3.9)

Theorem 3.2. Given k ≥ 0, there exists a continuous function c1 such that for all F, G ∈ Homk (D) and any compact K ⊂ D F ◦ GC k (K) ≤ F C k (K) c1 (GC k (K) ). 16 A

(3.10)

map d : X × X → R on a set X is called a pseudometric (or ´ ecart, or gauge) whenever

(i) d(x, y) ≥ 0, for all x and y, (ii) x = y ⇒ d(x, y) = 0, (iii) d(x, y) = d(y, x), (iv) d(x, z) ≤ d(x, y) + d(y, z) (J. Dugundji [1, p. 198]). This notion is analogous to the one of seminorm for topological vector spaces, but for a group the terminology semimetric (cf. footnote 7) is used for a metric without the triangle inequality.

3. Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

155

Proof. From Lemma 2.5 it readily follows that ψ = F ◦ G ∈ Homk (D). Moreover, ψC 0 (K) = F C 0 (K) , ψ

(1)

C 0 (K) ≤ F (1) C 0 (K) G(1) C 0 (K) ,

ψ (i) C 0 (K) ≤ F (1) C 0 (K) G(i) C 0 (K) +

i

F (j) C 0 (K) aj (G(1) C 0 (K) , . . . , G(i−1) C 0 (K) )

j=2

for i = 2, . . . , k, where aj is a polynomial. Hence ψC k (K) = max ψ i C(K) ≤ F C k (K) c1 (GC k (K) ) i=0,...,k

for some polynomial function c1 that depends only on k. Proof of Theorem 3.1. (i) Homk (D) is a group. The composition of C k k diﬀeomorphisms is again a C -diﬀeomorphism. By deﬁnition, each element has an inverse and the neutral element is I. (ii) dk is a metric. dk (F, G) ≥ 0, dk (F, G) = dk (G, F ). For the triangle inequality F ◦ H −1 = (F ◦ G−1 ) ◦ (G ◦ H −1 ) is a ﬁnite factorization of F ◦ H −1 . Consider arbitrary ﬁnite factorizations of F ◦ G−1 and G ◦ H −1 : F ◦ G−1 = H1 ◦ · · · ◦ Hm ,

Hi ∈ C k (D, D),

G ◦ H −1 = L1 ◦ · · · ◦ Ln ,

Lj ∈ C k (D, D).

This yields a new ﬁnite factorization of F ◦ H −1 . By deﬁnition of qKi , qKi (I, F ◦ H

−1

)≤

m

nKi (I, H ) +

nKi (I, H−1 )

+

n

nKi (I, L ) + nKi (I, L−1 )

=1

=1

and by taking inﬁma over each factorization, qKi (F, H) = qKi (I, F ◦ H −1 ) ≤qKi (I, F ◦ G−1 ) + qKi (I, G ◦ H −1 ) = qKi (F, G) + qKi (G, H). By deﬁnition of dk , since {Ki } is a monotonically increasing sequence of compact sets and {qKi } is a monotonically increasing sequence of pseudometrics, we then get the triangle inequality dk (F, H) ≤ dk (F, G) + dk (G, H). Since Homk (D) is a group and dk is right-invariant, it remains to show that dk (I, F ) = 0 if and only if F = I. Clearly if F = I, qKi (I, F ) = 0 since the qKi ’s are pseudometrics and dk (I, F ) = 0. Conversely, if def

dk (I, F ) =

∞ 1 qKi (I, F ) = 0, i 1 + q (I, F ) 2 Ki i=1

156

Chapter 3. Courant Metrics on Images of a Set

then for all Ki , qKi (I, F ) = 0. For each Ki and all ε, 0 < ε < 1, there exists a ﬁnite factorization F = F1 ◦ · · · ◦ Fn such that n

nKi (F − I) + nKi (F−1 − I) < ε < 1

=1

⇒ nKi (F − I) ≤

nKi (F ◦ · · · ◦ Fn − F+1 ◦ · · · ◦ Fn ).

From Lemma 3.1 ∀ε > 0,

nKi (F − I) < ε c(k, 1)

and (F − I)|Ki = 0. Since this is true for all i, F − I = 0 on D and dk is a metric. (iii) Completeness. Let {Fn } be a Cauchy sequence in Homk (D). (a) Boundedness of {Fn } and {Fn−1 } in C k (D, RN ). For all ε, 0 < ε < 1, there exists N such that, for all m, n > N , dk (Fm , Fn ) < ε. By the triangle inequality |dk (I, Fm ) − dk (I, Fn )| ≤ dk (Fm , Fn ) < ε. For all i, {qKi (I, Fn } is Cauchy and, a posteriori, bounded by some constant L. For ν(n) each n, there exists a factorization Fn = Fn1 ◦ · · · ◦ Fn such that

ν(n)

0≤

Fni − IC k (Ki ) + (Fni )−1 − IC k (Ki ) − dk (I, Fn ) < ε < 1

i=1

ν(n)

⇒

Fni − IC k (Ki ) + (Fni )−1 − IC k (Ki ) < L + ε < L + 1.

i=1

By Lemma 3.1 Fn − IC k (Ki ) + Fn−1 − IC k (Ki ) < 2(L + ε) c(k, L + 1) and the sequences {Fn − I} and {Fn−1 − I} and, a fortiori, {Fn } and {Fn−1 } are bounded in C k (Ki , RN ). (b) Convergence of {Fn } and {Fn−1 } in C k (D, RN ). For any m, n −1 − I) ◦ Fm Fn − Fm = (Fn ◦ Fm

and by Theorem 3.2 −1 − I) ◦ Fm C k (Ki ) Fm − Fn C k (Ki ) = (Fn ◦ Fm −1 ≤ Fn ◦ Fm − IC k (Ki ) c1 (Fm C k (Ki ) ) −1 ⇒ Fm − Fn C k (Ki ) ≤ c+ 1 Fn ◦ Fm − IC k (Ki ) , def

c+ 1 = sup c1 (Fm C k (Ki ) ) < ∞ m

3. Generalization to All Homeomorphisms and C k -Diﬀeomorphisms

157

since {Fm } is bounded in C k (Ki , RN ). For all ε, 0 < ε < c(k, 1) c+ 1 (where c(k, 1) is the constant of Lemma 3.1), ∃N, ∀n, m > N,

−1 dk (I, Fn ◦ Fm ) = dk (Fm , Fn ) < ε/(c(k, 1) c+ 1 ) < 1.

By Lemma 3.1 + −1 − IC k (Ki ) < c(k, 1)ε/(c(k, 1) c+ Fn ◦ Fm 1 ) = ε/c1

⇒ ∀n, m, > N,

−1 Fm − Fn )C k (Ki ) ≤ c+ 1 Fn ◦ Fm − IC k (Ki ) < ε

and, since this is true for all Ki , {Fn } is a Cauchy sequence in C k (D, RN ) that converges to some F ∈ C k (D, RN ) since C k (D, RN ) is a Fr´echet space for the family of seminorms {qKi }. Similarly, we get that {Fn−1 } is a Cauchy sequence in C k (D, RN ) that converges to some G ∈ C k (D, RN ). (c) F is bijective, G = F −1 , and F ∈ Homk (D). For all n, Fn , Fn−1 , F and G are continuous since C k (D, RN ) ⊂ C 0 (D, RN ). By continuity of F → F (x) : C k (D, RN ) → RN , Fn−1 (x) → G(x). By the same argument for Fn , x = Fn (Fn−1 (x)) → F (G(x)) = (F ◦ G)(x) and for all x ∈ RN , (F ◦ G)(x) = x. Similarly, starting from Fn−1 ◦ Fn = I, we get (G ◦ F )(x) = x. Hence, G ◦ F = I = F ◦ G. So F is bijective, F, F −1 ∈ C k (D, RN ), and F ∈ Homk (D). (d) Convergence of Fn → F in Homk (D). To check that Fn → F in Homk (D) recall that by Theorem 3.2 for each Ki qKi (Fn , F ) = qKi (I, F ◦ Fn−1 ) ≤ I − F ◦ Fn−1 C k (Ki ) + I − Fn ◦ F −1 C k (Ki ) = (Fn − F ) ◦ Fn−1 C k (Ki ) + (F − Fn ) ◦ F −1 C k (Ki ) ≤ c1 (Fn−1 C k (Ki ) )Fn − F C k (Ki ) + c1 (F −1 C k (Ki ) )F − Fn C k (Ki ) ≤ max c1 (Fn−1 C k (Ki ) ) + c1 (F −1 C k (Ki ) ) F − Fn C k (Ki ) n

that goes to zero as Fn goes to F in C k (D, RN ). This completes the proof. Remark 3.1. In section 2 it was possible to consider in F(Θ) a subgroup G of translations since Fa (x) = a + x

⇒ (Fa − I)(x) = a ∈ Θ

for spaces Θ that contain constants. Yet, since all of them were spaces of bounded functions in RN , it was not possible to include rotations or ﬂips. But this can be done now in Homk (RN ), and we can quotient out not only by the subgroup of translations but also by the subgroups of isometries or rotation that are important in image processing.

Chapter 4

Transformations Generated by Velocities 1

Introduction

In Chapter 3 we have constructed quotient groups of transformations F(Θ)/G and their associated complete Courant metrics. Such spaces are neither linear nor convex. In this chapter, we specialize the results of Chapter 3 to spaces of transformations that are generated at time t = 1 by the ﬂow of a velocity ﬁeld over a generic time interval [0, 1] with values in the tangent space Θ. The main motivation is to introduce a notion of semiderivatives in the direction θ ∈ Θ on such groups as well as a tractable criterion for continuity via C 1 or continuous paths in the quotient group endowed with the Courant metric. This point of view was adopted by J.´sio [2, 7] as early as 1973 and considerably expanded in his th`ese d’´etat in P. Zole 1979. One of his motivations was to solve a shape diﬀerential equation of the type A V (t) + G(Ωt (V )) = 0, t > 0, where G is the shape gradient of a functional and A a duality operator.1 At that time most people were using a simple perturbation of the identity to compute shape derivatives. The ﬁrst comprehensive book systematically promoting the velocity method was published in 1992 by J. Sokolowski and ´sio [9]. Structural theorems for the Eulerian shape derivative of smooth J.-P. Zole ´sio [7] and generalized to nonsmooth domains were ﬁrst given in 1979 by J.-P. Zole ´sio [14]. The velocity point domains in 1992 by M. C. Delfour and J.-P. Zole ´ [2], of view was also adopted in 1994 by R. Azencott [1], in 1995 by A. Trouve ´ and in 1998 by A. Trouve [3] and L. Younes [2] to construct complete metrics and geodesic paths in spaces of diﬀeomorphisms generated by a velocity ﬁeld with a broad spectrum of applications to imaging.2 The reader is referred to the forthcoming book of L. Younes [6] for a comprehensive exposition of this work and

1 Cf.

´sio [7] and the recent book by M. Moubachir and J.-P. Zole ´sio [1]. J.-P. Zole ﬁrst author would like to thank Robert Azencott, who pointed out his work and the contributions of his team during a visit in Houston early in 2010. Special thanks also to Alain Trouv´ e for an enlightening discussion shortly after in Paris. 2 The

159

160

Chapter 4. Transformations Generated by Velocities

to related papers such as the ones of P. W. Michor and D. Mumford [1, 2, 3] and L. Younes, P. W. Michor, I. Shah, and D. Mumford [1]. In view of the above motivations, this chapter begins with section 2, which specializes the results of Chapter 3 to transformations generated by velocity ﬁelds. It also explores the connections between the constructions of Azencott and Micheletti that implicitly use a notion of geodesic path with discontinuities. Section 3 motivates and adapts the deﬁnitions of Gateaux and Hadamard semiderivatives in topological vector spaces (cf. Chapter 9) to shape functionals deﬁned on shape spaces. The analogue of the Gateaux semiderivative for sets is obtained by the method of perturbation of the identity operator, while the analogue of the Hadamard semiderivative comes from the velocity (speed ) method. The ﬁrst notion does not extend to submanifolds and does not incorporate the chain rule for the semiderivative of the composition of functions (for instance, to get the semiderivative with respect to the parameters of an a priori parametrized geometry), while Hadamard does. In Chapter 9, ﬂows of velocity ﬁelds will be adopted as the natural framework for deﬁning shape semiderivatives. In section 3.3 the velocity and transformation viewpoints will be emphasized through a series of examples of commonly used families of transformations of sets. They include C k -domains, Cartesian graphs, polar coordinates, and level sets. The following two sections give technical results that will be used to characterize continuity and semidiﬀerentiability along local paths without restricting the analysis to the subgroup GΘ of F(Θ) of section 2. In section 4 we establish the equivalence between deformations obtained from a family of C 1 -paths and deformations obtained from the ﬂow of a velocity ﬁeld. Section 4.1 gives the equivalence under relatively general conditions. Section 4.2 shows that Lipschitzian perturbations of the identity operator can be generated by the ﬂow of a nonautonomous velocity ﬁeld. In section 4.3 the conditions of section 4.2 are sharpened for the special families of velocity ﬁelds in C0k (RN , RN ), C k (RN , RN ), and C k,1 (RN , RN ). The constrained case where the family of domains are subsets of a ﬁxed holdall is studied in section 5. In both sections 4 and 5 we show that, under appropriate conditions, starting from a family of transformations is locally equivalent to starting from a family of velocity ﬁelds. This key result bridges the two points of view. Section 6 builds on the results of section 4 to establish that the continuity of a shape function with respect to the Courant metric on F(Θ)/G is equivalent to its continuity along the ﬂows of all velocity ﬁelds V in C 0 ([0, τ ]; Θ) for Θ equal to C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0. This result is of both intrinsic and practical interest since it is generally easier to check the continuity along paths than directly with respect to the Courant metric. Finally, from the discussion in section 3, the technical results of section 4 will again be used in Chapter 9 to construct local C 1 -paths generated by velocity ﬁelds V in C 0 ([0, τ ]; Θ) to deﬁne the shape semiderivatives.

2. Metrics on Transformations Generated by Velocities

2

161

Metrics on Transformations Generated by Velocities

2.1

Subgroup GΘ of Transformations Generated by Velocities

From Chapter 3 recall the deﬁnition of the group def

F(Θ) = I + θ : θ ∈ Θ, (I + θ)−1 ∃, and (I + θ)−1 − I ∈ Θ for the Banach space Θ = C 0,1 (RN , RN ) with norm |θ(y) − θ(x)| def def , θΘ = max sup |θ(x)|, c(θ) , c(θ) = sup |y − x| N y=x x∈R

(2.1)

and the associated complete metric d on F(Θ). This section specializes to a subgroup of transformations I + θ of F(Θ) that are speciﬁed via a vector ﬁeld V on a generic interval [0, 1] with the following properties: (i) for all x ∈ RN , the function t → V (t, x) : [0, 1] → RN belongs to L1 (0, 1; RN ) and 1 1 V (t)C dt = sup |V (t, x)| dt < ∞; (2.2) 0

0 x∈RN

(ii) for almost all t ∈ [0, 1], the function x → V (t, x) : RN → RN is Lipschitzian and 1 1 |V (t, y) − V (t, x)| c(V (t)) dt = sup dt < ∞. (2.3) |y − x| 0 0 x=y In view of the assumptions on V , it is natural to introduce the norm 1 def V L1 (0,1;Θ) = V (t)Θ dt.

(2.4)

0

Associate with V ∈ L1 (0, 1; Θ) the ﬂow dTt (V ) = V (t) ◦ Tt (V ), dt

T0 (V ) = I.

(2.5)

In view of the assumptions on V , for each X ∈ RN , the diﬀerential equation x (t; X) = V (t, x(t; X)),

x(0; X) = X,

(2.6)

has a unique solution in W 1,1 (0, 1; RN ) ⊂ C([0, 1], RN ), the mapping X → x(·; X) : RN → C([0, 1], RN ) is continuous, and for all t ∈ [0, 1] the mapping X → Tt (V ) def (X) = x(t; X) : RN → RN is bijective. Deﬁne def

GΘ = T1 (V ) : ∀V ∈ L1 ([0, 1]; Θ) .

(2.7)

162

Chapter 4. Transformations Generated by Velocities

Theorem 2.1.

(i) The set GΘ with the composition ◦, ∀V1 , V2 ∈ L1 (0, 1; Θ),

T1 (V1 ) ◦ T1 (V2 ),

(2.8)

is a subgroup of F(Θ) and for all V ∈ L1 (0, 1; Θ), sup Tt (V ) − IΘ = sup θV (t)Θ ≤ V L1 exp2(1+V L1 ) , 0≤t≤1

(2.9)

0≤t≤1 def

where θV (t) = Tt (V ) − I and L1 stands for L1 (0, 1; Θ). In particular, for all t, Tt (V ) ∈ F(Θ) and t → Tt (V ) : [0, 1] → F(Θ) is a continuous path. (ii) For all V ∈ L1 (0, 1; Θ), (T1 (V ))−1 = T1 (V − ) for V − (t, x) = −V (1 − t, x), −θV ◦ (I + θV )−1 = (T1 (V ))−1 − I = T1 (V − ) − I = θV − , −

V L1 (0,1;Θ) = V L1 (0,1;Θ) ,

(2.10) (2.11)

and sup Tt (V )−1 − IΘ = sup θV− (t)Θ ≤ V L1 exp2(1+V L1 ) .

0≤t≤1

(2.12)

0≤t≤1

(iii) For all V ∈ L1 (0, 1; Θ), T1 (V ) − IΘ + (T1 (V ))−1 − IΘ ≤ 2 V L1 (0,1;Θ) exp2 (1+V L1 (0,1;Θ) ) (2.13) ⇒ d(T1 (V ), I) ≤ 2 V L1 (0,1;Θ) exp2 (1+V L1 (0,1;Θ) ) .

(2.14)

(iv) For all V, W ∈ L1 (0, 1; Θ), max sup θW (t) − θV (t)Θ , sup θW − (t) − θV − (t)Θ 0≤t≤1

≤V L1 exp

(1+V L1 )

0≤t≤1

(2.15) 1 + W L1 exp2(1+W L1 ) W (s) − V (s)L1 ,

where L1 stands for L1 (0, 1; Θ). Remark 2.1. Note that the condition (I − θ)−1 exists and (I − θ)−1 ∈ Θ in the deﬁnition of F(Θ) is automatically veriﬁed for the transformations I + θV generated by a velocity ﬁeld V ∈ L1 (0, 1; Θ). Proof. (i) Choosing V = 0, the identity I is in GΘ . To show that GΘ is closed under the composition, it is suﬃcient to construct a W ∈ L1 (0, 1; Θ) such that T1 (W ) = T1 (V1 ) ◦ T1 (V2 ). Given τ ∈ (0, 1) deﬁne the concatenation t 1 V , x , 0 ≤ t ≤ τ, τ 2 τ def (V1 ∗ V2 )(t, x) = (2.16) t−τ 1 V1 ,x , τ < t ≤ 1. 1−τ 1−τ

2. Metrics on Transformations Generated by Velocities

163

It is easy to check that W = V1 ∗ V2 ∈ L1 (0, 1; Θ) as the concatenation of the L1 -functions V1 and V2 so that the equation dTt (W ) = W (t) ◦ Tt (W ), dt

T0 (W ) = I

has a unique solution and T1 (W ) ∈ GΘ . On the interval [0, τ ]

s t/τ V2 (s , Ts τ (W )(X)) ds , Ts (W )(X) d = τ τ 0 0 t ⇒ Tt τ (W )(X) − X = V2 (s , Ts τ (W )(X)) ds , 0 ≤ t ≤ 1

Tt (W )(X) − X =

t

V2

s

0

⇒ Tt τ (W )(X) = Tt (V2 )(X)

and Tτ (W ) = T1 (V2 ).

On the interval [τ, 1]

t

Tt (W )(X) − T1 (V2 ) =

V1

τ t−τ 1−τ

=

s−τ , Ts (W )(X) 1−τ

d

s−τ 1−τ

V2 s , Tτ +s (1−τ ) (W )(X) ds

0 t

⇒ Tτ +t (1−τ ) t(W )(X) − T1 (V2 ) =

V2 s , Tτ +s (1−τ ) (W )(X) ds ,

0 ≤ t ≤ 1,

0

Tτ +t (1−τ ) (W )(X) = Tt (V1 ) ◦ T1 (V2 ) and T1 (W )(X) = T1 (V1 ) ◦ T1 (V2 ). To show that each T1 (V ) has an inverse in GΘ , consider the new function y(t; X) = [T1−t (V ) ◦ (T1 (V ))−1 ](X) = x(1 − t; (T1 (V ))−1 (X)), def

0 ≤ t ≤ 1,

for which y(0; X) = [T1 (V ) ◦ (T1 (V ))−1 ](X) = X and y(1; X) = (T1 (V ))−1 (X). Its equation is given by dy (t; X) = −V (1 − t, y(t; X)), dt def

y(0; X) = X,

and the associated vector ﬁeld in V − (t, x) = −V (1 − t, x) L1 (0, 1; Θ). Therefore GΘ is a group of homeomorphisms Hom(RN ). To show that GΘ ⊂ F(Θ), it is necessary to show T1 (V ) − I ∈ Θ. Given X ∈ RN , consider the functions

that clearly belongs to of RN , that is, GΘ ⊂ that for each V , θ = x(t) = Tt (V )(X) and

164

Chapter 4. Transformations Generated by Velocities

θ(t; X) = x(t) − X. By deﬁnition t V (s, X + θ(s; X)) ds, θ(t; X) = x(t) − X = 0 t t |V (s, X + θ(s; X)) − V (s, X)| ds + |V (s, X)| ds |θ(t; X)| ≤ 0 0 t t ≤ c(V (s)) |θ(s; X)| ds + |V (s, X)| ds, 0 0 t t c(V (s)) sup |θ(s; X)| ds + sup |V (s, X)| ds. sup |θ(t; X)| ≤ X

0

X

0

X

t Given 0 < α < 1 let gα (t) = exp( 0 c(V (s)) ds/α). Then it is easy to show that supX |θ(t; X)| 1 sup ≤ g (t) 1 − α α t∈[0,1] ⇒ sup θ(t)C ≤ t∈[0,1]

gα (1) 1−α

1

sup |V (s, X)| ds X

0

1

V (s)C ds. 0

To prove that θ is Lipschitzian, associate with X, Y ∈ RN , X = Y , the functions x(t) = Tt (V )(X), y(t) = Tt (V )(Y ), θ(t; X) = x(t) − X, and θ(t; Y ) = y(t) − Y . By deﬁnition t θ(t; X) − θ(t; Y ) = V (s, Y + θ(s; Y )) − V (s, X + θ(s; X)) ds, 0 t t c(V (s)) |θ(s; Y )) − θ(s; X)| ds + c(V (s)) |Y − X| ds |θ(t; X) − θ(t; Y )| ≤ 0 0 t t |θ(t; X) − θ(t; Y )| |θ(s; Y )) − θ(s; X)| ⇒ sup ≤ ds + c(V (s)) sup c(V (s)) ds |Y − X| |Y − X| X=Y X=Y 0 0 gα (1) 1 c(V (s)) ds. ⇒ sup c(θ(t)) ≤ 1−α 0 t∈[0,1] Combining the last inequality with the one of the ﬁrst part, for all α, 0 < α < 1, sup θ(t)Θ ≤ t∈[0,1]

gα (1) 1−α

1

V (t)Θ dt. 0

1 1 For V = 0, θ = 0; for V = 0, choose α = 0 V (t)Θ dt/( 0 V (t)Θ dt + 1). This yields (2.9). Since (I + θ)−1 = (T1 (V ))−1 = T1 (V − ) is of the same form for the velocity ﬁeld V − , (I + θ)−1 − I = (T1 (V ))−1 − I = T1 (V − ) − I ∈ Θ. Therefore GΘ is a subgroup of F(Θ). (ii) From the construction of the inverse in part (i) and the fact that (T1 (V ))−1 − I = −θV ◦ (I + θV )−1 . (iii) From inequality (2.9) applied to θV and θV − and identity (2.11).

2. Metrics on Transformations Generated by Velocities (iv) Let θV (t) = Tt (V ) − I and let θW (t) = Tt (W ) − I: t θW (t) − θV (t) = W (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds 0 t = (W (s) − V (s)) ◦ Ts (W ) ds 0 t V (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds, + 0 t W (s) − V (s)C ds θW (t) − θV (t)C ≤ 0 t + c(V (s)) θW (s) − θV (s)C ds 0

gα (1) 1−α

⇒ sup θW (t) − θV (t)C ≤ 0≤t≤1

for 0 < α < 1 and gα (t) = exp

t

1

W (s) − V (s)C ds 0

c(V (s) ds/α. Similarly,

0

t

θW (t) − θV (t) =

(W (s) − V (s)) ◦ Ts (W ) ds t V (s) ◦ Ts (W ) − V (s) ◦ Ts (V ) ds, + 0 t c(θW (t) − θV (t)) ≤ c(W (s) − V (s)) (1 + c(θW (s))) ds 0 t c(V (s)) c(θW (s) − θV (s)) ds + 0

0

⇒ sup c(θW (t) − θV (t)) ≤ 0≤t≤1

gα (1) 1−α

1

c(W (s) − V (s)) (1 + c(θW (s))) ds. 0

From part (i) sup c(θW (s)) ≤ W L1 (0,1;Θ) exp2(1+W L1 (0,1;Θ) ) 0≤s≤1

and ﬁnally sup θW (t) − θV (t)Θ 0≤t≤1

≤

gα (1) (1 + W L1 (0,1;Θ) exp2(1+W L1 (0,1;Θ) ) ) W (s) − V (s)L1 (0,1;Θ) ds. 1−α

By choosing α = V L1 (0,1;Θ) /(1 + V L1 (0,1;Θ) ) sup θW (t) − θV (t)Θ 0≤t≤1

≤ V L1 exp(1+V L1 ) (1 + W L1 (0,1;Θ) exp2(1+W L1 ) ) W (s) − V (s)L1 ds.

165

166

2.2

Chapter 4. Transformations Generated by Velocities

Complete Metrics on GΘ and Geodesics

Since GΘ is a subgroup of the complete group F(Θ), its closure GΘ with respect to the metric d of F(Θ) is a closed subgroup whose elements can be “approximated” by elements constructed from the ﬂow of a velocity ﬁeld. Is (GΘ , d) complete? Can a velocity ﬁeld be associated with a limit element? It is very unlikely that (GΘ , d) can be complete unless we strengthen the metric d to make the associated Cauchy sequence of velocities {Vn } converge in L1 (0, 1; Θ). We have seen that for each V ∈ L1 (0, 1; Θ), we have a continuous path t → Tt (V ) in GΘ . Since the part of the metric on I +θV is deﬁned as an inﬁmum over all ﬁnite factorizations (I + θV1 ) ◦ · · · ◦ (I + θVn ) of I + θV and (I + θV − ) ◦ · · · ◦ (I + θVn− ) 1 of (I + θV )−1 , T1 (V ) = T1 (V1 ) ◦ T1 (V2 ) ◦ · · · ◦ T1 (Vn ), n inf θVi Θ + θV − Θ , d(I, T1 (V )) = T1 (V )= T1 (V1 )◦···◦T1 (Vn ) i=1

i

is there a velocity V ∗ ∈ L1 (0, 1; Θ) that achieves that inﬁmum? Do we have a geodesic path between I and T1 (V ) that would be achieved by V ∗ ∈ L1 (0, 1; Θ)? Given a velocity ﬁeld we have constructed a continuous path t → Tt (V ) : [0, 1] → F(Θ). At each t, there exists δ > 0 suﬃciently small such that the path t → Tt+s (V ) ◦ Tt−1 (V ) : [0, δ] → F(Θ) is diﬀerentiable in t in the group F(Θ): Tt+s (V ) ◦ Tt−1 (V ) − I 1 t+s V (r) ◦ Tr (V ) dr ◦ Tt−1 (V ) = s s t (2.17) −1 → V (t) ◦ Tt (V ) ◦ Tt (V ) = V (t) dF Tt ⇒ = V (t) a.e. in Θ. (2.18) dt With this deﬁnition a jump from I + θ1 to [I + θ2 ] ◦ [I + θ1 ] at time 1/2 becomes −1 (V ) − I = [I + θ2 ] ◦ [I + θ1 ] ◦ [I + θ1 ]−1 − I = I + θ2 − I = θ2 , T1/2+s (V ) ◦ T1/2

that is, a Dirac delta function θ2 in the tangent space Θ to F(Θ) at 1/2. Now the norm of the velocity V = θ2 δ1/2 in the space of measures becomes V M1 ((0,1);Θ) =

1

θ2 Θ δ1/2 dt = θ2 Θ . 0

If we have n such jumps, V M1 ((0,1);Θ) =

θi Θ .

1=1,...,n

The metric of Micheletti is the inﬁmum of this norm over all ﬁnite factorizations. A consequence of this analysis is that a factorization of an element I + θ of the form (I +θ1 )◦(I +θ2 )◦· · ·◦(I +θn ) is in fact a path t → Tt : [0, 1] → F(Θ) of bounded

2. Metrics on Transformations Generated by Velocities

167

variations in F(Θ) between I and I +θ by assigning times 0 < t1 < t2 < · · · < tn < 1 to each jump in the group F(Θ): 0 ≤ t < t1 , I, , t1 ≤ t < t2 , I + θ 1 def t2 ≤ t < t3 , Tt (V ) = (I + θ1 ) ◦ (I + θ2 ), ... (I + θ1 ) ◦ (I + θ2 ) ◦ · · · ◦ (I + θn ), tn ≤ t ≤ 1, n dF Tt def = V (t), V M1 ((0,1);Θ) = θi δ(ti ) ⇒ θi Θ , V (t) = dt 1=1,...,n i=1

where the time-derivative is taken in the group sense (2.17) and V is a bounded measure. With this interpretation in mind, for V ∈ L1 (0, 1; Θ) the metric of Micheletti should reduce to 3 1 13 3 dF Tt (v) 3 3 3 d(I, T1 (V )) = inf v(t)Θ dt = inf 3 dt 3 dt, v∈L1 (0,1;Θ) 0 v∈L1 (0,1;Θ) 0 Θ T1 (v)=T1 (V )

T1 (v)=T1 (V )

where the derivative is in the group F(Θ). As a result we could talk of a geodesic path in F(Θ) between I and T1 (V ). If this could be justiﬁed, the next question is whether GΘ is complete in (F(Θ), d) or complete with respect to the metric constructed from the function 1 def dG (I, T1 (V )) = inf v(t)Θ dt, (2.19) 1 v∈L (0,1;Θ) T1 (v)=T1 (V )

0

dG (T1 (V ), T1 (W )) = d(T1 (V ) ◦ T1 (W )−1 , I). def

(2.20)

By deﬁnition dG (T1 (V )−1 , I) = dG (T1 (V ), I),

dG (T1 (V ), T1 (W )) = dG (T1 (W ), T1 (V ))

and dG is right-invariant. The inﬁmum is necessary since the velocity taking I to T1 (V ) is not unique. Theorem 2.2. Let Θ be equal to C k,1 (RN , RN ), C k+1 (RN , RN ), C0k+1 (RN , RN ), or Bk+1 (RN , RN ), k ≥ 0. Then dG is a right-invariant metric on GΘ . Remark 2.2. The theorem is also true for the right-invariant metric dG (T1 (V ), I) = d(T1 (W ), T1 (V )) + def

inf

W ∈L1 (0,1;Θ) T1 (W )=T1 (V )

W L1 (0,1;Θ) ,

dG (T1 (V ), T1 (W )) = dG (T1 (V ) ◦ T1 (W )−1 , I). def

(2.21)

(2.22)

168

Chapter 4. Transformations Generated by Velocities

Proof. We give only the proof for C 0,1 (RN , RN ). It remains to show that the triangle inequality holds and that dG (T1 (W ), T1 (V )) = 0 if and only if T1 (W ) = T1 (V ). Given U, V, W in L1 (0, 1; Θ), T1 (W ) ◦ T1 (U )−1 = T1 (W ) ◦ T1 (V )−1 ◦ T1 (V ) ◦ T1 (U )−1 . For all v ∈ L1 (0, 1; Θ) such that T1 (v) = T1 (W ) ◦ T1 (V )−1 and for all u ∈ L1 (0, 1; Θ) such that T1 (v) = T1 (V ) ◦ T1 (U )−1 , the concatenation v ∗ u is such that T1 (v ∗ u) = T1 (W ) ◦ T1 (U )−1 . By deﬁnition dG (T1 (W ) ◦ T1 (U )−1 , I) ≤ v ∗ uL1 = vL1 + uL1 . By taking inﬁma with respect to v and u we get the triangle inequality dG (T1 (W ) ◦ T1 (U )−1 , I) ≤ dG (T1 (W ) ◦ T1 (V )−1 , I) + dG (T1 (V ) ◦ T1 (U )−1 , I). If T1 (V ) = I, then inf w∈L1 (0,1;Θ) wL1 (0,1;Θ) = 0 and dG (T1 (V ), I) = 0. ConT1 (w)=I

versely, if dG (T1 (V ), I) = 0, then inf v∈L1 (0,1;Θ) vL1 (0,1;Θ) = 0 and there exists a T1 (v)=T1 (V )

sequence vn ∈ L1 (0, 1; Θ) such that vn → 0 in L1 (0, 1; Θ) and T1 (vn ) = T1 (V ). By continuity of T1 (vn ) − I with respect to vn from Theorem 2.1 (iv), T1 (V ) − I = T1 (vn ) − I → T1 (0) − I = 0 and T1 (V ) = I. Getting the completeness with the metric dG and even dG is not obvious. For a Cauchy sequence {T1 (Vn )}, the L1 -convergence of the velocities {Vn } to some V would yield the convergence of {T1 (Vn )} to T1 (V ) and even the convergence of the geodesics, but the metrics do not seem suﬃciently strong to do that. To appreciate this point, we ﬁrst establish an inequality that really follows from the triangle inequality for dG . Associate with W a w such that T1 (w) = T1 (W ), with V a v such that T1 (v) = T1 (V ), and let u be such that T1 (u) = T1 (W )◦T1 (V )−1 . Then the concatenation w ∗ v− is such that T1 (v ∗ u) = T1 (W ). From this we get the inequalities inf T1 (w)=T1 (W )

wL1 ≤ v ∗ uL1 = vL1 + uL1

and by taking inﬁma we get the inequality inf T1 (w)=T1 (W )

wL1 ≤

inf

T1 (u)=T1 (W )◦T1 (V )−1

uL1 +

inf T1 (v)=T1 (V )

vL1

inf ⇒ inf wL1 − inf vL1 ≤ uL1 . T1 (w)=T1 (W ) T1 (v)=T1 (V ) T1 (u)=T1 (W )◦T1 (V )−1 Pick any Cauchy sequence {T1 (Vn )}. For any ε > 0, there exists N such that for all n, m > N 1 1 w − inf v uL1 < ε. inf inf L L ≤ −1 T1 (w)=T1 (Vm )

T1 (v)=T1 (Vn )

T1 (u)=T1 (Vm )◦T1 (Vn )

Therefore, there exists a sequence {vn } such that T1 (vn ) = T1 (Vn ) and 0 ≤ vn L1 −

inf T1 (v)=T1 (Vn )

vL1 < ε

⇒ |vm L1 − vn L1 | < 3ε. The sequence {vn L1 } is bounded in L1 (0, 1; Θ), but this does not seem suﬃcient to get the convergence of the vn ’s in L1 (0, 1; Θ).

2. Metrics on Transformations Generated by Velocities

169

One could think of many ways to strengthen the metric. For instance, one could introduce the minimization of the whole W 1,1 (0, 1; Θ)-norm of the path t → Tt (v) with respect to v from which the minimizing velocity v(t) = ∂Tt /∂t ◦ Tt−1 could be recovered, but the triangle inequality would be lost.

2.3

Constructions of Azencott and Trouv´ e

In 1994 R. Azencott [1] deﬁned the following program. Given a smooth manifold M , consider the time continuous curve T = (Tt )0≤t≤1 in Aut(M ) solutions of ∂Tt = V (t) ◦ Tt , ∂t

T0 = I,

(2.23)

for a continuous time family V (t) of vector ﬁelds in a Hilbert space H ⊂ Aut(M ) and deﬁne 1 def l(φ) with l(φ) = |V (t)|I dt. (2.24) d(I, φ) = inf V ∈C([0,1];H) T1 (V )=φ

0

A fully rigorous construction of this distance, in the context of inﬁnite-dimensional ´ [2] in 1995 and [3] in 1998. He shows Lie groups, was performed by A. Trouve that the subset of φ ∈ Aut(M ) for which d(I, φ) can be deﬁned is a subgroup A of invertible mappings. He extends this distance between two arbitrary mappings def φ and ψ by right-invariance: d(φ, ψ) = d(I, ψ ◦ φ−1 ). Hence, in this extended framework, the new problem should be to ﬁnd in A 1 1 φˆ = argmin |f˜ − f ◦ φ|2 dx + d(Id, φ)2 , 2 M 2 where M is a smooth manifold, the function f : M → R is a high dimensional representation template, and f˜ is a new observed image belonging to this family (cf. ´ [3]). This approach was further developed by the group of Azencott A. Trouve (cf., for instance, L. Younes [1]). ´ [3] in 1998: Quoting from the introduction of A. Trouve . . . Hence, for a large deformation φ, one can consider φ as the concatenation of small deformations φui . More precisely, if φ = Φn , where the Φk are recursively deﬁned by Φ0 = Id and φk+1 = φuk+1 ◦ Φk , the family Φ = (Φk )0≤k≤n deﬁnes a polygonal ) line in Aut(M ) whose length l(Φ) can be approximated by l(φ) = nk=1 |uk |e . At a naive level, one could deﬁne the distance d(Id, φ) as the inﬁmum of l(Φn ) for all family Φ such that Φn = φ. For a more rigorous setting, one should consider the time continuous curve Φ = (Φt )0≤t≤1 in Aut(M ). . . . In fact what he describes as a naive approach is the very precise construction of A. M. Micheletti [1] in 1972. He is certainly not the ﬁrst author who has overlooked that paper in Italian, where the central completeness result that is of interest to us was just a lemma in her analysis of the continuity of the ﬁrst eigenvalue of the Laplacian. We brought it up only in the 2001 edition of this book.

170

Chapter 4. Transformations Generated by Velocities

By introducing a Hilbert space H ⊂ Θ = C 0,1 (RN , RN ) and minimizing over L (0, 1; H) rather than L1 (0, 1; Θ), the bounded sequence {vn } lives in L2 (0, 1; H), where there exist a v ∈ L2 (0, 1; H) and a weakly converging subsequence to v. By using the complete metric of Micheletti, we had the existence of a F ∈ F(Θ) such that T1 (vn ) → F , but now we have a candidate v for which F = T1 (v). This is very much analogous to the Hilbert uniqueness method of J.-L. Lions [1] in the context of the controllability of partial diﬀerential equations (cf. also R. Glowinski and J.-L. Lions [1, 2]). 2

Remark 2.3. It is important to understand that ﬁnding a complete metric of Courant type for the various spaces of diﬀeomorphisms is a much less restrictive problem than constructing a metric from geodesics. The existence of a geodesic is not required for the Courant metrics.

3 3.1

Semiderivatives via Transformations Generated by Velocities Shape Function

All spaces F(Θ) and F(Θ)/G(Ω0 ) in Chapter 3 associated with the family of sets X (Ω0 ) = {F (Ω0 ) : ∀F ∈ F(Θ)}

(3.1)

are nonlinear and nonconvex and the elements of F(Θ)/G(Ω0 ) are equivalence classes of transformations. Deﬁning a diﬀerential with respect to such spaces is similar to deﬁning a diﬀerential on an inﬁnite-dimensional manifold. Fortunately, the tangent space is invariant and equal to Θ in every point of F(Θ). This considerably simpliﬁes the analysis. Deﬁnition 3.1. Given a nonempty subset D of RN , consider the set P(D) = {Ω : Ω ⊂ D} of subsets of D. The set D will be referred to as the underlying holdall or universe. A shape functional is a map J :A→E (3.2) from some admissible family A of sets in P(D) into a topological space E. For instance, A could be the set X (Ω). D can represent some physical or mechanical constraint, a submanifold of RN , or some mathematical constraint. In most cases, it can be chosen large and as smooth as necessary for the analysis. In the unconstrained case, D is equal to RN .

3.2

Gateaux and Hadamard Semiderivatives

Consider the solution (ﬂow) of the diﬀerential equation dx (t, X) = V (t, x(t, X)), t ≥ 0, dt

x(t, X) = X,

(3.3)

3. Semiderivatives via Transformations Generated by Velocities

171

V x(t)

X Ωt = Tt (Ω)

Ω

Figure 4.1. Transport of Ω by the velocity ﬁeld V . def

for velocity ﬁelds V (t)(x) = V (t, x) (cf. Figure 4.1). It generates the family of transformations {Tt : RN → RN : t ≥ 0} deﬁned as follows: def

X → Tt (X) = x(t; X) : RN → RN .

(3.4)

Given an initial set Ω ⊂ RN , associate with each t > 0 the new set def

Ωt = Tt (Ω) = {Tt (X) : ∀X ∈ Ω} .

(3.5)

This perturbation of the initial set is the basis of the velocity (speed ) method . The choice of the terminology “velocity” to describe this method is accurate but may become ambiguous in problems where the variables involved are themselves “physical velocities”: this situation is commonly encountered in continuum mechanics. In such cases it may be useful to distinguish between the “artiﬁcial velocity” and the “physical velocity.” This is at the origin of the terminology speed method, which has often been used in the literature. The latter terminology is convenient, but is not as accurate as the velocity method. We shall keep both terminologies and use the one that is most suitable in the context of the problem at hand. It is important to work with the weakest notion of semiderivative that preserves the basic elements of the classical diﬀerential calculus such as the chain rule for the diﬀerentiation of the composition of functions. It should be able to handle the norm of a function or functions deﬁned as an upper or lower envelope of a family of diﬀerentiable functions. Since the spaces of geometries are generally nonlinear and nonconvex, we also need a notion of semiderivative that yields diﬀerentials in the tangent space as for manifolds. We are looking for a very general notion of semidiﬀerential but not more! In that context the most suitable candidate is the Hadamard semiderivative3 that will be introduced later. Important nondiﬀerentiable functions are Hadamard semidiﬀerentiable and the chain rule is veriﬁed. 3 In his 1937 paper, M. Fre ´chet [1] extends to function spaces the Hadamard derivative and promotes it as more general than his own Fr´echet derivative since it does not require the space to have a metric in inﬁnite-dimensional spaces. In the same paper, he also introduces a relaxation of the Hadamard derivative that is almost a semiderivative. A precise deﬁnition of the Hadamard semiderivative explicitly appears in J.-P. Penot [1] in 1978 and in A. Bastiani [1] in 1964 as a well-known notion.

172

Chapter 4. Transformations Generated by Velocities

We proceed step by step. A ﬁrst candidate for the shape semiderivative of J at Ω in the direction V would be def

dJ(Ω; V ) = lim

t0

J(Ωt (V )) − J(Ω) t

(when the limit exists).

(3.6)

This very weak notion depends on the history of V for t > 0 and will be too weak for most of our purposes. For instance, its composition with another function would not verify the chain rule. To make it compatible with the usual notion of semiderivative in a direction θ of the tangent space Θ to F(Θ) in F (Ω), it must be strengthened as follows. Deﬁnition 3.2. Given θ ∈ Θ def

dJ(Ω; θ) =

J(Ωt (V )) − J(Ω) , t ([0,τ ];Θ)

lim

0

V ∈C V (0)=θ t0

(3.7)

where the limit depends only on θ and is independent of the choice of V for t > 0. Equivalently, the limit dJ(Ω; θ) is independent of the way we approach Ω for a ﬁxed direction θ. This is precisely the adaptation of the Hadamard semiderivative.4 Deﬁnitions based on a perturbation of the identity def

Tt (X) = x(t) = X + t θ(X),

t ≥ 0,

(3.8)

can be recast in the velocity framework by choosing the velocity ﬁeld V (t) = θ◦Tt−1 , that is, ∀x ∈ RN , ∀t ≥ 0,

V (t, x) = θ(Tt−1 (x)) = θ ◦ (I + tθ)−1 (x), def

(3.9)

which requires the existence of the inverse for suﬃciently small t. With that choice for each X, x(t) = Tt (X) is the solution of the diﬀerential equation dx (t) = V t, x(t) , dt

x(0) = X.

(3.10)

Under appropriate continuity and diﬀerentiability assumptions, ∀x ∈ RN , ∀x ∈ RN ,

def

V (0)(x) = V (0, x) = θ(x), def ∂V ˙ (t, x) V (0)(x) = = − [Dθ(x)] θ(x), ∂t t=0

(3.11) (3.12)

where Dθ(x) is the Jacobian matrix of θ at the point x. In compact notation, V (0) = θ

and V˙ (0) = − [Dθ] θ.

(3.13)

This last computation shows that at time 0, the points of the domain Ω are simultaneously aﬀected by the velocity ﬁeld V (0) = θ and the acceleration ﬁeld 4 Cf. footnote 2 in section 2 of Chapter 9 for the deﬁnitions and properties and the comparison of the various notions of derivatives and semiderivatives in Banach spaces.

3. Semiderivatives via Transformations Generated by Velocities

173

V˙ (0) = −[Dθ]θ. The same result will be obtained without acceleration term by replacing I + tθ by the transformation Tt generated by the solutions of the diﬀerential equation dx (t) = θ(x(t)), dt

x(0) = X,

def

Tt (X) = x(t),

for which V (0) = θ and V˙ (0) = 0. Under suitable assumptions the two methods will produce the same ﬁrst-order semiderivative. The deﬁnition of a shape semiderivative def

dJ(Ω; θ) = lim

t0

J([I + tθ]Ω) − J(Ω) t

(3.14)

by perturbation of the identity is similar to a Gateaux semiderivative that will not verify the chain rule. Moreover, second-order semiderivatives will diﬀer by an acceleration term that will appear in the expression obtained by the method of perturbation of the identity.

3.3

Examples of Families of Transformations of Domains

In section 3.2 we have formally introduced a notion of semiderivative of a shape functional via transformations generated by ﬂows of velocity ﬁelds and perturbations of the identity operator as special cases of the velocity method. Before proceeding with a more abstract treatment, we present several examples of deﬁnitions of shape semiderivatives that can be found in the literature. We consider special classes of domains (C ∞ , C k , Lipschitzian), Cartesian graphs, polar coordinates, and level sets that provide classical examples of parametrized and/or constrained deformations. In each case we construct the associated underlying family (not necessarily unique) of transformations {Tt : 0 ≤ t ≤ τ }. 3.3.1

C ∞ -Domains

Let Ω be an open domain of class C ∞ in RN . Recall from section 5 of Chapter 2 that in any point x ∈ Γ the unit outward normal is given by def

∀y ∈ Γx = U(x) ∩ Γ,

n(y) =

mx (y) , |mx (y)|

(3.15)

where mx (y) = − ∗(Dhx )−1 (h−1 x (y)) eN in U(x) ⇒ n=−

∗

(3.16)

−1

(Dhx ) eN ◦ h−1 x . | ∗(Dhx )−1 eN |

(3.17)

When Γ is compact, it is possible to ﬁnd a ﬁnite sequence of points {xj : 1 ≤ j ≤ J} in Γ such that def

Γ⊂U =

J * j=1

Uj ,

def

Uj = U(xj ).

174

Chapter 4. Transformations Generated by Velocities

As in the deﬁnition of the boundary integral, associate with {Uj } a partition of unity {rj }: rj ∈ D(Uj ),

0 ≤ rj ≤ 1,

J

rj = 1 in U0

j=1

for some neighborhood U0 of Γ such that Γ ⊂ U0 ⊂ U0 ⊂ U. For the C ∞ -domain Ω the normal satisﬁes n=

J

rj n =

j=1

since

∀j,

J j=1

rj

mj ∈ C ∞ (Γ; RN ) |mj |

mj ◦ hj ∈ C ∞ (B; RN ). |mj |

Given any ρ ∈ C ∞ (Γ) and t ≥ 0, consider the following perturbation Γt of Γ along the normal ﬁeld n: def

Γt = {x ∈ RN : x = X + tρ(X)n(X), ∀X ∈ Γ}.

(3.18)

We claim that for τ suﬃciently small and all t, 0 ≤ t ≤ τ , the set Γt is the boundary of a C ∞ -domain Ωt by constructing a transformation Tt of RN which maps Ω onto Ωt and Γ onto Γt . First construct an extension N ∈ D(RN , RN ) of the normal ﬁeld n on Γ. Deﬁne J def rj mj ∈ D(U, RN ). (3.19) m = j=1

By construction

m ∈ C ∞ (Γ; RN ),

(3.20)

m = 0 on Γ, and there exists a neighborhood U1 of Γ contained in U0 where m = 0 since m is at least C 1 . Now construct a function r0 in D(U1 ), 0 ≤ r0 (x) ≤ 1, and a neighborhood V of Γ such that r0 = 1 in V and Γ ⊂ V ⊂ V ⊂ U1 . Deﬁne the vector ﬁeld ∀x ∈ RN ,

def

N (x) = r0 (x)

m(x) . |m(x)|

(3.21)

Hence N belongs to D(RN , RN ) since supp N ⊂ V is compact. Moreover m N= in V ⇒ N (x) = n(x) on Γ = Γ ∩ V. |m| For each j, ρ ◦ hj ∈ C ∞ (B0 ) and the extensions ρhj (ζ) = ρhj (ζ , ζN ) = ρ(hj (ζ , 0)), def

∀ζ ∈ B,

def

ρ˜j = ρhj ◦ gj ,

3. Semiderivatives via Transformations Generated by Velocities

175

belong, respectively, to C ∞ (B) and C ∞ (Uj ). Then def

ρ˜ =

J

rj ρ˜j ∈ D(RN )

j=1

is an extension of ρ from Γ to RN with compact support since supp rj ⊂ Uj ⊂ U. Deﬁne the following transformation of RN : def

Tt (X) = X + t ρ˜(X)N (X),

t ≥ 0.

(3.22)

By construction, ρ˜ N is uniformly Lipschitzian in RN and, by Theorem 4.2 in section 4.2, there exists 0 < τ such that Tt is bijective and bicontinuous from RN onto itself. As a result from J. Dugundji [1] for 0 ≤ t ≤ τ Ωt = Tt (Ω) = [I + t˜ ρ N ](Ω), ∂Ωt = Tt (∂Ω) = [I + t˜ ρ N ](∂Ω) = [I + tρ n](∂Ω) = Γt . Since the domain Ωt is speciﬁed by its boundary Γt , it depends only on ρ and not on its extension ρ˜. The special transformation Tt introduced here is of class C ∞ , that is, Tt ∈ C ∞ (RN , RN ), and 1t [Tt − I] is proportional to the normal ﬁeld n on Γ, but it is not proportional to the normal nt on Γt for t > 0. In other words, at t = 0 the deformation is along n, but at t > 0 the deformation is generally not along nt . If J(Ω) is a real-valued shape functional deﬁned on C ∞ -domains in D, the semiderivative (if it exists) is deﬁned as follows: for all ρ ∈ C ∞ (Γ) J (I + t˜ ρN )Ω − J(Ω) def dn J(Ω; ρ˜) = lim . (3.23) t0 t It turns out that this limit depends only on ρ and not on its extension ρ˜. 3.3.2

C k -Domains

When Ω is a domain of class C k with boundary Γ, the normal ﬁeld n belongs to C k−1 (Γ, RN ). Therefore, choosing deformations along the normal would yield transformations {Tt } mapping C k -domains Ω onto C k−1 -domains Ωt = Tt (Ω). The obvious way to deal with C k -domains is to relax the constraint that the perturbation ρ˜N be carried by the normal. Choose vector ﬁelds θ in D k (RN , RN ) and consider the family of transformations Tt = I + tθ,

Ωt = Tt (Ω),

t ≥ 0.

(3.24)

This is a generalization of the family of transformations (3.22) in section 3.3.1 from ρ˜N to θ. For k ≥ 1, θ is again uniformly Lipschitzian in RN and, by Theorem 4.2 in section 4.2, there exists τ > 0 such that Tt is bijective and bicontinuous from RN onto itself. Thus for 0 ≤ t ≤ τ , Ωt = Tt (Ω) = [I + tθ](Ω)

and ∂Ωt = Tt (∂Ω) = [I + tθ](∂Ω).

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Chapter 4. Transformations Generated by Velocities

A more restrictive approach to get around the lack of suﬃcient smoothness of the normal n to Γ would be to introduce a transverse ﬁeld p on Γ such that p ∈ C k (Γ; RN ),

∀x ∈ Γ,

p(x) · n(x) > 0.

Given p and ρ ∈ C k (Γ) deﬁne for t ≥ 0 def

Γt = x ∈ RN : x = X + tρ(X)p(X), ∀X ∈ Γ .

(3.25)

(3.26)

Choosing C k -extensions ρ˜ and p˜ of ρ and p we can go back to the case where θ = ρ˜p˜ ∈ Dk (RN , RN ).

(3.27)

For any θ ∈ Dk (RN , RN ) the semiderivative is deﬁned as J (I + tθ)(Ω) − J(Ω) def dk J(Ω; θ) = lim . t0 t 3.3.3

(3.28)

Cartesian Graphs

In many applications it is convenient to work with domains Ω which are the hypograph of some positive function γ in Cartesian coordinates. Such domains are typically of the form def (3.29) Ω = (x , xN ) ∈ RN : x ∈ U ⊂ RN −1 and 0 < xN < γ(x ) , where U is a connected open set in RN −1 and γ ∈ C 0 (U ; R+ ) is a positive function. Many free boundary and contact problems are formulated over such domains. Usually the domains Ω (and hence the functions γ) will be constrained. The function γ can be speciﬁed on U \U or not. In some examples the derivative of γ could also be speciﬁed, that is, ∂γ/∂ν = g on ∂U , where U is smooth and ν is the outward unit normal ﬁeld along ∂U . When γ (resp., ∂γ/∂ν) is speciﬁed along ∂U , the directions of deformation µ are chosen in C 0 (U ; R+ ) such that µ = 0 (resp., ∂µ/∂ν = 0) on ∂U.

(3.30)

For small t ≥ 0 and each such µ, deﬁne the perturbed domain

Ωt = (x , xN ) ∈ RN : 0 < xN < γ(x ) + tµ(x ) and the family of transformations (x , xN ) → Tt (x , xN ) =

x , xN +

(3.31)

xN tµ(x ) γ(x )

: Ω → Ωt ,

(3.32)

which can be extended to a neighborhood D of Ω containing the perturbed domains Ωt , 0 ≤ t ≤ t1 , for some small t1 > 0. In general, D will be such that D = U × [0, L] for some L > 0. This construction is also appropriate for domains that are Lipschitzian or of class C k , 1 ≤ k ≤ ∞, that is, when γ is a Lipschitzian or a C k -function. Again the

3. Semiderivatives via Transformations Generated by Velocities transformation Tt is equal to I + tθ, where µ(x ) def . θ(x , xN ) = 0, xN γ(x )

177

(3.33)

But of course this θ is not the only choice for that Ωt = Tt (Ω). For instance, let λ : R+ → R+ be any smooth increasing function such that λ(0) = 0 and λ(1) = 1. Then we could consider the transformations xN def ) . (3.34) µ(x Tt (x , xN ) = x , xN + tλ γ(x ) This example illustrates the general principle that the transformation Tt of D such that Ωt = Tt (Ω) is not unique. This means that, at least for smooth domains, only the trace Tt |Γt on Γt is important, while the displacement of the inner points does not contribute to the deﬁnition of Ωt . Nevertheless this statement is to be interpreted with caution. Here we implicitly assume that the objective function J(Ω) and the associated constraints are only a function of the shape of Ω. However, in some problems involving singularities at inner points of Ω (e.g., when the solution y(Ω) of the state equation has a singularity or when some constraints on the domain are active), the situation might require a ﬁner analysis. One such example is the internal displacement of the interior nodes of a triangularization τh when the solution y(Ω) of a partial diﬀerential equation is approximated by a piecewise polynomial solution over the triangularized domain Ωh in the ﬁnite element method. Such a displacement does not change the shape of Ωh , but it does change the solution yh of the problem. When the displacement of the interior nodes is a priori parametrized by the boundary nodes the solution yh will depend only on the position of the boundary nodes, but the interior nodes will contribute through the choice of the speciﬁed parametrization. 3.3.4

Polar Coordinates and Star-Shaped Domains

In some examples domains are star-shaped with respect to a point. Since a domain can always be translated, there is no loss of generality in assuming that this point is the origin. Then such domains Ω can be parametrized as follows: def

(3.35) Ω = x ∈ RN : x = ρζ, ζ ∈ SN −1 , 0 ≤ ρ < f (ζ) , where SN −1 is the unit sphere in RN , def

SN −1 = x ∈ RN : |x| = 1 ,

(3.36)

and f : SN −1 → R+ is a positive continuous mapping from SN −1 such that def

m = min {f (ζ) : ζ ∈ SN −1 } > 0.

(3.37)

Given any g ∈ C 0 (SN −1 ) and a suﬃciently small t ≥ 0 the perturbed domains are deﬁned as def

Ωt = x ∈ RN : x = ρζ, ζ ∈ SN −1 , 0 ≤ ρ < f (ζ) + tg(ζ) . (3.38)

178

Chapter 4. Transformations Generated by Velocities

For example, choose t, 0 ≤ t ≤ t1 , for some m t1 = >0 gC 0 (SN −1 )

(3.39)

and deﬁne the transformation Tt as Tt (X) = .0,

/

Tt (X) = ρ +

ρ t f (ζ) g(ζ)

if X = 0, ζ, if X = ρζ = 0.

(3.40)

As in the previous example, Tt is not unique and for any continuous increasing function λ : R+ → R+ such that λ(0) = 0 and λ(t) = 1 the transformation Tt (X) = .0,

Tt (X) = ρ + tλ

ρ f (ζ)

/

if X = 0,

g(ζ) ζ, if X = ρζ = 0,

(3.41)

yields the same domain Ωt . 3.3.5

Level Sets

In sections 3.3.1 to 3.3.4, the perturbed domain Ωt always appears in the form Ωt = Tt (Ω), where Tt is a bijective transformation of RN and Tt is of the form I +tθ. In some free boundary problems (e.g., plasma physics, propagation of fronts) the free boundary Γ is a level curve of a smooth function u deﬁned over an open domain D. Assume that D is bounded open with smooth boundary ∂D. Let u ∈ C 2 (D) be a positive function on D such that u ≥ 0 in D,

u = 0 on ∂D,

∃ a unique xu ∈ D such that ∀x ∈ D − {xu } , |∇u(x)| > 0.

If m = max u(x) : x ∈ D , then for each t in [0, m[ the level set Γt = u−1 (t) 2

(3.42)

(3.43)

N

is a C -submanifold of R in D, which is the boundary of the open set Ωt = {x ∈ D : u(x) > t} .

(3.44)

By deﬁnition Ω0 = D, for all t1 > t2 , Ωt1 ⊂ Ωt2 , and the domains Ωt converge in the Hausdorﬀ complementary topology5 to the point xu . The outward unit normal ﬁeld on Γt is given by x ∈ Γt ,

nt (x) = −|∇u(x)|−1 ∇u(x).

(3.45)

This suggests introducing the velocity ﬁeld ∀x ∈ D − {xu } ,

V (x) = |∇u(x)|−2 ∇u(x), def

(3.46)

which is continuous everywhere but at x = xu . If V were continuous everywhere, then for each X, the path x(t; X) generated by the diﬀerential equation dx (t) = V x(t) , x(0) = X (3.47) dt 5 Cf.

section 2 of Chapter 6.

3. Semiderivatives via Transformations Generated by Velocities would have the property that

u x(t) = u(X) + t

179

(3.48)

since formally

dx d u x(t) = ∇u x(t) · (t) = 1. (3.49) dt dt This means that the map X → Tt (X) = x(t; X) constructed from (3.47) would map the level set

(3.50) Γ0 = X ∈ D : u(X) = 0

onto the level set

Γt = x ∈ D : u(x) = t

(3.51)

and eventually Ω0 onto Ωt . Unfortunately, it is easy to see that this last property fails on the function u(x) = 1 − x2 deﬁned on the unit disk. To get around this diﬃculty, introduce for some arbitrarily small ε, 0 < ε < m/2, an inﬁnitely diﬀerentiable function ρε : RN → [0, 1] such that ρε (x) =

0, 1,

if |∇u(x)| < ε, if |∇u(x)| > 2ε,

(3.52)

x ∈ D.

(3.53)

and the velocity Vε (x) = ρε (x)V (x), As above, deﬁne the transformation X → Ttε (X) = x(t; X),

(3.54)

where x(t; X) is the solution of the diﬀerential equation dx (t) = Vε x(t) , t ≥ 0, x(0) = X ∈ D. (3.55) dt For 0 ≤ t < m − 2ε, Tt maps Γ0 onto Γt ; for 0 ≤ s < m − ε such that s + t < m − ε, Tt maps Γs onto Γt+s . However, for s > m − ε, Tt is the identity operator. As a result, for 0 ≤ t < m − 2ε, Tt (Ω0 ) = Ωt

and Tt (Γ0 ) = Γt .

(3.56)

Of course ε > 0 is arbitrary and we can make the construction for t’s arbitrary close to m. This is an example that can be handled by the velocity (speed) method and not by a perturbation of the identity. Here the domains Ωt are implicitly constrained to stay within the larger domain D. We shall see in section 5 how to introduce and characterize such a constraint. Another example of description by level sets is provided by the oriented distance function bΩ for some open domain Ω of class C 2 with compact boundary Γ (cf. Chapter 7). We shall see that there exists h > 0 and a neighborhood

Uh (Γ) = x ∈ RN : |bΩ (x)| < h such that bΩ ∈ C 2 (Uh (Γ)). Then for 0 ≤ t < h the ﬂow corresponding to the velocity ﬁeld V = ∇bΩ maps Ω and its boundary Γ onto

Tt (Ω) = Ωt = x ∈ RN : bΩ (x)| < t , Tt (Γ) = Γt = x ∈ RN : bΩ (x)| = t .

180

4

Chapter 4. Transformations Generated by Velocities

Unconstrained Families of Domains

´sio [12, Here we study equivalences between the velocity method (cf. J.-P. Zole 8]) and methods using a family of transformations. In section 4.1, we give some general conditions to construct a family of transformations of RN from a velocity ﬁeld. Conversely, we show how to construct a velocity ﬁeld from a family of transformations of RN . In section 4.2, this construction is applied to Lipschitzian perturbations of the identity. In section 4.3, the various equivalences of section 4.1 are specialized to velocities in C0k+1 (RN , RN ), C k+1 (RN , RN ), and C k,1 (RN , RN ), k ≥ 0.

4.1

Equivalence between Velocities and Transformations

Let the real number τ > 0 and the map V : [0, τ ] × RN → RN be given. If t is interpreted as an artiﬁcial time, the map V can be viewed as the (time-dependent) velocity ﬁeld {V (t) : 0 ≤ t ≤ τ } deﬁned on RN : def

x → V (t)(x) = V (t, x) : RN → RN . Assume that there exists τ = τ (V ) > 0 such that ∀x ∈ RN , V (·, x) ∈ C [0, τ ]; RN , (V) ∃c > 0, ∀x, y ∈ RN , V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|,

(4.1)

(4.2)

where V (·, x) is the function t → V (t, x). Note that V is continuous on [0, τ ] × RN . Hence it is uniformly continuous on [0, τ ] × D for any bounded open subset D of RN and V (·) ∈ C([0, τ ]; C(D; RN )).

(4.3)

Associate with V the solution x(t; V ) of the vector ordinary diﬀerential equation dx (t) = V t, x(t) , dt

t ∈ [0, τ ],

x(0) = X ∈ RN

(4.4)

and deﬁne the transformations def

X → Tt (V )(X) = xV (t; X) : RN → RN

(4.5)

and the maps (whenever the inverse of Tt exists) def

(t, X) → TV (t, X) = Tt (V )(X) : [0, τ ] × RN → RN ,

(4.6)

(t, x) → TV−1 (t, x) = Tt−1 (V )(x) : [0, τ ] × RN → RN .

(4.7)

def

Notation 4.1. In what follows we shall drop the V in TV (t, X), TV−1 (t, x), and Tt (V ) whenever no confusion arises.

4. Unconstrained Families of Domains

181

Theorem 4.1. (i) Under assumptions (V) the map T speciﬁed by (4.4)–(4.6) has the following properties: ∀X ∈ RN , T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1) ∀X, Y ∈ RN , T (·, Y ) − T (·, X)C 1 ([0,τ ];RN ) ≤ c|Y − X|, (T2) (T3)

∀t ∈ [0, τ ], X → Tt (X) = T (t, X) : RN → RN is bijective, ∀x ∈ RN , T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0, ∀x, y ∈ RN ,

(4.8)

T −1 (·, y) − T −1 (·, x)C([0,τ ];RN ) ≤ c|y − x|.

(ii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1) to (T3), the map def

(t, x) → V (t, x) =

∂T t, Tt−1 (x) : [0, τ ] × RN → RN ∂t

(4.9)

satisﬁes conditions (V), where Tt−1 is the inverse of X → Tt (X) = T (t, X). If, in addition, T (0, ·) = I, then T (·, X) is the solution of (4.4) for that V . (iii) Given a real number τ > 0 and a map T : [0, τ ] × RN → RN satisfying assumptions (T1) and (T2) and T (0, ·) = I, then there exists τ > 0 such that the conclusions of part (ii) hold on [0, τ ]. A more general version of this theorem for constrained domains (Theorem 5.1) will be given and proved in section 5.1. Proof. (i) Conditions (T1) follow by standard arguments. Condition (T2). Associate with X in RN the function y(s) = Tt−s (X), Then

dy (s) = −V (t − s, y(s)), ds

0 ≤ s ≤ t.

0 ≤ s ≤ t,

y(0) = Tt (X).

(4.10)

y(0) = x ∈ RN

(4.11)

For each x ∈ RN , the diﬀerential equation dy (s) = −V (t − s, y(s)), ds

0 ≤ s ≤ t,

has a unique solution in C 1 ([0, t]; RN ). The solutions of (4.11) deﬁne the map def

x → St (x) = y(t) : RN → RN such that ∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ RN ,

|St (y) − St (x)| ≤ c|y − x|.

(4.12)

182

Chapter 4. Transformations Generated by Velocities

In view of (4.10) and (4.11) St (Tt (X)) = y(t) = Tt−t (X) = X

⇒ St ◦ Tt = I on RN .

To obtain the other identity, consider the function z(r) = y(t − r; x), where y(·, x) is the solution of (4.11) for some arbitrary x in RN . By deﬁnition dz (r) = V (r, z(r)), dr

z(0) = y(t, x),

and necessarily x = y(0; x) = z(t) = Tt (y(t; x)) = Tt (St (x)) ⇒ Tt ◦ St = I on RN

⇒ St = Tt−1 : RN → RN .

Conditions (T3). The uniform Lipschitz continuity in (T3) follows from (4.12), and we need only show that ∀x ∈ RN ,

T −1 (·, x) ∈ C([0, τ ]; RN ).

Given t in [0, τ ] pick an arbitrary sequence {tn }, tn → t. Then for each x ∈ RN there exists X ∈ RN such that Tt (X) = x and Ttn (X) → Tt (X) = x from the ﬁrst condition (T1). But (x) − Tt−1 (x) = Tt−1 (Tt (X)) − Tt−1 (Tt (X)) Tt−1 n n (Tt (X)) − Tt−1 (Ttn (X)) . = Tt−1 n n By the uniform Lipschitz continuity of Tt−1 |Tt−1 (x) − Tt−1 (x)| = |Tt−1 (Tt (X)) − Tt−1 (Ttn (X)) | ≤ c|Tt (X) − Ttn (X)|, n n n and the last term converges to zero as tn goes to t. (ii) The ﬁrst part of conditions (V) is satisﬁed since, for each x ∈ RN and t, s in [0, τ ], |V (t, x) − V (s, x)| ∂T ∂T ∂T ∂T (t, Tt−1 (x)) − (t, Ts−1 (x)) + (t, Ts−1 (x)) − (s, Ts−1 (x)) ≤ ∂t ∂t ∂t ∂t ∂T ∂T −1 −1 −1 −1 (t, Ts (x)) − (s, Ts (x)) . ≤ c|Tt (x) − Ts (x)| + ∂t ∂t

4. Unconstrained Families of Domains

183

Thus from (T3) and (T1) t → V (t, x) is continuous at s = t, and hence for all x in RN , V (., x) ∈ C([0, τ ]; RN ). The Lipschitzian property follows directly from the Lipschitzian properties (T1) and (T3): for all x and y in RN , ∂T ∂T −1 −1 |V (t, y) − V (t, x)| = (t, Tt (y)) − (t, Tt (x)) ∂t ∂t −1 −1 ≤ c Tt (y) − Tt (x) ≤ cc |y − x|. This proves that V satisﬁes condition (V). (iii) From (T1) and (T2) for f (t) = Tt − I and t ≥ s, t ∂T ∂T (r, Tr (y)) − (r, Tr (x)) dr, f (t)(y) − f (t)(x) = ∂t ∂t 0 t c |Tr (y) − Tr (x)| dr ≤ c2 t |y − x|. |f (t)(y) − f (t)(x)| ≤ 0

For τ = min{τ, 1/(2c )} and 0 ≤ t ≤ τ , c(f (t)) ≤ 1/2, 2

g(t) = Tt−1 − I = (I − Tt ) ◦ Tt−1 = −f (t) ◦ [I + g(t)], (1 − c(f (t))) c(g(t)) ≤ c(f (t))

⇒ c(g(t)) ≤ 1 and c(Tt−1 ) ≤ 2,

and the second condition (T3) is satisﬁed on [0, τ ]. The ﬁrst one follows by the same argument as in part (i). Therefore the conclusions of part (ii) are true on [0, τ ]. This equivalence theorem says that we can start either from a family of velocity ﬁelds {V (t)} on RN or a family of transformations {Tt } of RN provided that the map V , V (t, x) = V (t)(x), satisﬁes (V) or the map T , T (t, X) = Tt (X), satisﬁes (T1) to (T3). Starting from V , the family of homeomorphisms {Tt (V )} generates the family def

Ωt = Tt (V )(Ω) = {Tt (V )(X) : X ∈ Ω}

(4.13)

of perturbations of the initial domain Ω. Interior (resp., boundary) points of Ω are mapped onto interior (resp., boundary) points of Ωt . This is the basis of the velocity method which will be used to deﬁne shape derivatives.

4.2

Perturbations of the Identity

In examples it is usually possible to show that the transformation T satisﬁes assumptions (T1) to (T3) and construct the corresponding velocity ﬁeld V deﬁned in (4.9). For instance, consider perturbations of the identity to the ﬁrst (A = 0) or second order: for t ≥ 0 and X ∈ RN , def

Tt (X) = X + tU (X) +

t2 A(X), 2

(4.14)

where U and A are transformations of RN . It turns out that for Lipschitzian transformations U and A, assumptions (T1) to (T3) are satisﬁed in some interval [0, τ ].

184

Chapter 4. Transformations Generated by Velocities

Theorem 4.2. Let U and A be two uniform Lipschitzian transformations of RN : there exists c > 0 such that for all X, Y ∈ RN , |U (Y ) − U (X)| ≤ c|Y − X|

and

|A(Y ) − A(X)| ≤ c|Y − X|.

There exists τ > 0 such that the map T given by (4.14) satisﬁes conditions (T1) to (T3) on [0, τ ]. The associated velocity V given by (t, x) → V (t, x) = U Tt−1 (x) + tA Tt−1 (x) : [0, τ ] × RN → RN (4.15) satisﬁes conditions (V) on [0, τ ]. Remark 4.1. Observe that from (4.14) and (4.15) V (0) = U,

∂V V˙ (0)(x) = (t, x) t=0 = A − [DU ]U, ∂t

(4.16)

where DU is the Jacobian matrix of U . The term V˙ (0) is an acceleration at t = 0 which will always be present even when A = 0, but it can be eliminated by choosing A = [DU ]U . Proof. (i) By the deﬁnition of T in (4.14), t → T (t, X) and t → ∂T (t, X) = U (X) + ∂t tA(X) are continuous on [0, ∞[. Moreover, for all X and Y , t2 |T (t, Y ) − T (t, X)| ≤ 1 + tc + c |Y − X| 2 and

∂T ∂T ≤ [c + tc] |Y − X|. (t, Y ) − (t, X) ∂t ∂t

Thus conditions (T1) are satisﬁed for any ﬁnite τ > 0. To check condition (T2), def

consider for any Y ∈ RN the mapping h(X) = Y −[Tt (X)−X]. For any X1 and X2 |h(X2 ) − h(X1 )| ≤ t|U (X2 ) − U (X1 )| +

t2 |A(X2 ) − A(X1 )| 2

t2 ≤ t c |X2 − X1 | + c |X2 − X1 | = t c [1 + t/2] |X2 − X1 |. 2

(4.17)

For τ = min{1, 1/(4c)}, and any t, 0 ≤ t ≤ τ , t c [1 + t/2] < 1/2 and h is a contraction. So for all 0 ≤ t ≤ τ and Y ∈ RN , there exists a unique X ∈ RN such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y and Tt is bijective. Therefore (T2) is satisﬁed in [0, τ ]. The last part of the proof is the uniform Lipschitzian property of Tt−1 . In view of (4.17), for all t, 0 ≤ t ≤ τ ,

4. Unconstrained Families of Domains

185

t c [1 + t/2] < 1/2 and |Tt (X2 ) − X2 − (Tt (X1 ) − X1 )| = |h(X2 ) − h(X1 )| ≤

1 |X2 − X1 | 2

1 |X2 − X1 | 2 ⇒ |X2 − X1 | ≤ 2 |Tt (X2 ) − Tt (X1 )|.

⇒ |X2 − X1 | − |Tt (X2 ) − Tt (X1 )| ≤

In view of condition (T2) for all x and y, |Tt−1 (y) − Tt−1 (x)| ≤ 2 |Tt (Tt−1 (y)) − Tt (Tt−1 (x))| = 2 |y − x|.

(4.18)

To complete our argument we prove the continuity with respect to t for each x. Let X = Tt−1 (x). For any s in [0, τ ] Ts−1 (x) − Tt−1 (x) = Ts−1 (Tt (X)) − Tt−1 (Tt (X)) = Ts−1 (Tt (X)) − Ts−1 (Ts (X)), and in view of (4.18) |Ts−1 (x) − Tt−1 (x)| ≤ 2|Tt (X) − Ts (X)|. The continuity of Ts−1 (x) at s = t now follows from the continuity of Ts (X) at s = t. Thus conditions (T3) are satisﬁed.

4.3

Equivalence for Special Families of Velocities

In this section we specialize Theorem 4.1 to velocities in C k,1 (RN , RN ), C0k+1 (RN , RN ), and C k+1 (RN , RN ), k ≥ 0. The following notation will be convenient: def

f (t) = Tt − I,

f (t) =

dTt , dt

g(t) = Tt−1 − I, def

whenever Tt−1 exists and the identities g(t) = −f (t) ◦ Tt−1 = −f (t) ◦ [I + g(t)], dTt ◦ Tt−1 = f (t) ◦ Tt−1 = f (t) ◦ [I + g(t)]. V (t) = dt Recall also for a function F : RN → RN the notation def

|F (y) − F (x)| and ∀k ≥ 1, |y − x| y=x

def

c(F ) = sup

ck (F ) =

c(∂ α F ).

|α|=k

Theorem 4.3. Let k ≥ 0 be an integer. (i) Given τ > 0 and a velocity ﬁeld V such that V ∈ C([0, τ ]; C k (RN , RN ))

and

ck (V (t)) ≤ c

(4.19)

186

Chapter 4. Transformations Generated by Velocities for some constant c > 0 independent of t, the map T given by (4.4)–(4.6) satisﬁes conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C k (RN , RN )) ∩ C([0, τ ]; C k,1 (RN , RN )),

ck (f (t)) ≤ c, (4.20)

for some constant c > 0 independent of t. Moreover, conditions (T3) are satisﬁed and there exists τ > 0 such that g ∈ C([0, τ ]; C k (RN , RN )),

ck (g(t)) ≤ ct,

(4.21)

for some constant c independent of t. (ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.20) and T (0, ·) = I, there exists τ > 0 such that the velocity ﬁeld V (t) = f (t) ◦ Tt−1 satisﬁes conditions (V) and (4.19) in [0, τ ]. Proof. We prove the theorem for k = 0. The general case is obtained by induction over k. (i) By assumption on V , the conditions (V) given by (4.2) are satisﬁed and by Theorem 4.1 the corresponding family T satisﬁes conditions (T1) to (T3). Conditions (4.20) on f . For any x and s ≤ t t Tt (x) − Ts (x) = V (r) ◦ Tr (x) dr, s t |Tt (x) − Ts (x)| ≤ c|Tr (x) − Ts (x)| + |V (r) ◦ Ts (x)| dr, s t c|f (r)(x) − f (s)(x)| + V (r)C dr. |f (t)(x) − f (s)(x)| ≤ s

By assumption on V and Gronwall’s inequality, ∀t, s ∈ [0, τ ],

f (t) − f (s)C ≤ c |t − s|

for another constant c independent of t. Moreover, |(f (t) − f (s))(y) − (f (t) − f (s))(x)| = |(Tt − Ts )(y) − (Tt − Ts )(x)| t ≤ |V (r) ◦ Tr (y) − V (r) ◦ Tr (x)| dr s t c|(Tr − Ts )(y) − (Tr − Ts )(x)| + c|Ts (y) − Ts (x)| dr ≤ s t ≤ c|(f (r) − f (s))(y) − (f (r) − f (s))(x)| + cc |y − x| dr s

for some other constant c by the second condition (T1). Again by Gronwall’s inequality there exists another constant c such that |(f (t) − f (s))(y) − (f (t) − f (s))(x)| ≤ c|t − s| |y − x| ⇒ c(f (t) − f (s)) ≤ c |t − s| ⇒ f ∈ C([0, τ ]; C 0,1 (RN , RN )) and f (t) − f (s)C0,1 ≤ c |t − s|.

(4.22)

4. Unconstrained Families of Domains

187

Moreover, f (t) = V (t) ◦ Tt and |f (t)(x) − f (s)(x)| ≤ |V (t)(Tt (x)) − V (s)(Tt (x))| + |V (s)(Tt (x)) − V (s)(Ts (x))| ≤ V (t) − V (s)C + c(V (s)) Tt − Ts C ≤ V (t) − V (s)C + c f (t) − f (s)C . Finally, |f (t)(y) − f (t)(x)| ≤ |V (t)(Tt (y)) − V (t)(Tt (x))| ≤ c(V (t)) |Tt (y) − Ts (x)| ≤ c c(Tt ) |y − x| and c(f (t)) ≤ c for some new constant c independent of t. Therefore, f ∈ C 1 ([0, τ ]; C(RN , RN )) and c(f (t)) ≤ c. Conditions (4.21) on g. Since conditions (T1) and (T2) are satisﬁed there exists τ > 0 such that conditions (T3) are satisﬁed by Theorem 4.1 (iii). Moreover, from conditions (4.20) |g(t)(y) − g(t)(x)| ≤ |f (t)(Tt−1 (y)) − f (t)(Tt−1 (x))| ≤ c(f (t))|Tt−1 (y) − Tt−1 (x)| ≤ c(f (t)) (|g(t)(y) − g(t)(x)| + |y − x|) ⇒ (1 − c(f (t))) |g(t)(y) − g(t)(x)| ≤ c(f (t)) |y − x| ≤ ct |y − x|. Choose a new τ = min{τ , 1/(2c)}. Then for 0 ≤ t ≤ τ , c(g(t)) ≤ 2ct. Now g(t) − g(s) = −f (t) ◦ [I + g(t)] + f (s) ◦ [I + g(s)], g(t) − g(s)C ≤f (t) ◦ [I + g(t)] − f (t) ◦ [I + g(s)]C + f (t) ◦ [I + g(s)] − f (s) ◦ [I + g(s)]C ≤c(f (t))g(t) − g(s)C + f (t) − f (s)C ≤ct g(t) − g(s)C + f (t) − f (s)C . For t in [0, τ ], ct ≤ 1/2, and g(t) − g(s)C ≤ 2f (t) − f (s)C ⇒ g ∈ C([0, τ ]; C(RN , RN )) and c(g(t)) ≤ 2ct. The conditions (4.20) on f are satisﬁed for k = 0. For k = 1 we start from the equation

t

DTt − DTs =

DV (r) ◦ Tr DTr dr s

188

Chapter 4. Transformations Generated by Velocities

and use the fact that DTt−1 = [DTt ]−1 ◦ Tt−1 in connection with the identity Dg(t) = −Df (t) ◦ Tt−1 DTt−1 = −(Df (t)[DTt ]−1 ) ◦ Tt−1 . (ii) From conditions (4.20) on f , the transformation T satisﬁes conditions (T1). To check condition (T2) we consider two cases: k ≥ 1 and k = 0. For k ≥ 1 the function t → Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t → det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ > 0 such that Tt is invertible for all t in [0, τ ] and (T2) is satisﬁed in [0, τ ]. In the case k = 0 consider for any Y the map h(X) = Y − f (t)(X). For any X1 and X2 , |h(X2 ) − h(X1 )| ≤ c(f (t)) |X2 − X1 |. But by assumption f ∈ C([0, τ ]; C 0,1 (RN )) and c(f (0)) = 0 since f (0) = 0. Hence there exists τ > 0 such that c(f (t)) ≤ 1/2 for all t in [0, τ ] and h is a contraction. So for all Y in RN there exists a unique X such that Y − [Tt (X) − X] = h(X) = X ⇐⇒ Tt (X) = Y, Tt is bijective, and condition (T2) is satisﬁed in [0, τ ]. By Theorem 4.1 (iii) from (T1) and (T2), there exists another τ > 0 for which conditions (T3) on g and (V) on V (t) = f (t) ◦ Tt−1 are also satisﬁed. Moreover, we have seen in the proof of part (i) that conditions (4.21) on g follow from (T2) and (4.20). Using conditions (4.20) and (4.21), |V (t)(y) − V (t)(x)| ≤ |f (t)(Tt−1 (y)) − f (t)(Tt−1 (x))| ≤ c(f (t)) |Tt−1 (y) − Tt−1 (x)| ≤ c(f (t)) [1 + c(g(t))] |y − x| ≤ c |y − x| and c(V (t)) ≤ c . Also |V (t)(x) − V (s)(x)| = |f (t)(Tt−1 (x)) − f (s)(Ts−1 (x))| ≤ |f (t)(Tt−1 (x)) − f (t)(Ts−1 (x))| + |f (t)(Ts−1 (x)) − f (s)(Ts−1 (x))| ≤ c(f (t)) |Tt−1 (x) − Ts−1 (x)| + f (t) − f (s)C ≤ c g(t) − g(s)C + f (t) − f (s)C . Therefore, since both g and f are continuous, V ∈ C([0, τ ]; C(RN , RN )) and c(V (t)) ≤ c for some constant c independent of t. This proves the result for k = 0. As in part (i), for k = 1 we use the identity DV (t) = Df (t) ◦ Tt−1 DTt−1 = (Df (t)[DTt ]−1 ) ◦ Tt−1 and proceed in the same way. The general case is obtained by induction over k.

4. Unconstrained Families of Domains

189

We now turn to the case of velocities in C0k (RN , RN ). As in Chapter 2, it will be convenient to use the notation C0k for the space C0k (RN , RN ), C k (RN ) for the space C k (RN , RN ), and C k,1 (RN ) for the space C k,1 (RN , RN ). Theorem 4.4. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity ﬁeld V such that V ∈ C([0, τ ]; C0k (RN , RN )),

(4.23)

the map T given by (4.4)–(4.6) satisﬁes conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C0k (RN , RN )).

(4.24)

Moreover, conditions (T3) are satisﬁed and there exists τ > 0 such that g ∈ C([0, τ ]; C0k (RN , RN )).

(4.25)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.24) and T (0, ·) = I, there exists τ > 0 such that the velocity ﬁeld V (t) = f (t) ◦ Tt−1 satisﬁes conditions (V) and (4.23) on [0, τ ]. Proof. As in the proof of Theorem 4.3, we prove only the theorem for k = 1. The general case is obtained by induction on k, the various identities on f , g, f , and V , and the techniques of Lemma 2.4, Theorems 2.11 and 2.12, and Lemmas 2.5 and 2.6 of section 2.5 in Chapter 3. (i) By the embedding C01 (RN , RN ) ⊂ C 1 (RN , RN ) ⊂ C 0,1 (RN , RN ), it follows from (4.23) that V ∈ C([0, τ ]; C 0,1 (RN , RN )) and condition (4.19) of Theorem 4.3 are satisﬁed. Therefore, conditions (4.20) and (4.21) of Theorem 4.3 are also satisﬁed in some interval [0, τ ], τ > 0. Conditions (4.24) on f . It remains to show that f (t) and f (t) belong to the subspace C0 (RN , RN ) of C(RN , RN ) and to prove the appropriate properties for Df (t) and Df (t). Recall from the proof of the previous theorems that there exists c > 0 such that t t |f (t)(x)| ≤ c |V (r)(x)| dr ≤ c |(V (r) − V (0))(x)| dr + c t |V (0)(x)|. 0

0

By assumption on V (0), for ε > 0 there exists a compact set K such that ∀x ∈ K,

|V (0)(x)| ≤ ε/(2c)

and there exists δ, 0 < δ < 1, such that ∀t, 0 ≤ t ≤ δ,

V (r) − V (0)C ≤ ε/(2c)

⇒ ∀t, 0 ≤ t ≤ δ, ∀x ∈ K,

|f (t)(x)| ≤ ε

⇒ f (t) ∈ C0 .

Proceeding in this fashion from the interval [0, δ] to the next interval [δ, 2δ] using the inequality t |(f (t) − f (s))(x)| ≤ c |(V (r) − V (δ))(x)| dr + c |t − δ| |V (δ)(x)|, s

190

Chapter 4. Transformations Generated by Velocities

the uniform continuity of V ∀t, s,

|t − s| < δ,

V (t) − V (s)C ≤ ε/(2c),

and the fact that V (δ) ∈ C0 , that is, that there exists a compact set K(δ) such that ∀x ∈ K(δ),

|V (δ)(x)| ≤ ε/(2c),

we get f (t) ∈ C0 , δ ≤ t ≤ 2δ, and hence f ∈ C([0, τ ]; C0 ). For f (t) we make use of the identity f (t) = V (t) ◦ Tt . Again by assumption for any ε > 0 there exists a compact set K(t) such that |V (t)(x)| ≤ ε on K(t). Thus by choosing the compact set Kt = Tt−1 (K(t)), |f (t)(x)| ≤ ε on Kt , and f ∈ C([0, τ ]; C0 ). In order to complete the proof, it remains to establish the same properties for Df (t) and Df (t). The matrix Df (t) is solution of the equations d Df (t) = DV (t) ◦ Tt DTt , dt

Df (0) = 0

⇒ Df (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt .

(4.26)

From the proofs of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, for each t the elements of the matrix def

A(t) = DV (t) ◦ Tt = DV (t) ◦ [I + f (t)] belong to C0 since DV (t) and f (t) do. By assumption, V ∈ C([0, τ ]; C0k ) and V and all its derivatives ∂ α V are uniformly continuous in [0, τ ] × RN . Therefore, for each ε > 0 there exists δ > 0 such that ∀ |t − s| < δ, ∀ |y − x | < δ,

|DV (t)(y ) − DV (s)(x )| < ε.

Pick 0 < δ < δ such that ∀ |t − s| < δ ,

Tt − Ts C = f (t) − f (s)C < δ ⇒ ∀x, ∀ |t − s| < δ , |Tt (x) − Ts (x)| < δ ⇒ |DV (t)(Tt (x)) − DV (s)(Ts (x))| < ε

⇒ A(t) − A(s)C < ε

⇒ A ∈ C([0, τ ]; (C0 )N ).

For each x, Df (t)(x) is the unique solution of the linear matrix equation (4.26). To show that Df (t) ∈ (C0 )N we ﬁrst show that Df (t)(x) is uniformly continuous for x in RN . For any x and y t |V (r, Tr (y)) − V (r, Tr (x))| dr |Df (t)(y) − Df (t)(x)| ≤ 0 t ≤ c|Tr (y) − Tr (x)| dr 0 t |f (r)(y) − f (r)(x)| + |y − x| dr. ≤c 0

4. Unconstrained Families of Domains

191

But f ∈ C([0, τ ]; C0 ) is uniformly continuous in (t, x): for each ε > 0 there exists δ, 0 < δ < ε/(2cτ ), such that ∀ |t − s| < δ, ∀ |y − x| < δ,

|f (t)(y) − f (s)(x)| < ε/(2cτ ).

Substituting in the previous inequality for each ε > 0, there exists δ > 0 such that ∀ t, ∀ |y − x| < δ,

|Df (t)(y) − Df (t)(x)| < ε.

Hence Df (t) is uniformly continuous in RN . Furthermore, from (4.26) we have the following inequality: t |Df (t)(x)| ≤ |DV (r)(Tr (x))| |Df (r)(x)| + |DV (r)(Tr (x))| dr t 0 (4.27) (|Df (r)(x)| + 1) dr ≤c 0

since V ∈ C([0, τ ]; C01 ). By Gronwall’s inequality |Df (t)(x)| ≤ c t for some other constant c independent of t. Hence Df (t) ∈ C(RN , RN )N . Finally to show that Df (t) vanishes at inﬁnity we start from the integral form of (4.26): t DV (r)(Tr (x)) DTr (x) dr, Df (t)(x) = t 0 |DV (r)(Tr (x)) − DV (r)(x)| + |DV (r)(x)| dr |Df (t)(x)| ≤ c t 0 ≤c |f (r)(x)| + |DV (r)(x)| dr. 0

By the same technique as for f (t), it follows that the elements of Df (t) belong to C0 since both f (s) and DV (r) do. Finally for the continuity with respect to t t A(r)Df (r) + A(r) dr, Df (t) − Df (s) = s t Df (t) − Df (s)C ≤ A(r)C Df (r) − Df (s)C s + A(r)C (1 + Df (r)C ) dr. Again, by Gronwall’s inequality, there exists another constant c such that Df (t) − Df (s)C ≤ c |t − s|. Therefore, Df ∈ C([0, τ ]; (C0 )N ) and f ∈ C([0, τ ]; C01 ). For Df we repeat the proof for f using the identity Df (t) = DV (t) ◦ Tt Df (t) + DV (t) ◦ Tt to get Df ∈ C([0, τ ]; (C0 )N )

⇒ f ∈ C([0, τ ]; C01 ).

192

Chapter 4. Transformations Generated by Velocities

Conditions (4.25) on g. From the remark at the beginning of part (i) of the proof, the conclusions of Theorem 4.3 are true for g, and it remains to check the remaining properties for g and Dg using the identities g(t) = −f (t) ◦ [I + g(t)],

Dg(t) = −Df (t) ◦ [I + g(t)] (I + Dg(t)).

From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, g(t) ∈ C0 since Df (t) and g(t) belong to C0 . Therefore, g(t) ∈ C01 . The continuity follows by the same argument as for f and g ∈ C([0, τ ]; C01 ). (ii) By assumption from conditions (4.24), conditions (T1) are satisﬁed. For (T2) observe that for k ≥ 1 the function t → Df (t) = DTt − I : [0, τ ] → C k−1 (RN , RN )N is continuous. Hence t → det DTt : [0, τ ] → R is continuous and det DT0 = 1. So there exists τ > 0 such that Tt is invertible for all t in [0, τ ] and (T2) is satisﬁed in [0, τ ]. Furthermore, from the proof of part (i) conditions (T3) and (4.25) on g are also satisﬁed in some interval [0, τ ], τ > 0. Therefore, the velocity ﬁeld V (t) = f (t) ◦ Tt−1 = f (t) ◦ [I + g(t)] satisﬁes the conditions (V) speciﬁed by (4.2) in [0, τ ]. From the proof of Theorems 2.11 and 2.12 in section 2.5 of Chapter 3, V (t) ∈ C0k since f (t) and g(t) belong to C0k . By assumption, f ∈ C 1 ([0, τ ]; C0k ). Hence f and all its derivatives ∂ α f , |α| ≤ k, are uniformly continuous on [0, τ ] × RN ; that is, given ε > 0, there exists δ > 0 such that ∀t, s, |t − s| < δ, ∀y , x , |y − x | < δ,

|∂ α f (t)(y ) − ∂ α f (s)(x )| < ε.

Similarly g ∈ C([0, τ ]; C0k ) and there exists 0 < δ ≤ δ such that ∀t, s, |t − s| < δ , ∀y, x, |y − x| < δ ,

|∂ α g(t)(y) − ∂ α g(s)(x)| < δ.

Therefore, for |t − s| < δ Tt−1 − Ts−1 C = g(t) − g(s)C < δ, and since δ < δ ∀x, |f (t)(Tt−1 (x)) − f (t)(Ts−1 (x))| < ε ⇒ V (t) − V (s)C < ε ⇒ V ∈ C([0, τ ]; C0 ). We then proceed to the ﬁrst derivative of V , DV (t) = Df (t) ◦ Tt−1 DTt−1 = Df (t) ◦ [I + g(t)] [I + Dg(t)], and by uniform continuity of the right-hand side V ∈ C([0, τ ]; C01 ). By induction on k, we ﬁnally get V ∈ C([0, τ ]; C0k ). The proof of the last theorem is based on the fact that the vector functions involved are uniformly continuous. The fact that they vanish at inﬁnity is not an essential element of the proof. Therefore, the theorem is valid with C k (RN , RN ) in place of C0k (RN , RN ).

5. Constrained Families of Domains

193

Theorem 4.5. Let k ≥ 1 be an integer. (i) Given τ > 0 and a velocity ﬁeld V such that V ∈ C([0, τ ]; C k (RN , RN )),

(4.28)

the map T given by (4.4)–(4.6) satisﬁes conditions (T1), (T2), and f ∈ C 1 ([0, τ ]; C k (RN , RN )).

(4.29)

Moreover, conditions (T3) are satisﬁed and there exists τ > 0 such that g ∈ C([0, τ ]; C k (RN , RN )).

(4.30)

(ii) Given τ > 0 and T : [0, τ ] × RN → RN satisfying conditions (4.29) and T (0, ·) = I, there exists τ > 0 such that the velocity ﬁeld V (t) = f (t) ◦ Tt−1 satisﬁes conditions (V) and (4.28) on [0, τ ].

5

Constrained Families of Domains

We now turn to the case where the family of admissible domains Ω is constrained to lie in a ﬁxed larger subset D of RN or its closure. For instance, D can be an open set or a closed submanifold of RN .

5.1

Equivalence between Velocities and Transformations

Given a nonempty subset D of RN , consider a family of transformations T : [0, τ ] × D → RN

(5.1)

for some τ = τ (T ) > 0 with the following properties: ∀X ∈ D, T (·, X) ∈ C 1 [0, τ ]; RN and ∃c > 0, (T1D ) ∀X, Y ∈ D, T (·, Y ) − T (·, X)C 1 ([0,τ ];RN ) ≤ c|Y − X|, (T2D ) (T3D )

X → Tt (X) = T (t, X) : D → D is bijective, ∀x ∈ D, T −1 (·, x) ∈ C [0, τ ]; RN and ∃c > 0,

∀t ∈ [0, τ ], ∀x, y ∈ D,

(5.2)

T −1 (·, y) − T −1 (·, x)C([0,τ ];RN ) ≤ c|y − x|,

where under assumption (T2D ) T −1 is deﬁned from the inverse of Tt as (t, x) → T −1 (t, x) = Tt−1 (x) : [0, τ ] × D → RN . def

(5.3)

These three properties are the analogue for D of the same three properties obtained for RN . In fact, Theorem 4.1 extends from RN to D by adding one assumption to (V). Speciﬁcally, consider a velocity ﬁeld V : [0, τ ] × D → RN

(5.4)

194

Chapter 4. Transformations Generated by Velocities

for which there exists τ = τ (V ) > 0 such that (V1D ) (V2D )

∀x ∈ D, ∀x, y ∈ D,

V (·, x) ∈ C [0, τ ]; RN ,

∃c > 0,

V (·, y) − V (·, x)C([0,τ ];RN ) ≤ c|y − x|,

∀x ∈ D, ∀t ∈ [0, τ ],

(5.5)

±V (t, x) ∈ TD (x),

where TD (x) is Bouligand’s contingent cone6 to D at the point x in D % % * 1 def TD (X) = (D − X) + εB h ε>0 α>0

(5.6)

0 0 and a map T : [0, τ ] × D → D satisfying assumptions (T1D ) and (T2D ) and T (0, ·) = I, then there exists τ > 0 such that the conclusions of part (ii) hold on [0, τ ]. Remark 5.1. Assumption (V 2D ) is a double viability condition. M. Nagumo [1]’s usual viability condition (5.11) V (t, x) ∈ TD (x), ∀t ∈ [0, τ ], ∀x ∈ D is a necessary and suﬃcient condition for a viable solution to (5.9): ∀t ∈ [0, τ ], ∀X ∈ D, x(t; X) ∈ D or Tt (D) ⊂ D

(5.12)

(cf. J.-P. Aubin and A. Cellina [1, p. 174 and p. 180]). Condition (V 2D ), ∀t ∈ [0, τ ], ∀x ∈ D,

±V (t, x) ∈ TD (x),

(5.13)

is a strict viability condition which says that Tt maps D into D and ∀t ∈ [0, τ ],

Tt : D → D

is a homeomorphism.

(5.14)

In particular it maps interior points onto interior points and boundary points onto boundary points (cf. J. Dugundji [1, pp. 87–88]). Remark 5.2. Condition (V 2D ) is a generalization to an arbitrary set D of the following condition ´sio [12] in 1979: for all x in ∂D, used by J.-P. Zole V (t, x) · n(x) = 0, 0,

if the outward normal n(x) exists, otherwise.

Proof of Theorem 5.1. (i) Existence and uniqueness of viable solutions to (5.9). Apply M. Nagumo [1]’s theorem to the augmented system on [0, τ ] dx (t) = V t, x(t) , x(0) = X ∈ D, dt (5.15) dx0 (t) = 1, x0 (0) = 0, dt that is,

x dˆ (t) = Vˆ x ˆ(t) , dt x ˆ def ˆ(0) = (0, X) ∈ D = R+ ×D,

(5.16)

196

Chapter 4. Transformations Generated by Velocities

x) = 1, V˜ (ˆ x) , and where x ˆ(t) = x0 (t), x(t) ∈ RN +1 , Vˆ (ˆ V (x0 , x), 0 ≤ x0 ≤ τ ˜ V (x0 , x) = , x ∈ D. V (τ, x), τ < x0

(5.17)

It is easy that systems (5.15) and (5.16) are equivalent on [0, τ ] and that to check x ˆ(t) = t, x(t) . The new velocity ﬁeld on Vˆ ⊂ RN +1 is continuous at each point ˆ by the ﬁrst assumption (V 1D ) since x ˆ∈D Vˆ (ˆ y ) − Vˆ (ˆ x) = 0, V˜ (y0 , y) − V˜ (x0 , x) , and for 0 ≤ x0 , y0 ≤ τ |V (y0 , y) − V (x0 , x)| ≤ |V (y0 , y) − V (y0 , x)| + |V (y0 , x) − V (x0 , x)| ≤ n|y − x| + |V (y0 , x) − V (x0 , x)|. In addition, TDˆ (ˆ x) = TR+ (x0 ) × TD (x) ⇒ Vˆ (ˆ x) = 1, V (ˆ x) ∈ TR+ (x0 ) × TD (x). ˆ is bounded and RN +1 is ﬁnite-dimensional. By using the version Moreover Vˆ (D) of M. Nagumo[1]’s theorem given in J.-P. Aubin and A. Cellina [1, Thm. 3, part b), pp. 182–183], there exists a viable solution xˆ to (5.16) for all t ≥ 0. In particular, ˆ = R+ ×D, ∀t ∈ [0, τ ], x ˆ(t) ∈ D which is necessarily of the form x ˆ(t) = (t, x(t)). Hence there exists a viable solution x, x(t) ∈ D on [0, τ ], to (5.9). The uniqueness now follows from the Lipschitz condition (V 1D ). The Lipschitzian continuity (T 1D ) can be established by a standard argument. Condition (T2D ). Associate with X in D the function y(s) = Tt−s (X),

0 ≤ s ≤ t.

Then

dy (s) = −V t − s, y(s) , 0 ≤ s ≤ t, ds For each x ∈ D, the diﬀerential equation dy (s) = −V t − s, y(s) , ds

0 ≤ s ≤ t,

y(0) = Tt (X).

(5.18)

y(0) = x ∈ D

(5.19)

has a unique viable solution in C 1 ([0, t]; RN ): ∀s ∈ [0, t],

y(s) ∈ D,

since by assumption (V 2D ) ∀t ∈ [0, τ ], ∀x ∈ D,

−V (t, x) ∈ TD (x).

(5.20)

5. Constrained Families of Domains

197

The proof is the same as above. The solutions of (5.19) deﬁne a Lipschitzian mapping x → St (x) = y(t) : D → D such that ∃c > 0, ∀t ∈ [0, τ ], ∀x, y ∈ D,

|St (y) − St (x)| ≤ c|y − x|.

Now in view of (5.18) and (5.19) St Tt (X) = y(t) = Tt−t (X) = X

⇒

(5.21)

St ◦ Tt = I on D.

To obtain the other identity, consider the function z(r) = y(t − r; x), where y(·, x) is the solution of (5.19). By deﬁnition, dz (r) = V r, z(r) , dr

z(0) = y(t, x)

and necessarily x = y(0; x) = z(t) = Tt y(t; x) = Tt St (x) ⇒ Tt ◦ St = I on D ⇒ St = Tt−1 : D → D. Condition (T3D ). The uniform Lipschitz continuity in (T3D ) follows from (5.21) and (T2D ), and we need only show that ∀x ∈ D, T −1 (·, x) ∈ C [0, τ ]; RN . Given t in [0, τ ] choose an arbitrary sequence {tn }, tn → t. Then for each x ∈ D there exists X ∈ D such that Tt (X) = x and Ttn (X) → Tt (X) = x from (T1D ). But Tt−1 Tt (X) − Tt−1 Tt (X) (x) − Tt−1 (x) = Tt−1 n n Tt (X) − Tt−1 Ttn (X) . = Tt−1 n n By the uniform Lipschitz continuity of Tt−1 Tt (X) − Tt−1 Ttn (X) | |Tt−1 (x) − Tt−1 (x)| = |Tt−1 n n n ≤ c|Tt (X) − Ttn (X)|, and the last term converges to zero as tn goes to t.

198

Chapter 4. Transformations Generated by Velocities ¯ and t, s in [0, τ ] (ii) The ﬁrst condition (V1D ) is satisﬁed since for each x ∈ D ∂T ∂T |V (t, x) − V (s, x)| ≤ t, Tt−1 (x) − t, Ts−1 (x) ∂t ∂t ∂T ∂T t, Ts−1 (x) − s, Ts−1 (x) + ∂t ∂t ≤ c|Tt−1 (x) − Ts−1 (x)| ∂T ∂T t, Ts−1 (x) − s, Ts−1 (x) . + ∂t ∂t

The second condition (V1D ) follows from (T1D ) and (T3D ) and the following inequality: for all x and y in D ∂T ∂T −1 −1 |V (t, y) − V (t, x)| = t, Tt (y) − t, Tt (x) ∂t ∂t ≤ c Tt−1 (y) − Tt−1 (x) ≤ cc |y − x|. To check condition (V2D ), recall deﬁnition (5.6) of the Bouligand contingent cone: % % * 1 TD (X) = (D − X) + εB , h ε>0 α>0 0 0, ∃h ∈ ]0, α[ , ∃u such that x (t) ∈ u + εB and x(t) + hu ∈ D. Choose δ, 0 < δ < α, such that ∀s,

|s − t| < δ

⇒

|x (s) − s (t)| < ε.

Then ﬁx t . For all t such that 0 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

dG ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Condition (6.3) on V coincides with condition (4.23) of Theorem 4.4, which implies conditions (4.24) and (4.25): f ∈ C 1 ([0, τ ]; C0k (RN , RN ))

and g ∈ C([0, τ ]; C0k (RN , RN ))

⇒ Tt − ICk (RN ) → 0 and Tt−1 − ICk (RN ) → 0 as t → 0. But by deﬁnition of the metric dG d([Tt ], [I]) ≤ Tt−1 − IC k + Tt − ICk → 0 as t → 0, and we get the convergence (6.2) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (6.3). (ii) Conversely, it is suﬃcient to prove that for any sequence {[Tn ]} such that dG ([Tn ], [I]) goes to zero, there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞. Indeed let = lim inf J(Tn (Ω)) n→∞

and L = lim sup J(Tn (Ω)). n→∞

By deﬁnition of the liminf, there is a subsequence, still indexed by n, such that = lim inf n→∞ J(Tn (Ω)). But since there exists a subsequence {Tnk } of {Tn } such that J(Tnk (Ω)) → J(Ω), then necessarily = J(Ω). The same reasoning applies to the limsup, and hence the whole sequence J(Tn (Ω)) converges to J(Ω) and we have the continuity of J at Ω. We prove that we can construct a velocity V associated with a subsequence of {Tn } verifying conditions (4.23) of Theorem 4.4 and hence conditions (6.3). By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequence {Tn } such that dG ([Tn ], [I]) → 0 a subsequence, still denoted by {Tn }, such that fn C k + gn Ck = Tn−1 − IC k + Tn − IC k ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Deﬁne the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ], def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

(Tn+1 (X) − Tn (X)),

def

T0 (X) = X,

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1).

204

Chapter 4. Transformations Generated by Velocities

Conditions on f . By deﬁnition, for all t, 0 ≤ t ≤ 1/2, f (t) = Tt − I ∈ C0k (RN ). Moreover, for 0 < t ≤ 1/2 ∂T ∂T (tn , X) = 0 = (tn+1 , X), ∂t ∂t ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t (t, X) = p , ∂t |tn − tn+1 | tn+1 − tn

Ttn (X) = Tn (X), Ttn+1 (X) = Tn+1 (X),

f (t) = ∂T /∂t(t, ·) ∈ C0k (RN ), and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By deﬁnition, f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and tn+1 − t (fn+1 − fn )C k f (t) − f (0)Ck = f (t)C k = fn + p tn+1 − tn ≤ 2fn Ck + fn+1 C k ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Deﬁne at t = 0, f (t) = 0. By the same technique, there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and f (t) − f (0)Ck = f (t)C k 3 3 3 ∂T 3 3 ≤ c Tn+1 − Tn Ck = c fn+1 − fn C k =3 (t, ·) 3 ∂t 3 k |tn+1 − tn | 2−(n+1) C

≤ c 2 2−2(n+2) /2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t ⇒ f (t)C k ≤ ct. So for each X the functions t → f (t)(X) and t → Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C0k (RN )) and the condition (4.24) of Theorem 4.4 is satisﬁed. Hence the corresponding velocity V satisﬁes conditions (4.23). Finally V satisﬁes conditions (6.3) and by (6.2) for all ε > 0 there exists δ > 0 such that ∀t, 0 ≤ t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ, and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε,

and this proves the d-continuity for the subsequence {Tn }. The case of the Courant metric associated with the space C k (RN ) = C k (RN , RN ) is a corollary to Theorem 6.1. Theorem 6.2. Let k ≥ 1 be an integer, B be a Banach space, and Ω be a nonempty open subset of RN . Consider a shape functional J : NΩ ([I]) → B deﬁned in a neighborhood NΩ ([I]) of [I] in F(C k (RN , RN ))/G(Ω).

6. Continuity of Shape Functions along Velocity Flows

205

Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(6.4)

t0

for all families of velocity ﬁelds {V (t) : 0 ≤ t ≤ τ } satisfying the condition V ∈ C([0, τ ]; C k (RN , RN )).

(6.5)

The proof of the theorem for the Courant metric topology associated with the space C k,1 (RN ) = C k,1 (RN , RN ) is similar to the proof of Theorem 6.1 with obvious changes. Theorem 6.3. Let k ≥ 0 be an integer, Ω be a nonempty open subset of RN , and B be a Banach space. Consider a shape functional J : NΩ ([I]) → B deﬁned in a neighborhood NΩ ([I]) of [I] in F(C k,1 (RN , RN ))/G(Ω). Then J is continuous at Ω for the Courant metric if and only if lim J(Tt (Ω)) = J(Ω)

(6.6)

t0

for all families {V (t) : 0 ≤ t ≤ τ } of velocity ﬁelds in C k,1 (RN , RN ) satisfying the conditions V ∈ C([0, τ ]; C k (RN , RN ))

and

ck (V (t)) ≤ c

(6.7)

for some constant c independent of t. Proof. As in the proof of Theorem 6.1, it is suﬃcient to prove the theorem for a real-valued function J. (i) If J is dG -continuous at Ω, then for all ε > 0 there exists δ > 0 such that ∀T, [T ] ∈ NΩ ([I]),

dG ([T ], [I]) < δ,

|J(T (Ω)) − J(Ω)| < ε.

Under condition (6.7) from Theorem 4.3 f = T − I ∈ C([0, τ ]; C k,1 (RN )) and Tt − IC k,1 → 0 as t → 0, g(t) = Tt−1 − I ∈ C k,1 (RN ) and Tt−1 − IC k,1 ≤ ct → 0 as t → 0. But by deﬁnition of the metric dG dG ([Tt ], [I]) ≤ Tt−1 − IC k,1 + Tt − IC k,1 → 0 as t → 0, and we get the convergence (6.6) of the function J(Tt (Ω)) to J(Ω) as t goes to zero for all V satisfying (6.7). (ii) Conversely, as in the proof of Theorem 6.1, it is suﬃcient to prove that given any sequence {[Tn ]} such that δ([Tn ], [I]) → 0 there exists a subsequence such that J(Tnk (Ω)) → J(I(Ω)) = J(Ω) as k → ∞.

206

Chapter 4. Transformations Generated by Velocities

By the same techniques as in Theorems 2.8 and 2.6 of Chapter 3, associate with a sequence {Tn } such that dG ([Tn ], [I]) → 0 a subsequence, still denoted by {Tn }, such that fn C k,1 + gn Ck,1 = Tn−1 − IC k,1 + Tn − ICk,1 ≤ 2−2(n+2) . For n ≥ 1 set tn = 2−n and observe that tn − tn+1 = −2−(n+1) . Deﬁne the following C 1 -interpolation in (0, 1/2]: for t in [tn+1 , tn ], def

Tt (X) = Tn (X) + p

tn+1 − t tn+1 − tn

(Tn+1 (X) − Tn (X)),

def

T0 (X) = X,

where p ∈ P 3 [0, 1] is the polynomial of order 3 on [0, 1] such that p(0) = 1 and p(1) = 0 and p(1) (0) = 0 = p(1) (1). Conditions on f . By deﬁnition for all t, 0 ≤ t ≤ 1/2, f (t) = Tt −I ∈ C k,1 (RN ). Moreover, for 0 < t ≤ 1/2 ∂T ∂T Ttn (X) = Tn (X), Ttn+1 (X) = Tn+1 (X), (tn , X) = 0 = (tn+1 , X), ∂t ∂t ∂T Tn+1 (X) − Tn (X) (1) tn+1 − t (t, X) = p , ∂t |tn − tn+1 | tn+1 − tn f (t) = ∂T /∂t ∈ C k,1 (RN ), and f (·)(X) = T (·, X) − I ∈ C 1 ((0, 1/2]; RN ). By deﬁnition, f (0) = 0. For each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and 3 3 3 3 tn+1 − t 3 (f f (t) − f (0)C k,1 = f (t)C k,1 = 3 f + p − f ) n+1 n 3 3 n tn+1 − tn C k,1 ≤ 2fn C k,1 + fn+1 C k,1 ≤ 2 2−2(n+2) + 2−2(n+3) ≤ 2−(n+1) ≤ t. Deﬁne at t = 0, f (t) = 0. By the same technique there exists a constant c > 0, and for each 0 < t ≤ 1/2 there exists n ≥ N such that tn+1 ≤ t ≤ tn and f (t) − f (0)C k,1 = f (t)C k,1 3 3 3 3 ∂T fn+1 − fn Ck,1 Tn+1 − Tn C k,1 3 (t, ·)3 =c =3 ≤c 3 ∂t |tn+1 − tn | 2−(n+1) k,1 C −2(n+2)

≤ c22

/2−(n+1) ≤ c 2−1 2−(n+1) ≤ c 2−(n+1) ≤ c t

⇒ f (t)C k,1 ≤ ct and ck (f (t)) ≤ ct and for each X the functions t → f (t)(X) and t → Tt (X) belong to C 1 ([0, 1/2]; RN ). By uniform C k -continuity of the Tn ’s and the continuity with respect to t for each X, it follows that f ∈ C 1 ([0, 1/2]; C k (RN )). Moreover, it can be shown that ck (f (t)) ≤ ct

⇒ ∀t, s ∈ [0, τ ],

ck (f (t) − f (s)) ≤ c |t − s|

for some c > 0. The result is straightforward for k = 0 and then the general case follows by induction on k. As a result f ∈ C([0, 1/2]; C k,1 (RN )) and the condition

6. Continuity of Shape Functions along Velocity Flows

207

(4.20) of Theorem 4.3 is satisﬁed. Hence the corresponding velocity V satisﬁes conditions (4.19). Finally the velocity ﬁeld V satisﬁes conditions (6.7), and by (6.6) for all ε > 0 there exists δ > 0 such that ∀t ≤ δ,

|J(Tt (Ω)) − J(Ω)| < ε.

In particular there exists N > 0 such that for all n ≥ N , tn ≤ δ and ∀n ≥ N,

|J(Tn (Ω)) − J(Ω)| = |J(Ttn (Ω)) − J(Ω)| < ε,

and we have the dG -continuity for the subsequence {Tn }. Remark 6.2. The conclusions of Theorems 6.1, 6.2, and 6.3 are generic. They also have their counterpart in the constrained case. The diﬃculty lies in the second part of the theorem, which requires a special construction to make sure that the family of transformations {Tt : 0 ≤ t ≤ τ } constructed from the sequence {Tn } are homeomorphisms of D.

Chapter 5

Metrics via Characteristic Functions 1

Introduction

The constructions of the metric topologies of Chapter 3 are limited to families of sets which are the image of a ﬁxed set by a family of homeomorphisms or diﬀeomorphisms. If it is connected, bounded, or C k , k ≥ 1, then the images will have the same properties under C k -transformations of RN . In this chapter we considerably enlarge the family of available sets by relaxing the smoothness assumption to the mere Lebesgue measurability and even just measurability to include Hausdorﬀ measures. This is done by identifying Ω ⊂ RN with its characteristic function & 1, if x ∈ Ω def def χΩ (x) = if Ω = ∅ and χ∅ = 0. (1.1) 0, if x ∈ /Ω We ﬁrst introduce an Abelian group structure1 on characteristic functions of measurable 2 subsets of a ﬁxed holdall D in section 2 and then construct complete metric spaces of equivalence classes of measurable characteristic functions via the Lp norms, 1 ≤ p < ∞, that turn out to be algebraically and topologically equivalent. Starting with section 3, we specialize to the Lebesgue measure in RN . The complete topology generated by the metric deﬁned as the Lp -norm of the diﬀerence of two characteristic functions is called the strong topology in section 3.1. In section 3.2 we consider the weak Lp -topology on the family of characteristic functions. Weak limits of sequences of characteristic functions are functions with values in [0, 1] which belong to the closed convex hull of the equivalence classes of measurable characteristic functions. This occurs in optimization problems where the function to be optimized depends on the solution of a partial diﬀerential equation on the variable domain, as was shown, for instance, by F. Murat [1] in 1971. It usually corresponds to the appearance of a microstructure or a composite material in mechanics. 1 Cf.

the group structure on subsets of a holdall D in section 2.2 of Chapter 2. in the sense of L. C. Evans and R. F. Gariepy [1]. It simultaneously covers Hausdorﬀ and Lebesgue measures. 2 Measurable

209

210

Chapter 5. Metrics via Characteristic Functions

One important example in that category was the analysis of the optimal thickness of a circular plate by K.-T. Cheng and N. Olhoff [2, 1] in 1981. The reader is referred to the work of F. Murat and L. Tartar [3, 1], initially published in 1985, for a comprehensive treatment of the calculus of variations and homogenization. Section 3.3 deals with the question of ﬁnding a nice representative in the equivalence class of sets. Can it be chosen open? We introduce the measure theoretic representative and characterize its interior, exterior, and boundary. It will be used later in section 6.4 to prove the compactness of the family of Lipschitzian domains verifying a uniform cone property. Section 3.4 shows that the family of convex subsets of a ﬁxed bounded holdall is closed in the strong topology. The use of the Lp -topologies is illustrated in section 4 by revisiting the optimal ´a and K. Malanowski [1] in 1970. By relaxing design problem studied by J. Ce the family of characteristic functions to functions with values in the interval [0, 1] a saddle point formulation is obtained. Functions with values in the interval [0, 1] can also be found in the modeling of a dam by H. W. Alt [1] and H. W. Alt and G. Gilardi [1], where the “liquid saturation” in the soil is a function with values in [0, 1]. Another problem amenable to that formulation is the buckling of columns as will be illustrated in section 5 using the work of S. J. Cox and M. L. Overton [1]. It is one of the very early optimal design problems, formulated by Lagrange in 1770. The Caccioppoli or ﬁnite perimeter sets of the celebrated Plateau problem are revisited in section 6. Their characteristic function is a function of bounded variation. They provide the ﬁrst example of compact families of characteristic functions in Lp (D)-strong, 1 ≤ p < ∞. This was developed mainly by R. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem of minimal surfaces. Section 6.3 exploits the embedding3 of BV(D) ∩ L∞ (D) into the Sobolev spaces W ε,p (D), 0 < ε < 1/p, p ≥ 1, to introduce a cascade of complete metrics between Lp (D) and BV(D) on characteristic functions. In section 6.4 we show that the family of Lipschitzian domains in a ﬁxed bounded holdall verifying the uniform cone property of section 6.4.1 in Chapter 2 is also compact. This condition naturally yields a uniform bound on the perimeter of the sets in the family and hence can be viewed as a special case of the ﬁrst compactness theorem. Section 7 gives an example of the use of the perimeter in the Bernoulli free boundary problem and in particular for the water wave. There the energy associated with the surface tension of the water is proportional to the perimeter, that is, the surface area of the free boundary.

2

Abelian Group Structure on Measurable Characteristic Functions

2.1

Group Structure on Xµ (RN )

From section 2.2 in Chapter 2, the Abelian group structure on the subsets of RN extends to the corresponding family of their characteristic functions. 3 BV(D)

is the space of functions of bounded variations in D.

2. Abelian Group Structure on Measurable Characteristic Functions

211

Theorem 2.1. Let ∅ = D ⊂ RN . The space of characteristic functions X(D) endowed with the multiplication and the neutral multiplicative element χ∅ = 0 def

χA χB = |χA − χB | = χ(A∩B)∪(B∩A) = χAB

(2.1)

is an Abelian group, that is, χ A χB = χ B χA ,

χA χ∅ = χA ,

χA χA = χ∅ ,

χ−1 A = χA .

(2.2)

Proof. Note that the underlying group structure on subsets of D is the one of section 2.2 of Chapter 2. The product is well-deﬁned in X(D) since it is the characteristic function of a subset of A ∪ B in D. The ﬁrst two identities (2.2) are obvious. As for the inverse χ A χB = χ ∅

⇔

|χA − χB | = 0

⇔

B = A,

and there is a unique multiplicative inverse χ−1 A = χA or χA χA = χ∅ . Remark 2.1. We chose the product terminology , but we could have chosen to call that algebraic operation an addition ⊕.

2.2

Measure Spaces

Deﬁnition 2.1. Let X be a set and P(X) be the collection of subsets of X. (i) A mapping µ : P(X) → [0, ∞] is called a measure 4 on X if (a) µ(∅) = 0, and ) ∞ (b) µ(A) ≤ ∞ k=1 µ(Ak ), whenever A ⊂ ∪k=1 Ak . (ii) A subset A ⊂ X is σ-ﬁnite with respect to µ if it can be written A = ∪∞ k=1 Bk , where Bk is µ-measurable and µ(Bk ) < ∞ for k = 1, 2, . . . . (iii) A measure µ on RN is a Radon measure if µ is Borel regular and µ(K) < ∞ for each compact set K ⊂ RN . The Lebesgue measure mN on RN is a Radon measure since RN is σ-ﬁnite with respect to mN . The s-dimensional Hausdorﬀ measure Hs , 0 ≤ s < N , on RN (cf. section 3.2.3 in Chapter 2) is Borel regular but not necessarily a Radon measure since bounded Hs -measurable subsets of RN are not necessarily σ-ﬁnite with respect to Hs . Yet, a Borel regular measure µ can be made σ-ﬁnite by restricting it to a µ-measurable set A ⊂ RN such that µ(A) < ∞ or, more generally, to a σ-ﬁnite µ-measurable set A ⊂ RN . For smooth submanifolds of dimension s, s an integer, Hs gives the same area as the integral of the canonical density in section 3.2.2 of Chapter 2. 4 A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outer measure in most texts.

212

Chapter 5. Metrics via Characteristic Functions

Given a µ-measurable subset D ⊂ RN , let Lp (D, µ) denote the Banach space of equivalence classes of µ-measurable functions such that |f |p dµ < ∞, 1 ≤ p < ∞, and let L∞ (D, µ) be the space of equivalence classes of µ-measurable functions such that ess supD |f | < ∞. Denote by [A]µ the equivalence class of µ-measurable subsets of D that are equal almost everywhere and def

Xµ (D) = {χA : A ⊂ D is µ-measurable} . Since A B is µ-measurable for A and B µ-measurable, {[A]µ : A ⊂ D is µ-measurable} is a subgroup of P(D) and Xµ (D) ∩ L1 (D, µ) = {χA : A ⊂ D is µ-measurable and µ(A) < ∞}

(2.3)

is a group in Lp (D, µ) with respect to . When µ = mN , we drop the subscript µ.

2.3

Complete Metric for Characteristic Functions in Lp -Topologies

Now that we have revealed the underlying group structure of Xµ (D) ∩ L1 (D, µ), we construct a metric on that group. We have a structure similar to the Courant metric of A. M. Micheletti [1] in Chapter 3 on groups of transformations of RN . We identify equivalence classes of µ-measurable subsets of D and characteristic functions via the bijection [A]µ ↔ χA . Theorem 2.2. Let 1 ≤ p < ∞, let µ be a measure on RN , and let ∅ = D ⊂ RN be µ-measurable. (i) Xµ (D) ∩ L1 (D, µ) is closed in Lp (D, µ). The function def

ρD ([A2 ]µ , [A1 ]µ ) = χA2 − χA1 Lp (D,µ) deﬁnes a complete metric structure on the Abelian group Xµ (D)∩L1 (D, µ) that makes it a topological group. If µ(D) < ∞, then Xµ (D) ∩ L1 (D, µ) = Xµ (D). (ii) If, in addition, D is σ-ﬁnite with respect to µ for a family {Dk } of µ-measurable subsets of D such that µ(Dk ) < ∞, for all k ≥ 1, then Xµ (D) is closed in Lploc (D, µ) and def

ρ([A2 ]µ , [A1 ]µ ) =

∞ χA2 − χA1 Lp (Dk ,µ) 1 2n 1 + χA2 − χA1 Lp (Dk ,µ) k=1

deﬁnes a complete metric structure on the Abelian group Xµ (D) that makes it a topological group. When µ is a Radon measure on RN , the assumption that D is σ-ﬁnite with respect to µ can be dropped. Remark 2.2. For the Lebesgue measure mN and D measurable (resp., measurable and bounded), X(D) is a topological Abelian group in Lploc (D, mN ) (resp., Lp (D, mN )). For Hausdorﬀ measures Hs , 0 ≤ s < N , the theorem does not say more.

2. Abelian Group Structure on Measurable Characteristic Functions

213

Proof. (i) Let {χAn } be a Cauchy sequence of µ-measurable characteristic functions in Xµ (D) ∩ L1 (D, µ) ⊂ Lp (D, µ) converging to some f ∈ Lp (D, µ). There exists a subsequence, still denoted by {χAn }, such that χAn (x) → f (x) in D except for a subset Z of zero µ-measure. Hence 0 = χAn (x) (1 − χAn (x)) → f (x) (1 − f (x)). Deﬁne the set def

A = {x ∈ D\Z : f (x) = 1}. Clearly A is µ-measurable and χA = f on D\Z since f (x) (1 − f (x)) = 0 on D\Z. Hence f = χA almost everywhere on D, χAn → χA in Lp (D, µ), and χA ∈ Xµ (D) ∩ L1 (D, µ). Finally, the metric is compatible with the group structure −1 ρD ([Ω2 ]µ , [Ω1 ]µ ) = χΩ2 − χΩ1 Lp = χΩ2 χ−1 Ω1 Lp = χΩ1 χΩ2 Lp def

as in Chapter 3 and Xµ (D) ∩ L1 (D, µ) is a topological group. (ii) From part (i) using the Fr´echet topology associated with the family of seminorms qk (f ) = f Lp (Dk ) , k ≥ 1. For p = 1 and µ-measurable subsets A1 and A2 in D, the metric ρD ([A2 ]µ , [A1 ]µ ) is the µ-measure of the symmetric diﬀerence A1 ∆A2 = (A1 ∩ A2 ) ∪ (A2 ∩ A1 ) and for 1 ≤ p < ∞ all the topologies are equivalent on Xµ (D) ∩ L1 (D, µ). Theorem 2.3. Let 1 ≤ p < ∞, let µ be a measure on RN , and let ∅ = D ⊂ RN be µ-measurable. (i) The topologies induced by Lp (D, µ) on Xµ (D) ∩ L1 (D, µ) are all equivalent for 1 ≤ p < ∞. (ii) If, in addition, D is σ-ﬁnite with respect to µ for a family {Dk } of µ-measurable subsets of D such that µ(Dk ) < ∞, for all k ≥ 1, then the topologies induced by Lploc (D, µ) on Xµ (D) are all equivalent for 1 ≤ p < ∞. Proof. For D µ-measurable and 1 ≤ p < ∞, Xµ (D) ∩ Lp (D, µ) = Xµ (D) ∩ L1 (D, µ), and for any χ and χ in Xµ (D) ∩ L1 (D, µ) χ − χpLp (D,µ) = |χ − χ(x)|p dµ = |χ − χ(x)| dµ = χ − χL1 (D,µ) , D

D

since χ(x) and χ(x) are either 0 or 1 almost everywhere in D. Therefore the linear (identity) mapping between Xµ (D)∩Lp (D, µ) and Xµ (D)∩L1 (D, µ) is bicontinuous and the topologies are equivalent. Remark 2.3. At this stage, it is not clear what the tangent space to this topological Abelian group is. If µ is a Radon measure, then for all ϕ ∈ D(RN )

1 1 χA χBε (x) χ−1 ϕ dµ = χB (x) ϕ dµ A µ(Bε (x)) RN µ(Bε (x)) RN ε

214

Chapter 5. Metrics via Characteristic Functions

and by the Lebesgue–Besicovitch diﬀerentiation theorem

1 N χA χBε (x) χ−1 A ϕ dµ → ϕ(x), µ a.e. in R . µ(Bε (x)) RN

3

Lebesgue Measurable Characteristic Functions

In this section we specialize to the Lebesgue measure and the corresponding family of characteristic functions def

X(RN ) = χΩ : ∀Ω Lebesgue measurable in RN . (3.1) Clearly X(RN ) ⊂ L∞ (RN ), and for all p ≥ 1, X(RN ) ⊂ Lploc (RN ). Also associate with ∅ = D ⊂ RN the set def

X(D) = {χΩ : ∀Ω Lebesgue measurable in D} .

(3.2)

The above deﬁnitions are special cases of the deﬁnitions of section 2.3, and Theorems 2.2 and 2.3 apply to mN as a Radon measure.

3.1

Strong Topologies and C ∞ -Approximations

In view of the equivalence Theorem 2.3, we introduce the notion of strong convergence. Deﬁnition 3.1. (i) If D is a bounded measurable subset of RN , a sequence {χn } in X(D) is said to be strongly convergent in D if it converges in Lp (D)-strong for some p, 1 ≤ p < ∞. (ii) A sequence {χn } in X(RN ) is said to be locally strongly convergent if it converges in Lploc (RN )-strong for some p, 1 ≤ p < ∞. The following approximation theorem will also be useful Theorem 3.1. Let Ω be an arbitrary Lebesgue measurable subset of RN . There exists a sequence {Ωn } of open C ∞ -domains in RN such that χΩn → χΩ

in L1loc (RN ).

Proof. The construction of the family of C ∞ -domains {Ωn } can be found in many places (cf., for instance, E. Giusti [1, sect. 1.14, p. 10 and Lem. 1.25, p. 23]). Associate with χ = χΩ the sequence of convolutions fn = χ ∗ ρn for a sequence {ρn } of symmetric molliﬁers. By construction, 0 ≤ fn ≤ 1. For t, 0 < t < 1, deﬁne def

Fnt = x ∈ RN : fn (x) > t . By deﬁnition, fn − χ > t in Fnt \Ω, χ − fn > 1 − t in Ω\Fnt , and 1 |χ − fn | a.e. in RN . |χ − χFnt | ≤ min{t, 1 − t}

3. Lebesgue Measurable Characteristic Functions

215

By Sard’s Theorem 4.3 of Chapter 2, the set Fnt is a C ∞ -domain for almost all t in (0, 1). Fix α, 0 < α < 1/2, and choose a sequence {tn } such that, for all n, def

α ≤ tn ≤ 1 − α, and deﬁne Ωn = Fntn . Therefore |χΩ − χΩn | ≤

1 1 |χΩ − fn | ≤ |χΩ − fn | a.e. in RN min{tn , 1 − tn } α

and for any bounded measurable subset D of RN χΩ − χΩn L1 (D) ≤ α−1 χΩ − fn L1 (D) , which goes to zero as n goes to inﬁnity.

3.2

Weak Topologies and Microstructures

Some shape optimization problems lead to apparent paradoxes. Their solution is no longer a geometric domain associated with a characteristic function, but a fuzzy domain associated with the relaxation of a characteristic function to a function with values ranging in [0, 1]. The intuitive notion of a geometric domain is relaxed to the notion of a probability distribution of the presence of points of the set. When the underlying problem involves two diﬀerent materials characterized by two constants k1 = k2 , the occurrence of such a solution can be interpreted as the mixing or homogenization of the two materials at the microscale. This is also referred to as a composite material or a microstructure. Somehow this is related to the fact that the strong convergence needs to be relaxed to the weak Lp -convergence and the space X(D) needs to be suitably enlarged. Even if X(D) is strongly closed and bounded in Lp (D), it is not strongly compact. However, for 1 < p < ∞ its closed convex hull co X(D) is weakly compact in the reﬂexive Banach space Lp (D). In fact, co X(D) = {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D} .

(3.3)

Indeed, by deﬁnition co X(D) ⊂ {χ ∈ Lp (D) : χ(x) ∈ [0, 1] a.e. in D}. Conversely any χ that belongs to the right-hand side of (3.3) can be approximated by a sequence of convex combinations of elements of X(D). Choose χn =

n 1 χBnm , n m=1

m Bnm = x : χ(x) ≥ , n

for which |χn (x) − χ(x)| < 1/n. The elements of co X(D) are not necessarily characteristic functions of a domain; that is, the identity χ(x) (1 − χ(x)) = 0 a.e. in D is not necessarily satisﬁed. We ﬁrst give a few basic results and then consider a classical example from the theory of homogenization of diﬀerential equations.

216

Chapter 5. Metrics via Characteristic Functions

Lemma 3.1. Let D be a bounded open subset of RN , let K be a bounded subset of R, and let def

K = {k : D → R : k is measurable and k(x) ∈ K a.e. in D} . (i) For any p, 1 ≤ p < ∞, and any sequence {kn } ⊂ K the following statements are equivalent: (a) {kn } converges in L∞ (D)-weak; (b) {kn } converges in Lp (D)-weak; (c) {kn } converges in D(D) , where D(D) is the space of scalar distributions on D. (ii) If K is bounded, closed, and convex, then K is convex and compact in L∞ (D)weak, Lp (D)-weak, and D(D) . The above results remain true in the vectorial case when K is the set of mappings k : D → K for some bounded subset K ⊂ Rp and a ﬁnite integer p ≥ 1. Proof. (i) It is clear that (a) ⇒ (b) ⇒ (c). To prove that (c) ⇒ (a) recall that since K is bounded, there exists a constant c > 0 such that K ⊂ cB, where B denotes the unit ball in RN . By density of D(D) in L1 (D), any ϕ in L1 (D) can be approximated by a sequence {ϕm } ⊂ D(D) such that ϕm → ϕ in L1 (D). So for each ε > 0 there exists M > 0 such that ∀m ≥ M,

ϕm − ϕL1 (D) ≤

ε . 4c

Moreover, there exists N > 0 such that ε ∀n ≥ N, ∀ ≥ N, ϕM (kn − k ) dx ≤ . 2 D Hence for each ε > 0, there exists N such that for all n ≥ N and ≥ N ϕ(kn − k ) dx ≤ ϕM (kn − k ) dx + (ϕ − ϕM )(kn − k ) dx D D D ε ≤ + 2cϕ − ϕM L1 (D) ≤ ε. 2 (ii) When K is bounded, closed, and convex, K is also bounded, closed, and convex. Since L2 (D) is a Hilbert space, K is weakly compact. In view of the equivalences of part (i), K is also sequentially compact in all the other weak topologies.5 5 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E , any closed ball is a compact metrizable space for the weak topology. Since K is a bounded subset of the normed reﬂexive separable Banach space Lp (D), 1 ≤ p < ∞, the weak compactness of K coincides ´ [1, Vol. 2, Chap. XII, sect. 12.15.9, p. 75]). with its weak sequential compactness (cf. J. Dieudonne

3. Lebesgue Measurable Characteristic Functions

217

In view of this equivalence we adopt the following terminology. Deﬁnition 3.2. Let ∅ = D ⊂ RN be bounded open. A sequence {χn } in X(D) is said to be weakly convergent if it converges for some topology between L∞ (D)-weak and D(D) . It is interesting to observe that working with the weak convergence makes sense only when the limit element is not a characteristic function. Theorem 3.2. Let the assumptions of the theorem be satisﬁed. Let {χn } and χ be elements of X(RN ) (resp., X(D)) such that χn χ weakly in L2 (D) (resp., L2loc (RN )). Then for all p, 1 ≤ p < ∞, χn → χ strongly in Lp (D) resp., Lploc (RN ) . Proof. It is suﬃcient to prove the result for D. The strong L2 (D)-convergence follows from the property |χn |2 dx = χn 1 dx → χ1 dx = |χ|2 dx D D D D 2 2 2 |χn − χ| dx = |χn | − 2 χχn + |χ| dx → |χ|2 − 2 χχ+ |χ|2 dx = 0. ⇒ D

D

D

The convergence for any p ≥ 1 now follows from Theorem 2.2. Working with the weak convergence creates new phenomena and diﬃculties. For instance when a characteristic function is present in the coeﬃcient of the higherorder term of a diﬀerential equation the weak convergence of a sequence of characteristic functions {χn } to some element χ of co X(D), χn χ in L2 (D)-weak

⇒ χn χ in L∞ (D)-weak,

does not imply the weak convergence in H 1 (D) of the sequence {y(χn )} of solutions to the solution of the diﬀerential equation corresponding to y(χ), y(χn ) y(χ) in H 1 (D)-weak. By compactness6 of the injection of H 1 (D) into L2 (D) this would have implied convergence in L2 (D)-strong: y(χn ) → y(χ) in L2 (D)-strong. This fact was pointed out in 1971 by F. Murat [1] in the following example, which will be rewritten to emphasize the role of the characteristic function. 6 This is true since D is a bounded Lipschitzian domain. Examples of domains between two spirals can be constructed where the injection is not compact.

218

Chapter 5. Metrics via Characteristic Functions

Example 3.1. Consider the following boundary value problem: d dy − k + ky = 0 in D = ]0, 1[ , dx dx y(0) = 1 and y(1) = 2,

(3.4)

where K = {k = k1 (x)χ + k2 (x)(1 − χ) : χ ∈ X(D)} with

4 k1 (x) = 1 −

1 x2 − , 2 6

4 k2 (x) = 1 +

1 x2 − . 2 6

(3.5)

(3.6)

This is equivalent to K = {k ∈ L∞ (0, 1) : k(x) ∈ {k1 (x), k2 (x)} Associate with the integer p ≥ 1 the function 4 m 2m + 1 1 x2 1 − 2 − 6 , p < x ≤ 2p , k p (x) = 4 1 x2 2m + 1 m+1 1 + − , <x≤ , 2 6 2p p and the characteristic function m 2m + 1 <x≤ , 1, p 2p χp (x) = m+1 2m + 1 0, <x≤ , 2p p

a.e. in [0, 1]} .

0 ≤ m ≤ p − 1,

0 ≤ m ≤ p − 1.

(3.7)

(3.8)

(3.9)

It is readily seen that kp = k1 χp + k2 (1 − χp ). He shows that for each p, k p ∈ K (resp., χp ∈ X(D)) and that 1 k p k∞ = 1 in L∞ (0, 1)-weak, resp., χp (3.10) 2 1 1 1 1 1 = in L∞ (0, 1)-weak . + (3.11) p k 2 k1 k2 1/2 + x2 /6 Moreover, yp y in H 1 (0, 1)-weak,

(3.12)

where yp denotes the solution of (3.4) corresponding to k = k p and y the solution of the boundary value problem 1 x2 dy − d + + y = 0 in ]0, 1[ , dx 2 6 dx (3.13) y(0) = 1, y(1) = 2.

3. Lebesgue Measurable Characteristic Functions

219

Deﬁne the function kH (x) = which corresponds to

1 x2 + , 2 6

4 1 1 x2 χH (x) = 1+ − ∈ co X(D). 2 2 6

(3.14)

(3.15)

Notice that kH appears in the second-order term and k∞ in the zeroth-order term in (3.13): dy d (3.16) kH + k∞ y = 0 in ]0, 1[ . − dx dx It is easy to check that y(x) = 1 + x2 in [0, 1],

(3.17)

which is not equal to the solution y∞ of (3.4) for the weak limit k∞ = 1: y∞ (x) =

2(ex − e−x ) + e1−x − e−(1−x) . e − e−1

(3.18)

To our knowledge this was the beginning of the theory of homogenization in France. This is only one part of F. Murat [1]’s example. He also constructs an objective function for which the lower bound is not achieved by an element of K. This is a nonexistence result. The above example uses space varying coeﬃcients k1 (x) and k2 (x). However, it is still valid for two positive constants k1 > 0 and k2 > 0. It is easy to show that 1 k1 + k2 1 1 1 1 k2 , χH = k∞ = , χ∞ = , and = + , 2 2 kH 2 k1 k2 k1 + k2 and the solution y of the boundary value problem (3.16) is given by k1 + k2 2 sinh cx + sinh c(1 − x) ≥ 1, , where c = √ sinh c 2 k1 k2 and the solution y∞ by 2 sinh x + sinh(1 − x) . y∞ (x) = sinh 1 Thus for k1 = k2 , or equivalently c > 1, y = y∞ . In fact, using the same sequence of χp ’s, the sequence yp = y(χp ) of solutions of (4.5) weakly converges to yH = y(χH ), which is diﬀerent from the solution y∞ = y(χ∞ ) for k1 = k2 . This is readily seen by noticing that since χ∞ and χH are constant y(x) =

−∆(kH yH ) = χ∞ f = −∆(k∞ y∞ ) ⇒ kH yH = k∞ y∞

⇒ yH =

(k1 + k2 )2 y∞ = y∞ , for k1 = k2 . 4k1 k2

220

Chapter 5. Metrics via Characteristic Functions

Despite this example, we shall see in the next section that in some cases we obtain the existence (and uniqueness) of a maximizer in co X(D) which belongs to X(D).

3.3

Nice or Measure Theoretic Representative

Since the strong and weak Lp -topologies are deﬁned on equivalence classes [Ω] of (Lebesgue) measurable subsets Ω of RN , it is natural to ask if there is a nice representative that is generic of the class [Ω]. For instance we have seen in Chapter 2 that within the equivalence class of a set of class C k there is a unique open and a unique closed representative and that all elements of the class have the same interior, boundary, and exterior. The same question will again arise for ﬁnite perimeter sets. As an illustration of what is meant by a nice representative, consider the smiling and the expressionless suns in Figure 5.1. The expressionless sun is obtained by adding missing points and lines “inside” Ω and removing the rays “outside” Ω. This “restoring/cleaning” operation can be formalized as follows.

Ω

Ω

Figure 5.1. Smiling sun Ω and expressionless sun Ω. Deﬁnition 3.3. Associate with a Lebesgue measurable set Ω in RN the sets def

Ω0 = x ∈ RN : ∃ρ > 0 such that m(Ω ∩ B(x, ρ) = 0 , def

Ω1 = x ∈ RN : ∃ρ > 0 such that m Ω ∩ B(x, ρ) = m B(x, ρ) , def

Ω• = x ∈ RN : ∀ρ > 0 such that 0 < m Ω ∩ B(x, ρ) < m B(x, ρ) and the measure theoretic exterior O, interior I, and boundary ∂∗ Ω m(B(x, r) ∩ Ω) def O = x ∈ RN : lim =0 , r0 m(B(x, r)) m(B(x, r) ∩ Ω) def I = x ∈ RN : lim =1 , r0 m(B(x, r)) def

∂∗ Ω =

x∈R

N

& m(B(x, r) ∩ Ω) m(B(x, r) ∩ Ω) : 0 ≤ lim inf < lim sup ≤1 . r0 m(B(x, r)) m(B(x, r)) r0

We shall say that I is the nice or measure theoretic representative of Ω.

3. Lebesgue Measurable Characteristic Functions

221

The six sets Ω0 , Ω1 , Ω• , O, I, and ∂∗ Ω are invariant for all sets in the equivalence class [Ω] of Ω. They are two diﬀerent partitions of RN Ω0 ∪ Ω 1 ∪ Ω • = R N

and

O ∪ I ∪ ∂∗ Ω = RN

(3.19)

like int Ω ∪ int Ω ∪ ∂Ω = RN . The next theorem links the six invariant sets and describes some of the interesting properties of the measure theoretic representative. Theorem 3.3. Let Ω be a Lebesgue measurable set in RN and let Ω0 , Ω1 , Ω• , O, I, and ∂∗ Ω be the sets constructed from Ω in Deﬁnition 3.3. (i) The sets O, I, and ∂∗ Ω are invariants of the equivalence class [Ω]. Moreover, they are Borel measurable and χI = χΩ ,

χO = χΩ ,

and

χ∂∗ Ω = 0

a.e. in RN .

(ii) The sets Ω0 , Ω1 , and Ω• are invariants of the equivalence class [Ω]. The sets Ω0 and Ω1 are open, Ω• is closed, and int Ω ⊂ Ω1 = int I, int Ω ⊂ Ω0 = int O, ∂Ω ⊃ I ∩ O = Ω• ⊃ ∂∗ Ω, (3.20) ∂∗ Ω ∪ ∂O ∪ ∂I ⊂ Ω• ⊂ ∂Ω. (3.21) (iii) The condition m(∂Ω) = 0 implies m(∂I) = 0. When m(∂I) = 0, then int I and I can be chosen as the respective open and closed representatives of Ω. In particular, this is true for Lipschitzian sets and hence for convex sets. Proof. (i) All the sets involved depend only on χΩ almost everywhere. They are invariants of the equivalence class [Ω]. From L. C. Evans and R. F. Gariepy [1, Lem. 2, sect. 5.11, p. 222, Cor. 3, sect. 1.7.1, p. 45], I, O, and ∂∗ Ω are Borel measurable and m(I\Ω ∪ Ω\I) = 0, and lim

r0

m(B(x, r) ∩ Ω) = 1 a.e. in Ω, m(B(x, r))

lim

r0

m(B(x, r) ∩ Ω) = 0 a.e. in Ω. m(B(x, r))

Therefore, m(∂∗ Ω) = 0, χI = χΩ , and χO = χΩ almost everywhere in RN . (ii) We use the proof given by E. Giusti [1, Prop. 4.1, pp. 42–43] for Ω0 and Ω1 . To show that Ω0 is open, pick any point x in Ω0 . By construction, there exists ρ > 0 such that m Ω ∩ B(x, ρ) = 0. For any y ∈ B(x, ρ) (that is, |y − x| < ρ), ρ0 = ρ − |x − y| > 0. Then B(y, ρ0 ) ⊂ B(x, ρ) and choose m B(y, ρ0 ) ∩ Ω ≤ m B(x, ρ) ∩ Ω = 0. So B(x, ρ) ⊂ Ω0 and Ω0 is open. Similarly, by repeating the argument for Ω, we obtain that Ω1 is open. By complementarity, Ω• is closed. By deﬁnition, int Ω ⊂ Ω1 ⊂ I and, since Ω1 is open, Ω1 ⊂ int I. For all x ∈ int I, there exists r > 0 such that B(x, r) ⊂ I and m B(x, r) = m B(x, r) ∩ I = m B(x, r) ∩ Ω . Hence int I ⊂ Ω1 and necessarily int I = Ω1 . Similarly, int Ω ⊂ Ω0 = int O. By taking the union term by term of the two chains int Ω ⊂ int I ⊂ Ω1 = int I ⊂ I

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Chapter 5. Metrics via Characteristic Functions

and int Ω ⊂ Ω0 = int O ⊂ O and then the complement, we get ∂Ω ⊃ I ∩ O = Ω• ⊃ ∂∗ Ω. From the ﬁrst two relations (3.20), Ω1 = int I implies Ω1 = I ⊂ O and Ω0 = int O implies Ω0 = O ⊂ I. Therefore, Ω• = Ω1 ∩ Ω0 ⊃ I ∩ I = ∂I and similarly Ω• = Ω1 ∩ Ω0 ⊃ O ∩ O = ∂O. (iii) From part (ii). The inclusions (3.20) indicate that the measure theoretic interior and exterior are enlarged and that the measure theoretic boundary is reduced. In general those operations do not commute with the set theoretic operations. For instance, they do not commute with the closure, as can be seen from the following simple example. Example 3.2. Consider the set Ω of all rational numbers in [0, 1]. Then O = Ω0 = R I = Ω1 = ∅ ⇒ I(Ω) = ∅, ∂∗ Ω = Ω• = ∅ O(Ω) = (Ω)0 = R \[0, 1] I(Ω) = (Ω)1 = ]0, 1[ ⇒ I(Ω) = ]0, 1[ . ∂ (Ω) = (Ω) = {0, 1} ∗

•

However, the complement operation commutes with the other set of operations and determines the same sets (Ω)0 = Ω1 , (Ω)1 = Ω0 , I(Ω) = O(Ω), I(Ω) = O(Ω),

(Ω)• = Ω• , ∂∗ (Ω) = ∂∗ (Ω).

(3.22) (3.23)

The next theorem is a companion to Lemma 5.1 in Chapter 2. Theorem 3.4. The following conditions are equivalent: (i) int Ω = Ω (ii) int Ω = Ω1 ,

and

int Ω = Ω.

int Ω = Ω0 ,

∂Ω = ∂Ω0 = ∂Ω1 (= Ω• ).

If either of the above two conditions is satisﬁed, then int I = int Ω,

int O = int Ω,

I ∩ O = ∂Ω = ∂Ω = ∂Ω.

Proof. (i) ⇒ (ii). First note that Ω1 = (Ω0 ∪ Ω• ) = Ω0 ∩ Ω• ⊂ Ω0 . Similarly, Ω0 ⊂ Ω1 . From the ﬁrst two relations (3.20), int Ω ⊂ Ω1 and int Ω ⊂ Ω0 . Combining them with the two identities in (i), Ω = int Ω ⊂ Ω1 ⊂ Ω0 and Ω = int Ω ⊂ Ω0 ⊂ Ω1 . By taking the intersection of the two chains, ∂Ω ⊂ Ω0 ∩ Ω1 = (Ω0 ∪ Ω1 ) = Ω• . But int Ω ∪ int Ω ∪ ∂Ω and Ω1 ∪ Ω0 ∪ Ω• are two disjoint partitions of RN such that int Ω ⊂ Ω1 , int Ω ⊂ Ω0 , and ∂Ω ⊂ Ω• . We conclude that int Ω = Ω1 , int Ω = Ω0 , ∂Ω = Ω• , and ∂Ω = Ω• = ∂Ω0 = ∂Ω1 since,

3. Lebesgue Measurable Characteristic Functions

223

by Lemma 5.1 in Chapter 2, int Ω = Ω and int Ω = Ω imply ∂ int Ω = ∂Ω = ∂ int Ω. (ii) ⇒ (i). By assumption, Ω = int Ω ∪ ∂Ω = Ω1 ∪ ∂Ω1 = Ω1 = int Ω, Ω = int Ω ∪ ∂Ω = Ω0 ∪ ∂Ω0 = Ω0 = int Ω ⇒ ∂Ω = Ω ∩ Ω = int Ω ∩ Ω = Ω ∩ Ω = ∂ Ω ⇒ ∂Ω = Ω ∩ Ω = Ω ∩ int Ω = Ω ∩ Ω = ∂ Ω.

3.4

The Family of Convex Sets

Convex sets will play a special role here and in the subsequent chapters. We shall say that an equivalence class [Ω] of Lebesgue measurable subsets of D is convex if there exists a convex Lebesgue measurable subset Ω∗ of D such that [Ω] = [Ω∗ ]. We also introduce the notation def

C(D) = {χΩ : Ω convex subset of D} .

(3.24)

Theorem 3.5. Let Ω = ∅ be a convex subset of RN . (i) Ω and int Ω are convex.7 (ii) (int Ω) = Ω = Ω = int Ω. If int Ω = ∅, then int Ω = Ω. If int Ω = ∅ and Ω = RN , then ∂Ω = ∅ and int Ω = ∅. (iii) Given a measurable (resp., bounded measurable) subset D in RN ∀1 ≤ p < ∞,

C(D) is closed in Lploc (D) (resp., Lp (D)).

Proof. (i) For all x and y in Ω, there exist {xn } and {yn } in Ω such that xn → x and yn → y. Then for all λ ∈ [0, 1] Ω λ xn + (1 − λ) yn → λ x + (1 − λ) y ∈ Ω. If int Ω = ∅, then for all x and y in int Ω, there exist rx > 0 and ry > 0 such that B(x, rx ) ⊂ Ω and B(y, ry ) ⊂ Ω. So for all λ ∈ [0, 1] def

xλ = λx + (1 − λ)y ∈B(xλ , λrx + (1 − λ)ry ) ⊂ λB(x, rx ) + (1 − λ)B(y, ry ) ⊂ Ω and xλ ∈ int Ω. (ii) By deﬁnition, int Ω = Ω and int Ω = Ω. If Ω = RN , Ω ⊂ Ω = ∅. This implies int Ω = Ω = Ω = ∅ and int Ω = Ω. If Ω = RN and int Ω = ∅, 7 By

convention ∅ is convex.

224

Chapter 5. Metrics via Characteristic Functions

from Theorem 5.6 in Chapter 2, Ω is locally Lipschitzian, and from Theorem 5.4 in Chapter 2, ∂Ω = ∅, int Ω = ∅, int Ω = Ω, and Ω = Ω. Finally, if int Ω = ∅, then Ω = RN and Ω belongs to a linear subspace L of RN of dimension strictly less than N . Therefore Ω ⊃ Ω ⊃ L, RN = Ω ⊃ Ω ⊃ L = RN , and Ω = Ω. (iii) It is suﬃcient to prove the result for D bounded. For any Cauchy sequence {χn } in C(D), there exist a sequence of convex sets {Ωn } and χΩ in X(D) such that χΩn → χΩ in Lp (D). In particular, there exists a subsequence χk = χΩnk such that χk (x) → χΩ (x) almost everywhere in D. Deﬁne Ω∗ = {x ∈ D : χk (x) → 1 as k → ∞}. def

To show that Ω∗ is convex consider x and y in Ω∗ . There exists K ≥ 1 such that ∀k ≥ K,

|χk (x) − 1| < 1/2,

|χk (y) − 1| < 1/2,

and, since χk is either 0 or 1, for all k ≥ K, χk (x) = 1 = χk (y). By convexity for all λ ∈ [0, 1], xλ = λx + (1 − λ)y ∈ Ωnk and ∀k ≥ K,

χk (λx + (1 − λ)y) = 1 → 1

⇒ x λ ∈ Ω∗ .

But χk (x) → χΩ (x) almost everywhere and lim χk (x) ∈ {0, 1} a.e.

k→∞

So for x ∈ D\Ω∗ , χk (x) → 0 = χΩ∗ (x) almost everywhere in D. By the Lebesguedominated convergence theorem, χk → χΩ∗ in Lp (D) and necessarily χΩ = χΩ∗ . This means that [Ω] is a convex equivalence class. We shall show in section 6.1 (Corollary 1) that for bounded domains D, C(D) is also compact for the Lp (D) topology, 1 ≤ p < ∞.

3.5

Sobolev Spaces for Measurable Domains

The lack of a priori smoothness on Ω may introduce technical diﬃculties in the formulation of some boundary value problems. However, it is possible to relax such boundary value problems from smooth bounded open connected domains Ω to ´sio [7]). For instance consider the homogeneous measurable domains (cf. J.-P. Zole Dirichlet boundary value problem −y = f in Ω,

y = 0 on Γ

(3.25)

over a bounded open connected domain Ω with a boundary Γ of class C 1 and associate with its solution y = y(Ω) the objective function and volume constraint 1 2 J(Ω) = |y − g| dx, dx = π. (3.26) 2 Ω Ω There is a priori no reason to assume that an optimal (minimizing) domain Ω∗ is of class C 1 or is connected. So the problem must be suitably relaxed to a large

3. Lebesgue Measurable Characteristic Functions

225

enough class of domains, which preserves the meaning of the underlying function spaces, the well-posedness of the original problem, and the volume. To extend problem (3.25)–(3.26) to Lebesgue measurable sets, we ﬁrst have to make sense of the Sobolev space for measurable subsets Ω of D. Theorem 3.6. Let D be an open domain in RN . For any Lebesgue measurable subset Ω of RN , the spaces def

H•1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )∇ϕ = 0 a.e. in D , (3.27)

def H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D (3.28) are closed subspaces of H01 (D) and hence Hilbert spaces.8 Similarly, for any χ ∈ co X(D), def

H•1 (χ; D) = ϕ ∈ H01 (D) : (1 − χ)∇ϕ = 0 a.e. in D , (3.29)

def H1 (χ; D) = ϕ ∈ H01 (D) : (1 − χ)ϕ = 0 a.e. in D (3.30) are also closed subspaces of H01 (D) and hence Hilbert spaces. Furthermore9 H1 (Ω; D) ⊂ H•1 (Ω; D),

H1 (χ; D) ⊂ H•1 (χ; D).

Proof. We give only the proof for H•1 (Ω; D). Let {ϕn } in H•1 (Ω; D) be a Cauchy sequence. It converges to an element ϕ in the H01 (D)-topology. Hence {∇ϕn } converges to ∇ϕ in L2 (D). But for all n (1 − χΩ )∇ϕn = 0 in L2 (D). By Schwarz’s inequality the map ϕ → (1 − χΩ )∇ϕ is continuous and (1 − χΩ )∇ϕ = 0 in L2 (D). So ﬁnally ϕ ∈ H•1 (Ω; D) and H•1 (Ω; D) is a closed subspace of H01 (D). Assuming that D is bounded, the variational problems 1 ﬁnd y = y(Ω) ∈ H• (Ω; D) such that ∀ϕ ∈ H•1 (Ω; D), ∇y · ∇ϕ dx = χΩ f ϕ dx, D D 1 ﬁnd y = y(Ω) ∈ H (Ω; D) such that ∀ϕ ∈ H1 (Ω; D), ∇y · ∇ϕ dx = χΩ f ϕ dx D

(3.31)

(3.32)

D

8 Observe that for two open domains in the same equivalence class, the spaces deﬁned by (3.27)– (3.28) coincide. Therefore their functions do not see cracks in the underlying domain. The case of cracks will be handled in Chapter 8 by capacity methods. 9 From L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 130], ∇ϕ = 0 a.e. on {ϕ = 0}.

226

Chapter 5. Metrics via Characteristic Functions

now make sense and have unique solutions for measurable subsets Ω of D (or even χ ∈ co X(D)), and the associated objective function J(Ω) = h χΩ , y(Ω) ,

def

h(χ, ϕ) =

1 2

χ|ϕ − g|2 dx

(3.33)

D

can be minimized over all measurable subsets Ω of D (or all χ ∈ co X(D)) with ﬁxed measure m(Ω) = π. The above problems are now well-posed, and their restriction to smooth bounded open connected domains coincides with the initial problem (3.25)–(3.26). Indeed, if Ω is a connected open domain with a boundary Γ of class C 1 , then Γ has a zero Lebesgue measure and the deﬁnition (3.28) of H•1 (Ω; D) coincides with H1 (Ω; D). Therefore, the problem speciﬁed by (3.31)–(3.33) is a well-deﬁned extension of problem (3.25)–(3.26). In general, when Ω contains holes, the elements of H•1 (Ω; D) are not necessarily equal to zero on the boundary of each hole and can be equal to diﬀerent constants from hole to hole, as in physical problems involving a potential. Example 3.3. Let D = ]−2, 2[ , Ω = ]−2, −1[ ∪ ]1, 2[ , and f = 1 on D. The solution of (3.31) is given by d2 y − 2 =1 dx y(−2) = 0,

on ]−2, −1[ ,

d2 y − 2 = 1 on ]1, 2[ , dx y(2) = 0, dy (1) = 0, dx

dy (−1) = 0, dx 1 y= on [−1, 1], 2 x −(x + 2) 2 in ]−2, −1[ , ⇒ y(x) = 12 in [−1, 1], −(x − 2) x2 in ]1, 2[ .

Example 3.4. Let D = B(0, 2), the open ball of radius 2 centered in 0 in R2 , H = B(0, 1), Ω = D H, and f = 1. In polar coordinates the solution of (3.31) is given by 1 d dy r = 1 in ]1, 2[ , − 1 1 3 r dr dr andy(r) = ln + 2 2 4 y(2) = 0, dy (1) = 0, dr or explicitly y(r) =

1 2 1 2

ln 2r + 1 − ln 12 + 34

r2 4

in ]1, 2[ , in [0, 1].

in [0, 1],

3. Lebesgue Measurable Characteristic Functions

227

The last example is a special case that retains the characteristics of the onedimensional example. In higher dimensions the normal derivative is not necessarily zero on the “internal boundary” of Ω. Example 3.5. Let D = B(0, 1) in R2 , let H be a bounded open connected hole in D such that H ⊂ D, and let Ω = D H. Then it can be checked that the solution of (3.31) is of the form −∆y = f in Ω,

y = 0 on ∂D,

where the constant c on H is determined by the condition ∂y ∀ϕ ∈ H•1 (Ω; D), ϕ dγ = 0, ∂Ω ∂n or equivalently

∂H

∂y dγ = 0. ∂n

Example 3.6. Let D = ]−2, 2[ × ]−2, 2[ and let Ω1 and Ω2 be two open squares in G = ]−1, 1[ × ]−1, 1[ such that Ω1 ⊂ G, Ω2 ⊂ G, and Ω1 ∩ Ω2 = ∅. Deﬁne the domain as Ω = Ω0 ∪ Ω1 ∪ Ω2 ,

Ω0 = D\G

(cf. Figure 5.2).

Then the solution of (3.31) is characterized by −∆y = f in Ω,

y = 0 on ∂D,

y = c on G\ Ω1 ∪ Ω2 ,

where the constant c is determined by the condition ∂y ∂y ∂y dγ + dγ + dγ = 0 int ∂n ∂n ∂n ∂Ω0 ∂Ω1 ∂Ω2 and ∂Ωint 0 = ∂G, the interior boundary of Ω0 .

Ω2

Ω0 = D\G

Ω1

Figure 5.2. Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 .

228

4

Chapter 5. Metrics via Characteristic Functions

Some Compliance Problems with Two Materials

As a ﬁrst example consider the optimal compliance problem where the optimization variable is the distribution of two materials with diﬀerent physical characteristics within a ﬁxed domain D. It cannot a priori be assumed that the two regions are separated by a smooth boundary and that each region is connected. The optimal solution may lead to a nonsmooth interface and even to the mixing of the two materials. This type of solution occurs in control or optimization problems over a bounded nonconvex subset of a function space. In general, their relaxed solution lies in the closed convex hull of that subset. In control theory, the phenomenon is known as chattering control. To illustrate this approach it is best to consider a generic example. In section 4.1, we consider a variation of the optimal design problem ´a and K. Malanowski [1] in 1970 and then discuss its original studied by J. Ce version in section 4.2. The variation of the problem is constructed in such a way that the form of the associated variational equation is similar to the one of Example 3.1, which provided a counterexample to the weak continuity of the solution. Yet the maximization of the minimum energy yields as solution a characteristic function in both cases. However, this is no longer true for the minimization of the same minimum energy as was shown by F. Murat and L. Tartar [1, 3].

4.1

Transmission Problem and Compliance

Let D ⊂ RN be a bounded open domain with Lipschitzian boundary ∂D. Assume that the domain D is partitioned into two subdomains Ω1 and Ω2 separated by a smooth boundary ∂Ω1 ∩ ∂Ω2 as illustrated in Figure 5.3. Domain Ω1 (resp., Ω2 ) is made up of a material characterized by a constant k1 > 0 (resp., k2 > 0). Let y be the solution of the transmission problem − k1 y = f in Ω1 , −k2 y = 0 in Ω2 , (4.1) ∂y ∂y y = 0 on ∂D, k1 + k2 = 0 on ∂Ω1 ∩ ∂Ω2 , ∂n1 ∂n2 where n1 (resp., n2 ) is the unit outward normal to Ω1 (resp., Ω2 ) and f is a given function in L2 (D). Our objective is to maximize the equivalent of the compliance J(Ω1 ) = − f y dx (4.2) Ω1

Ω1

Ω2

Figure 5.3. Fixed domain D and its partition into Ω1 and Ω2 .

4. Some Compliance Problems with Two Materials

229

over all domains Ω1 in D. In mechanics the compliance is associated with the total work of the body forces f . Denote by Ω the domain Ω1 . By complementarity Ω2 = D Ω and let χ = χΩ . Problem (4.1) can be rewritten in the following variational form: ﬁnd y = y(χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D), [k1 χ + k2 (1 − χ)] ∇y · ∇ϕ dx = χf ϕ dx. D

(4.3)

D

For k1 > 0 and k2 > 0 and all χ in X(D) def

k(x) = k1 χ(x) + k2 (1 − χ(x)), 0 < min{k1 , k2 } ≤ k(x) ≤ max{k1 , k2 } a.e. in D.

(4.4)

By the Lax–Milgram theorem, the variational equation (4.3) still makes sense and has a unique solution y = y(χ) in H01 (D) that coincides with the solution of the boundary value problem −div (k ∇y) = χf in D,

y = 0 on ∂D.

(4.5)

Notice that the high-order term has the same form as the term in Example 3.1. As for the objective function, it can be rewritten as J(χ) = − χf y(χ) dx. (4.6) D

Thus the initial boundary value problem (4.1) has been transformed into the variational problem (4.3), and the initial objective function (4.2) into (4.6). Both make sense for χ in X(D) and even in co X(D). The family of characteristic functions X(D) and its closed convex hull co X(D) have been deﬁned and characterized in (3.2) and (3.3). The objective function (4.6) can be further rewritten as a minimum J(χ) = for the energy function def

min

ϕ∈H01 (D)

E(χ, ϕ)

(4.7)

(k1 χ + k2 (1 − χ)) |∇ϕ|2 − 2 χf ϕ dx

E(χ, ϕ) =

(4.8)

D

associated with the variational problem (4.3). The initial optimal design problem becomes a max-min problem max J(χ) = max χ∈X(D)

min

χ∈X(D) ϕ∈H01 (D)

E(χ, ϕ),

where X(D) can be considered as a subset of Lp (D) for some p, 1 ≤ p < ∞.

(4.9)

230

Chapter 5. Metrics via Characteristic Functions This problem can also be relaxed to functions χ with value in [0, 1] max

J(χ) =

χ∈co X(D)

max

min

χ∈co X(D) ϕ∈H01 (D)

E(χ, ϕ).

(4.10)

We shall refer to this problem as the relaxed problem. ´a and K. Malanowski These formulations have been introduced by J. Ce [1], who used the variable k(x) and the following equality constraint on its integral: γ − k2 m(D) k(x) dx = γ or χ(x) dx = k1 − k 2 D D for some appropriate γ > 0. Moreover the force f in (4.1)–(4.3) was exerted everywhere in D and not only in Ω. So the objective function (4.2)–(4.6) was the integral of f y over all of D. We shall see in section 4.2 that the fact that the support of f is Ω or all of D does not aﬀect the nature of the results. In both maximization problems (4.1)–(4.2) and (4.3)–(4.6) it is necessary to introduce a volume constraint in order to avoid a trivial solution. Notice that by using (4.3) with ϕ = y(χ) the objective function (4.6) becomes (k1 χ + k2 (1 − χ)) |∇y(χ)|2 dx ≤ 0. J(χ) = − D

So maximizing J(χ) is equivalent to minimizing the integral of k(x) |∇y|2 . Therefore, χ = 0 (only material k2 ) is a maximizer since the corresponding solution of (4.3) is y = 0. In order to eliminate this situation we introduce the following constraint on the volume of material k1 : χ dx ≥ α > 0 (4.11) D

for some α, 0 < α ≤ m(D). The case α = m(D) yields the unique solution χ = 1 (only material k1 ). So we can further assume that α < m(D). For 0 < α < m(D), the optimal design problem becomes max J(χ) = max χ∈X(D) χ∈X(D) D

χ dx≥α

D

min

E(χ, ϕ)

(4.12)

min

E(χ, ϕ).

(4.13)

ϕ∈H01 (D)

χ dx≥α

and its relaxed version max

χ∈co X(D) χ dx≥α D

J(χ) =

max

χ∈co X(D) ϕ∈H01 (D) χ dx≥α D

We shall now show that problem (4.13) has a unique solution χ∗ in co X(D) and that χ∗ is a characteristic function, χ∗ ∈ X(D), for which the inequality constraint is saturated: ∗ χ∗ dx = α. (4.14) χ ∈ X(D) and D

4. Some Compliance Problems with Two Materials

231

At this juncture it is advantageous to incorporate the volume inequality constraint into the problem formulation by introducing a Lagrange multiplier λ ≥ 0. The formulation of the relaxed problem becomes max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ),

def

χ dx − α

G(χ, ϕ, λ) = E(χ, ϕ) + λ

(4.15)

D

and E(χ, ϕ) is the energy function given in (4.7). We ﬁrst establish the existence of saddle point solutions to problem (4.15). We use a general result in I. Ekeland and R. Temam [1, Prop. 2.4, p. 164]. The set co X(D) is a nonempty bounded closed convex subset of L2 (D) and the set H01 (D)×[R+ ∪{0}] is trivially closed and convex. The function G is concave-convex with the following properties: ∀χ ∈ co X(D), (ϕ, λ) → G(χ, ϕ, λ) is convex, continuous, and (4.16) G(χ0 , ϕ, λ) = +∞; ∃χ0 ∈ co X(D) such that ϕ 1lim +|λ|→∞ H

∀ϕ ∈

H01 (D),

∀λ ≥ 0,

0

the map χ → G(χ, ϕ, λ) is aﬃne and

2

continuous for L (D)-strong.

(4.17)

For the ﬁrst condition recall that 0 < α < m(D) and pick χ0 = 1 on D. To check the second condition pick any sequence {χn } in co X(D) which converges to some χ in L2 (D)-strong. Then the sequence also converges in L2 (D)-weak and by Lemma 3.1 in L∞ (D)-weak∗ . Hence G(χn , ϕ, λ) converges to G(χ, ϕ, λ). ˆ is of the form X × Y ⊂ co X(D) × {H 1 (D) × The set of saddle points (χ, ˆ y, λ) 0 + [R ∪ {0}]} and is completely characterized by the following variational equation and inequalities (cf. I. Ekeland and R. Temam [1, Prop. 1.6, p. 157]): 1 [k2 + (k1 − k2 )χ] ˆ ∇y · ∇ϕ − χf ˆ ϕ dx = 0, (4.18) ∀ϕ ∈ H0 (D), D. / ˆ (χ − χ) (k1 − k2 )|∇y|2 − 2f y + λ ˆ dx ≤ 0, (4.19) ∀χ ∈ co X(D), D ˆ = 0, ˆ ≥ 0. χ ˆ dx − α λ χ ˆ dx − α ≥ 0, λ (4.20) D

D

But for each χ ∈ co X(D) there exists a unique y(χ) solution of (4.18) and then

∀χ ˆ ∈ X, {χ} ˆ × Y ⊂ {χ} ˆ × {y(χ)} ˆ × [R+ ∪ {0}] . Therefore, y(χ) ˆ is independent of χ ˆ ∈ X, that is, Y = {y} × Λ and ˆ ∈ Λ, ∀χ ˆ ∈ X, λ ˆ inequality (4.19) is For each λ, terization of the maximizer χ ˆ ∈ X: if 1, χ(x) ˆ = ∈ [0, 1], if 0, if

y(χ) ˆ = y.

completely equivalent to the following characˆ > 0, (k1 − k2 )|∇y|2 − 2f y + λ 2 ˆ = 0, (k1 − k2 )|∇y| − 2f y + λ 2 ˆ < 0. (k1 − k2 )|∇y| − 2f y + λ

(4.21)

232

Chapter 5. Metrics via Characteristic Functions

Associate with an arbitrary λ ≥ 0 the sets D+ (λ) = {x ∈ D : (k1 − k2 )|∇y|2 − 2f y + λ > 0}, D0 (λ) = {x ∈ D : (k1 − k2 )|∇y|2 − 2f y + λ = 0}.

(4.22)

In order to complete the characterization of the optimal triplets, we need the following general result. Lemma 4.1. Consider a function G:A×B →R

(4.23)

for some sets A and B. Deﬁne def

g = inf sup G(x, y), x∈A y∈B

def

A0 =

x ∈ A : sup G(x, y) = g ,

def

h = sup inf G(x, y), y∈B x∈A

def

B0 =

(4.24)

y∈B

y ∈ B : inf G(x, y) = h . x∈A

(4.25)

When g = h the set of saddle points (possibly empty) will be denoted by def

S = {(x, y) ∈ A × B : g = G(x, y) = h} .

(4.26)

Then the following hold. (i) In general h ≤ g and ∀(x0 , y0 ) ∈ A0 × B0 ,

h ≤ G(x0 , y0 ) ≤ g.

(4.27)

(ii) If h = g, then S = A0 × B0 . Proof. (i) If A0 × B0 = ∅, there is nothing to prove. If there exist x0 ∈ A0 and y0 ∈ B0 , then by deﬁnition h = inf G(x, y0 ) ≤ G(x0 , y0 ) ≤ sup G(x0 , y) = g. x∈A

(4.28)

y∈B

(ii) If h = g, then in view of (4.26)–(4.27), A0 × B0 ⊂ S. Conversely if there exists (x0 , y0 ) ∈ S, then h = G(x0 , y0 ) = g and by the deﬁnitions of A0 and B0 , (x0 , y0 ) ∈ A0 × B0 . ˆ the characteristic function Associate with an arbitrary solution (χ, ˆ y, λ), χλˆ =

χD+ (λ) ˆ , 0,

ˆ = ∅, if D+ (λ) ˆ = ∅. if D+ (λ)

(4.29)

Then from (4.21) ˆ = χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ ˆ a.e. in D χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ λ

4. Some Compliance Problems with Two Materials and

233

ˆ [χk ˆ 1 + (1 − χ)k ˆ 2 ] |∇y| − 2χf ˆ y dx + λ

ˆ = G(χ, ˆ y, λ) D

.

χ ˆ dx − α

2

/

D

ˆ dx − λα ˆ k2 |∇y|2 + χ ˆ (k1 − k2 )|∇y|2 − 2f y + λ

= D = D

. / ˆ dx − λα ˆ k2 |∇y|2 + χλˆ (k1 − k2 )|∇y|2 − 2f y + λ

ˆ = G(χλˆ , y, λ). ˆ is also a saddle point. So there exists a maximizer From Lemma 4.1, (χλˆ , y, λ) χλˆ ∈ X(D) that is a characteristic function and necessarily max

min

χ∈X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ) =

max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ).

ˆ ∈ Λ such that λ ˆ > 0, then by But we can show more than that. If there exists λ construction χ ˆ ≥ χλˆ and by (4.20) α= χ ˆ dx ≥ χλˆ dx = α. D

D

ˆ then χ ˆ = χλˆ . But, always by Since χ ˆ = χλˆ = 1 almost everywhere in D+ (λ), construction, χλˆ is independent of χ. ˆ Thus the maximizer is unique, it is a characteristic function, and its integral is equal to α. The case Λ = {0} is a degenerate one. Set ϕ = y in (4.18) and regroup the terms as follows: (k2 + (k1 − k2 )χ) ˆ |∇y|2 − f y χ ˆ dx 0= D k1 − k2 k1 − k2 2 2 χ ˆ |∇y| dx + χ ˆ |∇y| − f y χ ˆ dx. = k2 + 2 2 D D The integrand of the ﬁrst integral is positive. From the characterization of χ, ˆ the integrand of the second one is also positive. Hence they are both zero almost everywhere in D. As a result m(D+ (0)) = 0

and ∇y = 0.

ˆ = Therefore y = 0 in D. Hence the saddle points are of the general form (χ, ˆ y, λ) (χ, ˆ 0, 0), χ ˆ ∈ X. In particular, D+ (0) = ∅ and from our previous considerations χ∅ ; that is, χ ˆ = 0 is a solution. But this is impossible since D χ ˆ dx ≥ α > 0. ˆ = 0 cannot occur. Therefore λ In conclusion, there exists a (unique for α > 0) maximizer χ∗ in co X(D) that is in fact a characteristic function, and necessarily max χ∈X(D) D

χ dx≥α

min

ϕ∈H01 (D)

E(χ, ϕ) =

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α D

E(χ, ϕ).

(4.30)

234

Chapter 5. Metrics via Characteristic Functions

Moreover, for 0 < α ≤ m(D),

χ∗ dx = α

(4.31)

D

and χ∗ is the unique solution of the problem with an equality constraint: max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ) =

χ dx=α

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α

E(χ, ϕ).

(4.32)

D

As a numerical illustration of the theory consider (4.1) over the diamond-shaped domain D = {(x, y) : |x| + |y| < 1}

(4.33)

with the function f of Figure 5.4. This function has a sharp peak in (0, 0) which has been scaled down in the picture. The variational form (4.3) of the boundary value problem was approximated by continuous piecewise linear ﬁnite elements on each triangle, and the function χ by a piecewise constant function on each triangle. The constant on each triangle was constrained to lie between 0 and 1 together with the global constraint on its integral over the whole domain D. Figure 5.5 shows the optimal partition (the domain has been rotated by 45 degrees to save space). The ˆ m ), where χ grey triangles correspond to the region D0 (λ ˆ ∈ [0, 1]. The presence of this grey zone in the approximated problem is due to the fact that equality for the total area where χ ˆ = 1 could not be exactly achieved with the chosen triangulation of the domain. Thus the problem had to adjust the value of χ ˆ between 0 and 1 in

60 50 40 30 20

1 0.8

10

0.6 0.4

0 1

0.2 0.8

0 0.6

0.4

−0.2 0.2

−0.4 0

−0.2

−0.6 −0.4

−0.6

−0.8 −0.8

−1

−1

Figure 5.4. The function f (x, y) = 56 (1 − |x| − |y|)6 .

4. Some Compliance Problems with Two Materials

235

Figure 5.5. Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1 (white) for the problem of section 4.1. a few triangles in order to achieve equality for the integral of χ. ˆ For this example the Lagrange multiplier associated with the problem is strictly positive.

4.2

The Original Problem of C´ ea and Malanowski

We now put the force f everywhere in the ﬁxed domain D. The same technique and the same conclusion can be drawn: there exists at least one maximizer of the compliance, which is a characteristic function. For completeness we give the main elements below. Fix the bounded open Lipschitzian domain D in RN and let Ω be a smooth subset of D. Let y be the solution of the transmission problem − k1 y = f in Ω, −k2 y = f in D\Ω, y = 0 on ∂D, (4.34) ∂y ∂y k1 + k2 = 0 on ∂Ω ∩ D, ∂n1 ∂n2 where n1 (resp., n2 ) is the unit outward normal to Ω (resp., D Ω) and f is a given function in L2 (D). Again problem (4.34) can be reformulated in terms of the characteristic function χ = χΩ : ﬁnd y = y(χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D), [k1 χ + k2 (1 − χ)] ∇y · ∇ϕ dx = f ϕ dx, D

(4.35)

D

with the objective function

J(χ) = −

f y(χ) dx D

(4.36)

236

Chapter 5. Metrics via Characteristic Functions

to be maximized over all χ ∈ X(D). As in the previous case, the function J(χ) can be rewritten as the minimum of the energy function def [k1 χ + k2 (1 − χ)] |∇ϕ|2 − 2f ϕ dx (4.37) E(χ, ϕ) = D

over H01 (D), J(χ) =

min

ϕ∈H01 (D)

E(χ, ϕ),

(4.38)

and we have the relaxed max-min problem max

min

χ∈co X(D) ϕ∈H01 (D)

E(χ, ϕ).

(4.39)

Without constraint on the integral of χ, the problem is trivial and χ = 1 (resp., χ = 0) if k1 > k2 (resp., k2 > k1 ). In other words it is optimal to use only the strong material. In order to make the problem nontrivial assume that the strong material is k1 , that is, k1 > k2 , and put an upper bound on the volume of material k1 which occupies the part Ω of D: χ dx ≤ α, 0 < α < m(D). (4.40) D

The case α = 0 trivially yields χ = 0. Under assumption (4.40) the case χ = 1 with only the strong material k1 is no longer admissible. Thus we consider for 0 < α < m(D) the problem max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≤α

E(χ, ϕ).

(4.41)

D

We shall now show that problem (4.41) has a unique solution χ∗ in co X(D) and that in fact χ∗ is a characteristic function for which the inequality constraint is saturated: χ∗ dx = α. (4.42) D

This solves the original problem of C´ea and Malanowski with the equality constraint: max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ).

(4.43)

χ dx=α

As in section 4.1 it is convenient to reformulate the problem with a Lagrange multiplier λ ≥ 0 for the constraint inequality (4.40): max min G(χ, ϕ, λ), G(χ, ϕ, λ) = E(χ, ϕ) − λ χ dx − α . (4.44) 1 χ∈X(D) ϕ∈H0 (D) λ≥0

D

4. Some Compliance Problems with Two Materials

237

We then relax the problem to co X(D): max

min

χ∈co X(D) ϕ∈H01 (D) λ≥0

G(χ, ϕ, λ).

(4.45)

By the same arguments as the ones used in section 4.1 we have the existence of ˆ which are completely characterized by saddle points (χ, ˆ y, λ) ∀ϕ ∈ H01 (D), [k2 + (k1 − k2 )χ]∇y ˆ · ∇ϕ − f ϕ dx = 0, (4.46) D . / ˆ (χ − χ) (k1 − k2 )|∇y|2 − λ ˆ dx ≤ 0, (4.47) ∀χ ∈ co X(D), D ˆ = 0, ˆ ≥ 0. χ ˆ dx − α λ χ ˆ dx − α ≤ 0, λ (4.48) D

D

As before y is unique and the set of saddle points of G is the closed convex set

X × {y} × Λ ⊂ X(D) × {y} × {λ : λ ≥ 0} . ˆ m ≥ 0. The closed convex set Λ has a minimal element λ ˆ each maximizer χ For each λ, ˆ ∈ X is necessarily of the form ˆ > 0, if (k1 − k2 )|∇y|2 − λ 1, 2 ˆ = 0, χ(x) ˆ = ∈ [0, 1], if (k1 − k2 )|∇y| − λ 2 ˆ < 0. 0, if (k1 − k2 )|∇y| − λ Associate with each λ > 0 the sets

D+ (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ > 0 ,

D0 (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ = 0 ,

D− (λ) = x ∈ D : (k1 − k2 )|∇y|2 − λ < 0 .

(4.49)

(4.50) (4.51) (4.52)

Deﬁne the characteristic function χm = χD+ (λm ) . By construction all χ ˆ ∈ X are of the form ˆ χ ˆ dx ≤ α. (4.53) χm ≤ χ, D

ˆ m ) is also a saddle point of G. Therefore, we Again it is easy to show that (χm , y, λ have a maximizer χm ∈ X(D) over co X(D) which is a characteristic function and χm dx ≤ χ ˆ dx ≤ α. D

D

ˆ > 0 in Λ, then for all χ If there exists λ ˆ∈X χm dx = α = χ ˆ dx. D

D

238

Chapter 5. Metrics via Characteristic Functions

As a result the maximizer χm is unique, it is a characteristic function, and its integral is equal to α. The case Λ = {0} cannot occur since the triplet (1, y1 , 0) would be a saddle point, where y1 is the solution of the variational equation (4.34) for χ = 1. To see this, ﬁrst observe that for k1 > k2 ∀χ, ϕ,

G(χ, ϕ, 0) = E((χ, ϕ) ≤ E((1, ϕ) = G(1, ϕ, 0).

As a result inf G(χ, ϕ, λ) ≤ inf G(χ, ϕ, 0) ≤ inf G(1, ϕ, 0), ϕ

ϕ,λ

ϕ

inf G(1, ϕ, 0) = G(1, y1 , 0) ≤ sup inf G(χ, ϕ, 0) ϕ

χ

ϕ

⇒ sup inf G(χ, ϕ, λ) ≤ sup inf G(χ, ϕ, 0) = G(1, y1 , 0). χ ϕ,λ

χ

ϕ

But we know that there exists a unique y ∈ H01 (D) such that sup inf G(χ, ϕ, 0) ≤ sup G(χ, y, 0) = inf sup G(χ, ϕ, λ). χ

ϕ

ϕ,λ χ

χ

So from the above two inequalities, sup inf G(χ, ϕ, λ) = G(1, y1 , 0) = inf sup G(χ, ϕ, λ), χ ϕ,λ

ϕ,λ χ

(1, y1 , 0) is a saddle point of G, and α≥ χ dx = m(D). D

This contradicts the fact that α < m(D). In conclusion in all cases there exists a (unique when 0 ≤ α < m(D)) maximizer χ∗ in co X(D), which is in fact a characteristic function, and max χ∈X(D) D

min

ϕ∈H01 (D)

E(χ, ϕ) =

χ dx≥α

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α

E(χ, ϕ).

D

Moreover, for 0 ≤ α < m(D)

χ∗ dx = α. D

This is precisely the solution of the original problem of C´ea and Malanowski and max χ∈X(D) D

χ dx=α

min

ϕ∈H01 (D)

E(χ, ϕ) =

max

min

χ∈co X(D) ϕ∈H01 (D) χ dx≥α

E(χ, ϕ).

D

Again, as an illustration of the theoretical results, consider (4.34) over the diamondshaped domain D deﬁned in (4.33) with the function f of Figure 5.4 in section 4.1.

4. Some Compliance Problems with Two Materials

239

The variational form (4.35) of the boundary value problem was approximated in the same way as the variational form (4.3) of (4.1) in section 4.1. Figure 5.6 shows the optimal partition of the domain (rotated 45 degrees). The grey triangles ˆ m ), where χ ˆ ∈ [0, 1]. The black region corresponds correspond to the region D0 (λ ˆ m /(k1 − k2 ) and the white region to the ones where to the points where |∇y|2 > λ ˆ m /(k1 − k2 ), as can be readily seen in Figure 5.6. For this example the |∇y|2 < λ Lagrange multiplier associated with the problem is strictly positive. It is interesting to compare this computation with the one of Figure 5.5 in section 4.1, where the support of the force was restricted to Ω.

Figure 5.6. Optimal distribution and isotherms with k1 = 2 (black) and k2 = 1 (white) for the problem of C´ea and Malanowski. Remark 4.1. The formulation and the results remain true if the generic variational equation is replaced by a variational inequality (unilateral problem) max χ∈X(D) D

4.3

min

ϕ∈{ψ∈H01 (D):ψ≥0 in D}

E(χ, ϕ).

χ dx=α

Relaxation and Homogenization

In sections 4.1 and 4.2 the possible homogenization phenomenon predicted in Example 3.1 of section 3.2 did not take place, and in both examples the solution was a characteristic function. Yet, it occurs when the maximization is changed to a ´a and K. Malanowski [1] of section 4.2: minimization in the problem of J. Ce inf χ∈X(D) D

χ dx=α

min

ϕ∈H01 (D)

E(χ, ϕ).

240

Chapter 5. Metrics via Characteristic Functions

In 1985 F. Murat and L. Tartar [1] gave a general framework to study this class of problems by relaxation. It is based on the use of L. C. Young [2]’s generalized functions (measures). They present a fairly complete analysis of the homogenization theory of second-order elliptic problems of the form ∂ ∂u aij (x) = f. − ∂xi ∂xj ij

They give as examples the maximization and minimization versions of the problem of section 4.2. This material, which would have deserved a whole chapter in this book, is fortunately available in English (cf. F. Murat and L. Tartar [3]). More results on composite materials can be found in the book edited by A. Cherkaev and R. Kohn [1], which gathers a selection of translations of key papers originally written in French and Russian.

5

Buckling of Columns

One of the very early optimal design problem was formulated by Lagrange in 1770 (cf. I. Todhunter and K. Pearson [1]) and later studied by T. Clausen in 1849. It consists in ﬁnding the best proﬁle of a vertical column to prevent buckling. This problem and other problems related to columns have been revisited in a series of papers by S. J. Cox [1], S. J. Cox and M. L. Overton [1], S. J. Cox [2], and S. J. Cox and C. M. McCarthy [1]. Since Lagrange many authors have proposed solutions, but a complete theoretical and numerical solution for the buckling of a column was given only in 1992 by S. J. Cox and M. L. Overton [1]. Consider a normalized column of unit height and unit volume. Denote by the magnitude of the normalized axial load and by u the resulting transverse displacement. Assume that the potential energy is the sum of the bending and elongation energies 1 1 2 EI |u | dx − |u |2 dx, 0

0

where I is the second moment of area of the column’s cross section and E is its Young’s modulus. For suﬃciently small load the minimum of this potential energy with respect to all admissible u is zero. Euler’s buckling load λ of the column is the largest for which this minimum is zero. This is equivalent to ﬁnding the following minimum: 1 EI |u |2 dx def 0 λ = inf , (5.1) 1 0=u∈V |u |2 dx 0 where V = H02 (0, 1) corresponds to the clamped case, but other types of boundary conditions can be contemplated. This is an eigenvalue problem with a special Rayleigh quotient. Assume that E is constant and that the second moment of area I(x) of the column’s cross section at the height x, 0 ≤ x ≤ 1, is equal to a constant c times its

5. Buckling of Columns

241

cross-sectional area A(x),

I(x) = c A(x)

1

⇒

A(x) dx = 1. 0

Normalizing λ by cE and taking into account the engineering constraints ∃0 < A0 < A1 , ∀x ∈ [0, 1],

0 < A0 ≤ A(x) ≤ A1 ,

we ﬁnally get

1

A |u |2 dx , 1 0=u∈V A∈A |u |2 dx 0 1 def 2 A(x) dx = 1 . A = A ∈ L (0, 1) : A0 ≤ A ≤ A1 and sup λ(A),

def

λ(A) =

inf

0

(5.2) (5.3)

0

This problem can also be reformulated by rewriting 1 def 1 − A0 A(x) = A0 + χ(x) (A1 − A0 ), χ(x) dx = α = A1 − A0 0 for some χ ∈ co X([0, 1]). Clearly the problem makes sense only for 0 < A0 ≤ 1. Then 1 [A0 + (A1 − A0 ) χ] |u |2 dx def 0 ˜ ˜ λ(χ), λ(χ) = inf . (5.4) sup 1 0=u∈V |u |2 dx χ∈co X([0,1]) 0 1

0

χ(x) dx=α

Rayleigh’s quotient is not a nice convex concave function with respect to (v, A), and its analysis necessitates tools diﬀerent from the ones of section 4. One of the original elements of the paper of S. J. Cox and M. L. Overton [1] was to replace Rayleigh’s quotient by G. Auchmuty [1]’s dual variational principle for the eigenvalue problem (5.1). We ﬁrst recall the existence of solution to the minimization of Rayleigh’s quotient. In what follows we shall use the norm u L2 for the space H01 (0, 1) and u L2 for the space H02 (0, 1). Theorem 5.1. There exists at least one nonzero solution u ∈ V to the minimization problem 1 A |u |2 dx def 0 λ(A) = inf . (5.5) 1 0=u∈V |u |2 dx 0 Then λ(A0 ) > 0 and for all A ∈ A, λ(A) ≥ λ(A0 ), and 1 1 ∀v ∈ V, |v |2 dx ≤ λ(A0 )−1 A |v |2 dx. 0

(5.6)

0

The solutions are completely characterized by the variational equation: 1 1 A u v dx = λ(A) u v dx. ∃u ∈ V, ∀v ∈ V, 0

0

(5.7)

242

Chapter 5. Metrics via Characteristic Functions

Proof. The inﬁmum is bounded below by 0 and is necessarily ﬁnite. Let {un } be a minimizing sequence such that un L2 = 1. Then the sequence un L2 (D) is bounded. Hence {un } is a bounded sequence in H 2 (0, 1) and there exist u ∈ H 2 (0, 1) and a subsequence, still indexed by n, such that un u in H02 (0, 1)-weak. Therefore the subsequence strongly converges in H01 (0, 1) and 1 1 1 1 2 2 2 |un | dx → |u | dx and A |u | dx ≤ lim inf A |un |2 dx 1= 0

1 ⇒

0

0

A |u |2 dx

1 0

n→∞

0

|u |2

dx

1

A |un |2 dx

≤ lim inf 0 1 n→∞

0

|un |2 dx

0

= λ(A),

and, by the deﬁnition of λ(A), u ∈ H02 (0, 1) is a minimizing element. Notice that & 1 2 A |u | dx 0 0 ∀A ∈ A, λ(A0 ) = inf : ∀u ∈ H02 (0, 1), u = 0 1 |2 dx |u 0 & 1 A |u |2 dx 2 0 : ∀u ∈ H0 (0, 1), u = 0 = λ(A) ≤ inf 1 |u |2 dx 0 and similarly λ(A) ≤ λ(A1 ). If λ(A0 ) = 0, we repeat the above construction and end up with an element u ∈ H02 (0, 1) such that 1 1 1 |u |2 dx = 1 and A0 |u |2 dx = A0 |u |2 dx = 0, 0

0

0

which is impossible in H02 (0, 1) since A0 = 0 and u = 0 implies u = 0. If u = 0, then Rayleigh’s quotient is diﬀerentiable and its directional semiderivative in the direction v is given by 1 1 1 A u v dx u v dx 2 0 −2 A |u | dx 0 4 . 2 u 2L2 u L2 0 A nonzero solution u of the minimization problem is necessarily a stationary point, and for all v ∈ H02 (0, 1) 1 1 A0 |u |2 dx 1 A u v dx = 0 1 u v dx = λ(A) u v dx. |2 dx |u 0 0 D 0 Conversely any nonzero solution of (5.7) is necessarily a minimizer of Rayleigh’s quotient. The dual variational principle of G. Auchmuty [1] for the eigenvalue problem (5.5) can be chosen as

def µ(A) = inf L(A, v) : v ∈ H02 (0, 1) , (5.8) 1 1/2 1 def 1 L(A, v) = A |v |2 dx − |v |2 dx . (5.9) 2 0 0

5. Buckling of Columns

243

Theorem 5.2. For each A ∈ A, there exists at least one minimizer of L(A, v), µ(A) = −

1 , 2λ(A)

(5.10)

and the set of minimizers of (5.8) is given by u is solution of (5.7) and def 1/2 E(A) = u ∈ H02 (0, 1) : 1 . |u |2 dx = 1/λ(A)

(5.11)

0

Proof. The existence of the solution follows from the fact that v → L(A, v) is weakly lower semicontinuous and coercive. From Theorem 5.1 the set E(A) is not empty, and for any u ∈ E(A) µ(A) ≤ L(A, u) = −

1 < 0. 2λ(A)

(5.12)

Therefore, the minimizers of (5.7) are diﬀerent from the zero functions. For u = 0, the function L(A, u) is diﬀerentiable and its directional semiderivative is given by

1

dL(A, u; v) = 0

1 A u v dx − u L2

1

u v dx,

(5.13)

0

and any minimizer u of L(A, v) is a stationary point of dL(A, u; v), that is,

1

∀v ∈ H02 (0, 1),

A u v dx −

0

1 u L2

1

u v dx = 0.

(5.14)

0

Therefore, u is a solution of the eigenvalue problem with λ =

1 u

L2

⇒ −

1 1 = µ(A) ≤ − 2λ 2λ(A)

from inequality (5.12). By the minimality of λ(A), we necessarily have λ = λ(A), and this concludes the proof of the theorem. Theorem 5.3. (i) The set A is compact in the L2 (0, 1)-weak topology. (ii) The function A → µ(A) is concave and upper semicontinuous with respect to the L2 (0, 1)-weak topology. (iii) There exists A in A which maximizes µ(A) over A and, a fortiori, which maximizes λ(A) over A. Proof. (i) A is convex, bounded, and closed in L2 (0, 1). Hence it is compact in L2 (0, 1) weak.

244

Chapter 5. Metrics via Characteristic Functions (ii) Concavity. For all λ in [0, 1] and A and A in A, L(λA + (1 − λ)A , v) = λL(A, v) + (1 − λ)L(A , v) ≥ λ inf L(A, v) + (1 − λ) inf L(A , v) v∈V

v∈V

⇒ µ(λA + (1 − λ)A ) = inf L(λA + (1 − λ)A , v) ≥ λµ(A) + (1−λ)µ(A ). v∈V

Upper semicontinuity. Let uA ∈ A be a minimizer of L(A, v) and {An } be a sequence converging to A in the L2 (0, 1)-weak topology. By Lemma 3.1, {An } converges to A in the L∞ (0, 1)-weak topology. Then 1 1 µ(An ) − µ(A) ≤ L(An , uA ) − L(A, uA ) = (An − A) |uA |2 dx 1 2 0 1 (An − A) |uA |2 dx = 0. ⇒ lim sup µ(An ) − µ(A) ≤ lim n→∞ 2 0 n→∞ (iii) The existence of solution follows from the concavity and upper semicontinuity of µ(A). Hence it is weakly upper semicontinuous over the weakly compact subset A of L2 (0, 1). The existence of a minimizer of λ(A) follows from identity 5.10.

6

Caccioppoli or Finite Perimeter Sets

The notion of a ﬁnite perimeter set has been introduced and developed mainly by R. Caccioppoli [1] and E. De Giorgi [1] in the context of J. A. F. Plateau [1]’s problem, named after the Belgian physicist and professor (1801–1883), who did experimental observations on the geometry of soap ﬁlms. A modern treatment of this subject can be found in the book of E. Giusti [1]. One of the diﬃculties in studying the minimal surface problem is the description of such surfaces in the usual language of diﬀerential geometry. For instance, the set of possible singularities is not known. Finite perimeter sets provide a geometrically signiﬁcant solution to Plateau’s problem without having to know ahead of time what all the possible singularities of the solution can be. The characterization of all the singularities of the solution is a diﬃcult problem which can be considered separately. This was a very fundamental contribution to the theory of variational problems, where the optimization variable is the geometry of a domain. This point of view has been expanded, and a variational calculus was developed by F. J. Almgren, Jr. [1]. This is the theory of varifolds. Perhaps the one most important virtue of varifolds is that it is possible to obtain a geometrically signiﬁcant solution to a number of variational problems, including Plateau’s problem, without having to know ahead of time what all the possible singularities of the solution can be. (cf. F. J. Almgren, Jr. [1, p. viii]). The treatment of this topic is unfortunately much beyond the scope of this book, and the interested reader is referred to the above reference for more details.

6. Caccioppoli or Finite Perimeter Sets

6.1

245

Finite Perimeter Sets

Given an open subset D of RN , consider L1 -functions f on D with distributional gradient ∇f in the space M 1 (D)N of (vectorial) bounded measures; that is, def f div ϕ

dx : D1 (D; RN ) → R ϕ

→ ∇f, ϕ

D = − D

is continuous with respect to the topology of uniform convergence in D: def

∇f M 1 (D)N = def

def

sup 1

N

∇f, ϕ

D < ∞, ϕC = sup | ϕ(x)|RN ,

ϕ ∈D (D;R ) ϕC ≤1 0

(6.1)

x∈D

where M 1 (D) = D (D) is the topological dual of D0 (D), and N ∇f ∈ L D0 (D; RN ), R ≡ L D0 (D), R = M 1 (D)N . Such functions are known as functions of bounded variation. The space def

BV(D) = f ∈ L1 (D) : ∇f ∈ M 1 (D)N

(6.2)

endowed with the norm f BV(D) = f L1 (D) + ∇f M 1 (D)N

(6.3)

is a Banach space. Deﬁnition 6.1 (L. C. Evans and R. F. Gariepy [1]). A function f ∈ L1loc (U ) in an open subset U of RN has locally bounded variation if for each bounded open subset V of U such that V ⊂ U , f ∈ BV(V ). The set of all such functions will be denoted by BVloc (U ). Theorem 6.1. Given an open subset U of RN , f belongs to BVloc (U ) if and only if for each x ∈ U there exists ρ > 0 such that B(x, ρ) ⊂ U and f ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x. Proof. If f ∈ BVloc (U ), then the result is true by specializing to balls in each point of U . Conversely, by assumption, the open set U is covered by a family F(U ) of balls B(x, ρx ) such that B(x, ρx ) ⊂ U and f ∈ BV(B(x, ρx )). Given a bounded open subset V of U such that V ⊂ U , the compact subset V can be covered by a ﬁnite number of balls {B(xi , ρi )}, 1 ≤ i ≤ n, in F(U ) for points xi ∈ V , 1 ≤ i ≤ n, V ⊂

n *

Bi ,

def

Bi = B(xi , ρxi ).

i=1

Denote by {ψi ∈ D(Bi )}ni=1 a partition of unity for the family {Bi } such that ∀i, 0 ≤ ψi ≤ 1 in Bi , and

n i=1

ψi = 1 on V .

246

Chapter 5. Metrics via Characteristic Functions

Then for each ϕ

∈ V 1 (V )N ϕ

=

n

ϕ

i on V,

def

ϕ

i = ψi ϕ

,

i=1

∈ V 1 (V )N such that and by construction ϕ

i ∈ V (Bi ∩ V )N . Therefore, for all ϕ ϕC(V ) ≤ 1 n n − f div ϕ

dx = − f div ϕ

i dx = − f div ϕ

i dx 1

V

i=1

V

i=1

n f div ϕ

dx ≤ ⇒ V

i=1

Bi ∩V

V ∩Bi

f div ϕ

i dx

n n ⇒ f div ϕ

dx ≤ ∇f M 1 (Bi ∩V ) ≤ ∇f M 1 (Bi ) < ∞, V

i=1

since V (Bi ∩ V ) ⊂ V (Bi ) and f ∈ BVloc (RN ). 1

N

1

N

i=1

and ϕi C(Bi ) ≤ 1. Therefore, ∇f M 1 (V ) is ﬁnite

For more details and properties see also F. Morgan [1, p. 117], H. Federer [5, sect. 4.5.9], L. C. Evans and R. F. Gariepy [1], W. P. Ziemer [1], and R. Temam [1]. As in the previous section, consider measurable subsets Ω of a ﬁxed bounded open subset D of RN . Their characteristic functions χΩ ∈ X(D) are L1 (D)-functions χΩ dx = m(Ω) ≤ m(D) < ∞ (6.4) χΩ L1 (D) = D

with distributional gradient

∀ ϕ ∈ D(D)

N

,

def

∇χΩ , ϕ

D = −

χΩ div ϕ

dx.

(6.5)

D

When Ω is an open domain with boundary Γ of class C 1 , then by the Stokes divergence theorem div ϕ

dx = − ϕ

· n dΓ = − ϕ

· n dΓ, − Ω∩D

∂(Ω∩D)

Γ∩D

where n is the outward normal ﬁeld along ∂(Ω ∩ D). Since Γ is of class C 1 the normal ﬁeld n along Γ belongs to C 0 (Γ), and from the last identity the maximum is def 2 |n| dΓ = dΓ = HN −1 (Γ ∩ D), PD (Ω) = ∇χΩ M 1 (D) = Γ∩D

Γ∩D

the (N − 1)-dimensional Hausdorﬀ measure of Γ ∩ D. As a result HN −1 (Γ) = PD (Ω) + HN −1 (Γ ∩ ∂D).

(6.6)

Thus the norm of the gradient provides a natural relaxation of the notion of perimeter to the following larger class of domains.

6. Caccioppoli or Finite Perimeter Sets

247

Deﬁnition 6.2. Let Ω be a Lebesgue measurable subset of RN . (i) The perimeter of Ω with respect to an open subset D of RN is deﬁned as def

PD (Ω) = ∇χΩ M 1 (D)N .

(6.7)

Ω is said to have ﬁnite perimeter with respect to D if PD (Ω) is ﬁnite. The family of measurable characteristic functions with ﬁnite measure and ﬁnite relative perimeter in D will be denoted as def

BX(D) = {χΩ ∈ X(D) : χΩ ∈ BV(D)} . (ii) Ω is said to have locally ﬁnite perimeter if for all bounded open subsets D of RN , χΩ ∈ BV(D), that is, χΩ ∈ BVloc (RN ). (iii) Ω is said to have ﬁnite perimeter if χΩ ∈ BV(RN ). Theorem 6.2. Let Ω be a Lebesgue measurable subset of RN . Ω has locally ﬁnite perimeter, that is, χΩ ∈ BVloc (RN ), if and only if for each x ∈ ∂Ω there exists ρ > 0 such that χΩ ∈ BV(B(x, ρ)), B(x, ρ) the open ball of radius ρ in x. Proof. We use Theorem 6.1. In one direction the result is obvious. Conversely, if x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ

∈ D 1 (B(x, ρ)) − χΩ div ϕ

dx = − div ϕ

dx = − ϕ

· n dHN −1 = 0 B(x,ρ)

B(x,ρ)

∂B(x,ρ)

since ϕ

= 0 on the boundary of B(x, ρ). Thus χΩ ∈ BV(B(x, ρ)). If x ∈ int Ω, then there exists ρ > 0 such that B(x, ρ) ⊂ int Ω, and for all ϕ

∈ D1 (B(x, ρ)) − χΩ div ϕ

dx = 0. B(x,ρ)

So again χΩ ∈ BV(B(x, ρ)). Finally, if x ∈ ∂Ω, then by assumption there exists ρ > 0 such that χΩ ∈ BV(B(x, ρ)). Therefore, by Theorem 6.1, χΩ ∈ BVloc (RN ). The interest behind this construction is twofold. First, the notion of perimeter of a set is extended to measurable sets; second, this framework provides a ﬁrst compactness theorem which will be useful in obtaining existence of optimal domains. Theorem 6.3. Assume that D is a bounded open domain in RN with a Lipschitzian boundary ∂D. Let {Ωn } be a sequence of measurable domains in D for which there exists a constant c > 0 such that ∀n,

PD (Ωn ) ≤ c.

(6.8)

Then there exist a measurable set Ω in D and a subsequence {Ωnk } such that χΩnk → χΩ in L1 (D) as k → ∞

and

PD (Ω) ≤ lim inf PD (Ωnk ) ≤ c. k→∞

(6.9)

248

Chapter 5. Metrics via Characteristic Functions

in Moreover ∇χΩnk “converges in measure” to ∇χΩ in M 1 (D)N ; that is, for all ϕ D0 (D, RN ), lim ∇χΩnk , ϕ

M 1 (D)N → ∇χΩ , ϕ

M 1 (D)N .

k→∞

(6.10)

Proof. This follows from the fact that the injection of the space BV(D) endowed with the norm (6.3) into L1 (D) is continuous and compact (cf. E. Giusti [1, Thm. 1.19, p. 17], V. G. Maz’ja [1, Thm. 6.1.4, p. 300, Lem. 1.4.6, p. 62], C. B. Morrey, Jr. [1, Thm. 4.4.4, p. 75], and L. C. Evans and R. F. Gariepy [1]). Example 6.1 (The staircase). In (6.9) the inequality can be strict, as can be seen from the following example (cf. Figure 5.7). For each n ≥ 1, deﬁne the set n * j j+1 j Ωn = , × 0, 1 − . n n n j=1 Its limit is the set Ω = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ x}. It is easy to check that the sets Ωn are contained in the holdall D = ]−1, 2[ × ]−1, 2[ and that √ ∀n ≥ 1, PD (Ωn ) = 4, PD (Ω) = 2 + 2, χΩn → χΩ in Lp (D), 1 ≤ p < ∞. Each Ωn is uniformly Lipschitzian, but the Lipschitz constant and the two neighborhoods of Deﬁnition 3.2 in Chapter 2 cannot be chosen independently of n.

1/n

Ωn

Ω Figure 5.7. The staircase.

Corollary 1. Let D be open bounded and Lipschitzian. There exists a constant c > 0 such that for all convex domains Ω in D PD (Ω) ≤ c,

m(Ω) ≤ c

and the set C(D) of convex subsets of D is compact in Lp (D) for all p, 1 ≤ p < ∞.

6. Caccioppoli or Finite Perimeter Sets

249

Proof. If Ω is a convex set with zero volume, m(Ω) = 0, then its perimeter PD (Ω) = 0 (cf. E. Giusti [1, Rem. 1.7 (iii), p. 6]). If m(Ω) > 0, then int Ω = ∅. For convex sets with a nonempty interior the perimeter and the volume enjoy a very nice monotonicity property. If C • denotes the set of nonempty closed convex subsets of RN , then the map Ω → m(Ω) : C • → R is strictly increasing: ∀A, B ∈ C • ,

A B =⇒ m(A) < m(B)

(cf. M. Berger [1, Vol. 3, Prop. 12.9.4.3, p. 141]). Similarly if P (Ω) denotes the perimeter of Ω in RN , the map Ω → P (Ω) : C • → R is strictly increasing (cf. M. Berger [1, Vol. 3, Prop. 12.10.2, p. 144]). As a result ∀n,

m(Ωn ) ≤ m(D) < ∞ and PD (Ωn ) ≤ P (Ωn ) ≤ P (co D) < ∞,

since the perimeter of a bounded convex set is ﬁnite. Then the conditions of the theorem are satisﬁed and the conclusions of the theorem follow. But we have seen in Theorem 3.5 (iii) that the set C(D) is closed in L1 (D). Therefore, Ω can be chosen convex in the equivalence class. This completes the proof. This theorem and the lower (resp., upper) semicontinuity of the shape function Ω → J(Ω) will provide existence results for domains in the class of ﬁnite perimeter sets in D. One example is the transmission problem (4.1) to (4.3) of section 4.1 with the objective function 1 J(Ω) = |y(Ω) − g|2 dx + αPD (Ω), α > 0, (6.11) 2 Ω for some g ∈ L2 (D). We shall come back later to the homogeneous Dirichlet boundary value problem (3.25)–(3.26). It is important to recall that even if a set Ω in D has a ﬁnite perimeter PD (Ω), its relative boundary Γ ∩ D can have a nonzero N -dimensional Lebesgue measure. To illustrate this point consider the following adaptation of Example 1.10 in E. Giusti [1, p. 7]. Example 6.2. Let D = B(0, 1) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let {xi } be an ordered sequence of all points in D with rational coordinates. Associate with each i the open ball Bi = {x ∈ D : |x − xi | < ρi },

0 < ρi ≤ min{2−i , 1 − |xi |}.

Deﬁne the new sequence of open subsets of D, Ωn = and notice that for all n ≥ 1, m(∂Ωn ) = 0,

n *

Bi ,

i=1

PD (Ωn ) ≤ 2π,

250

Chapter 5. Metrics via Characteristic Functions

where m = m2 is the Lebesgue measure in R2 and ∂Ωn is the boundary of Ωn . Moreover, since the sequence of sets {Ωn } is increasing, χΩn → χΩ in L1 (D),

Ω=

∞ *

Bi ,

PD (Ω) ≤ lim inf PD (Ωn ) ≤ 2π. n→∞

i=1

Observe that Ω = D and that ∂Ω = Ω ∩ Ω ⊃ D ∩ Ω. Thus 2π m(∂Ω) = m(D ∩ Ω) ≥ m(D) − m(Ω) ≥ 3 since ∞ π π2−2i = . m(D) = π and m(Ω) ≤ 3 1=1 Recall that m(Ωn ) ≤

n i=1

m(Bi ) ≤

n

π2−2i

⇒ m(D) ≤

i=1

∞ i=1

m(Bi ) ≤

∞

π2−2i .

i=1

For p, 1 ≤ p < ∞, the sequence of characteristic functions {χΩn } converges to χΩ in Lp (D)-strong. However, for all n, m(∂Ωn ) = 0, but m(∂Ω) > 0. In fact we can associate with the perimeter PD (Ω) the reduced boundary, ∂ ∗ Ω, which is the set of all x ∈ ∂Ω for which the normal n(x) exists. We quote the following interesting theorems from W. H. Fleming [1, p. 455] (cf. also E. De Giorgi [2], L. C. Evans and R. F. Gariepy [1, Thm. 1, sect. 5.1, p. 167, Notation, pp. 168–169, Lem. 1, p. 208], and H. Federer [2]). Theorem 6.4. Let χΩ ∈ BVloc (RN ). There exist a Radon measure ∂Ω on RN and a ∂Ω-measurable function νΩ : RN → RN such that (i) |νΩ (x)| = 1, ∂Ω almost everywhere and (ii) RN χΩ div ϕ dx = − RN ϕ · νΩ d∂Ω, for all ϕ ∈ Cc1 (RN ; RN ). Deﬁnition 6.3. Let χΩ ∈ BVloc (RN ). The point x ∈ RN belongs to the reduced boundary ∂ ∗ Ω if (i) ∂Ω(Br (x)) > 0, for all r > 0, (ii) limr0 m(B1r (x)) Br (x) νΩ d∂Ω = νΩ (x), and (iii) |νΩ (x)| = 1. Theorem 6.5. Let Ω have ﬁnite perimeter P (Ω). Let ∂ ∗ Ω denote the reduced boundary of Ω. Then (i) ∂ ∗ Ω ⊂ ∂∗ Ω ⊂ Ω• ⊂ ∂Ω and ∂ ∗ Ω = ∂Ω, (ii) P (Ω) = HN −1 (∂ ∗ Ω) (cf. E. Giusti [1, Chap. 4]),

6. Caccioppoli or Finite Perimeter Sets

251

(iii) the Gauss–Green theorem holds with ∂ ∗ Ω, and (iv) m(∂ ∗ Ω) = m(∂∗ Ω) = 0,

HN −1 (∂∗ Ω\∂ ∗ Ω) = 0.

We quote the following density theorem (cf. E. Giusti [1, Thm. 1.24 and Lem. 1.25, p. 23]), which complements Theorem 3.1. Theorem 6.6. Let Ω be a bounded measurable domain in RN with ﬁnite perimeter. Then there exists a sequence {Ωj } of C ∞ -domains such that as j goes to ∞ |χΩj − χΩ | dx → 0 and P (Ωj ) → P (Ω). RN

6.2

Decomposition of the Integral along Level Sets

We complete this section by giving some useful theorems on the decomposition of the integral along the level sets of a function. The ﬁrst one was used by J.´sio [6] (Grad’s model in plasma physics) in 1979 and [9, sect. 4.4, p. 95] in P. Zole 1981, by R. Temam [1] (monotone rearrangements) in 1979, by J. M. Rakotoson and R. Temam [1] in 1987, and more recently in 1995 by M. C. Delfour and ´sio [19, 20, 21, 25] (intrinsic formulation of models of shells). We quote J.-P. Zole the version given in L. C. Evans and R. F. Gariepy [1, Prop. 3, p. 118]. The original theorem can be found in H. Federer [3] and L. C. Young [1]. Theorem 6.7. Let f : RN → R be Lipschitz continuous with |∇f | > 0

a.e.

For any Lebesgue summable function g : RN → R, we have the following decomposition of the integral along the level sets of f :

∞

0

g dx = {f >t}

t

{f =s}

g dHN −1 |∇f |

1 ds.

The second theorem (W. H. Fleming and R. Rishel [1]) uses a BV function instead of a Lipschitz function. Theorem 6.8. Let D be an open subset of RN . For any f ∈ BV(D) and real t, let def

Et = {x : f (x) < t}. Then ∇f M 1 (D)N =

∞ −∞

P (Et ) dt

(cf., for instance, H. Whitney [1, Chap. 11]). This is known as the co-area formula (see also E. Giusti [1, Thm. 1.23, p. 20] and E. De Giorgi [4]).

252

6.3

Chapter 5. Metrics via Characteristic Functions

Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade of Complete Metric Spaces

There is a general property enjoyed by functions in BV(D). Theorem 6.9. Let D be a bounded open Lipschitzian domain in RN . (i) BV(D) ⊂ W ε,1 (D),

0 ≤ ε < 1.

(ii) BV(D) ∩ L∞ (D) ⊂ W ε,p (D),

0 ≤ ε < 1p , 1 ≤ p < ∞.

This theorem says that for a Caccioppoli set Ω in D, ∀p ≥ 1, 0 ≤ ε

0 small we break the above integral into two parts: |f (y) − f (x)| |f (y) − f (x)| dx dy , I = dx dy . I1 = 2 N +ε |y − x| |y − x|N +ε D D∩B(x,α) D D B(x,α) The second integral I2 is bounded since |y − x| ≥ α. For the ﬁrst one we change the variable y to t = y − x, |(χD f )(x + t) − (χD f )(x)| dx dt , I1 = |t|N +ε D B(0,α) and after a change in the order of integration, 1 I1 = dt N +ε dx |(χD f )(x + t) − (χD f )(x)|. |t| B(0,α) D But the integral over D is bounded by |t| ∇f M 1 (D) . The result is true for smooth functions (cf. V. G. Maz’ja [1, Lem. 1.4.6, p. 62]). Pick a sequence of smooth functions such that fn → f in L1 (D),

∇fn → ∇f in M 1 (D)-weak;

then ∇f M 1 (D) ≤ lim sup ∇fn M 1 (D) = lim sup ∇fn L1 (D) n→∞

n→∞

6. Caccioppoli or Finite Perimeter Sets

253

is bounded, and by going to the limit on both sides of the inequality, dx |(χD fn )(x + t) − (χD fn )(x)| ≤ |t|∇fn L1 (D) , D

we obtain the desired result. Now coming back to the estimate of I1 , α1−ε ∇f M 1 (D) , 0 < ε < 1. I1 ≤ |t|1−N −ε dt ∇f M 1 (D) ≤ c(α) 1−ε B(x,α) (ii) The function f belongs to W ε,p (D) if the integral |f (y) − f (x)|p dx dy < +∞. I= |y − x|N +εp D D For each f ∈ BV(D) ∩ L∞ (D), there exists M > 0 such that |f (x)| ≤ M

a.e. in D.

Then

I = (2 M )p

dx D

and since

dy D

f (y)−f (x) p 2M |y − x|N +εp

,

f (y) − f (x) ≤ 1, 2M

then for each p, 1 ≤ p < ∞, f (y) − f (x) p f (y) − f (x) ≤ . 2M 2M As a result

I ≤ (2 M )

p−1

dx D

dy D

|f (y) − f (x)| . |y − x|N +εp

But we have seen in part (i) that for f ∈ BV(D), the double integral is ﬁnite if 0 ≤ εp < 1 ⇒ 0 ≤ ε

0, we have 0 ≤ ε < 1/p for all p ≥ 1 and |χΩ (x) − χΩ (y)| dx dy + χΩ pLp (D) . (6.12) χΩ pW ε,p (D) = |x − y|N +pε D D

254

Chapter 5. Metrics via Characteristic Functions

The W ε,p (D)-norm is equivalent to the W ε ,p (D) for all pairs (p , ε ), p ≥ 1, 0 ≥ ε < 1/p , such that p ε = p ε. At this stage, it is not clear whether that norm is related to some Hausdorﬀ measure of the relative boundary Γ ∩ D of the set. For p = 2 and a characteristic function, χΩ , of a measurable set Ω, we get 2 χΩ H ε (D) = 2 |x − y|−(N +2ε) dx dy + m(Ω ∩ D). (6.13) Ω

D Ω

ε,2

The space X(D)∩W (D) is a closed subspace of the Hilbert space W ε,2 (D) and the square of its norm is diﬀerentiable. A direct consequence for optimization problems is that a penalization term of the form χΩ 2W ε,2 (D) is now diﬀerentiable and can be used in various minimization problems or to regularize the objective function to obtain existence of approximate solutions. For instance, to obtain existence results when minimizing a function J(Ω) (deﬁned for all measurable sets Ω in D), we can consider a regularized problem in the following form: Jα (Ω) = J(Ω) + αχΩ 2H ε (D) ,

α > 0.

(6.14)

Section 6.1 provided the important compactness Theorem 6.3 for Caccioppoli or ﬁnite perimeter sets. For a bounded open holdall D, this also provides a new complete metric topology on X(D) ∩ BV(D) when endowed with the metric def

ρBV (χ1 , χ2 ) = χ2 − χ1 BV(D) = χ2 − χ1 L1 (D) + ∇χ2 − ∇χ1 M 1 (D) . Similarly, the intermediate spaces of Theorem 6.9 between BV(D) and Lp (D) provide little rougher metrics for p ≥ 1 and ε < 1/p on X(D) ∩ W ε,p (D) when endowed with the metric def

ρW ε,p (χ1 , χ2 ) = χ2 − χ1 W ε,p (D) . Theorem 6.10. Given a bounded open holdall D in RN , X(D)∩BV(D) and X(D)∩ W ε,p (D), p ≥ 1, ε < 1/p, are complete metric spaces and X(D) ∩ BV(D) ⊂ X(D) ∩ W ε,p (D) ⊂ X(D).

6.4

Compactness and Uniform Cone Property

In Theorem 6.3 of section 6.1 we have seen a ﬁrst compactness theorem for a family of sets with a uniformly bounded perimeter. In this section we present a second compactness theorem for measurable domains satisfying a uniform cone property due to D. Chenais [1, 4, 6]. We provide a proof of this result that emphasizes the fact that the perimeter of sets in that family is uniformly bounded. It then becomes a special case of Theorem 6.3. This result will also be obtained as Corollary 2 to Theorem 13.1 in section 13 of Chapter 7. It will use completely diﬀerent arguments and apply to families of sets that do not even have a ﬁnite boundary measure. Theorem 6.11. Let D be a bounded open holdall in RN with uniformly Lipschitzian boundary ∂D. For r > 0, ω > 0, and λ > 0 consider the family def Ω is Lebesgue measurable and satisﬁes L(D, r, ω, λ) = Ω ⊂ D : the uniform cone property for (r, ω, λ) . (6.15)

6. Caccioppoli or Finite Perimeter Sets

255

For p, 1 ≤ p < ∞, the set def

X(D, r, ω, λ) = {χΩ : ∀Ω ∈ L(D, r, ω, λ)}

(6.16)

is compact in Lp (D) and there exists a constant p(D, r, ω, λ) > 0 such that ∀Ω ∈ L(D, r, ω, λ),

HN −1 (∂Ω) ≤ p(D, r, ω, λ).

(6.17)

We ﬁrst show that the perimeter of the sets of the family L(D, r, ω, λ) is uniformly bounded and use Theorem 6.8 to show that any sequence of X(D, r, ω, λ) has a subsequence converging to the characteristic function of some ﬁnite perimeter set Ω. We use the full strength of the compactness of the injection of BV(D) into L1 (D) rather than checking directly the conditions under which a subset of Lp (D) is relatively compact. The proof will be completed by showing that the nice representative (Deﬁnition 3.3) I of Ω satisﬁes the same uniform cone property. The proof uses some elements of D. Chenais [4, 6]’s original proof. Proof of Theorem 6.11. Each Ω ∈ L(D, r, ω, λ) satisﬁes the same uniform cone property (cf. Deﬁnition 6.3 (ii)) of Chapter 2 with parameters r, λ, and rλ . By Theorem 6.8 of Chapter 2 it is uniformly Lipschitzian in the sense of Deﬁnition 5.2 (iii) of Chapter 2. As a result ρ, the largest radius such that BH (0, ρ) ⊂ {PH (y − x) : ∀y ∈ B(x, rλ ) ∩ ∂Ω}, and the neighborhoods V = BH (0, ρ) and U = B(0, rλ ) ∩ {y : PH y ∈ BH (0, ρ)} can be chosen as speciﬁed in Theorem 6.6 (i) of Chapter 2. They are the same for all Ω ∈ L(D, r, ω, λ): for all Ω ∈ L(D, r, ω, λ) and all x ∈ ∂Ω, Vx = V , and there exists Ax ∈ O(N) such that U(x) = x + Ax U . By construction B(x, ρ) ⊂ B(x, rλ ) ∩ {y : PH A−1 x (y − x) ∈ BH (0, ρ)} = x + Ax U = U(x).

The family B (z, ρ/2) : z ∈ D is an open cover of D. Since D is compact, there exists a ﬁnite subcover {Bi }m i=1 , Bi = B (zi , ρ/2), of D. Now for any Ω ∈ L(D, r, ω, λ), m ∂Ω ⊂ D and there is a subcover {Bik }K k=1 of {Bi }i=1 such that ∂Ω ⊂

K *

Bik and ∀k,

1 ≤ k ≤ K,

∂Ω ∩ Bik = ∅.

k=1

Pick a sequence {xk }K k=1 such that xk ∈ ∂Ω ∩ Bik , 1 ≤ k ≤ K, and notice that ρ Bik = B zik , ⊂ B(xk , ρ) 2 or ∃K ≤ m,

∃{xk }K k=1 ⊂ ∂Ω,

∂Ω ⊂

K * k=1

B(xk , ρ).

256

Chapter 5. Metrics via Characteristic Functions

Thus from the estimate (5.37) in Theorem 5.7 of Chapter 2 with cx = tan ω, HN −1 (∂Ω) ≤

K

K HN −1 ∂Ω ∩ B(xk , ρ) ≤ HN −1 ∂Ω ∩ U(xk )

k=1

≤

K

k=1

ρN −1 αN −1

'

1 + (tan ω)2 ≤ m ρN −1 αN −1

' 1 + (tan ω)2 ,

k=1

where αN −1 is the volume of the unit (N − 1)-dimensional ball. So the right-hand side of the above inequality is a constant that is equal to p = p(D, r, ω, λ) > 0 and is independent of Ω in L(D, r, ω, λ): ∀Ω ∈ L(D, r, ω, λ),

PD (Ω) = HN −1 (∂Ω) ≤ p.

Now from Theorem 6.3 for any sequence {Ωn } ⊂ L(D, r, ω, λ), there exists Ω such that χΩ ∈ BV(D) and a subsequence, still denoted by {Ωn }, such that χΩn → χΩ in L1 (D) and PD (Ω) ≤ p. (ii) To complete the proof we consider the representative I of Ω (cf. Deﬁnition 3.3) and show that I ∈ L(D, r, ω, λ). We need the following lemma. Lemma 6.1. Let χΩn → χΩ in Lp (D), 1 ≤ p < ∞, for Ω ⊂ D, Ωn ⊂ D, and let I be the measure theoretic representative of Ω. Then ∀x ∈ I, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), m B(x, R) ∩ Ωn > 0. Moreover, ∀x ∈ ∂I, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), m B(x, R) ∩ Ωn > 0 and m B(x, R) ∩ Ωn > 0 and B(x, R) ∩ ∂Ωn = ∅. Proof. We proceed by contradiction. Assume that ∃x ∈ I,

∃R > 0,

∀N > 0,

∃n ≥ N,

m B(x, R) ∩ Ωn = 0.

So there exists a subsequence {Ωnk }, nk → ∞, such that m B(x, R) ∩ Ω = lim m B(x, R) ∩ Ωnk = 0 k→∞ ⇒ ∃R > 0, m B(x, R) ∩ Ω = 0 =⇒ x ∈ Ω0 = int I = I. But this is in contradiction with the fact that x ∈ I. For x ∈ ∂I = I ∩ I, simultaneously apply the ﬁrst statement to I and χΩn → χΩ and to I and χΩn → χΩ since χΩ = χI almost everywhere by Theorem 3.3 (i) and choose the larger of the two integers. As for the last property it follows from the fact that the open ball B(x, R) cannot be partitioned into two nonempty disjoint open subsets.

6. Caccioppoli or Finite Perimeter Sets

257

We wish to prove that ∀x ∈ ∂I,

∃Ax ∈ O(N),

∀y ∈ I ∩ B(x, r),

y + Ax C(λ, ω) ⊂ int I.

Since Ωn is Lipschitzian, Ωn is also Lipschitzian and χΩn = χΩn

and χΩ = χΩn n

almost everywhere. So we apply the second part of the lemma to x ∈ ∂I, χΩn → χΩ and χΩ → χΩ . So for each x ∈ ∂I, for all k ≥ 1, there exists nk ≥ k such that n

r B x, k ∩ ∂Ωnk = ∅. 2

Denote by xk an element of that intersection: r ∀k ≥ 1, xk ∈ B x, k ∩ ∂Ωnk . 2 By construction, xk → x. Next consider y ∈ B(x, r) ∩ I. By Lemma 6.1, there exists a subsequence of {Ωnk }, still denoted by {Ωnk }, such that r ∀k ≥ 1, B y, k ∩ Ωnk = ∅. 2 For each k ≥ 1 denote by yk a point of that intersection. By construction, yk ∈ Ωnk → y ∈ I ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this note that y ∈ B(x, r) and that ρ ∃ρ > 0, B(y, ρ) ⊂ B(x, r) and |y − x| + < r. 2 Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | . r r ρ ρ/ r 2 + . 2 log 2

So we have constructed a subsequence {Ωnk } such that for k ≥ K, xk ∈ ∂Ωnk → x ∈ ∂I

and yk ∈ Ωnk ∩ B(xk , r) → y ∈ I ∩ B(x, r).

For each k, there exists Ak ∈ O(N) such that yk + Ak C(λ, ω) ⊂ int Ωnk .

258

Chapter 5. Metrics via Characteristic Functions

Pick another subsequence of {Ωnk }, still denoted by {Ωnk }, such that ∃A ∈ O(N),

Ak → A,

since O(N) is compact. Now consider z ∈ y + AC(λ, ω). Since z is an interior point ∃ρ > 0,

B(z, ρ) ⊂ y + AC(λ, ω).

So there exists K ≥ K such that ρ ∀k ≥ K , B z, ⊂ yk + Ak C(λ, ω) ⊂ int Ωnk . 2 Therefore, for k ≥ K ρ ρ m B z, = m B z, ∩ Ωnk , 2 2 and, as k → ∞, ρ ρ m B z, = m B z, ∩Ω , 2 2 and by Deﬁnition 3.3, z ∈ Ω1 and y + AC(λ, ω) ⊂ Ω1 = int I. This proves that I ⊂ D satisﬁes the uniform cone property and I ∈ L(D, λ, ω, r).

7 7.1

Existence for the Bernoulli Free Boundary Problem An Example: Elementary Modeling of the Water Wave

Consider a ﬂuid in a domain Ω in R3 and assume that the velocity of the ﬂow u satisﬁes the Navier–Stokes equations ut + Du u − ν ∆u + ∇p = −ρ g in Ω,

div u = 0 in Ω,

(7.1)

where ν > 0 and ρ > 0 are the respective viscosity and density of the ﬂuid, p is the pressure, and g is the gravity constant. The second equation characterizes the incompressibility of the ﬂuid. A standard example considered by physicists is the water wave in a channel. The boundary conditions on ∂Ω are the sliding conditions at the bottom S and on the free boundary Γ, that is, u · n = 0 on S ∪ Γ.

(7.2)

Assume that a stationary regime has been reached so that the velocity of the ﬂuid is no longer a function of the time. Furthermore assume that the motion of the ﬂuid is irrotational. By the classical Hodge decomposition, the velocity can be written in the form u = ∇ϕ + curl ξ.

(7.3)

7. Existence for the Bernoulli Free Boundary Problem

259

As curl u = curl curl ξ = 0 we conclude that ξ = 0, since curl curl is a good isomorphism. Then u = ∇ϕ and the incompressibility condition becomes ∆ϕ = 0 in Ω.

(7.4)

Therefore the Navier–Stokes equation reduces to D(∇ϕ) ∇ϕ + ν ∇(∆ϕ) + ∇p = −∇(ρgz),

(7.5)

but D(∇ϕ) ∇ϕ = so that

∇

1 ∇(|∇ϕ|2 ), 2

1 |∇ϕ|2 + p + ρgz 2

(7.6)

= 0 in Ω.

(7.7)

Then if Ω is connected, there exists a constant c such that 1 |∇ϕ|2 + p + ρgz = c in Ω. 2

(7.8)

In the domain Ω, that Bernoulli condition yields explicitly the pressure p as a function of the velocity |∇ϕ| and the height z in the ﬂuid. The ﬂow is now assumed to be independent of the transverse variable y so that the initial three-dimensional problem in a perfect channel reduces to a two-dimensional one. Since the problem has been reduced to a two-dimensional one, introduce the harmonic conjugate ψ, the so-called stream function, so that the boundary condition ∂ϕ/∂n = 0 takes the form ψ = constant on each connected component of the boundary ∂Ω. On the free boundary at the top of the wave, the pressure is related to the existing atmospheric pressure pa through the surface tension σ > 0 and the mean curvature H of the free boundary. Actually p − pa = −σ H

(7.9)

on the free boundary of the stationary wave, where H is the mean curvature associated with the ﬂuid domain. Also, from the Cauchy conditions, we have |∇ψ| = |∇ϕ| so that the boundary condition (7.8) on the free boundary of the wave becomes 1 |∇ψ|2 + ρgz + σH = pa . 2

(7.10)

In order to simplify the presentation we replace the equation ∆ψ = 0 by ∆ψ = f in Ω to avoid a forcing term on a part of the boundary, and we show that the resulting free boundary problem has the following shape variational formulation. Let D be a ﬁxed, suﬃciently large, smooth, bounded, and open domain in R2 . Let a be a real number such that 0 < a < m(D). To ﬁnd Ω ⊂ D, m(Ω) = a and ψ ∈ H01 (D) such that −∆ψ = f in Ω

(7.11)

260

Chapter 5. Metrics via Characteristic Functions

and ψ = constant and satisﬁes the boundary condition (7.10) on the free part ∂Ω ∩ D of the boundary. For a ﬁxed Ω, the solution of this problem is a minimizing element of the following variational problem: 1 def 2 |∇ϕ| J(Ω) = inf − f ϕ + ρgz dx + σ PD (Ω), (7.12) 2 ϕ∈H1 (Ω;D) Ω where def

(7.13) H1 (Ω; D) = u ∈ H01 (D) : u(x) = 0 a.e. x in D \ Ω is the relaxation of the deﬁnition of the Sobolev space H01 (Ω)10 for any measurable subset Ω of D. Its properties were studied in Theorem 3.6. With that formulation there exists a measurable Ω∗ ⊂ D, |Ω∗ | = a, such that ∀Ω ⊂ D, |Ω| = a,

J(Ω∗ ) ≤ J(Ω).

(7.14)

By using the methods of Chapter 10 it follows that if ∂Ω∗ is suﬃciently smooth, the shape Euler condition dJ(Ω∗ ; V ) = 0 yields the original free boundary problem and the free boundary condition (7.10). The existence of a solution will now follow from Theorem 6.3 in section 6.1. The case without surface tension is physically important. It occurs in phenomena with “evaporation ” (cf. M. Souli and J.´sio [2]). In that case the previous Bernoulli condition becomes P. Zole 2 ∂ξ = g 2 (g ≥ 0) ∂n on the free boundary, so that if we assume (in the channel setting) that there is no cavitation or recirculation in the ﬂuid, then ∂ξ/∂n > 0 on the free boundary and we get the Neumann-like condition ∂ξ/∂n = g together with the Dirichlet condition. In section 7.4 we shall consider the case with surface tension.

7.2

Existence for a Class of Free Boundary Problems

´sio [25, 26, Consider the following free boundary problem, studied in J.-P. Zole 29]: ﬁnd Ω in a ﬁxed holdall D and a function y on Ω such that −∆y = f in Ω,

y = 0 and

∂y = Q2 on ∂Ω, ∂n

(7.15)

where f and Q are appropriate functions deﬁned in D. To study this type of problem H. W. Alt and L. A. Caffarelli [1] have introduced the following function: 1 def Q2 χϕ>0 dx, (7.16) J(ϕ) = |∇ϕ|2 − f ϕ dx + 2 D D 10 In Chapter 8 the space H 1 (Ω; D) of extensions by zero to D of elements of H 1 (Ω) is deﬁned 0 0 for Ω open. It is then characterized in Lemma 6.1 of Chapter 8 by a capacity condition on the complement D\Ω. This characterization extends to quasi-open sets Ω as deﬁned in section 6 of Chapter 8.

7. Existence for the Bernoulli Free Boundary Problem

261

which is minimized over def

K = {u ∈ H01 (D) : u(x) ≥ 0 a.e. in D},

(7.17)

where χϕ>0 (resp., χϕ=0 ) is the characteristic function of the set {x ∈ D : ϕ(x) > 0}11 (resp., {x ∈ D : ϕ(x) = 0}). The existence of a solution is based on the following lemma. Lemma 7.1. Let {un } and {χn } be two converging sequences such that un → u in L2 (D)-strong, and let the χn ’s be characteristic functions, χn (1 − χn ) = 0, which converge to some function λ in L2 (D)-weak. Then ∀n,

(1 − χn )un = 0

⇒

λ ≥ χu=0 .

(7.18)

Proof. We have (1 − χn )un = 0, and in the limit (1 − λ)u = 0. Thus on the set {x : u(x) = 0} we have λ = 1; elsewhere λ lies between 0 and 1 as the weak limit of a sequence of characteristic functions. Proposition 7.1. Let f and Q be two elements of L2 (D) such that f ≥ 0 almost everywhere. There exists u in K which minimizes the function J over the positive cone K of H01 (D). Proof. Let {un } ⊂ K be a minimizing sequence for the function J over the convex set K. Denote by χn the characteristic function of the set {x ∈ D : un (x) > 0}, which is in fact equal to the subset {x ∈ D : un (x) = 0}. It is easy to verify that the sequence {un } remains bounded in H01 (D). Still denote by {un } a subsequence that weakly converges in H01 (D) to an element u of K. That convergence holds in L2 (D)-strong so that Lemma 7.1 applies and we get λ ≥ χu=0 for any weak limiting element λ of the sequence {χn } (which is bounded in L2 (D)). Denote by j the minimum of J over K. Then J(un ) weakly converges to j. We get 1 1 2 (7.19) |∇u| − f u dx ≤ lim inf |∇un |2 − f un dx, n→∞ D 2 D 2 χu=0 Q2 dx ≤ λQ2 dx = lim χn Q2 dx. (7.20) D

D

n→∞

D

Finally, by adding these two estimates we get J(u) ≤ j. Obviously in the upper bound in (7.20), Q2 is a positive function. Moreover (i) the set Ω is given by {x ∈ D : u(x) > 0}, and (ii) the restriction u|Ω of u to Ω is a weak solution of the free boundary problem −∆u = f in Ω,

u = 0 and

∂u = Q2 on ∂Ω. ∂n

(7.21)

11 This set deﬁned up to a set of zero measure is a quasi-open set in the sense of section 6 in Chapter 8.

262

Chapter 5. Metrics via Characteristic Functions

Eﬀectively, the minimization problem (7.16)–(7.17) can be written as a shape optimization problem. First introduce for any measurable Ω ⊂ D the positive cone def

H01 (Ω)+ = u ∈ H1 (Ω; D) : u(x) ≥ 0 a.e. x ∈ D in the Hilbert space H1 (Ω; D). Then consider the following shape optimization problem: inf {E(Ω) : Ω is measurable subset in D} , where the energy function E is given by 1 def 2 E(Ω) = |∇ϕ| min − f ϕ dx + Q2 dx. ϕ∈H01 (Ω)+ Ω 2 Ω

(7.22)

(7.23)

The necessary condition associated with the minimum could be obtained by the techniques introduced in section 2.1 of Chapter 10. The important diﬀerence with the previous formulation (7.16)–(7.17) is that the independent variable in the objective function is no longer a function but a domain. The shape formulation (7.22)– (7.23) now makes it possible to handle constraints on the volume or the perimeter of the domain, which were diﬃcult to incorporate in the ﬁrst formulation. As we have seen in Theorem 3.6, H1 (Ω; D) endowed with the norm of H01 (D) is a Hilbert space, and H01 (Ω)+ a closed convex cone in H1 (Ω; D), so that for any measurable subset Ω in D, problem (7.23) has a unique solution y in H01 (Ω)+ . Thus we have the following equivalence between problems (7.22)–(7.23) and the minimization (7.16)–(7.17) of J over K. Proposition 7.2. Let u be a minimizing element of J over K. Then def

Ω = {x ∈ D : u(x) > 0} is a solution of problem (7.22) and y = u|Ω is a solution of (7.23). Conversely, if Ω is a measurable subset of D and y is a solution of (7.22)–(7.23) in H01 (Ω)+ , then the element u deﬁned by def

u(x) =

y(x), 0,

if x ∈ Ω, if x ∈ D \ Ω,

belongs to K and minimizes J over K.

7.3 7.3.1

Weak Solutions of Some Generic Free Boundary Problems Problem without Constraint

Problem (7.22)–(7.23) can be relaxed as follows: given any f in L2 (D), G in L1 (D), (P0 )

inf {E(Ω) : Ω ⊂ D measurable},

(7.24)

7. Existence for the Bernoulli Free Boundary Problem

263

where for any measurable subset Ω of D the energy function is now deﬁned by 1 def def min E (ϕ) + G dx, E (ϕ) = (7.25) E(Ω) = |∇ϕ|2 − f ϕ dx, D D 1 ϕ∈H (Ω;D) Ω D 2 where H1 (Ω; D) is deﬁned in (7.13). By Theorem 3.6 H1 (Ω; D) is contained in H•1 (Ω; D), and for any element u in H1 (Ω; D) we have ∇u(x) = 0 for almost all x in D \ Ω so that 1 1 (7.26) ∀ϕ ∈ H (Ω; D), ED (ϕ) = |∇ϕ|2 − f ϕ dx Ω 2 1 G dx . (7.27) ⇒ E(Ω) = min |∇ϕ|2 − f ϕ dx + 1 ϕ∈H (Ω;D) Ω 2 Ω We have the following existence result for problem (P0 ). Theorem 7.1. For any f in L2 (D), G = Q2 in L1 (D), there exists at least one solution to problem (P0 ). Proof. Let {Ωn } be a minimizing sequence for problem (P0 ), and for each n let un be the solution to problem (7.25) with Ω = Ωn . If χn is the characteristic function of the measurable set Ωn , we have un in H1 (Ωn ; D), which implies that (1 − χn )un = 0. On the other hand, the sequence {un } remains uniformly bounded in H01 (D). Taking ϕ = 0 in (7.25) we get 1 |∇un |2 − f un dx ≤ 0, D 2 and the conclusion follows from the equivalence of norms in H01 (D). We can assume that the sequence {χn } weakly converges in L2 (D) to an element λ and that the sequence {un } weakly converges in H01 (D) to an element u. From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Deﬁne def

Ω(u) = {x ∈ D : u(x) = 0}. Then u belongs to H01 Ω(u) and we have m(Ω(u)) = m({x ∈ D : λ(x) = 1}) ≤ α, since we have α = m({x : λ(x) = 1}) + m({x : 0 ≤ λ(x) < 1}). In the limit with Ω = Ω(u) we get 1 1 1 2 2 |∇u| − f u dx = |∇u| − f u dx ≤ lim inf |∇un |2 − f un dx, (7.28) 2 2 Ω D D 2 G dx ≤ λG dx = lim G dx. (7.29) Ω

D

n→∞

Ωn

By adding (7.28) and (7.29) we get that Ω minimizes E and u minimizes E(Ω).

264 7.3.2

Chapter 5. Metrics via Characteristic Functions Constraint on the Measure of the Domain Ω

Consider an important variation of problem (7.22)–(7.23). Given any f in L2 (D), G in L1 (D), and a real number α, 0 < α < m(D), (P0α )

inf {E(Ω) : Ω ⊂ D measurable and m(Ω) = α} .

(7.30)

We have the following existence result for problem (P0α ). Theorem 7.2. For any f in L2 (D), G = 0, and any real number α, 0 < α < m(D), there exists at least one solution to problem (P0α ). Proof. Let {Ωn } be a minimizing sequence for problem (P0 ), and for each n let un be the unique solution to problem (7.25). If χn is the characteristic function of the measurable set Ωn , we have un in H1 (Ωn ; D), which implies that (1 − χn )un = 0. On the other hand, by picking ϕ = 0 in (7.25), {un } remains bounded in H01 (D), 1 |∇un |2 − f un dx ≤ 0, D 2 and the conclusion follows from the equivalence of norms in H01 (D). We can assume that {χn } weakly converges in L2 (D) to an element λ and that {un } weakly converges in H01 (D) to an element u ∈ H01 (D). In the limit we get λ(x) dx = lim χn (x) dx = α. n→∞

D

D

From Lemma 7.1 we get λ ≥ χu=0 almost everywhere in D. Deﬁne def

Ω(u) = {x ∈ D : u(x) = 0}. Then u belongs to H1 (Ω(u); D) and we have m(Ω(u)) = m({x ∈ D : λ(x) = 1}) ≤ α (as we have α = m({x : λ(x) = 1}) + m({x : 0 ≤ λ(x) < 1})). In the limit we get for Ω = Ω(u) 1 1 1 |∇u|2 − f u dx = |∇u|2 − f u dx ≤ lim inf |∇un |2 − f un dx, (7.31) n→∞ 2 2 2 Ω D D λG dx = lim G dx, (7.32) n→∞

D

so that

Ω

1 |∇u|2 − f u dx + 2

Ωn

λG dx ≤ D

inf

Ω ⊂D m(Ω )=α

If G ≥ 0 almost everywhere in D, we get E(Ω) ≤

inf

Ω ⊂D m(Ω )=α

{E(Ω )} ,

E(Ω ).

7. Existence for the Bernoulli Free Boundary Problem

265

but Ω does not necessarily satisfy the constraint m(Ω) = α. Note that for any measurable set Ω such that Ω ⊂ Ω ⊂ D, we have 1 1 2 |∇u| − f u dx = |∇u|2 − f u dx 2 Ω Ω 2 so that in expression (7.31) Ω can be enlarged to any such Ω . The inclusion of Ω in Ω implies the inclusion of H1 (Ω; D) in H1 (Ω ; D). So if G ≤ 0 almost everywhere in D, then it is readily seen that from (7.25) E(Ω ) ≤ E(Ω). In view of the previous assumption on G, now G = 0 almost everywhere in D. To conclude the proof we just have to select Ω with m(Ω ) = α. That measurable set Ω is admissible and minimizes the objective function in (7.30), and we have E(Ω ) = E(Ω) =

inf

{E(Ω )}.

Ω ⊂D m(Ω )=α

Corollary 1. Assume that G = 0 and f in L2 (D) and f = Q2 in L1 (D). Then the following problem has an optimal solution: (P0α− )

inf

Ω⊂D m(Ω)≤α

E(Ω).

(7.33)

Proof. The proof is similar to the proof of the theorem where the minimizing sequence is chosen such that m(Ωn ) ≤ α, so that in the weak limit we get λ(x) dx ≤ α. m(Ω) ≤ D

7.4

Weak Existence with Surface Tension

Problems (P0 ), (P0α ), and (P0α− ) have optimal solutions, but as they are associated with the homogeneous Dirichlet boundary condition, u in H1 (Ω; D), the optimal domain Ω is in general not allowed to have holes, that is to say, roughly speaking, that the topology of Ω is a priori speciﬁed. In many examples it turns out that the solution u physically corresponds to a potential and the classical homogeneous Dirichlet boundary condition is not the appropriate one. The physical condition is that the potential u should be constant on each connected component of the boundary ∂Ω in D. When Ω is a simply connected domain in R2 , then ∂Ω has a single connected component so the constant can be taken as zero. In general this constant can be ﬁxed only in one connected component; in the others the constant is an unknown of the problem. The minimization problems (P0 ), (P0α ), and (P0α− ) associated with the Hilbert space H•1 (Ω; D) fail (in the sense that the previous techniques for existence of an optimal Ω fail). The main reason is that Lemma 7.1 is no longer true when un is replaced by ∇un weakly converging in L2 (D)N . The key idea is to recover the equivalent of Lemma 7.1 by imposing the strong L2 (D)-convergence of the sequence {un }. In practice {un } corresponds to the sequence of characteristic functions

266

Chapter 5. Metrics via Characteristic Functions

χΩn of a minimizing sequence {Ωn }. To obtain the strong L2 (D)-convergence of a subsequence we add a constraint on the perimeters. Consider the family of ﬁnite perimeter sets in D of Deﬁnition 6.2 (ii) in section 6.1, def

BPS(D) = {Ω ⊂ D : χΩ ∈ BV(D)}, where BV(D) is deﬁned in (6.2). Introduce the following problem indexed by σ > 0: (Pσα )

inf

Ω⊂D m(Ω)=α def

E(Ω) =

Eσ (Ω),

min

ϕ∈H•1 (Ω;D)

Ω

def

Eσ (Ω) = E(Ω) + σPD (Ω), 1 |∇ϕ|2 − f ϕ dx + 2

(7.34)

G dx .

(7.35)

Ω

Theorem 7.3. Let f in L2 (D), G in L1 (D), σ > 0, and 0 ≤ α < m(D). Then problem (Pσα ) has at least one optimal solution Ω in BPS(D). Proof. Let {Ωn } be a minimizing sequence for (Pσα ) and let {χn } be the corresponding sequence of characteristic functions associated with {Ωn }. By picking ϕ = 0 in (7.35), −1 2 |G| dx + σPD (D) . cf L2 (D) + PD (Ωn ) ≤ σ D

By Theorem 6.3 there exist a subsequence of {Ωn }, still indexed by n, and Ω ⊂ D with ﬁnite perimeter such that χn → χ = χΩ in L1 (D)-strong and α = m(Ωn ) = m(Ω). For each n, let un in H•1 (Ωn ; D) be the unique minimizer of (7.35). That sequence remains bounded in H01 (D). Pick a subsequence, still indexed by n, such that {un } weakly converges to an element u in H01 (D). From the identity (1 − χn )∇un = 0 almost everywhere in D, we get in the limit (1 − χΩ )∇u = 0 almost everywhere in D, so that the limiting element u belongs to H•1 (Ω; D). Hence we have 1 1 f u dx |∇u|2 − f u dx = |∇u|2 dx − Ω 2 D 2 Ω (7.36) 1 2 ≤ lim inf |∇un | dx − f un dx, n→∞ D 2 Ωn f u dx = lim f un dx, G dx = lim G dx, (7.37) Ω

so that

Ω

n→∞

Ωn

1 |∇u|2 − f u dx + 2

Ω

n→∞

Ωn

G dx + σPD (Ω) ≤ Ω

inf

Ω⊂D m(Ω)=α

Eσ (Ω).

(7.38)

Chapter 6

Metrics via Distance Functions 1

Introduction

In Chapter 5 the characteristic function was used to embed the equivalence classes of measurable subsets of D into Lp (D) or Lploc (D), 1 ≤ p < ∞, and induce a complete metric on the equivalence classes of measurable sets. This construction is generic and extends to other set-dependent functions embedded in an appropriate function space. The Hausdorﬀ metric is the result of such a construction, where the distance function plays the role of the characteristic function. The distance function embeds equivalence classes of subsets A of a closed holdall D with the same closure A into the space C(D) of continuous functions. When D is bounded the C 0 -norm of the diﬀerence of the distance functions of two subsets of D is the Hausdorﬀ metric. The Hausdorﬀ topology has many much desired properties. In particular, for D bounded, the set of equivalence classes of nonempty subsets A of D is compact. Yet, the volume and perimeter are not continuous with respect to the Hausdorﬀ topology. This can be ﬁxed by changing the space C(D) to the space W 1,p (D) since distance functions also belong to that space. With that metric the volume is now continuous. The price to pay is the loss of compactness when D is bounded. But other sequentially compact families can easily be constructed. By analogy with Caccioppoli sets we introduce the sets for which the elements of the Hessian matrix of second-order derivatives of the distance function are bounded measures. They are called sets of bounded curvature. Their closure is a Caccioppoli set, and they enjoy other interesting properties. Convex sets belong to that family. General compactness theorems are obtained for such families under global or local conditions. This chapter also deals with the family of open sets characterized by the distance function to their complement. They are discussed in parallel along with the distance functions to a set. The properties of distance functions and Hausdorﬀ and Hausdorﬀ complementary metric topologies are studied in section 2. The projections, skeletons, cracks, and diﬀerentiability properties of distance functions are discussed in section 3. W 1,p topologies are introduced in section 4 and related to characteristic functions. The 267

268

Chapter 6. Metrics via Distance Functions

compact families of sets of bounded and locally bounded curvature are characterized in section 5. The notion of reach and Federer’s sets of positive reach are studied in section 6. The smoothness of smooth closed submanifolds is related to the smoothness of the square of the distance function in a neighborhood. Approximation of distance functions by dilated sets/tubular neighborhoods and their critical points are presented in section 7. Convex sets are characterized in section 8 along with the special family of convex sets. Both sets of positive reach and convex sets will be further investigated in Chapter 7. Finally section 9 gives several compactness theorems under global and local conditions on the Hessian matrix of the distance function.

2

Uniform Metric Topologies

2.1

Family of Distance Functions Cd (D)

In this section we review some properties of distance functions and present the general approach that will be followed in this chapter. Deﬁnition 2.1. Given A ⊂ RN the distance function from a point x to A is deﬁned as def

dA (x) =

inf |y − x|,

A = ∅,

+∞,

A = ∅,

y∈A

(2.1)

and the family of all distance functions of nonempty subsets of D as def

Cd (D) = dA : ∀A, ∅ = A ⊂ D .

(2.2)

When D = RN the family Cd (RN ) is denoted by Cd . By deﬁnition dA is ﬁnite in RN if and only if A = ∅. Recall the following properties of distance functions. Theorem 2.1. Assume that A and B are nonempty subsets of RN . (i) The map x → dA (x) is uniformly Lipschitz continuous in RN : ∀x, y ∈ RN ,

|dA (y) − dA (x)| ≤ |y − x|

(2.3)

0,1 (RN ).1 and dA ∈ Cloc

(ii) There exists p ∈ A such that dA (x) = |p − x| and dA = dA in RN . (iii) A = {x ∈ RN : dA (x) = 0}. 1 A function f belongs to C 0,1 (RN ) if for all bounded open subsets D of RN its restriction to loc D belongs to C 0,1 (D).

2. Uniform Metric Topologies

269

(iv) dA = 0 in RN ⇐⇒ A = RN . (v) A ⊂ B ⇐⇒ dA ≥ dB . (vi) dA∪B = min{dA , dB }. (vii) dA is (Fr´echet) diﬀerentiable almost everywhere and |∇dA (x)| ≤ 1

a.e. in RN .

(2.4)

Proof. (i) For all z ∈ A and x, y ∈ RN |z − y| ≤ |z − x| + |y − x|, dA (y) = inf |z − y| ≤ inf |z − x| + |y − x| = dA (x) + |y − x| y∈A

y∈A

and hence the Lipschitz continuity. (ii) As the function y → |y − x| is continuous (hence upper semicontinuous), dA (x) = inf |y − x| = inf |y − x| = dA (x). y∈A

y∈A

Since the function |y − x| ≥ 0 is bounded below, its inﬁmum over y ∈ A is ﬁnite. Let {yn } ⊂ A be a minimizing sequence dA (x) = inf |y − x| = lim |yn − x|. y∈A

n→∞

Then {yn − x} and hence {yn } are bounded sequences. Hence there exists a subsequence converging to some y ∈ A and by continuity dA (x) = |y − x|. (iii) By the continuity of dA , d−1 A {0} is closed and A ⊂ d−1 A {0}

⇒ A ⊂ d−1 A {0}.

Conversely, if dA (x) = 0, there exists y in A such that 0 = dA (x) = |y − x| and necessarily x = y ∈ A. (iv) follows from (iii). (v) From (ii) for x ∈ A, dA (x) = 0 and dB (x) ≤ dA (x) = 0 necessarily implies dB (x) = 0 and x ∈ B. Conversely, if A ⊂ B, then dB (x) = dB (x) = inf |y − x| ≤ inf |y − x| = dA (x) = dA (x). y∈B

y∈A

(vi) is obvious. (vii) follows from Rademacher’s theorem.

2.2

Pomp´ eiu–Hausdorﬀ Metric on Cd (D)

Let D be a nonempty subset of RN and associate with each nonempty subset A of D the equivalence class def

[A]d = B : ∀B, B ⊂ D and B = A ,

270

Chapter 6. Metrics via Distance Functions

since from Theorem 2.1 (v) dA = dB if and only if A = B. So A is the invariant closed representative of the class [A]d . Consider the set def

Fd (D) = [A]d : ∀A, ∅ = A ⊂ D that can be identiﬁed with the set

A : ∀A, ∅ = A closed ⊂ D . In general, there is no open representative in the class [A]d since dA = dA ≤ dint A , where int A = A denotes the interior of A. By the deﬁnition of [A]d the map [A]d → dA : Fd (D) → Cd (D) ⊂ C(D) is injective. So Fd (D) can be identiﬁed with the subset of distance functions Cd (D) in C(D). The distance function plays the same role as the set X(D) in Lp (D) of equivalence classes of characteristic functions of measurable sets. When D is bounded, C(D) is a Banach space when endowed with the norm f C(D) = sup |f (x)|.

(2.5)

x∈D

As for the characteristic functions of Chapter 5, this induces a complete metric def

ρ([A]d , [B]d ) = dA − dB C(D) = sup |dA (x) − dB (x)|

(2.6)

x∈D

on Fd (D), which turns out to be equal to the classical Pomp´eiu–Hausdorﬀ metric2

def

ρH ([A]d , [B]d ) = max sup dA (x), sup dB (y) x∈B

y∈A

(cf. J. Dugundji [1, Chap. IX, Prob. 4.8, p. 205] for the deﬁnition of ρH ). Indeed for x ∈ D, xA ∈ A, and xB ∈ B |x − xA | ≤ |x − xB | + |xB − xA | ⇒ dA (x) ≤ |x − xB | + dA (xB ) ≤ |x − xB | + sup dA (y) y∈B

⇒

dA (x) ≤ dB (x) + sup dA (y) y∈B

⇒

dA (x) − dB (x) ≤ sup dA (y). y∈B

2 The “´ ´iu [1] in his thesis presented ecart mutuel” between two sets was introduced by D. Pompe in Paris in March 1905. This is the ﬁrst example of a metric between two sets in the literature. It was studied in more detail by F. Hausdorff [2, “Quellenangaben,” p. 280, and Chap. VIII, sect. 6] in 1914.

2. Uniform Metric Topologies

271

By interchanging the roles of A and B ∀x ∈ D,

|dA (x) − dB (x)| ≤ max sup dA (z), sup dB (y) . z∈B

y∈A

The converse inequality follows from the fact that A ⊂ D and B ⊂ D. 0 When D is open but not necessarily bounded, we use Cloc (D) (cf. (2.19) in section 2.6.1 of Chapter 2), the space C(D) of continuous functions on D endowed with the Fr´echet topology of uniform convergence on compact subsets K of D, which is deﬁned by the family of seminorms ∀K compact ⊂ D,

def

qK (f ) = max |f (x)|. x∈K

(2.7)

The Fr´echet topology on C(D) is metrizable since the topology induced by the family of seminorms {qK } is equivalent to the one generated by the subfamily {qKk }k≥1 , where the compact sets {Kk }k≥1 are chosen as follows: 1 def Kk = x ∈ D : dD (x) ≥ and |x| ≤ k , k ≥ 1. (2.8) k It is equivalent to the topology deﬁned by the metric def

δ(f, g) =

∞ 1 qKk (f − g) . 2k 1 + qKk (f − g)

(2.9)

k=1

It will be shown below that Cd (D) is a closed subset of Cloc (D) and that this will induce the following complete metric on Fd (D): def

ρδ ([A]d , [B]d ) = δ(dA , dB ) =

∞ 1 qKk (dA − dB ) , 2k 1 + qKk (dA − dB )

(2.10)

k=1

which is a natural extension of the Hausdorﬀ metric to an unbounded domain D. Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN . (i) The set Cd (D) is closed in Cloc (D) (resp., C(D)) and ρδ (resp., ρ) deﬁnes a complete metric topology on Fd (D). (ii) When D is a bounded open subset of RN , the set Cd (D) is compact in C(D) and the metrics ρ and ρH are equal. Proof. (i) We give the proof of C. Dellacherie [1, Thm. 2, p. 42 and Rem. 1, p. 43]. The basic constructions will again be used in Chapter 7. Consider a sequence {An } of nonempty subsets of D such that dAn converges to some element f of Cloc (D). We wish to prove that f = dA for def

A = {x ∈ D : f (x) = 0}

272

Chapter 6. Metrics via Distance Functions

and that the closed subset A of D is nonempty. Fix x in D; then ∀n, ∃yn ∈ An ,

|yn − x| = inf |z − x| = dAn (x) z∈An

⇒ lim |yn − x| = lim dAn (x) = f (x). n→∞

n→∞

Hence {yn − x} and {yn } ⊂ D are bounded and there exists a subsequence, still indexed by n, which converges to some y ∈ D yn → y, and |y − x| = f (x). In particular, f (y) = 0, since in the inequality f (y) ≤ f (y) − dAn (y) + dAn (y) − dAn (yn ) + dAn (yn ), the last term is zero and dAn is Lipschitz continuous with constant equal to 1: |f (y)| ≤ |f (y) − dAn (y)| + |y − yn |, and both terms go to zero. By the deﬁnition of A, y ∈ A and A is not empty. Therefore for each x ∈ D, there exists y ∈ A such that f (x) = |y − x| ≥ inf |z − x| = dA (x). z∈A

Next we prove the inequality in the other direction. By construction for any An ⊂ D ∀x, y ∈ D,

|dAn (x) − dAn (y)| ≤ |x − y|

and |f (x) − f (y)| ≤ |f (x) − dAn (x)| + |dAn (x) − dAn (y)| + |dAn (y) − f (y)|. By uniform convergence the ﬁrst and last terms converge to zero, and by Lipschitz continuity of dAn , ∀x, y ∈ D, |f (x) − f (y)| ≤ |x − y|. Hence for all x ∈ D and y ∈ A, f (y) = 0 and f (x) ≤ f (y) + |x − y| = |x − y| ⇒ ∀x ∈ D,

f (x) ≤ inf |x − y| = dA (x). y∈A

This proves the reverse equality. (ii) Observe that on a compact set D and for any A ⊂ D def

dA (x) = inf |y − x| ≤ sup |y − x| ≤ c = sup |y − x| < ∞, y∈A

∀x, y ∈ D,

y∈A

y,x∈D

|dA (y) − dA (x)| ≤ |y − x|.

2. Uniform Metric Topologies

273

The compactness of Cd (D) now follows by Ascoli–Arzel`a Theorem 2.4 of Chapter 2 and part (i). By construction, ρ and ρδ are metrics on Fd (D). By deﬁnition, for A and B in the compact D, ρ(A, B) = max |dA (x) − dB (x)| x∈D ≥ max max |dA (x) − dB (x)|, max |dA (x) − dB (x)| x∈B x∈A ≥ max max dA (x), max dB (x) = ρH (A, B). x∈B

x∈A

Conversely, for any x ∈ D and y in A, dA (x) − dB (x) ≤ |y − x| − dB (x) and there exists xB ∈ B such that dB (x) = |x − xB |. Therefore, ∀y ∈ A, dA (x) − dB (x) ≤ |y − x| − |x − xB | ≤ |y − xB | ⇒ dA (x) − dB (x) ≤ inf |y − xB | = dA (xB ) ≤ max dA (x). y∈A

x∈B

Similarly dB (x) − dA (x) ≤ max dB (x), x∈A

and for all x in D

|dB (x) − dA (x)| ≤ max max dA (x), max dB (x) ⇒ ρ(A, B) ≤ ρH (A, B). x∈B

x∈A

When D is bounded the family Fd (D) enjoys more interesting properties. Theorem 2.3. Let D be a nonempty open (resp., bounded open) subset of RN . Deﬁne for a subset S of RN the sets def

H(S) = dA ∈ Cd (D) : S ⊂ A , def

I(S) = dA ∈ Cd (D) : A ⊂ S , def

J(S) = dA ∈ Cd (D) : A ∩ S = ∅ . (i) Let S be a subset of RN . Then H(S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then I(S) is closed in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then J(S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then J(S) is open in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then I(S) is open in Cloc (D) (resp., C(D)). (iv) For a nonempty bounded holdall D, the map def

[A]d → #(A) : F# (D) = [A]d : ∀A, ∅ = A ⊂ D and #(A) < ∞ → R

274

Chapter 6. Metrics via Distance Functions is lower semicontinuous, where #(A) is the number of connected components as deﬁned in Deﬁnition 2.2 of Chapter 2. For a ﬁxed number c > 0 the subset {dA ∈ Cd (D) : #(A) ≤ c} of Cd (D) is compact in C(D). In particular, the subset {dA ∈ Cd (D) : A is connected} of Cd (D) is compact in C(D).

Proof. (i) If S ⊂ D, then H(S) = ∅ and there is nothing to prove. Assume that S ⊂ D. From Theorem 2.1 (v), S ⊂ A ⇒ dS ≥ dA . So for any sequence {dAn } in H(S) converging to dA in Cloc (D), ∀n,

dS ≥ dAn

⇒ dS ≥ dA

⇒ S⊂S⊂A

⇒ dA ∈ H(S).

(ii) We shall use the same technique for I(S) as for H(S), but here we need S = S to conclude. For D ∩ S = ∅, J(S) = ∅ and there is nothing to prove. Assume that D ∩ S = ∅ and consider a sequence {dAn } in J(S) converging to dA in Cloc (D). Assume that A ∩ S = ∅. Then A ⊂ D implies A ∩ [S ∩ D] = A ∩ S = ∅. By assumption, S ∩ D is compact and ∃δ > 0, ∀x ∈ S ∩ D, dA (x) ≥ δ, ∃N ≥ 1, ∀n ≥ N, dAn − dA C(D) ≤ δ/2. So for all x ∈ D ∩ S dAn (x) ≥ dA (x) − |dAn (x) − dA (x)| ≥ δ − δ/2 > 0 ⇒ An ∩ S = An ∩ [D ∩ S] = ∅, and this contradicts the fact that An ∈ J(S). (iii) By deﬁnition, for any S, J(S) = I(S),

J(S) = dA ∈ Cd (D) : A ∩ S = ∅ = dA ∈ Cd (D) : A ⊂ S = I(S). Since S is closed, J(S) is closed from part (ii) and J(S) is open. Similarly, replacing S by S in the previous identity, I(S) = J(S). So from part (ii) if D ∩ S is compact, then I(S) is closed and I(S) is open. From part (ii) if D ∩ S is compact, then I(S) is closed and I(S) is open. (iv) Let {An } and A be nonempty subsets of D such that dAn converges to dA in C(D). Assume that #(A) = k is ﬁnite. Then there exists a family of disjoint open sets G1 ,. . . , Gk such that A ⊂ G = ∪ki=1 Gi

and ∀i, A ∩ Gi = ∅.

In view of the deﬁnitions of I(S) and J(S) in part (i) A ∈ U = ∩ki=1 J(Gi ) ∩ I(G).

2. Uniform Metric Topologies

275

But U is not empty and open as the ﬁnite intersection of k + 1 open sets. As a result there exist ε > 0 and an open neighborhood of [A]d : Nε ([A]d ) = {[B]d : dB − dA < ε} ⊂ U. Hence since dAn converges to dA , there exists n > 0 such that, for all n ≥ n, [An ]d ∈ U, and necessarily An ⊂ G,

∀n ≥ n,

An ∩ Gi = ∅, ∀i

⇒ #(An ) ≥ #(A).

Therefore, [A]d → #(A) is lower semicontinuous. This theorem has many interesting corollaries. For instance part (iii) says something about the function that gives the number of connected components of A (cf. T. J. Richardson [1] for an application to image segmentation).

2.3

Uniform Complementary Metric Topology and Cdc (D)

In the previous section we dealt with a theory of closed sets since the equivalence class of the subsets of D was completely determined by their unique closure. In partial diﬀerential equations the underlying domain is usually open. To accommodate this point of view, consider the set of open subsets Ω of a ﬁxed nonempty open holdall D in RN , endowed with the Hausdorﬀ topology generated by the distance functions dΩ to the complement of Ω. This approach has been used in ´sio [6, sect. 1.3, p. 405] in the conseveral contexts, for instance, in J.-P. Zole ˇ ´ k [2] uses the family Cdc (D) for text of free boundary problems. Also, V. Sver a open sets Ω such that #([Ω]) ≤ c for some ﬁxed c > 0. His main result is that in dimension 2 the convergence of a sequence {Ωn } to Ω of such sets implies the convergence of the corresponding projection operators {PΩn : H01 (D) → H01 (Ωn )} to PΩ : H01 (D) → H01 (Ω), where the projection operators are directly related to homogeneous Dirichlet linear boundary value problems on the corresponding domains {Ωn } and Ω. In dimension 1 the constraint on the number of components can be dropped. This result will be discussed in Theorem 8.2 of section 8 in Chapter 8. By analogy with the constructions of the previous section, consider for a nonempty subset D of RN the family

dA : ∅ = A and A ⊂ D . By deﬁnition, A ⊃ D

⇒ dA = 0 in D,

and, by Lipschitz continuity, dA = 0 in D and dA = 0 on ∂D = D ∩ D. If int D = ∅, then dA = 0 in RN and for all sets A in the family int A = ∅. If int D = ∅, associate with each A the open set def

Ω = int A = A

⇒ Ω = A and dΩ = dA .

276

Chapter 6. Metrics via Distance Functions

By deﬁnition and the previous considerations, A ⊃ D

⇒ Ω = A ⊂ D = int D.

So ﬁnally, for int D = ∅,

dA : ∅ = A and A ⊂ D = dΩ : Ω = RN and Ω open ⊂ int D . The invariant set int A is the open representative in the equivalence class def [A]cd = B : B = A = ∅ , but, in general, there is no closed representative in that class since dint A = dA = dA ≤ dA , as in the case of dA , where dA = dA ≤ dint A . From this analysis it will be suﬃcient to consider the family of open subsets of an open holdall D. Deﬁnition 2.2. Let D be a nonempty open subset of RN . Deﬁne the family of functions def

Cdc (D) = dΩ : ∀Ω open subset of D and Ω = RN corresponding to the family of open sets def

G(D) = Ω ⊂ D : ∀Ω open and Ω = RN .

(2.11)

(2.12)

For D bounded deﬁne the Hausdorﬀ complementary metric ρcH on G(D): def

ρcH (Ω2 , Ω1 ) = dΩ2 − dΩ1 C(D) .

(2.13)

Note that for Ω1 ⊂ D and Ω2 ⊂ D, dΩ1 = dΩ2 = 0 in D. Therefore, dΩ1 = dΩ2 in D ⇐⇒ Ω1 = Ω2 ⇐⇒ Ω1 = Ω2 . Theorem 2.4. Let D be a nonempty open subset of RN . (i) The set Cdc (D) is closed in Cloc (D). (ii) If, in addition, D is bounded, then Cdc (D) is compact in C0 (D) and (G(D), ρcH ) is a compact metric space. (iii) (Compactivorous property) Let {Ωn } and Ω be sets in G(D) such as dΩn → dΩ

in Cloc (D).

Then for any compact subset K ⊂ Ω, there exists an integer N (K) > 0 such that ∀n ≥ N (K), K ⊂ Ωn .

2. Uniform Metric Topologies

277

Proof. (i) Let {Ωn } be a sequence of open subsets in D such that {dΩn } is a Cauchy sequence in Cloc (D). For all n, dΩn = 0 in D and {dΩn } is also a Cauchy sequence in Cloc (RN ). By Theorem 2.2 (i) the sequence {dΩn } converges in Cloc (RN ) to some distance function dA . By construction Ωn ⊂ D =⇒ Ωn ⊃ D =⇒ dΩn ≤ dD =⇒ dA ≤ dD =⇒ A ⊃ D =⇒ A ⊂ D. By choosing the open set Ω = A in D we get dΩn → dΩ

in Cloc (RN ) and C0,loc (D).

(ii) To prove the compactness we use the compactness of Cd (D) from Theorem 2.2 (ii) and the fact that Cdc (D) is closed in C0 (D). Observe that since Ωn ⊃ D, dΩn = 0 in D, dΩn ∈ C0 (D), and Ωn = [Ωn ∩ D] ∪ [Ωn ∩ D] ⊂ [Ωn ∩ D] ∪ D ⊂ Ωn ⇒ dΩn = d[Ωn ∩D]∪D = min{dΩn ∩D , dD }. There exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that dΩn ∩D → dA

in C0 (D)

⇒

dΩn = min{dΩn ∩D , dD } → min{dA , dD } = dA∪D

⇒

dΩn → dA∪D

in C(D)

in C(RN )

since dΩn = 0 = dA∪D in D. The result now follows by choosing the open set def

Ω = [A ∪ D] = A ∩ D ⊂ D. (iii) Deﬁne

def

m = inf inf |x − y| = inf dΩ (x). x∈K y∈Ω

x∈K

Since K ⊂ Ω ⊂ D, there exist x ˆ ∈ K, yˆ ∈ ∂Ω such that x − yˆ|. m = inf inf |x − y| = |ˆ x∈K y∈∂Ω

Necessarily, m > 0 since Ω is open, x ˆ ∈ K ⊂ Ω, and yˆ ∈ ∂Ω. Now ∃N > 0, ∀n ≥ N, dΩn − dΩ C(K) < m/2 ⇒ ∀x ∈ K, dΩn (x) ≥ dΩ (x) − |dΩn (x) − dΩ (x)| ≥ m − m/2 > 0 ⇒ x = Ωn

⇒ x ∈ Ωn = int Ωn = Ωn

Notation 2.1. It will be convenient to write

⇒ K ⊂ Ωn .

Hc

Ωn → Ω for the Hausdorﬀ complementary convergence of open sets of G(D) dΩn → dΩ

in Cloc (D).

278

Chapter 6. Metrics via Distance Functions

We have the analogue of Theorem 2.3 for the distance function to the complement of an open subset of D. Theorem 2.5. Let D be a nonempty open (resp., bounded open) subset of RN . Deﬁne for a subset S of RN and open subsets Ω of D the following families: def

H c (S) = dΩ ∈ Cdc (D) : S ⊂ Ω , def

I c (S) = dΩ ∈ Cdc (D) : Ω ⊂ S , def

J c (S) = dΩ ∈ Cdc (D) : Ω ∩ S = ∅ . (i) Let S be a subset of RN . Then H c (S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then I c (S) is closed in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then J c (S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then J c (S) is open in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then I c (S) is open in Cloc (D) (resp., C(D)). (iv) For a nonempty bounded open holdall D, the map def

Ω → #(Ω) : G# (D) = Ω : Ω open ⊂ D and # Ω < ∞ → R is lower semicontinuous, where #(Ω) is the number of connected components of Ω as deﬁned in Deﬁnition 2.2 of Chapter 2. For a ﬁxed number c ≥ 0, the subset {dΩ ∈ Cdc (D) : # Ω ≤ c} is compact in C(D). In particular, the subset {dΩ ∈ Cdc (D) : Ω is hole-free} is compact in C(D), where hole-free is in the sense of Deﬁnition 2.2 of Chapter 2.

2.4

c Families Cdc (E; D) and Cd,loc (E; D)

Now and then, it is desirable to avoid having the empty set as a solution of a shape optimization problem. This can occur for the C(D)-topology on Cdc (D). One way to get around this is to assume that each set of the family contains a ﬁxed nonempty set or, more generally, that each set of the family contains a translated and rotated image of a ﬁxed nonempty set. Theorem 2.6. Let D, ∅ = D ⊂ RN , be bounded open, let E, ∅ = E ⊂ D, and let def

Cdc (E; D) = {dΩ : E ⊂ Ω open ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open ⊂ D N

dΩ :

(2.14)

& .

(2.15)

3. Projection, Skeleton, Crack, and Diﬀerentiability

279

c (E; D) are compact in C(D). (i) If E is open, Cdc (E; D) and Cd,loc c (ii) If E is closed, Cdc (E; D) and Cd,loc (E; D) are open in C(D).

Proof. (a) We ﬁrst deal with Cdc (E; D) using the fact that

Cdc (E; D) = dΩ : Ω ⊂ E and Ω open ⊂ D = I c (E) and Theorem 2.5. If E is open, E is closed and I c (E) is closed and hence compact since D is bounded. If E is closed, E is open, E ∩ D is compact, and I c (E) is open. c (D) (b) Assume that E is open. Let {dΩn } be a Cauchy sequence in CD converging to some dΩ . Denote by xn ∈ RN and An ∈ O(N) the translation and rotation of E such that xn + An E ⊂ Ωn and dxn +An E ≤ dΩ . Since, for all n, xn + An E ⊂ D and D is bounded, there exist x ∈ RN and A ∈ O(N) and subsequences, still indexed by n, such that xn → x and An → A. But dxn +An E (z) = dE (An (z − xn )) since A−1 = A in O(N) and dE (An (z − xn )) → dE (A(z − x)). Finally, dx+AE = dE (A(z − x)) ≤ dΩ and, since x + AE and Ω are both closed, x + AE ⊂ Ω, c c which implies x + AE ⊃ Ω and dΩ ∈ Cd,loc (E; D). Therefore Cd,loc (E; D) as c a closed subset of the compact set Cd (D) is compact. We leave the proof that c (E; D) is open when E is closed to the reader. Cd,loc

3

Projection, Skeleton, Crack, and Diﬀerentiability

In this section we study the connection between the gradient of dA , the set of projections onto A, and the characteristic function of A. We partition the set of singularities of the gradient of dA into the skeleton 3 and set of cracks of A. Deﬁnition 3.1. Let A ⊂ RN , ∅ = A, and denote by def

Sing (∇dA ) = x ∈ RN : ∇dA (x)

(3.1)

the set of singularities of the gradient of dA . (i) Given x ∈ RN , a point p ∈ A such that |p − x| = dA (x) is called a projection onto A. The set of all projections onto A will be denoted by def

ΠA (x) = p ∈ A : |p − x| = dA (x) . (3.2) When ΠA (x) is a singleton, its element will be denoted by pA (x). 3 Our

deﬁnition of a skeleton does not exactly coincide with the one used in morphological mathematics, where it is deﬁned as the closure Sk (A) of our skeleton Sk (A) (cf., for instance, `re [1], and J. Serra [1] for the pioneering applications in mining G. Matheron [1] or A. Rivie engineering in 1968 and later in image processing).

280

Chapter 6. Metrics via Distance Functions

(ii) The skeleton of A is the set of all points of RN whose projection onto A is not unique. It will be denoted by def

Sk (A) = x ∈ RN : ΠA (x) is not a singleton . (3.3) (iii) The set of cracks is deﬁned as def

Ck (A) = Sing ∇dA \Sk (A).

(3.4)

Note that Sk (A) = Sk (A), Ck (A) = Ck (A), and Sing (A) = Sing (A) since dA = dA by Theorem 2.1 (ii). Even for a set A with smooth boundary ∂A, the gradient ∇dA (x) may not exist far from the boundary as shown in Figure 6.1, where ∇dA (x) exists everywhere outside A except on Sk (A), a semi-inﬁnite line. Yet, ∇dA (x) exists close to the boundary. An example of the set of cracks is given by the eyes, nose, mouth, and rays of the smiling sun in Figure 5.1 of Chapter 5 that belong to the boundary of the set as opposed to the skeleton Sk (A) that lies outside of A. Sk (A)

Sk (A) A

Figure 6.1. Skeletons Sk (A), Sk (A), and Sk (∂A) = Sk (A) ∪ Sk (A). The level sets of dA generate dilations/tubular neighborhoods of A. Deﬁnition 3.2. Let ∅ = A ⊂ RN and let h > 0. (i) The open h-tubular neighborhood , the closed h-tubular neighborhood , and the h-boundary of A are respectively deﬁned as def

def

Uh (A) = x ∈ RN : dA (x)| < h , Ah = x ∈ RN : dA (x)| ≤ h , and d−1 A {h}. We shall also use the terminology open or closed dilated sets. (ii) For 0 < s < h < ∞ def

def

Us,h (A) = x ∈ RN : s < dA (x)| < h and As,h = x ∈ RN : s ≤ dA (x)| ≤ h . This terminology is justiﬁed by the fact that a dilated set is a relatively nice set.

3. Projection, Skeleton, Crack, and Diﬀerentiability

281

Theorem 3.1. Let ∅ = A ⊂ RN . For r > 0, Ar = Ur (A), ∂Ar = ∂Ur (A) = d−1 A {r}, Ar = Ur (A), int Ar = Ur (A).

(3.5)

In particular, for 0 < s < r, Us,r (A) = As,r . Proof. By deﬁnition A ⊂ Ur (A) ⊂ Ar and Ur (A) ⊂ Ar . For any x ∈ Ar , dA (x) ≤ r. / A and there If dA (x) < r, then x ∈ Ur (A) ⊂ Ur (A). If dA (x) = r > 0, then x ∈ exists p ∈ ∂A such that dA (x) = |x − p| = r > 0. Consider the points x−p x−p 1 def r →p+r = x. xn = p + 1 − n |x − p| |x − p| Then p ∈ ΠA (xn ), and dA (xn ) = r (1 − 1/n) < r, dA (xn ) → dA (x) = r, and x ∈ Ur (A). Therefore, since Ur (A) is open, Ar = Ur (A) = Ur (A) ∪ ∂Ur (A) implies that ∂Ur (A) = d−1 A {r}. Moreover, since Ur (A) is closed, Ur (A) = int Ur (A) ∪ ∂Ur (A) = Ar ∪ ∂Ur (A) implies that Ar = Ur (A). The proof is the same for Us,r (A) = As,r . We now give several important technical results that will be used later for sets of positive reach and in Chapter 7. We ﬁrst give a few additional deﬁnitions. Deﬁnition 3.3. (i) A function f : U → R deﬁned in a convex subset U of RN is convex if for all x and y in U and all λ ∈ [0, 1], f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y). It is concave if the function −f is convex. A function f : U → R deﬁned in a convex subset U of RN is semiconvex (resp., semiconcave) if ∃c ≥ 0,

fc (x) = c |x|2 + f (x)

(resp., fc (x) = c |x|2 − f (x))

is convex in U . (ii) A function f : U → R deﬁned in a subset U of RN is locally convex (resp., locally concave) if it is convex (resp., concave) in every convex subset of U . A function f : U → R deﬁned in a subset U of RN is locally semiconvex (resp., locally semiconcave) if it is semiconvex (resp., semiconcave) in every convex subset of U . When U is convex the local deﬁnitions coincide with the global ones. The next theorem is a slight extension of a result in J. H. G. Fu [1, Prop. 1.2]. Theorem 3.2. Let A ⊂ RN be such that A = ∅. (i) Given h > 0, the function def

kA,h (x) =

|x|2 − 2 h dA (x), |x|2 − d2A (x) − h2 ,

if dA (x) ≥ h, if dA (x) < h,

282

Chapter 6. Metrics via Distance Functions is convex (and continuous) in RN and the function x →

kA,h (x) |x|2 = − dA (x) : RN \Uh (A) → R 2h 2h

is locally convex (and continuous) in RN \Uh (A). (ii) The function def

x → fA (x) =

1 2 |x| − d2A (x) : RN → R 2

is convex and continuous, d2A (x) is the diﬀerence of two convex functions d2A (x) = |x|2 − |x|2 − d2A (x) , (3.6) N ∇fA and ∇d2A belong to BVloc (RN )N , and ∇dA ∈ BVloc RN \∂A . Proof. (i) For all p ∈ A deﬁne the convex function p (x) =

|x − p| − h, 0,

|x − p| ≥ h, |x − p| < h.

Since p is nonnegative, 2p is convex and |x|2 − 2 h |x − p| + |p|2 + h2 − 2 x · p, 0,

2p (x) =

|x − p| ≥ h, |x − p| < h.

By subtracting the constant term |p|2 + h2 and the linear term −2 x · p from 2p , we get the new convex function mp (x) =

|x|2 − 2 h |x − p|, |x|2 − |p − x|2 − h2 ,

|x − p| ≥ h, |x − p| < h.

Then the function def

k(x) = sup mp (x) = p∈A

|x|2 − 2 h dA (x),

& dA (x) ≥ h,

|x|2 − d2A (x) − h2 ,

dA (x) < h

is convex in RN . Moreover, it is ﬁnite for each x ∈ RN and hence continuous in x. Indeed, if x is such that dA (x) ≥ h, by Theorem 2.1 (ii) there exists p¯ ∈ A such that |x − p¯| = dA (x) ≥ h and |x − p¯| = dA (x) =

inf p∈A |x−p|≥h

|x − p|

3. Projection, Skeleton, Crack, and Diﬀerentiability

283

and k(x) = |x|2 − 2 h dA (x). If dA (x) < h, then there exists p¯ ∈ A such that |x − p¯| = dA (x) < h and |x − p¯| = dA (x) =

inf

|x − p|

p∈A |x−p| 0, |x|2 − d2A (x) is locally convex in Uh (A) it is locally convex and continuous in RN = ∪h>0 Uh (A) and hence convex in RN . Therefore, from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240], ∇fA ∈ BVloc (RN )N and ∇d2A ∈ BVloc (RN )N . Finally, RN \∂A is the union of the two disjoint open sets RN \A and int A. On int A, ∇dA = 0. For any point x in RN \A, there exists ρ, 0 < 3 ρ < dA (x), such that the open ball B(x, 2 ρ) is contained in RN \Uρ (A), where kA,ρ is locally convex by part (i). Hence kA,ρ is convex in B(x, 2 ρ) and ∇kA,ρ , and ∇dA belong to BVloc (B(x, 2ρ))N and, by Deﬁnition 6.1 in Chapter 5, ∇dA belong to BV(B(x, ρ))N . Finally, by Theorem 6.1 in Chapter 5, ∇dA belongs to BVloc (RN \A)N . We now give some basic results. Theorem 3.3. Let A, ∅ = A ⊂ RN , and let x ∈ RN . (i) The set ΠA (x) is nonempty, compact, and ∀x ∈ /A

ΠA (x) ⊂ ∂A

and

∀x ∈ A

ΠA (x) = {x}.

(ii) For all x and v in RN , the Hadamard semiderivative4 of d2A always exists, d2A (x + tv) − d2A (x) = min 2(x − p) · v t0 t p∈ΠA (x)

dH d2A (x; v) = lim

= 2(x · v − σΠA (x) (v)), fA (x + tv) − fA (x) = σΠA (x) (v) = σco ΠA (x) (v), dH fA (x; v) = lim t0 t where σB is the support function of the set B, σB (v) = sup z · v, z∈B

and co B is the convex hull of B. In particular,

Sk (A) = x ∈ RN : ∇d2A (x) ⊂ RN \A. 4A

function f : RN → R has a Hadamard semiderivative in x in the direction v if def

dH f (x; v) = lim

t0 w→v

(cf. Chapter 9, Deﬁnition 2.1 (ii)).

f (x + tw) − f (x) exists t

(3.7)

284

Chapter 6. Metrics via Distance Functions Given v ∈ RN , for all x ∈ RN \∂A the Hadamard semiderivative of dA exists, dH dA (x; v) =

min 2 p∈ΠA (x)

x−p · v, |x − p|

(3.8)

and for x ∈ ∂A, dH dA (x; v) exists if and only if lim

t0

dA (x + tv) t

exists.

(3.9)

(iii) The following statements are equivalent: (a) d2A (x) is (Fr´echet) diﬀerentiable at x. (b) d2A (x) is Gateaux diﬀerentiable at x. (c) ΠA (x) is a singleton. (iv) ∇fA (x) exists if and only if ΠA (x) = {pA (x)} is a singleton. In that case 1 pA (x) = ∇fA (x) = x − ∇d2A (x). 2 For all x and y in RN , ∀p(x) ∈ ΠA (x),

1 2 1 2 |y| − d2A (y) ≥ |x| − d2A (x) + p(x) · (y − x), (3.10) 2 2

or equivalently, ∀p(x) ∈ ΠA (x),

d2A (y) − d2A (x) − 2 (x − p(x)) · (y − x) ≤ |x − y|2 . (3.11)

For all x and y in RN ∀p(x) ∈ ΠA (x), ∀p(y) ∈ ΠA (y),

(p(y) − p(x)) · (y − x) ≥ 0.

(3.12)

(v) The functions pA : RN \Sk (A) → RN

and

∇d2A : RN \Sk (A) → RN

are continuous. For all x ∈ RN \Sk (A) 1 pA (x) = x − ∇d2A (x) = ∇fA (x). 2 In particular, for all x ∈ A, ΠA (x) = {x} and ∇d2A (x) = 0,

Sk (A) = x ∈ RN : ∇d2A (x) and ∇dA (x) ⊂ RN \A,

Ck (A) = x ∈ RN : ∇d2A (x) ∃ and ∇dA (x) ⊂ ∂A, and Sing (∇dA ) = Sk (A) ∪ Ck (A).

(3.13)

(3.14)

3. Projection, Skeleton, Crack, and Diﬀerentiability

285

(vi) The function ∇dA : RN \ Sk (A) ∪ ∂A → RN is continuous. For all x ∈ int A, ∇dA (x) = 0 and for all x ∈ RN \ Sk (A) ∪ A ∇dA (x) =

∇d2A (x) x − pA (x) = 2 dA (x) |x − pA (x)|

|∇dA (x)| = 1.

and

(3.15)

If ∇dA (x) exists for x ∈ A, then ∇dA (x) = 0. In particular, ∇dA = 0 almost everywhere in A. (vii) The functions d2A and dA are diﬀerentiable almost everywhere and m(Sk (A)) = m(Ck (A)) = m(Sing (∇dA )) = 0. For ∂A = ∅ Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A).

(3.16) (3.17)

If ∂A = ∅, then either A = RN or A = ∅. (viii) Given A ⊂ RN , ∅ = A (resp., ∅ = A), χA (x) = 1 − |∇dA (x)|,

χint A (x) = |∇dA (x)| in RN \Sing (∇dA )

(resp., χA (x) = 1 − |∇dA (x)|, χint A (x) = |∇dA (x)| in RN \Sing (∇dA )) and the above identities hold for almost all x in RN . def

(ix) Given x ∈ RN , α ∈ [0, 1], p ∈ ΠA (x), and xα = p + α (x − p), then dA (xα ) = |xα − p| = α |x − p| = α dA (x)

and

∀α ∈ [0, 1], ΠA (xα ) ⊂ ΠA (x).

In particular, if ΠA (x) is a singleton, then ΠA (xα ) is a singleton and ∇d2A (xα ) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = A, then for all 0 < α ≤ 1 ∇dA (xα ) exists and ∇dA (xα ) = ∇dA (x). Remark 3.1. In general, for each v ∈ RN ∃ˆ p = p(v) ∈ ΠA (x),

dH d2A (x; v) = 2(x − pˆ) =

inf p∈ΠA (x)

2(x − p) · v,

since ΠA (x) is compact. However, when ΠA (x) is not a singleton, pˆ is not necessarily unique since 2(x − p) depends on the direction v. For points x outside of A and Sk (A) the norm of ∇dA (x) is equal to 1. When A is suﬃciently smooth, ∇dA (x) coincides with the outward unit normal to A at the point pA (x). When A is not smooth, this normal is not always unique as shown in Figure 6.2.

286

Chapter 6. Metrics via Distance Functions x pA (x) = pA (y) ∇dA (x) A

∇dA (y)

y Figure 6.2. Nonuniqueness of the exterior normal.

It is useful to consider examples to illustrate the subtleties of the theorem. The set of cracks Ck (A) of a set A has zero measure, but its boundary ∂A can have a nonempty interior and a nonzero measure. Example 3.1. Let B1 (0) be the open ball of radius 1 at the origin in RN and let A = {x ∈ B1 (0) : x has rational coordinates}. Then A = B1 (0), A = RN , ∂A = B1 (0), ∂A = ∂B1 (0), ∇dA (x) = ∇d2A (x) = 0 in B1 (0)

x = pA (x) in B1 (0), |x| Ck (A) = ∂B1 (0).

and ∇dA (x) =

Sk (A) = ∅ and

In general ∂A ⊂ ∂A and this inclusion can be strict as illustrated by this example. Also note that the boundary ∂A has a nonempty interior int ∂A = B1 (0) and a nonzero Lebesgue measure (nonzero volume in dimension N = 3). Thus the boundary of two members of the equivalence class [A] can be diﬀerent. The boundary ∂A of a closed or an open set A has no interior; that is, ∂A is nowhere dense. Yet, its boundary ∂A still does not necessarily have a zero measure. If A is closed, then A = int A ∪ ∂A. If int A = ∅, there exist x ∈ ∂A and r > 0 such that Br (x) ⊂ ∂A ⊂ A and x ∈ int A, a contradiction. If A is open, A is closed, A = int A ∪ ∂A, and ∂A has no interior. Example 3.2 (Modiﬁed version of Example 1.10 in E. Giusti [1, p. 7]). Let B1 (0) in R2 be the open ball in 0 of radius 1. For i ≥ 1, let {xi : i ∈ N} be the sequence of all points in B1 (0) with rational coordinates for some choice of ordering. Associate with each i the open ball Bi = {x ∈ R2 : |x − xi | < ρi },

0 < ρi ≤ min{2−i , 1 − |xi |}

⇒ Bi ⊂ B1 (0).

3. Projection, Skeleton, Crack, and Diﬀerentiability

287

Consider the increasing sequence of open subsets of B1 (0) def

Ωn =

n *

def

Bi Ω =

i=1

∞ *

Bi

i=1

def

and the closed set A = Ω. We get A = A = Ω,

A = Ω,

A = Ω = B1 (0),

∂A = B1 (0) ∩ Ω.

As a result, we have the following decreasing sequence of closed subsets of B1 (0): def

Dn = B1 (0) ∩ Ωn ∂A = B1 (0) ∩ Ω 2 n n 1 m(Bi ) = π − π ⇒ m(Dn ) ≥ m(B1 (0)) − 2i i=1 i=1 2 ∞ 2 1 ⇒ m(∂A) ≥ π − π = π. i 2 3 i=1 Proof of Theorem 3.3. (i) Existence. The function z → |z − x|2 : RN → R

(3.18)

is continuous and, a fortiori, upper semicontinuous, and hence inf |a − x|2 = inf |a − x|2 .

a∈A

a∈A

If A is bounded, there exists p ∈ A such that |p − x|2 = inf a∈A |a − x|2 . If A is unbounded, the function (3.18) is coercive with respect to A, |a − x|2 = +∞, |a| a∈A, |a|→∞ lim

and we also have the existence of p ∈ A such that |p − x|2 = inf a∈A |a − x|2 . By deﬁnition, ΠA (x) ⊂ A. For x ∈ RN \A, dA (x) > 0. Assume that there exists p ∈ ΠA (x) such that p ∈ int A. Therefore, there exists r, 0 < r < dA (x)/2, such that Br (p) ⊂ A. Choose the point r x−p 2 |x − p|

⇒ |y − p| = r/2 ⇒ y ∈ Br (p) ⊂ A r r x−p = 1− (p − x) ⇒ y−x=p−x+ 2 |x − p| 2dA (x) r ⇒ ∃y ∈ A such that |y − x| = 1 − |p − x| < dA (x); 2dA (x) y =p+

this contradicts the minimality of dA (x). Hence x ∈ ∂A.

288

Chapter 6. Metrics via Distance Functions

(ii) Semidiﬀerentiability of fA . The semiderivative in a direction v can be readily computed by using a theorem on the diﬀerentiability of the min with respect to a parameter (cf. Chap. 10, sect. 2.3, Thm. 2.1). It is suﬃcient to prove the semidiﬀerentiability of d2A in any direction v, that is, for each x and v the existence of the limit d2 (x + tv) − d2A (x) def . dd2A (x; v) = lim A t0 t Recall that for t ≥ 0,

ΠA (x + t v) = pt ∈ A¯ : |x + tv − pt |2 = d2A (x + tv) and consider for t > 0 the quotient qt =

d2A (x + tv) − d2A (x) . t

For all p ∈ ΠA (x) and pt ∈ ΠA (x + t v) |x + tv − pt |2 − |x − p|2 t |x + tv − p|2 − |x − p|2 = t |v|2 + 2 v · (x − p), ≤ t

qt =

and for all p ∈ ΠA (x) lim sup qt ≤ 2 v · (x − p) t0

⇒ q¯ = lim sup qt ≤ 2 t0

inf p∈ΠA (x)

v · (x − p).

In the other direction choose a sequence tk > 0 such that tk → 0 and qtk → q = lim inf t0 . The corresponding sequence ptk in A is uniformly bounded since |ptk | ≤ |x + tk v − ptk | + |x + tk v| ≤ dA (x + tk v) − dA (x) + dA (x) + |x + tk v| ≤ tk |v| + dA (x) + |x + tk v|, and we can ﬁnd p0 ∈ A¯ and a subsequence of {ptk }, still denoted by {ptk }, such that ptk → p0 . But by continuity dA (x + tk v) → dA (x) and |x + tk v − ptk | → |x − p0 | ⇒ ∃p0 ∈ A¯ such that dA (x) = |x − p0 | ⇒ p0 ∈ ΠA (x). Therefore, |x + tk v − ptk |2 − |x − p0 |2 tk |x + tk v − ptk |2 − |x − ptk |2 ≥ = tk |v|2 + 2 v · (x − ptk ) tk ⇒ q ≥ 2 v · (x − p0 ) ≥ 2 inf v · (x − p).

qtk =

p∈ΠA (x)

3. Projection, Skeleton, Crack, and Diﬀerentiability

289

Hence dd2A (x; v) = 2

inf p∈ΠA (x)

v · (x − p).

Finally, by Theorem 2.4 (ii), dH d2A (x; v) = dd2A (x; v) since d2A is locally Lipschitzian. The same remark applies to ddA (x; v) in RN \∂A. (iii) From part (ii), d2A (x) is Gateaux diﬀerentiable at x if and only if ΠA (x) is a singleton. The (Fr´echet) diﬀerentiability is a standard part of Rademacher’s theorem, but we reproduce it here for completeness as a lemma. Lemma 3.1. Let f : RN → R be a locally Lipschitzian function. Then f is (Fr´echet) diﬀerentiable at x, that is, lim

y→x

f (y) − f (x) − ∇f (x) · (y − x) =0 |y − x|

if and only if it is Gateaux diﬀerentiable at x, that is, ∀v ∈ RN ,

f (x + tv) − f (x) exists t0 t

df (x; v) = lim

and the map v → df (x; v) is linear and continuous. Proof. It is suﬃcient to prove that the Gateaux diﬀerentiability implies that f (y) − f (x) − ∇f (x) · (y − x) → 0. y→x |y − x| lim

The function f is locally Lipschitzian and there exist δ1 > 0 and c1 > 0 such that f is uniformly Lipschitzian in the open ball B(x, δ1 ) with Lipschitz constant c1 . Therefore, |∇f (x)| ≤ c1 . The unit sphere S(0, 1) = {v ∈ RN : |v| = 1} is compact and can be covered by the family of open balls {B(v, ε/(4c1 )) : v ∈ S(0, 1)} for an arbitrary ε > 0. Hence there exists a ﬁnite subcover {B(vn , ε/(4c1 )) : 1 ≤ n ≤ N } of S(0, 1). As a result, given any v ∈ S(0, 1), there exists n, 1 ≤ n ≤ N , such that |v − vn | ≤ ε/(4c1 ). Deﬁne the function g(x, v, t) =

f (x + tv) − f (x) − ∇f (x) · v t

for x ∈ RN , v ∈ S(0, 1), and t > 0. Since f (x) is Gateaux diﬀerentiable ∃δ, 0 < δ < δ1 , ∀n, 1 ≤ n ≤ N, ∀t < δ,

|g(x, vn , t)| < ε/2.

Hence for |y − x| < δ g x, y − x , |y − x| |y − x| y−x ≤ |g(x, vn , |y − x|)| + g x, , |y − x| − g(x, vn , |y − x|) |y − x| y−x ε ε − vn ≤ + 2 c1 ≤ |g(x, vn , |y − x|)| + 2 c1 = ε, |y − x| 2 (4c1 ) and we conclude that f (x) is diﬀerentiable at x.

290

Chapter 6. Metrics via Distance Functions

(iv) From part (ii), ∇fA (x) exists if and only if ΠA (x) is a singleton. Inequality (3.10) follows directly from the inequality ∀x and y ∈ RN , ∀ p(x) ∈ ΠA (x),

d2A (y) ≤ |p(x) − y|2

since d2A (y) − |y|2 ≤ |p(x) − x + x − y|2 − |y|2 ≤ |p(x) − x|2 + |x − y|2 + 2 (p(x) − x) · (x − y) − |y|2 ≤ |p(x) − x|2 − |x|2 + 2 (p(x) − x) · (x − y) + |x − y|2 + |x|2 − |y|2 and − 2fA (y) ≤ −2fA (x) + 2 p(x) · (x − y) − 2 x · (x − y) + |x − y|2 + |x|2 − |y|2 ≤ −2fA (x) − 2 p(x) · (y − x) ⇒ fA (y) ≥ fA (x) + 2 p(x) · (y − x). Inequality (3.11) is (3.10) rewritten. Inequality (3.12) follows by adding inequality (3.10) to the same inequality with x and y permuted. (v) From the equivalences in (ii), when d2A is diﬀerentiable at x, then ΠA (x) = {pA (x)} is a singleton, and from part (i) ∇d2A (x) = 2(x − pA (x)), which yields the expression for pA (x). When x ∈ A, ΠA (x) = {x}, and by substitution ∇d2A (x) = 0. Finally, it is suﬃcient to prove the continuity of pA . Given x ∈ RN \Sk (A), consider a sequence {xn } ⊂ RN \Sk (A) such that xn → x and the associated sequence {pA (xn )} ⊂ A. For all n |pA (xn ) − x| ≤ |pA (xn ) − xn | + |xn − x| ≤ |pA (x) − xn | + |xn − x| ≤ |pA (x) − x| + 2|xn − x| < dA (x) + 2r,

def

r = sup |xn − x| < ∞. n

So the sequence {pA (xn )} ⊂ A is bounded and there exists p ∈ A and a subsequence, still denoted by {pA (xn )}, such that pA (xn ) → p. Therefore dA (x) ← dA (xn ) = |xn − pA (xn )| → |x − p| ⇒ ∃p ∈ A,

dA (x) = |x − p|

⇒ pA (x) = p

and pA is continuous at x since p = pA (x) is independent of the choice of the converging subsequence. To complete the proof of identities (3.14), recall that the projection of each point of A onto A being a singleton, Sk (A) ⊂ RN \A. Therefore, for any x ∈ Sk (A), we have dA (x) > 0 and if ∇d2A (x) does not exist, then ∇dA (x) cannot exist. This sharpens the characterization (3.7) of Sk (A). From this we readily get the characterization of Ck (A) = Sing (∇dA )\Sk (A) and Ck (A) ⊂ A. This last

3. Projection, Skeleton, Crack, and Diﬀerentiability

291

property can be improved to Ck (A) ⊂ ∂A by noticing that for x ∈ int A, dA (x) = 0 and ∇d2A (x) = ∇dA (x) = 0. (vi) For all x ∈ int A, dA (x) = 0 and hence ∇dA (x) = 0 is constant and, a fortiori, continuous in int A. For x ∈ RN \(Sk (A) ∪ A), dA (x) > 0 and for all v ∈ RN and t > 0 suﬃciently small dA (x + tv)2 − dA (x)2 1 dA (x + tv) − dA (x) = t t dA (x + tv) + dA (x) dd2 (x; v) 1 1 ⇒ ddA (x; v) = A = ∇d2 (x) · v ⇒ ∇dA (x) = ∇d2 (x). 2dA (x) 2dA (x) A 2dA (x) A By continuity of ∇d2A (x) from part (iv) and continuity of dA , ∇dA (x) is continuous since dA (x) > 0. Moreover, from part (iv), ∇dA (x) = 2(x − pA (x))/2dA (x) and |∇dA (x)| = 1. If ∇dA (x) exists in some x ∈ A, then dA (x) = 0 and for all v ∈ RN , the diﬀerential quotient converges: dA (x + tv) dA (x + tv) − dA (x) = lim = ∇dA (x) · v t0 t t ⇒ ∀v, 0 ≤ ±∇dA (x) · v ⇒ ∇dA (x) = 0.

0 ≤ lim

t0

Since ∇dA exists almost everywhere, ∇dA = 0 almost everywhere in A. (vii) The function dA is Lipschitzian and the function d2A is locally Lipschitzian. By Rademacher’s theorem they are both diﬀerentiable almost everywhere. The other properties follow from parts (iii), (iv), and (v). By deﬁnition Sk (∂A) = {x ∈ RN : ∇d2∂A (x)} ⊂ RN \∂A = RN \A∪RN \A. If x ∈ Sk (∂A) ∩ RN \A, then d∂A (x) = dA (x), ∇d2A (x), and x ∈ Sk (A); if x ∈ Sk (∂A) ∩ RN \A, then d∂A (x) = dA (x), ∇d2A (x), and x ∈ Sk (A). Conversely, if x ∈ Sk (A) ⊂ RN \A, then d∂A (x) = dA (x), ∇d2∂A (x), and x ∈ Sk (∂A); if x ∈ Sk (A) ⊂ RN \A, then d∂A (x) = dA (x), ∇d2∂A (x), and x ∈ Sk (∂A). Finally, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ ∂A ∪ ∂A = ∂(∂A). The property that Ck (∂A) = Ck (A) ∪ Ck (A) is a consequence of the following property for x ∈ ∂A: ∇d∂A (x) exists if and only if ∇dA (x) and ∇dA (x) exist. Indeed, for x ∈ ∂A, d∂A (x) = dA (x) = dA (x) = 0. From part (v), if ∇d∂A (x) exists, it is 0. But ∂A ⊂ A implies d∂A ≥ dA and for all v ∈ RN and t > 0 0≤

dA (x + tv) − dA (x) d∂A (x + tv) − d∂A (x) ≤ →0 t t

⇒ ∇dA (x) = 0

and similarly ∇dA (x) = 0. Conversely, from part (v), if ∇dA (x) and ∇dA (x) exist, they are 0. Moreover, d∂A (x) = max{dA (x), dA (x)}, where dA (x + tv) − dA (x) dA (x + tv) − dA (x) d∂A (x + tv) − d∂A (x) 0≤ = max , . t t t Since both ∇dA (x) = 0 and ∇dA (x) = 0, then ∇d∂A (x) = 0. Finally, it is easy to check that ∂A ∪ ∂A = ∂(∂A). The other properties for ∂A = ∅ are obvious.

292

Chapter 6. Metrics via Distance Functions

(viii) From part (i), if ∇dA (x) exists for x ∈ A, it is equal to 0. Hence χA (x) = 1 = 1 − |∇dA (x)| on A\Ck (A). From the deﬁnition of Sk (A), ∇dA (x) exists and |∇dA (x)| = 1 for all points in int A\Sk (A). Hence from part (i), χA (x) = 0 = 1 − |∇dA (x)| = 0 on A\Sk (A). From this, we get the ﬁrst line of identities in RN \Sing (∇dA ). Since m(Sing (∇dA )) = 0, the above identities are satisﬁed almost everywhere in RN . We get the same results with A in place of A. (ix) If x ∈ A, ΠA (x) = {x} and there is nothing to prove. If x ∈ / A, then dA (x) = |x − p| > 0 and p = x for all p ∈ ΠA (x). First dA (xα ) ≤ |xα − p| = α |x − p|. If the inequality is strict, then there exists pα ∈ A such that |xα − pα | = dA (xα ) and |x − pα | ≤ |x − xα | + |xα − pα | = (1 − α)|x − p| + dA (xα ) < |x − p| = dA (x). This contradicts the minimality of dA (x) with respect to A. Therefore, dA (xα ) = α |x − p| = |xα − p|

⇒ p ∈ ΠA (xα ).

Now for any pα ∈ ΠA (xα ), pα ∈ A and |pα − x| ≤ |pα − xα | + |xα − x| = α |x − p| + (1 − α) |p − x| = |p − x| = dA (x) and pα ∈ ΠA (x). Hence ΠA (xα ) ⊂ ΠA (x).

4 4.1

W 1,p -Metric Topology and Characteristic Functions Motivations and Main Properties

0,1 Since distance functions are locally Lipschitzian, they belong to Cloc (RN ) and hence 1,p N to Wloc (R ) for all p ≥ 1. Thus the constructions of section 2.1 can be repeated 1,p with Wloc (D) in place of Cloc (D) to generate new W 1,p -metric topologies on the family Cd (D). One big advantage is that the W 1,p -convergence of sequences will imply the Lp -convergence of the corresponding characteristic functions of the closure of the sets and hence the convergence of volumes (cf. Theorem 3.3 (viii)). In the uniform Hausdorﬀ topology, the convergence of volumes and perimeters is usually lost. In general, the measure of a closed set is only upper semicontinuous with respect to that topology. This includes Lebesgue and Hausdorﬀ measures. The next example shows that the Hausdorﬀ metric convergence is not suﬃcient to get the Lp -convergence of the characteristic functions of the closure of the corresponding sets in the sequence. The volume function is only upper semicontinuous with respect to the Hausdorﬀ topology. The perimeters do not converge either as illustrated by the example of the staircase of Example 6.1 and Figure 5.7 in Chapter 5, where the volumes converge but not the perimeters.

4. W 1,p -Metric Topology and Characteristic Functions

293

Example 4.1. Denote by D = ] − 1, 2[ × ] − 1, 2[ the open unit square in R2 , p ≥ 1 a real number, and for each n ≥ 1 deﬁne the sequence of closed sets 2k 2k + 1 def ≤ x1 ≤ , 0 ≤ k < n, 0 ≤ x2 ≤ 1 . An = (x1 , x2 ) ∈ D : 2n 2n

set A4

set A8

set S

Figure 6.3. Vertical stripes of Example 4.1. This deﬁnes n vertical stripes of equal width 1/2n each distant of 1/2n (cf. Figure 6.3). Let S = [0, 1] × [0, 1] be the closed unit square. Clearly, for all n ≥ 1, m(An ) = ∀x ∈ S,

1 , 2

dAn (x) ≤

PD (An ) = 2n + 1, 1 , 4n

∇dAn Lp (D) ≥ 2−1/p ,

where m(An ) is the surface and PD (An ) the perimeter of An . Hence dAn → dS

in C(D),

m(S) = 1,

PD (S) = 4.

But m(An ) = m(An ) =

1 1 = m(S) 2

PD (An ) PD (S),

χAn

⇒ χAn χS in Lp (D), 1 χS in Lp (D)-weak. 2

Since the characteristic functions do not converge, the sequence {∇dAn } does not converge in Lp (D)N and {dAn } does not converge in W 1,p (D). Theorem 4.1. Let µ be a measure5 and let D ⊂ RN be bounded open such that µ(D) < ∞. (i) The map dA → µ(A) : Cd (D) → R

(4.1)

is upper semicontinuous with respect to the topology of uniform convergence (Hausdorﬀ topology on Cd (D)). 5 A measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1]. It is called an outer measure in most texts.

294

Chapter 6. Metrics via Distance Functions

(ii) The map dA → µ(int A) : Cdc (D) → R

(4.2)

is lower semicontinuous with respect to the topology of uniform convergence (Hausdorﬀ complementary topology on Cdc (D)). Note that µ can be the Lebesgue measure but also any of the Hausdorﬀ measures. Proof. We prove (ii). The proof of (i) is similar and simpler. It is suﬃcient to work with open sets Ω = int A = A. Let {Ωn } be a sequence of open subsets of D such that dΩn → dΩ in C(D) for some open Ω ⊂ D. Given ε > 0, there exists N such that for all n > N dΩn − dΩ }C(D) ≤ ε

⇒ Ωn ⊂ (Ω)ε .

By deﬁnition of a measurable set µ(Ωn ) = µ(Ωn ∩ D) = µ(D) − µ(D ∩ Ωn ) ≥ µ(D) − µ(D ∩ (Ω)ε ) ⇒ lim inf µ(Ωn ) ≥ µ(D) − µ(D ∩ (Ω)ε ). n→∞

But D ∩ (Ω)ε is monotonically decreasing to ∩ε>0 D ∩ (Ω)ε = D ∩ Ω and hence lim inf µ(Ωn ) ≥ µ(D) − lim µ(D ∩ (Ω)ε ) = µ(D) − µ(D ∩ Ω) = µ(Ω) n→∞

ε→0

since D ∩ Ω = Ω. Theorem 4.2. Let D be an open (resp., bounded open) subset of RN . 1,p (D) (resp., W 1,p (D)) on Cd (D) and Cdc (D) (i) The topologies induced by Wloc are all equivalent for p, 1 ≤ p < ∞. 1,p (D) (resp., W 1,p (D)) for p, 1 ≤ p < ∞, and (ii) Cd (D) is closed in Wloc def

ρD ([A2 ], [A1 ]) =

∞ dA2 − dA1 W 1.p (B(0,n)) 1 n 2 1 + dA2 − dA1 W 1.p (B(0,n)) n=1 def

(resp., ρD ([A2 ], [A1 ]) = dA2 − dA1 W 1,p (D) ) deﬁnes a complete metric structure on F(D). For p, 1 ≤ p < ∞, the map 1,p dA → χA = 1 − |∇dA | : Cd (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and nonempty subsets A1 and A2 of D χA2 − χA1 Lp (K) ≤ ∇dA2 − ∇dA1 Lp (K) ≤ dA2 − dA1 W 1,p (K) .

4. W 1,p -Metric Topology and Characteristic Functions

295

1,p (D) (resp., W01,p (D)) for p, 1 ≤ p < ∞, and (iii) Cdc (D) is closed in Wloc def

ρD (Ω2 , Ω1 ) =

∞ dΩ2 − dΩ1 W 1.p (B(0,n)) 1 n 2 1 + dΩ2 − dΩ1 W 1.p (B(0,n)) n=1 def

(resp., ρD (Ω2 , Ω1 ) = dΩ2 − dΩ1 W 1,p (D) ) deﬁnes a complete metric structure on the family G(D) of open subsets of D. For p, 1 ≤ p < ∞, the map 1,p dΩ → χΩ = |∇dΩ | : Cdc (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and open subsets Ω1 = RN and Ω2 = RN of D χΩ2 − χΩ1 Lp (K) ≤ ∇dΩ2 − ∇dΩ1 Lp (K) ≤ dΩ2 − dΩ1 W 1,p (K) . Proof. (i) The proof is similar to the proof of Theorem 2.3 in Chapter 5. It is suﬃcient to prove it for D bounded open. Since D is bounded it is contained in a suﬃciently large ball of radius c. Therefore, dA (x) = inf |x − y| ≤ |x| + |y| ≤ 2c y∈A

since y ∈ A ⊂ D, and by Theorem 2.1 (vii) |∇dA (x)| ≤ 1 a.e. in D. For all p > 1 the injection of W 1,p (D) into W 1,1 (D) is continuous since dA W 1,1 (D) ≤ dA W 1,p (D) m(D)1/q with 1/p + 1/q = 1. Conversely, we have the continuity in the other direction. For p > 1 and any dA and dB in Cd (D), |dA − dB |p + |∇dA − ∇dB |p dx D |dA − dB | |dA − dB |p−1 + |∇dA − ∇dB | |∇dA − ∇dB |p−1 dx = D p−1 p−1 |dA − dB | + |∇dA − ∇dB | dx ≤ max{(2c) , 2 } D

⇒ dA − dB pW 1,p (D) ≤ (2 max{c, 1})p−1 dA − dB W 1,1 (D) . Therefore, for any ε > 0, pick δ = εp /(2 max{c, 1})p−1 and dA − dB W 1,1 (D) ≤ δ

⇒ dA − dB W 1,p (D) ≤ ε.

(ii) It is suﬃcient to prove it for D bounded open. Let {dAn } be a Cauchy sequence in Cd (D) which converges to some f in W 1,p (D)-strong. By Theorem 2.2 (ii),

296

Chapter 6. Metrics via Distance Functions

Cd (D) is compact in C(D) and there exist a subsequence, still denoted by {dAn }, and ∅ = A ⊂ D such that dAn → dA in C(D) and, a fortiori, in Lp (D)-strong since D is bounded. By uniqueness of the limit in Lp (D)-strong, f = dA and dAn converges to dA in W 1,p (D)-strong. Therefore Cd (D) is closed in W 1,p (D). For the Lipschitz continuity, recall that the distance function dA is diﬀerentiable almost everywhere in RN for A = ∅. In view of Theorem 3.3 (viii) χA = 1 − |∇dA (x)| almost everywhere in RN . Given two nonempty subsets A1 and A2 of D

⇒ χ A1 ≤ χ A2

|∇dA2 | ≤ |∇dA1 | + |∇dA2 − ∇dA1 | + |∇dA2 − ∇dA1 | ⇒ |χA1 − χA2 |p dx ≤ dA2 − dA1 pW 1,p (D) D

for 1 ≤ p < ∞ and with the ess-sup norm for p = ∞. (iii) Again it is suﬃcient to prove the result for D bounded. In that case Ω = ∅ for all open subsets Ω of D. Let {Ωn } be a sequence of open subsets of D such that {dΩn } is Cauchy in W 1,p (D). By assumption Ωn ⊂ D, Ωn ⊃ D, ∀n ≥ 1,

dΩn = 0 in D

⇒ dΩn ∈ W01,p (D),

and the Cauchy sequence converges to some f ∈ W01,p (D). By Theorem 2.4 (ii), Cdc (D) is compact in C(D) and there exist a subsequence, still denoted by {dΩn }, and an open set Ω ⊂ D such that dΩn → dΩ in C(D) and hence in Lp (D)-strong since D is bounded. By uniqueness of the limit in Lp (D), f = dΩ and the Cauchy sequence dΩn converges to dΩ in W01,p (D). The other part of the proof is similar to that of part (ii).

4.2

Weak W 1,p -Topology

We have the following general result. Theorem 4.3. Let D be a bounded open domain in RN . (i) If {dAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. (ii) If {dAn } converges in C(D), then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. Conversely if {dAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, it converges in C(D). (iii) Cd (D) is compact in W 1,p (D)-weak for all p, 1 ≤ p < ∞.6 (iv) Parts (i) to (iii) also apply to Cdc (D). 6 In a metric space the compactness is equivalent to the sequential compactness. For the weak topology we use the fact that if E is a separable normed space, then, in its topological dual E , any closed ball is a compact metrizable space for the weak topology. Since Cd (D) is a bounded subset of the normed reﬂexive separable Banach space W 1,p (D), 1 ≤ p < ∞, the weak compactness of ´ [1, Vol. II, Chap. XII, Cd (D) coincides with the weak sequential compactness (cf. J. Dieudonne sect. 12.15.9, p. 75]).

4. W 1,p -Metric Topology and Characteristic Functions

297

Proof. (i) For D bounded there exists a constant c > 0 such that for all dA ∈ Cd (D) dA (x) ≤ c and |∇dA (x)| ≤ 1 a.e. in D. If {dAn } weakly converges in W 1,p (D), then {dAn } weakly converges in Lp (D),

{∇dAn } weakly converges in Lp (D)N .

By Lemma 3.1 (iii) in Chapter 5 both sequences weakly converge for all p ≥ 1, and hence {dAn } weakly converges in W 1,p (D) for all p ≥ 1. (ii) If {dAn } converges in C(D), then by Theorem 2.2 (i) there exists dA ∈ Cb (D) such that dAn → dA in C(D) and hence in Lp (D). So for all ϕ ∈ D(D)N , ∇dAn · ϕ dx = − dAn div ϕ dx → − dA div ϕ dx = ∇dA · ϕ dx. D

D

D

D

By density of D(D) in L2 (D), ∇dAn → ∇dA in L2 (D)N -weak and hence dAn → dA in W 1,2 (D)-weak. From part (i) it converges in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Conversely, the weakly convergent sequence converges to some f in W 1,p (D). By compactness of Cb (D) there exist a subsequence, still indexed by n, and dA such that dAn → dA in C(D) and hence in W 1,p (D)-weak. By uniqueness of the limit dA = f . Therefore, all convergent subsequences in C(D) converge to the same limit, so the whole sequence converges in C(D). (iii) Consider an arbitrary sequence {dAn } in Cd (D). From Theorem 2.2 (ii) Cd (D) is compact and there exist a subsequence {dAnk } and dA ∈ Cd (D) such that dAnk → dA in C(D). From part (ii) the subsequence weakly converges in W 1,p (D) and hence Cd (D) is compact in W 1,p (D)-weak. Theorem 4.4. Let D be a closed subset of RN and {An }, ∅ = An ⊂ D, be a sequence such that dAn → dA in Cloc (D) for some A, ∅ = A ⊂ D. Then ∀x ∈ RN \A, lim χAn (x) = χA (x) = 0 and ∀x ∈ RN , lim sup χAn (x) ≤ χA (x), n→∞

n→∞

and for any compact subset K of D χAn dx ≤ χA dx. lim sup n→∞

K

K

Corollary 1. Let D = ∅ be a bounded open subset of RN and {Ωn }, ∅ = Ωn ⊂ D, be a sequence of open subsets of D converging to an open subset Ω, ∅ = Ω ⊂ D, of ¯ Then D in the Hausdorﬀ complementary topology: dΩn → dΩ in C(D). ∀x ∈ Ω, lim χΩn (x) = χΩ (x) = 1, n→∞

and

and

n→∞

χΩn dx ≥

lim inf n→∞

∀x ∈ RN , lim inf χΩn (x) ≥ χΩ (x),

D

χΩ dx. D

298

Chapter 6. Metrics via Distance Functions

Proof of Theorem 4.4. For all x ∈ / A, dA (x) > 0 and ∃Nx > 0, ∀n ≥ Nx , dA − dAn ≤ dA (x)/2 ⇒ ∃Nx > 0, ∀n ≥ Nx , dAn (x) ≥ dA (x)/2 > 0 ⇒ ∃Nx > 0, ∀n ≥ Nx ,

x∈ / An and χAn (x) = 0.

So for all x ∈ /A lim χAn (x) = χA (x) = 0.

n→∞

Finally, for all x ∈ A and all n ≥ 1 χAn (x) ≤ 1 = χA (x) and the result follows trivially by taking the limsup of each term. We conclude that lim sup χAn ≤ χA , n→∞

and by using the analogue of Fatou’s lemma for the limsup we get for all compact subsets K of D lim sup χAn dx ≤ χA dx. lim sup χAn dx ≤ n→∞

K n→∞

K

K

In order to get the Lp -convergence of the characteristic functions of the closure of the sets in the sequence, we need the Lp -convergence of the gradients of the distance functions which are related to the characteristic functions of the closure of the sets (cf. Theorem 3.3 (viii)). We have seen in Example 4.1 that the weak convergence of the characteristic functions is not suﬃcient to obtain the strong convergence of the sequence {dAn } to dA in W 1,2 (D). However, if we assume that {χAn } is strongly convergent, it converges to the characteristic function χB of some measurable subset B of D. Is this suﬃcient to conclude that χB = χA ? The answer is negative. The counterexample is provided by Example 6.2 in Chapter 5, where dAn dD in W 1,2 (D)-weak

and

χAn → χB in L2 (D)-strong

for some B ⊂ D such that m(D) = π >

π ≥ m(B) 3

⇒ χD = χB .

Remark 4.1. In view of part (ii) of Theorem 4.2 an optimization problem with respect to the characteristic functions χΩ of open sets Ω in D for which we have the continuity with respect to χΩ can be transformed into an optimization problem with respect to dΩ in W 1,1 (D) since χΩ = |∇dΩ |. This would apply to the transmission problem (4.3)–(4.6) in section 4.1 of Chapter 5.

5. Sets of Bounded and Locally Bounded Curvature

5

299

Sets of Bounded and Locally Bounded Curvature

From Theorem 6.1 in Chapter 5 and Theorem 3.2, it is readily seen that N ∇dA ∈ BVloc RN

⇐⇒

∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N .

The global properties of ∇dA depend only on its local properties around ∂A. This local property is fairly general and is veriﬁed for sets with corners, that is, sets with discontinuities in the orientation of the normal along the boundary. This suggests introducing a new family of sets and motivates the study of their general properties, We shall later see that convex sets have this property and later in Chapter 7 that the larger family of sets with positive reach (cf. section 6) also does. One remarkable property is that such sets are Caccioppoli sets (cf. section 6 in Chapter 5). We give the deﬁnitions for an arbitrary set A, but they can be specialized to A and ∂A (associated with the distance functions dA and d∂A ) that would require a ﬁner terminology such as exterior, interior, or boundary curvature to distinguish them. Deﬁnition 5.1. (i) Given a bounded open holdall D ⊂ RN and A, ∅ = A ⊂ D, the set A is said to be of bounded curvature with respect to D if ∇dA ∈ BV(D)N .

(5.1)

The family of sets with bounded curvature will be denoted by def

BCd (D) = dA ∈ Cd (D) : ∇dA ∈ BV(D)N . (ii) A set A, A = ∅, in RN is said to be of locally bounded curvature if ∇dA ∈ BVloc (RN )N . The family of sets with locally bounded curvature will be denoted by def

BCd = dA ∈ Cd (RN ) : ∇dA ∈ BVloc (RN )N . As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, it is suﬃcient to satisfy the BV property in a neighborhood of each point of the boundary ∂A. Theorem 5.1. A, ∅ = A ⊂ RN , is of locally bounded curvature if and only if ∀x ∈ ∂A, ∃ρ > 0 such that ∇dA ∈ BV(B(x, ρ))N ,

(5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x. Proof. From Theorem 6.1 and Deﬁnition 6.1 in Chapter 5, a function belongs to BVloc (RN ) if and only if for each x ∈ RN it belongs to BV(B(x, ρ)) for some ρ > 0. The equivalence now follows from Theorem 3.2.

300

Chapter 6. Metrics via Distance Functions

Recall that the relaxation of the perimeter of a set was obtained from the norm of the gradient of its characteristic function for Caccioppoli sets. For distance functions χA = 1 − |∇dA | almost everywhere and the gradient of dA also has a jump discontinuity along the boundary ∂A of magnitude at most 1. So it is not too surprising that a set A with bounded curvature is a Caccioppoli set. Theorem 5.2. (i) Let D ⊂ RN be a bounded open holdall and let A, ∅ = A ⊂ D. If ∇dA ∈ BV(D)N , then ∇χA M 1 (D) ≤ 2 D2 dA M 1 (D) ,

χA ∈ BV(D),

and A has ﬁnite perimeter with respect to D. (ii) For any subset A of RN , A = ∅, ∇dA ∈ BVloc (RN )N

⇒

χA ∈ BVloc (RN )

and A has locally ﬁnite perimeter. Proof. Given ∇dA in BV(D)N , there exists a sequence {uk } in C ∞ (D)N such that uk → ∇dA in L1 (D)N

and Duk M 1 (D) → D 2 dA M 1 (D)

as k goes to inﬁnity, and since |∇dA (x)| ≤ 1, this sequence can be chosen in such a way that |uk (x)| ≤ 1 for all k ≥ 1. This follows from the use of molliﬁers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N − χA div V dx = (|∇dA |2 − 1)div V dx = |∇dA |2 div V dx. D

D

D

For each uk |uk |2 div V dx = −2 D

∗

D

[Duk ]uk · V dx = −2

uk · [Duk ]V dx, D

where ∗ Duk is the transpose of the Jacobian matrix Duk and |uk |2 div V dx ≤ 2 |uk | |Duk | |V | dx D

D

≤ 2 Duk L1 V C(D) ≤ 2 Duk M 1 V C(D) since for W 1,1 (D)-functions ∇f L1 (D)N = ∇f M 1 (D)N . Therefore, χ div V dx = |∇dA |2 div V dx ≤ 2D2 dA M 1 V A C(D) D

D

as k goes to inﬁnity, where D2 dA is the Hessian matrix of second-order partial derivatives of dA . Finally, ∇χA ∈ M 1 (D)N .

5. Sets of Bounded and Locally Bounded Curvature

5.1

301

Examples

It is useful to consider the following three simple illustrative examples (cf. Figure 6.4). By convexity, they are all of locally bounded curvature. Example 5.1 (Half-plane in R2 ). Consider the domain A = {(x1 , x2 ) : x1 ≤ 0},

∂A = {(x1 , x2 ) : x1 = 0}.

It is readily seen that (0, 0), x1 < 0, dA (x1 , x2 ) = max{x1 , 0}, ∇dA (x1 , x2 ) = (1, 0), x1 > 0, ϕ dH1 , ∂12 dA = ∂21 dA = ∂22 dA = 0, ∂11 dA , ϕ = ∂A ∆dA , ϕ = ϕ dH1 ⇒ ∆dA = H1 . ∂A

Thus ∆dA is the one-dimensional Hausdorﬀ measure of ∂A.

Figure 6.4. ∇dA for Examples 5.1, 5.2, and 5.3. Example 5.2 (Ball of radius R > 0 in R2 ). Consider the domain A = {x ∈ R2 : |x| ≤ R},

∂A = {x ∈ R2 : |x| = R}.

Clearly dA (x) = max{0, |x| − R},

∇dA (x) =

2π

∂11 dA , ϕ =

2

R cos (θ)ϕ dθ + 0

2π

A

x/|x|, (0, 0),

|x| > R, |x| < R,

x22 ϕ dx, (x21 + x22 )3/2

x21 ϕ dx, 2 2 3/2 0 A (x1 + x2 ) 2π x2 x1 R cos(θ) sin(θ)ϕ dθ + ϕ dx, ∂12 dA , ϕ = ∂21 dA , ϕ = 2 + x2 )3/2 (x 0 A 1 2 1 ϕ dH1 + ϕ dx, ∆dA , ϕ = 2 2 1/2 ∂A A (x1 + x2 ) ∂22 dA , ϕ =

R sin2 (θ)ϕ dθ +

302

Chapter 6. Metrics via Distance Functions

where H1 is the one-dimensional Hausdorﬀ measure. ∆dA contains the onedimensional Hausdorﬀ measure of the boundary ∂A plus a term that corresponds to the volume integral of the mean curvature over the level sets of dA in A. Example 5.3 (Unit square in R2 ). Consider the domain A = {x = (x1 , x2 ) : |x1 | ≤ 1, |x2 | ≤ 1}. Since A is symmetrical with respect to both axes, it is suﬃcient to specify dA in the ﬁrst quadrant. We use the notation Q1 , Q2 , Q3 , and Q4 for the four quadrants in the counterclockwise order and c1 , c2 , c3 , and c4 for the four corners of the square in the same order. We also divide the plane into three regions: D1 = {(x1 , x2 ) : |x2 | ≤ min{1, |x1 |}} , D2 = {(x1 , x2 ) : |x1 | ≤ min{1, |x2 |}} , D3 = {(x1 , x2 ) : |x1 | ≥ 1 and |x2 | ≥ 1} . Hence

min{x2 − 1, 0}, x ∈ D2 ∩ Q1 , dA (x) = |x − c1 |, x ∈ D3 ∩ Q1 , min{x1 − 1, 0}, x ∈ D1 ∩ Q1 ,

∇dA (x) =

(0, 1), x−c1

x ∈ D2 ∩ Q1 and x2 > 1, |x−c1 | , x ∈ D3 ∩ Q1 ,

(1, 0), (0, 0),

x ∈ D1 ∩ Q1 and x1 > 1, x ∈ Q1 , x1 < 1 and x2 < 1,

(x2 − ci,2 )2 ϕ dx + ϕ dH1 , |x − ci |2 ∂A∩Qi ∩D1 i=1 D3 ∩Qi 4 (x1 − ci,1 )2 ∂22 dA , ϕ ϕ dx + ϕ dH1 , |x − ci |2 ∂A∩Qi ∩D2 i=1 D3 ∩Qi 4 (x2 − ci,2 )(x1 − ci,1 ) ∂12 dA , ϕ = ∂21 dA , ϕ = ϕ dx, |x − ci |2 i=1 D3 ∩Qi 4 1 ϕ dx + ∆dA , ϕ = ϕ dH1 . |x − ci | ∂A i=1 D3 ∩Qi ∂11 dA , ϕ =

4

Notice that the structure of the Laplacian is similar to that observed in the previous examples. The norm D2 dA M 1 (Uh (A)) is decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary ∂A.

6. Reach and Federer’s Sets of Positive Reach

303

Example 5.4. Let N = 2. For the ﬁnite square and the ball of ﬁnite radius, lim ∆dA M 1 (Uh (∂A)) = H1 (∂A),

h0

where H1 is the one-dimensional Hausdorﬀ measure.

6

Reach and Federer’s Sets of Positive Reach

The material in this section is mostly taken from the pioneering 1959 work of H. Federer [3, sect. 4], where he introduced the notion of reach and the sets of positive reach.

6.1

Deﬁnitions and Main Properties

By deﬁnition, A∩Sk (A) = ∅. For a ∈ int A, there exists r > 0 such that Br (a) ⊂ A. Thus Sk (A) ∩ Br (a) = ∅ and sup {r > 0 : Sk (A) ∩ Br (a) = ∅} > 0.

(6.1)

/ Sk (A) or a ∈ Sk (A). If a ∈ / Sk (A), there exists r > 0 If a ∈ ∂A, then either a ∈ such that Br (a) ∩ Sk (A) = ∅. Thus Br (a) ∩ Sk (A) = ∅ and (6.1) is veriﬁed. If a ∈ Sk (A), then for all r > 0, Br (a) ∩ Sk (A) = ∅. This leads to the following deﬁnition. Deﬁnition 6.1. Given A, ∅ = A ⊂ RN , and a point a ∈ A def

reach (A, a) =

0, if a ∈ ∂A ∩ Sk (A), sup {r > 0 : Sk (A) ∩ Br (a) = ∅} , otherwise, def

reach (A) = inf {reach (A, a) : a ∈ A} . The set A is said to have positive reach if reach (A) > 0. Remark 6.1. (1) Since Sk (A) = Sk (A), reach (A) = reach (A). (2) For any set A, Sk (A) ⊂ Sk (∂A) and ∀a ∈ ∂A,

0 ≤ reach (∂A, a) ≤ reach (A, a) ≤ +∞.

(3) For all a ∈ / ∂A ∩ Sk (A), reach (A, a) > 0. Lemma 6.1. Let ∅ = A ⊂ RN . The following statements are equivalent: (i) There exists a ∈ A such that reach (A, a) = +∞. (ii) Sk (A) = ∅.

304

Chapter 6. Metrics via Distance Functions

(iii) For all a ∈ A, reach (A, a) = +∞. (iv) reach (A) = +∞. Proof. (i) ⇒ (ii) If there exists a ∈ A such that reach (A, a) = +∞, then RN = ∪r>0 Br (a) ⊂ RN \Sk (A) and Sk (A) = ∅. (ii) ⇒ (iii) If Sk (A) = ∅, for all a ∈ A reach (A, a) = +∞. (iii) ⇒ (iv) If for all a ∈ A reach (A, a) = +∞, then reach (A) = inf a∈A reach (A, a) = +∞. (iv) ⇒ (i) is obvious. Remark 6.2. We shall see later that all nonempty convex sets A have positive reach and that reach (A) = +∞. Bounded domains and submanifolds of class C 2 also have positive reach. Theorem 6.1. Let A, ∅ = A ⊂ RN , be such that Sk (A) = ∅. (i) The function a → reach (A, a) : A → R is continuous. (ii) If reach (A) > 0, then for all h, 0 < h < reach (A), Uh (A) ∩ Sk (A) = ∅. Conversely, if there exists h > 0 such that Uh (A) ∩Sk (A) = ∅, reach (A) ≥ h, and A has positive reach. (iii) If reach (A) > 0, then A ∩ Sk (A) = ∅. If A ∩ Sk (A) = ∅ and A is bounded, then reach (A) > 0. Remark 6.3. When A is unbounded and A ∩ Sk (A) = ∅, we can possibly have reach (A) = 0 even if for all a ∈ A, reach (A, a) > 0 as readily seen from the following example: def

A = {(x1 , x2 ) : |x2 | ≥ ex1 } , where Sk (A) = {(x1 , 0) : x1 ∈ R} . Proof. (i) We prove that the function is both lower and upper semicontinuous. For h, reach (A, a) > h, there exists r, h < r < reach (A, a), such that Br (a) ∩ Sk (A) = ∅. Let a ∈ Bδ (a), δ = (r − h)/2 > 0. Then r − δ = (r + h)/2 > h and for all a ∈ Bδ (a)∩A, Br−δ (a ) ⊂ Br (a). Thus Br−δ (a )∩Sk (A) ⊂ Br (a)∩Sk (A) = ∅, reach (A, a ) ≥ r − δ > h, and reach (A, a) is lower semicontinuous. Similarly, for k, reach (A, a) < k, there exists r, reach (A, a) > r > k, such that Br (a) ∩ Sk (A) = ∅. Let a ∈ Bδ (a), δ = (k − r)/2 > 0. Then r + δ = (r + k)/2 < k and for all a ∈ Bδ (a) ∩ A, Br (a) ⊂ Br+δ (a ). Thus ∅ = Br (a) ∩ Sk (A) ⊂ Br+δ (a ) ∩ Sk (A), reach (A, a ) ≤ r + δ < k, and reach (A, a) is upper semicontinuous. (ii) By deﬁnition of reach (A) as an inﬁmum. Conversely, for all 0 < r < h and all x ∈ A, Br (x) ∩ Sk (x) = ∅, and reach (A, x) ≥ r. Since this is true for all 0 < r < h, reach (A, x) ≥ h and reach (A) = inf x∈A reach (A, x) ≥ h > 0. (iii) By separation of a closed set and a compact set. Theorem 6.2. Let ∅ = A ⊂ RN . (i) Associate with a ∈ A the sets def

P (a) = v ∈ RN : ΠA (a + v) = {a} ,

def

Q(a) = v ∈ RN : dA (a + v) = |v| .

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta A)∗ .

6. Reach and Federer’s Sets of Positive Reach

305

(ii) Given a ∈ A and v ∈ RN , assume that def

0 < r(a, v) = sup {t > 0 : ΠA (a + tv) = {a}} . Then for all t, 0 ≤ t < r(a, v), ΠA (a + tv) = {a} and dA (a + tv) = t |v|. Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (A). (iii) If there exists h such that 0 < h ≤ reach (A), then for all x ∈ Uh (A)\A and 0 < t < h, ΠA (x(t)) = {pA (x)} and dA (x(t)) = t, where def

x(t) = pA (x) + t ∇dA (x) = pA (x) + t

x − pA (x) . dA (x)

(6.2)

(iv) Given b ∈ A, x ∈ RN \Sk (A), a = pA (x), such that reach (A, a) > 0, then (x − a) · (a − b) ≥ −

|a − b|2 |x − a| . 2 reach (A, a)

(6.3)

(v) Given 0 < r < q < ∞ and x ∈ RN \Sk (A) such that dA (x) ≤ r and reach (A, pA (x)) ≥ q, y ∈ RN \Sk (A) such that dA (y) ≤ r and reach (A, pA (y)) ≥ q, then |pA (y) − pA (x)| ≤

q |y − x|. q−r

(6.4)

(vi) If 0 < reach (A) < +∞, then for all h, 0 < h < reach (A), reach (A) |y − x|, ∀x, y ∈ Ah = {x ∈ RN : dA (x) ≤ h}, reach (A) − h 2 ∇dA (y) − ∇d2A (x) ≤ 2 1 + reach (A) |y − x|, ∀x, y ∈ Ah ; (6.5) reach (A) − h

for all 0 < s < h < reach (A) and all x, y ∈ As,h = x ∈ RN : s ≤ dA (x) ≤ h , reach (A) 1 2+ |y − x| . |∇dA (y) − ∇dA (x)| ≤ s reach (A) − h

|pA (y) − pA (x)| ≤

If reach (A) = +∞, ∀x, y ∈ RN , |pA (y) − pA (x)| ≤ |y − x|, 2 ∇d (y) − ∇d2 (x) ≤ 4 |y − x|; A A for all 0 < s and all x, y ∈ {x ∈ RN : s ≤ dA (x)}, |∇dA (y) − ∇dA (x)| ≤

3 |y − x| . s

(6.6)

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(vii) If reach (A) > 0, then, for all 0 < r < reach (A), Ur (A) is a set of class C 1,1 , Ur (A) = Ar , ∂Ur (A) = {x ∈ RN : dA (x) = r}, and Ar = Ur (A) = {x ∈ RN : dA (x) ≥ r}. (viii) For all a ∈ A,

dA (a + th) =0 . h ∈ R : lim inf t0 t

Ta A =

N

In particular, if a ∈ int A, Ta A = RN . Remark 6.4. For part (viii), more properties of Ta A and the normal cone (−Ta A)∗ (cf. Deﬁnitions 2.3 and 2.4 in Chapter 2) can be found in H. Federer [3, sect. 4]. Proof. (i) By deﬁnition for all a ∈ A, a ∈ P (A) a ∈ Q(A)

⇐⇒ ⇐⇒

∀b ∈ A\{a}, ∀b ∈ A\{a},

|a + v − b| > |v|, |a + v − b| ≥ |v|

and P (a) ⊂ Q(a). To prove the convexity, we need the identity |a + v − b|2 − |v|2 = (a − b) · (2v + a − b). The convexity of P (a) and Q(a) now follow from the following identities: for all λ ∈ [0, 1] and v, w ∈ RN , |a + (λv + (1 − λ)w) − b|2 − |(λv + (1 − λ)w|2 = (a − b) · (2((λv + (1 − λ)w) + a − b) = λ (a − b) · (2v + a − b) + (1 − λ) (a − b) · (2w + a − b) = λ |a + v − b|2 − |v|2 + (1 − λ) |a + w − b|2 − |v|2 . Finally, for h ∈ Ta A, there exist sequences {bn } ⊂ A and {εn > 0}, εn 0, such that (bn − a)/εn → h and bn → a. For all v ∈ Q(a), 2

bn − a |bn − a|2 ·v ≤ |bn − a| ≤ 0 εn εn

⇒ h·v ≤0

⇒ h ∈ [−Ta A]∗ .

def

(ii) Let S = {t > 0 : ΠA (a + tv) = {a}}. Since, by assumption, r(a, v) = sup S > 0, then S = ∅ and v = 0. For each t ∈ S, ΠA (a + tv) = {a} and dA (a + tv) = t |v|. In particular for all 0 < s < t, dA (a + sv) ≤ s |v|. For any p ∈ ΠA (a + sv) ⊂ A, dA (a + sv) = |a + sv − p| ≤ s |v| and dA (a + tv) ≤ |a + tv − p| = |a + sv − p + (t − s)v| ≤ |a + sv − p| + (t − s)|v| ≤ s |v| + (t − s)v| = t |v| = dA (a + tv). As a result p ∈ ΠA (a + tv) = {a}, p = a, and necessarily ΠA (a + sv) = {a} and dA (a + sv) = s |v|. This proves that for all t ∈ S, {s : 0 < s ≤ t} ⊂ S and hence for all 0 ≤ t < r(a, v).

6. Reach and Federer’s Sets of Positive Reach

307

Assume that when r = r(a, v) < ∞, a + rv ∈ / Sk (A). Then there exists δ > 0 such that Bδ (a+rv)∩(Sk (A)∪A) = ∅. By continuity of the projection, ΠA (a+rv) = = r |v|> 0 so that a+rv ∈ / A. By Theorem 3.3 (vi), the function {a} and dA (a+rv) ∇dA : RN \ Sk (A) ∪ ∂A → R is continuous and for all x ∈ RN \ Sk (A) ∪ A , ∇dA (x) = (x − pA (x))/|x − pA (x)|. For 0 < t < r, pA (a + tv) = a, dA (a + tv) = t|v|, and ∇dA (a + tv) = v/|v|. By continuity, ∇dA (a + rv) = v/|v|. Since ∇dA is continuous in Bδ (a+rv), the Caratheodory conditions are veriﬁed and there exists ε, 0 < ε < δ, such that the diﬀerential equation x (t) = ∇dA (x(t)),

x(0) = a + rv = a + r |v|

v |v|

has a solution in (−ε, ε). This yields d dA (x(t)) = ∇dA (x(t)) · x (t) = ∇dA (x(t)) · ∇dA (x(t)) = 1 dt ⇒ dA (x(t)) = r|v| + t in (−ε, ε). For −ε < p < q < ε, the length of the curve x(t) is given by q q d dA (x(t)) dt = dA (x(q)) − dA (x(p)) ≤ |x(q) − x(p)| |x (t)| dt = p p dt so that the length of the curve x(t) between the points x(p) and x(q) is less than or equal to the distance |x(q) − x(p)| between x(p) and x(q). Therefore x(t) is a line between x(p) and x(q): x(q) − x(p) , q−p

x(q) − x(p) q−p v x(q) − x(p) ⇒ = x (0) = ∇dA (x(0)) = ∇dA (a + rv) = q−p |v| v v = a + rv + t ⇒ ∀t, 0 < t < ε, x(t) = x(0) + (t − 0) |v| |v| ⇒ |x(t) − a| = r|v| + t = dA (x(t)) ⇒ pA (x(t)) = a and ΠA (a + r|v| + t) = {a} x(t) = x(p) + (t − p)

x (t) =

and we get the contradiction sup S ≥ r + t/|v| > r = sup S. Therefore, a + rv ∈ / Sk (A). (iii) If there exists 0 < h ≤ reach (A), then for all a ∈ A and all 0 < s < h, Bs (a) ∩ Sk (A) = ∅. For x ∈ Uh (A)\A, 0 < dA (x) < h and for all p ∈ ΠA (x) (dA (x) + h)/2 < h ≤ reach (A)

⇒ B(dA (x)+h)/2 (p) ∩ Sk (A) = ∅. def

Since x ∈ B(dA (x)+h)/2 (p), ΠA (x) is a singleton and p = pA (x). From part (ii), S = def

{t > 0 : ΠA (pA (x) + t ∇dA (x)) = {pA (x)}} = ∅ and rA = r(pA (x), ∇dA (x)) ≥ dA (x) > 0. If rA = +∞, then for all t ≥ 0 we have ΠA (x(t)) = {pA (x)} and dA (x(t)) = t. If rA < +∞, then x(rA ) ∈ Sk (A) and for all ρ > 0, BrA +ρ (pA (x)) ∩ Sk (A) = ∅. This implies that, for all ρ > 0, reach (pA (x), A) ≤ rA + ρ and

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reach (A) ≤ reach (pA (x), A) ≤ rA . Thus, for all 0 < t < h, t < h ≤ reach (A) ≤ rA and we have ΠA (x(t)) = {pA (x)} and dA (x(t)) = t. (iv) If b = a, the inequality is veriﬁed. For b = a, let def

v =

x−a , |x − a|

def

S = {t > 0 : ΠA (a + tv) = {a}} .

By assumption pA (x) = a and x = a + |x − a| v

⇒ |x − a| ∈ S,

sup S ≥ |x − a| > 0,

and, from part (ii), a + sup S v ∈ Sk (A). If reach (A, a) > sup S, then for ρ, reach (A, a) > ρ > sup S, Bρ (a) ∩ Sk (A) = ∅

⇒ Bsup S (a) ∩ Sk (A) = ∅

and this contradicts the fact that a + sup S v ∈ Sk (A). Therefore, reach (A, a) ≤ sup S. Given t, 0 < t < reach (A, a), deﬁne xt = a + tv. Then t ∈ S and t < sup S. Thus ΠA (xt ) = {a}, pA (xt ) = a, and dA (xt ) = |xt − a| = t. For b ∈ A, |xt − b| ≥ |xt − a| = t > 0 and 2 a + t x − a − b ≥ t2 |x − a|

⇒ (a − b) · (x − a) ≥ −|a − b|2

|x − a| . 2t

To conclude, choose a sequence {tn }, 0 < tn < reach (A, a), such that tn → reach (A, a) in the above inequality. (v) Apply the inequality of part (iii) twice to the triplets (a, x, b) and (b, y, a): |x − a| r ≥ −|a − b|2 , 2 reach (A, a) 2q |y − b| r ≥ −|a − b|2 . (b − a) · (y − b) ≥ −|a − b|2 2 reach (A, b) 2q

(a − b) · (x − a) ≥ −|a − b|2

By adding the two inequalities r (x − a + b − y) · (a − b) ≥ − |a − b|2 q

⇒ (x − y) · (a − b) ≥

⇒ |pA (x) − pA (y)| = |a − b| ≤

r 1− q

|a − b|2

q |x − y|. q−r

(vi) Given 0 < h < r < reach (A), for all a ∈ A, 0 < h < r < reach (A) ≤ reach (A, a)

and Br (a) ∩ Sk (A) = ∅.

Therefore Ah ⊂ Ur (A) = ∪a∈A Br (a) ⊂ RN \Sk (A) and from (iv) ∀x, y ∈ Ah , ∀r, h < r < reach (A),

|pA (x) − pA (y)| ≤

r |x − y|. r−h

6. Reach and Federer’s Sets of Positive Reach

309

The function r → r/(r − h) : (h, ∞) → R is strictly decreasing to 1. If reach (A) = +∞, then for all a ∈ A, reach (A, a) = +∞ and by letting r go to inﬁnity ∀x, y ∈ Ah ,

|pA (x) − pA (y)| ≤ |x − y|.

If reach (A) is ﬁnite, then for all a ∈ A, reach (A, a) ≥ reach (A) > r and by letting r go to reach (A) ∀x, y ∈ Ah ,

|pA (x) − pA (y)| ≤

reach (A) |x − y|. reach (A) − h

The inequalities for ∇d2A follow from identity (3.13) in Theorem 3.3 (v). Finally, the Lipschitz continuity of ∇dA follows from identity (3.15) in Theorem 3.3 (vi). For all pairs 0 < s < h < reach (A) and all x, y ∈ {x ∈ RN : s ≤ dA (x) ≤ h} y − pA (y) x − pA (x) |∇dA (y) − ∇dA (x)| = − dA (y) dA (x) |(y − pA (y)) dA (x) − (x − pA (x)) dA (y)| = dA (x) dA (y) |(y − pA (y)) (dA (x) − dA (y))| |[(y − pA (y)) − (x − pA (x))] dA (y)| + ≤ dA (x) dA (y) dA (x) dA (y) 1 ≤ (|dA (x) − dA (y)| + |y − x| + |pA (y) − pA (x)|) dA (x) 1 reach (A) ≤ 2+ |y − x| . s reach (A) − h When reach (A) = +∞, for all s > 0 and all x, y ∈ {x ∈ RN : s ≤ dA (x)} |∇dA (y) − ∇dA (x)| ≤

3 |y − x| . s

(vii) For 0 < r < reach (A), there exists h such that 0 < r < h < reach (A). N The open set Ur (A) is deﬁned via the r-level set d−1 A {r} = {x ∈ R : dA (x) = r} of −1 the function dA . From part (vi), for each x ∈ dA {r}, there exists a neighborhood V (x) = Bρ (x), ρ = min{r, h − r}/2, of x in Uh (A)\A such that dA ∈ C 1,1 (V (x)). Therefore, ∇dA exists and is not zero on the level set ∂Ar = d−1 A {r} since |∇dA | = 1. From Theorem 4.2 in Chapter 2, the set Ur (A) is of class C 1,1 , Ur (A) = Ar , and ∂Ur (A) = d−1 A {r}, and Ar = Ur (A). (viii) Given h ∈ Ta A (cf. Deﬁnition 2.3 in Chapter 2), there exist sequences {an } ⊂ A and {εn }, εn 0, such that (an − a)/εn → h. For each n, dA (a + th) dA (a + εn h) |a + εn h − an | an − a → 0 ⇒ lim inf = 0. 0≤ ≤ = h − t0 εn εn εn t

310

Chapter 6. Metrics via Distance Functions

Conversely, there exists {tn > 0}, tn 0, such that dA (a + tn h)/tn → 0. This means that there exists a sequence of projections {an } ⊂ A such that h − an − a = |a + tn h − an | = dA (a + tn h) → 0 tn tn tn and h ∈ Ta A. The next theorem summarizes several characterizations of sets of positive reach. Note that condition (ii) is a global condition on the smoothness of d2A in the tubular neighborhood Uh (A). Theorem 6.3. Given ∅ = A ⊂ RN , the following conditions are equivalent: (i) A has positive reach, that is, reach (A) > 0. (ii) There exists h > 0 such that d2A ∈ C 1,1 (Uh (A)). (iii) There exists h > 0 such that for all s, 0 < s < h, dA ∈ C 1,1 (Us,h (A)). (iv) There exists h > 0 such that dA ∈ C 1 (Uh (A)\A). (v) There exists h > 0 such that for all x ∈ Uh (A), ΠA (x) is a singleton. Proof. (i) ⇒ (ii) For all h, 0 < h < reach (A), and all a ∈ A, Bh (a) ∩ Sk (A) = ∅. Hence for all x ∈ Uh (A) ΠA (x) is a singleton and ∇d2A (x) exists. The functions d2A and ∇d2A are bounded and uniformly continuous in Uh (A): ∀x ∈ Uh (A),

d2A (x) ≤ h2 ,

∇d2A (x) = |2(x − pA (x)| = 2 dA (x) ≤ 2h,

|d2A (y) − d2A (x)| = (|dA (y) − dA (x)) |dA (y) − dA (x)| ≤ 2h |y − x|, 2 reach (A) 2 ∀x, y ∈ Uh (A), ∇dA (y) − ∇dA (x) ≤ 2 1 + |y − x| reach (A) − h

∀x, y ∈ Uh (A),

from inequality (6.5) of Theorem 6.2 (vi). By deﬁnition, d2A ∈ C 1,1 (Uh (A)). (ii) ⇒ (iii) From the proof of Theorem 6.2 (vi). (iii) ⇒ (iv) By assumption, for all 0 < s < h, dA ∈ C 1 (Us,h (A)). Since Uh (A)\A = ∪0<s 0 and ∇d2A (x) = 2 dA (x)∇dA (x) exists; from Theorem 3.3 (v), for all x ∈ A, ∇d2A (x) = 0. This means that ΠA (x) is a singleton in Uh (A) = Uh (A)\A ∪ A. (v) ⇒ (i) By assumption, for all a ∈ A and all x ∈ Bh (a), Bh (a) ∩ Sk (A) = ∅. This implies that for all a ∈ A, reach (A, a) ≥ h, which ﬁnally implies that reach (A) ≥ h > 0.

6.2

C k -Submanifolds

For arbitrary closed submanifolds M of RN of codimension greater than or equal to one, ∇dM has a discontinuity along M and generally does not exist. In that case the smoothness of M is related to the existence and the smoothness of ∇d2M in a neighborhood of M that is, a fortiori, locally of positive reach. The following analysis of the smoothness of M was given by J.-B. Poly and G. Raby [1] in 1984.

6. Reach and Federer’s Sets of Positive Reach

311

Theorem 6.4. Let A be a closed nonempty subset of RN and k ≥ 2 be an integer (k = ∞ and ω included7 ). Then sing k A = A ∩ sing k d2A , where sing k d2A = RN \reg k d2A , sing k A = RN \reg k A, and def

reg k d2A = x ∈ RN : d2A is C k in a neighborhood of x , def

reg k A = x ∈ A : A is a C k -submanifold of RN in a neighborhood of x . It is more interesting to rephrase their theorem in terms of regularities than singularities. Theorem 6.5. Let ∅ = A ⊂ RN and let k ≥ 2 be an integer (k = ∞ and ω included), and x ∈ A.8 (i) d2A is C k in a neighborhood of a point x ∈ A if and only if A is a C k submanifold in a neighborhood of x. (ii) Under conditions (i), the dimension of A in x is equal to the rank of DpA (x) (dimension of the subspace DpA (RN )) and DpA (x) is the orthogonal projector onto the tangent space to A in x. (iii) If ∂A = ∅, then A = RN . If ∂A = ∅ and d2A is C 2 in a neighborhood of x ∈ A, then there exists an open neighborhood V (x) of x such that (a) if x ∈ int A, then dA = dRN = 0 in V (x); (b) if x ∈ ∂A, then dA = d∂A in V (x), int A ∩ V (x) = ∅, dA = 0 in V (x), A is a C 2 -submanifold of dimension less than or equal to N − 1 in V (x), and m(∂A ∩ V (x)) = 0. Proof. (i) This follows from Theorem 6.4. (ii) For y ∈ V (x) 1 ∇fA (y) = y − ∇d2A (y) = pA (y), def 1 2 2 2 |y| − dA (y) fA (y) = ⇒ 1 2 2 D fA (y) = I − D2 d2A (y) = DpA (y). 2 Since pA is a projection, pA (pA (y)) = pA (y) implies that DpA ◦ pA DpA = DpA and (DpA )2 = DpA on A. Therefore, if R(x) denotes the image of DpA (x), DpA (x) is an orthogonal projector onto R(x) and I − DpA (x) is an orthogonal projector onto R(x)⊥ and ∀v ∈ RN , 7ω

v = DpA (x)v + [I − DpA (x)]v ∈ R(x) ⊕ R(x)⊥ .

indicates the analytical case. that A can have several connected components with the same smoothness k but diﬀerent dimension. When A = RN , dA is identically zero. Another set of results for H¨ olderian sets was obtained by G. M. Lieberman [1] (see also V. I. Burenkov [1]) by introducing a regularized distance function. 8 Note

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Chapter 6. Metrics via Distance Functions

Consider the C 1 -mapping Tx (y) = (DpA (x)(pA (y) − x), [I − DpA (x)](y − pA (y))) : W (x) → R(x) ⊕ R(x)⊥ . def

Tx is a C 1 -diﬀeomorphism from a neighborhood U (x) ⊂ V (x) of x onto a neighborhood W (0) of 0 since DTx is continuous in V (x) and def

DTx (y) = (DpA (x)DpA (y), [I − DpA (x)](I − DpA (y))), def

DTx (x) = (DpA (x)DpA (x), [I − DpA (x)](I − DpA (x))) = (DpA (x)DpA (x), [I − DpA (x)](I − DpA (x)) = (DpA (x), I − DpA (x)) = I. By construction Tx (U (x) ∩ A) = V (0) ∩ R(x), ⊥

V (0) ∩ R(x) = {(0, [I − DpA (x)](y − pA (y))) : ∀y ∈ U (x)} , and U (x) ∩ A is a C 1 -submanifold in RN of codimension r, where N − r is the rank of DpA (x). (iii) If ∂A = ∅, A ⊂ A and A is both open and closed. Since A = ∅, then A = RN . Let ∂A = ∅. If x ∈ ∂A, then A = ∅. We claim that int A = ∅. If this is not true, there exists a neighborhood V (x) of x where d2A is C 2 and int A ∩ V (x) = ∅ where dA = 0 and D2 d2A = 0. This implies that DpA = I in int A ∩ V (x). In A ∩ V (x), D2 d2A is continuous and dA > 0. Therefore, ∇dA and D2 dA exist, |∇dA | = 1, and 1 2 2 D dA = ∇dA ∗∇dA + dA D 2 dA 2 ⇒ DpA = I − ∇dA ∗∇dA − dA D 2 dA and DpA ∇dA = 0 since |∇dA |2 = 1 implies that D2 dA ∇dA = 0. As a result the rank of DpA is less than or equal to N − 1 in A ∩ V (x) and equal to N in int A ∩ V (x). This contradicts the continuity of D2 d2A and DpA in x. Therefore, A∩V (x) = ∂A∩V (x), and d∂A = dA in V (x) and, by the previous argument in A ∩ V (x), A is a C 2 submanifold of dimension less than or equal to N − 1 in a neighborhood of x. At this juncture, several interesting remarks can be made. (a) The equivalence in (i) fails in the direction (⇒) for d2A ∈ C 1,1 as seen from Example 6.1 of J.-B. Poly and G. Raby [1]. (b) The equivalence in (i) also fails in the direction (⇐) for a C 1, submanifold, 0 ≤ < 1, as seen from Example 6.2 of S. G. Krantz and H. R. Parks [1]. (c) The square d2A is really pertinent for C k -submanifolds of codimension greater than or equal to one and not for sets of class C k . Indeed, when ∂A = ∅ and d2A is C k , k ≥ 2, in a neighborhood of each point of A, then A is necessarily a C k -submanifold of codimension greater than or equal to one. In particular, A = RN , A = ∂A, and m(A) = 0.

6. Reach and Federer’s Sets of Positive Reach

313

Example 6.1. Consider the half-space in RN def

A = x ∈ RN : x · eN ≤ 0 , dA (x) =

0, x · eN ,

x · eN ≤ 0, x · eN > 0,

∇d2A (x) =

x · eN ≤ 0, x · eN > 0,

0, 2 (x · eN ) eN ,

and d2A is locally C 1,1 . For any point x ∈ ∂A = {x ∈ RN : x · eN = 0}, A ∩ V (x) is not even a manifold in any neighborhood V (x) of x. The equivalence in (i) also fails in the direction (⇐) for C 1, , 0 ≤ < 1, as seen from Example 6.2, that shows that for a set of class C 1, , 0 ≤ < 1, we do not get bA ∈ C 1,λ in a neighborhood of the boundary ∂A. Example 6.2. Consider the two-dimensional domain Ω deﬁned as the epigraph of the function f : def

Ω = {(x, z) : f (x) > z, ∀x ∈ R} ,

def

1

f (x) = |x|2− n

for some arbitrary integer n ≥ 1. Ω is a set of class C 1,1−1/n and ∂Ω is a C 1,1−1/n submanifold of dimension 1. In view of the presence of the absolute value in (0, 0), it is the point where the smoothness of ∂Ω is minimum. We claim that Sk (∂Ω) = Sk (Ω) = {(0, y) : y > 0}

and ∂Ω ∩ Sk (∂Ω) = (0, 0) = ∅.

Because the skeleton is a line touching ∂Ω in (0, 0), d2∂Ω and d2Ω cannot be C 1 in any neighborhood of (0, 0). Indeed, for each (0, y)

d2∂Ω (0, y) = inf x2 + |f (x) − y|2 . x∈R

The point (0, y), y > 0, belongs to Sk (∂Ω) since there exist two diﬀerent points x ˆ that minimize the function def

F (x, y) = x2 + +|f (x) − y|2 .

(6.7)

Since f is symmetric with respect to the y-axis it is suﬃcient to show that there exists a strictly positive minimizer x ˆ > 0. A locally minimizing point x ˆ ≥ 0 must satisfy the conditions 1 1 1 2− n x ˆ1− n = 0, x, y) = 2ˆ x + 2 |ˆ x| −y 2− (6.8) Fx (ˆ n 2 1 2 1 1 1 1 F (ˆ 3− x ˆ2− n − 2 − 1− yx ˆ− n ≥ 0. (6.9) x, y) = 1 + 2 − 2 x n n n n Equation (6.8) can be rewritten as . 1 / 1 1 2ˆ x1− n x ˆn + x ˆ2− n − y (2 − 1/n) = 0,

314

Chapter 6. Metrics via Distance Functions

and x ˆ = 0 is a solution. The second factor can be written as an equation in the def 1/n new variable X = x ˆ and def

g(X) =

n X + X 2n−1 − y = 0. 2n − 1

ˆ since It has exactly one solution X ∀X,

n dg (X) = + (2n − 1)X 2n−2 > 0, dX 2n − 1

ˆ > 0 for g(0) = −y, and g(X) goes to inﬁnity as X goes to inﬁnity. In particular, X n ˆ ˆ y > 0 and X = 0 for y = 0. For y > 0 and x ˆ=X 1 x, y) = 0 F (ˆ 2 x and

1

⇒ y=x ˆ2− n +

1 n x ˆn 2n − 1

(6.10)

2 1 2 1 F (ˆ 3− x ˆ2− n x, y) =1 + 2 − 2 x n n 2 1 1 n 2− n 1− x ˆ − 2− + n n 2n − 1 2 2 2 2n − 1 2n − 1 1− 1 n−1 1 + x ˆ n > 0. =1 − x ˆ2− n = + n n n n

Therefore, x ˆ is a local minimum. To complete the proof for y > 0 we must compare ˆ = 0 with F (0, y) = y 2 for the solution corresponding to x 1

F (ˆ x, y) = x ˆ2 + |ˆ x2− n − y|2 for the solution x ˆ > 0. Again using identity (6.10) 2 2 n F (ˆ x, y) = x ˆ + x ˆn, 2n − 1 2 2 n 2 2 2− 1 1 1 n n x ˆ n = x x ˆ2 ˆ2− n + x ˆn + 2 F (0, y) = x ˆ n+ 2n − 1 2n − 1 2n − 1 2 2 1 1 2n 1 −1 x ˆ2 + x x ˆ2 + x ⇒ F (0, y) − F (ˆ x, y) = ˆ2− n = ˆ2− n > 0. 2n − 1 2n − 1

2

This proves that for y > 0, x ˆ > 0 is the minimizing positive solution. Therefore, for all n ≥ 1, f (x) = |x|2−1/n yields a domain Ω for which the strictly positive part of the y-axis is the skeleton. It is interesting to note that, for each point x, the points (x, f (x)) ∈ ∂Ω are the projections of the point 1 1 n (0, y) = 0, |x|2− n + |x| n 2n − 1

6. Reach and Federer’s Sets of Positive Reach

315

and that the square of the distance function is equal to 2 n 2 xn , b2Ω (0, y) = x2 + 2n − 1 1 ∇bΩ (x, f (x)) = ' (f (x), −1) 1 + f (x)2 1 1 −n = 2n−1 2 2− 2 ((2 − 1/n) x |x| , −1). 1+ |x| n n Finally for all n ≥ 1, f ∈ / C 1,1 . For n = 1, f ∈ C 0,1 , and for n > 1, f ∈ C 1,1−1/n . This gives an example in dimension 2 of a domain Ω of class C 1,λ , 0 ≤ λ < 1, with skeleton converging to the boundary.

6.3

A Compact Family of Sets with Uniform Positive Reach

The following family of sets of positive reach and the associated compactness theorem were ﬁrst given in H. Federer [3, Thm. 4.13, Rem. 4.14]. Theorem 6.6. Given an open (resp., bounded open) holdall D in RN and h > 0, the family def

Cd,h (D) = dA : ∅ = A ⊂ D, reach (A) ≥ h

(6.11)

is closed in Cloc (D) (resp., compact in C(D)). Proof. It is suﬃcient to prove the theorem for a bounded open D. Given any C(D)Cauchy sequence {dAn } ⊂ Cd,h (D), there exists dA ∈ Cd (D) such that dAn → dA in C(D). Choose W open such that W ⊂ Uh (A). Then supx∈W dA (x) < h. Choose r, supx∈W dA (x) < r < h. Then W ⊂ Ur (A) ⊂ Ar ⊂ Uh (A). Choose N such that ∀n > N,

sup dAn (x) < r < h.

x∈W

Then for all n > N , W ⊂ Ur (An ). From Theorem 6.2 (v), for all n > N and ∀x, y ∈ (An )r ,

2 ∇dA (y) − ∇d2A (x) ≤ 2 1 + n n

h h−r

|y − x|.

(6.12)

But W ⊂ Ur (A) ⊂ Ar ⊂ Uh (A), d2An → d2A in C(D), and ∀n > N,

d2An ∈ C 1,1 (W ) ⊂ C 1 (W ).

From Theorem 2.5 in Chapter 2, the embedding C 1,1 (W ) ⊂ C 1 (W ) is compact. Therefore, for all open subsets such that W ⊂ Uh (A), d2An → d2A ∈ C 1 (W ), d2A ∈ C 1 (Ur (A)), and reach (A) ≥ r. Since this is true for all r < h, we ﬁnally get reach (A) ≥ h and dA ∈ Cd,h (D).

316

Chapter 6. Metrics via Distance Functions The compactness of Theorem 6.7 can also be expressed as follows.

Theorem 6.7. Let D be a ﬁxed bounded open subset of RN . Let {An }, An = ∅, be a sequence of subsets of D. Assume that there exists h > 0 such that ∀n,

d2An ∈ C 1,1 (Uh (An )).

(6.13)

Then there exist a subsequence {Ank } and A ⊂ D, A = ∅, such that reach (A) ≥ h and for all r, 0 < r < h, d2A ∈ C 1,1 (Ur (A)), d2An → d2A in C 1 (Ur (A)) k

and

dAnk → dA in C(D).

(6.14)

This condition in an h-tubular neighborhood is generic of other such conditions that will be expressed in diﬀerent function spaces.

7

Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points

The W 1,p -topology turns out to be an appropriate setting for the approximation of the closure of a set by its dilated sets as deﬁned in Deﬁnition 3.2. Theorem 7.1. Let ∅ = A ⊂ RN . For r > 0, Ar = Ur (A), ∂Ar = ∂Ur (A) = d−1 A {r}, Ar = Ur (A), int Ar = Ur (A), dUr (A) = dAr (x) = ∀x ∈ RN ,

dA (x) − r, 0,

if dA (x) ≥ r, if dA (x) < r,

0 ≤ dA (x) − dAr (x) ≤ r,

dUr (A) = dAr (x) → dA (x) uniformly in R

N

dUr (A) = dAr → dA in

1,p Wloc (RN )

and

(7.1) (7.2) (7.3)

as r → 0,

χAr → χA in Lploc (RN )

(7.4)

for all p, 1 ≤ p < ∞. Proof. The ﬁrst line of properties follows from Theorem 3.1. For r > 0, A ⊂ Ur (A) ⊂ Ar and dA (x) ≥ dAr (x) for all x ∈ RN . For x ∈ RN such that dA (x) < r, dAr (x) = 0 and 0 ≤ dA (x) − dAr (x) < r. For x ∈ RN such that dA (x) ≥ r and all a ∈ A and ar ∈ Ar , |x − a| ≤ |x − ar | + |x − ar | ⇒ dA (x) ≤ |x − ar | + dA (ar ) ≤ |x − ar | + r ⇒ dA (x) ≤ dAr (x) + r ⇒ 0 ≤ dA (x) − dAr (x) ≤ r. We show that dA (x) − dAr (x) = r. Assume that dA (x) − dAr (x) < r. There exists a unique p ∈ A such that dA (x) = |x − p| ≥ r > 0. Consider the point def

xr = p + r

x−p |x − p|

⇒ |x − xr | = dA (x) − r

and |xr − p| = r.

7. Approximation by Dilated Sets/Tubular Neighborhoods and Critical Points

317

By deﬁnition, dA (xr ) ≤ |xr − p| = r and xr ∈ Ar . As a result, dA (x) < dAr (x) + r ≤ |x − xr | + r = dA (x) − r + r = dA (x) yields a contradiction. This establishes identity (7.2) and for all x ∈ RN , |dAr (x) − dA (x)| ≤ r → 0 as r goes to zero. When ∇dAr (x) exists at a point ∈ Ar , ∇dAr (x) = 0 (cf. Theorem 3.3 (vi)). This means that ∇dAr = 0 almost everywhere in Ar and that the gradient of dAr is almost everywhere equal to ∇dAr (x) =

∇dA (x), 0,

dA (x) > r, dA (x) ≤ r.

For any compact K ⊂ RN χAr − χA dx. |∇dAr − ∇dA | dx = |∇dA | χAr dx = (1 − χA ) χAr dx = K

K

K

K

Since {K ∩ Ar } is a decreasing family of closed sets as r → 0 and ∩r>0 Ar = A, m(K ∩ Ar ) → m(K ∩ A) (cf. W. Rudin [1, Thm. 1.1.9 (e), p. 16]). Finally, 1,1 ∇dAr − ∇dA L1 → 0 and dAr → dA in Wloc (RN ). S. Ferry [1] proved in 1975 that if N = 2 or 3, then d−1 A {r} is an (N − 1)manifold for almost all r, and that if A is a ﬁnite polyhedron in RN , then d−1 A {r} is an (N − 1)-manifold for all suﬃciently small r. On the other hand, there is a Cantor set K in R4 such that d−1 K {r} is not a 3-manifold for any r between 0 and 1. The set K generalizes to higher dimensions. Generalizing those results, J. H. G. Fu [1] investigated the regularity properties of ∂Ar and proved the following theorem involving the notion of critical point of dA and of Lipschitz manifold. In order to make things simple we use a speciﬁc characterization of the critical points of dA as a deﬁnition (cf. J. H. G. Fu [1, 2] and Figure 6.5). Deﬁnition 7.1. Given ∅ = A ⊂ RN , x ∈ RN \A is said to be a critical point of dA if x ∈ co ΠA (x). The set of all critical points of dA will be denoted by crit (dA ). Deﬁnition 7.2. An m-dimensional Lipschitz manifold is a paracompact metric space M such that there is a system of open sets {Uα } covering M and, for each α, a bi-Lipschitzian homeomorphism ϕα of Uα onto an open subset of Rm . Theorem 7.2. Let A ⊂ RN be compact. The subset C(A) = dA (crit (dA )) ⊂ R is compact: 1/2 N C(A) ⊂ 0, diam (A) and H(N −1)/2 (C(A)) = 0 2N + 2 (the assertion about the measure of C(A) is nontrivial only for N ≤ 3). For all r∈ / C(A), Ar has positive reach and ∂Ar is a Lipschitz manifold.

318

Chapter 6. Metrics via Distance Functions

critical points of A

Figure 6.5. Set of critical points of A.

8

Characterization of Convex Sets

8.1

Convex Sets and Properties of dA

In the convex case the squared distance function is diﬀerentiable everywhere and this property can be used to characterize the convexity of a set. Theorem 8.1. Let ∅ = A ⊂ RN . The following statements are equivalent: (i) There exists a ∈ A such that reach (A, a) = +∞. (ii) Sk (A) = ∅. (iii) For all a ∈ A, reach (A, a) = +∞. (iv) reach (A) = +∞. (v) For all x ∈ RN , ΠA (x) is a singleton.9 (vi) A is convex. (vii) dA is convex. In particular, ∀x, ∀y ∈ RN ,

|pA (y) − pA (x)| ≤ |y − x|

(8.1)

1,1 2,∞ (RN ) (and, a fortiori, to Wloc (RN )). Moreover, if A is and d2A belongs to Cloc convex, then A = A and dA = dA .

Remark 8.1. The convexity of dA implies the convexity of A but not of A. The set A of all points of the unit open ball with rational coordinates has a convex closure but is not convex. 9 This part of the theorem is related to deeper results on the convexity of Chebyshev sets in metric spaces. A subset A of a metric space X is called a Chebyshev set provided that every point x of X has a unique projection pA (x) in A. The reader is referred to V. L. Klee [1] for details and background material.

8. Characterization of Convex Sets

319

Proof. The equivalence of properties (i) to (v) follows from Lemma 6.1. (v) ⇒ (vi) Given x, y ∈ A and λ ∈ [0, 1], consider the point xλ = λx + (1 − λ)y and the continuous function λ → dA (xλ ) : [0, 1] → R is continuous. There exists µ ∈ [0, 1] such that dA (xµ ) = supλ∈[0,1] dA (xλ ). If supλ∈[0,1] dA (xλ ) = 0, then xλ ∈ A. If supλ∈[0,1] dA (xλ ) > 0, then x = y, dA (xµ ) > 0, µ ∈ (0, 1), and xµ ∈ / A. Since Sk (A) = ∅, ∇dA (xµ ) exists and the maximum is characterized by d = 0 ⇒ ∇dA (xµ ) · (x − y) = 0. dA (xλ ) dλ λ=µ By assumption there exists a unique projection pµ = pA (xµ ) ∈ A such that dA (xµ ) = |xµ − pµ | and by Theorem 3.3 (vi) ∇dA (xµ ) =

xµ − pµ . |xµ − pµ |

From inequality (6.6) in Theorem 6.2 (vi) |pµ − x| = |pA (xµ ) − pA (x)| ≤ |xµ − x| = (1 − µ) |x − y|, |pµ − y| = |pA (xµ ) − pA (y)| ≤ |xµ − y| = µ |x − y| ⇒ |x − y| ≤ |pµ − x| + |pµ − y| ≤ |x − y|. Therefore the point pµ lies on the segment between x and y: there exists µ ∈ [0, 1], µ = µ, such that pµ = xµ and xµ − pµ = xµ − xµ = (µ − µ ) (x − y). Finally, µ − µ xµ − xµ xµ − pµ |x − y| = · (x − y) = · (x − y) = ∇dA (xµ ) · (x − y) = 0 |µ − µ | |xµ − xµ | |xµ − pµ | and we get the contradiction x = y. (vi) ⇒ (v) By deﬁnition for each x ∈ RN d2A (x) = inf |z − x|2 = inf |z − x|2 , z∈A

z∈A

and since A is closed and convex and z → |z − x|2 is strictly convex and coercive, there exists a unique minimizing point pA (x) in A and ΠA (x) is a singleton. (vi) ⇒ (vii) Given x and y in RN , there exist x and y in A such that dA (x) = |x−x| and dA (y) = |y−y|. By convexity of A, for all λ, 0 ≤ λ ≤ 1, λ x+(1−λ) y ∈ A and dA (λ x + (1 − λ) y) ≤|λ x + (1 − λ) y − (λ x + (1 − λ) y)| ≤λ |x − x| + (1 − λ) |y − y| = λ dA (x) + (1 − λ) dA (y) and dA is convex in RN . (vii) ⇒ (vi) If dA is convex, then ∀λ ∈ [0, 1], ∀x, y ∈ A,

dA (λx + (1 − λ)y) ≤ λdA (x) + (1 − λ)dA (y).

320

Chapter 6. Metrics via Distance Functions

But x and y in A imply that dA (x) = dA (y) = 0 and hence ∀λ ∈ [0, 1], dA (λx + (1 − λ)y) = 0. Thus λx + (1 − λ)y ∈ A and A is convex. Properties (8.1) follow from Theorem 6.2 and the identity on the complements follows from Theorem 3.5 (ii) in Chapter 5. The next theorem is a consequence of the fact that the gradient of a convex function is locally of bounded variations. Theorem 8.2. For all subsets A of RN such that ∂A = ∅ and A is convex, (i) ∇dA belongs to BVloc (RN )N , (ii) the Hessian matrix D2 dA of second-order derivatives is a matrix of signed Radon measures that are nonnegative on the diagonal, and (iii) dA has a second-order derivative almost everywhere and for almost all x and y in RN , dA (y) − dA (x) − ∇dA (x) · (y − x) − 1 (y − x) · D2 dA (x) (y − x) = o(|y − x|2 ) 2 as y → x. Proof. From the equivalence of (vi) and (vii) in Theorem 8.1 dA is convex and continuous and the result follows from L. C. Evans and R. F. Gariepy [1, Thm. 3, p. 240, Thm. 2, p. 239, and Aleksandrov’s Thm., p. 242].

8.2

Semiconvexity and BV Character of dA

Since the BV properties in Theorem 8.2 arise solely from the convexity of the distance function dA , the BV property will extend to sets A such that dA is semiconvex. Theorem 8.3. Let A be a nonempty subset of RN such that ∀x ∈ ∂A, ∃ρ > 0 such that dA is semiconvex in B(x, ρ). Then ∇dA ∈ BVloc (RN )N . Proof. For all x ∈ int A, dA = 0 and there exists ρ > 0 such that B(x, ρ) ⊂ int A and the result is trivial. For each x ∈ RN \A, dA (x) > 0. Pick h = dA (x)/2. Then for all y ∈ B(x, h) and all a ∈ A, |y − a| ≥ |x − a| − |y − x| ≥ dA (x) − |y − x| > 2h − h = h and B(x, h) ⊂ RN \Ah . By Theorem 3.2 (i) the function |x|2 /(2h) − dA (x) is locally convex in RN \Ah and, a fortiori, convex in B(x, h). Finally, by assumption for all x ∈ ∂A, there exists ρ > 0 such that dA is semiconvex. Hence for each x ∈ RN , ∇dA belongs to BV(B(x, ρ/2)) for some ρ > 0. Therefore, ∇dA belongs to BVloc (RN )N .

8. Characterization of Convex Sets

321

However, it is never semiconcave in RN for nontrivial sets. Theorem 8.4. (i) Given a subset A of RN , A = ∅, ∃c ≥ 0,

fc (x) = c |x|2 − dA (x) is convex in RN

if and only if ∃h > 0, ∃c ≥ 0,

fc (x) = c |x|2 − dA (x)

is locally convex in Uh (A). (ii) Given a subset A of RN , ∅ = A = RN , c ≥ 0,

fc (x) = c |x|2 − dA (x) is convex in RN .

Proof. (i) If fc is convex in RN , it is locally convex in Uh (A). Conversely, assume that there exist h > 0 and c ≥ 0 such that fc is locally convex in Uh (A). From Theorem 3.2 (i), 1 2 |x| − dA (x) h is locally convex in RN \Ah/2 . Therefore, for c = max{c, 1/h}, the function fc is locally convex in RN and hence convex on RN . (ii) Assume the existence of a c ≥ 0 for which fc is convex in RN . For each y ∈ A, the function x → Fc (x, y) = c |x − y|2 − dA (x) is also convex since it diﬀers from fc (x) by the linear term x → c (|y|2 − 2 x · y). Since ∅ = A = RN , there exist x ∈ A and p ∈ A such that 1 . 2c For any t > 0, deﬁne xt = p − t (x − p) and λ = t/(1 + t) ∈ ]0, 1[ and observe that 0 < dA (x) = |x − p| ≤

def

xλ = λ x + (1 − λ) xt = p and Fc (xλ , p) = 0. But λ Fc (x, p) + (1 − λ) Fc (xt , p) 1 t = [c |x − p|2 − dA (x)] + [c |xt − p|2 − dA (xt )] 1+t 1+t t ≤ [c dA (x)2 − dA (x) + c tdA (x)2 ] 1+t t ≤ dA (x) [(1 + t) c dA (x) − 1] 1+t dA (x) t t − 1 t 1 dA (x) (1 + t) − 1 = , ≤ 1+t 2 1+t 2

322

Chapter 6. Metrics via Distance Functions

since by construction dA (x) > 0 and c dA (x) < 1/2. Therefore for t, 0 < t < 1, the above quantity is strictly negative and we have constructed two points x and xt and a λ, 0 < λ < 1, such that Fc (λ x + (1 − λ) xt , p) = 0 > λ Fc (x, p) + (1 − λ) Fc (xt , p). This contradicts the convexity of the function x → Fc (x, p) and, a fortiori, of fc .

8.3

Closed Convex Hull of A and Fenchel Transform of dA

Another interesting property is that the closed convex hull of A is completely characterized by the convex envelope of dA that can be obtained from the double Fenchel transform of dA . Theorem 8.5. Let A be a nonempty subset of RN : (i) The Fenchel transform d∗A (x∗ ) = sup x∗ · x − dA (x) x∈RN

of dA is given by

d∗A (x∗ ) = σA (x∗ ) + IB(0,1) (x∗ ),

(8.2)

where σA (x∗ ) is the support function of A and IB(0,1) (x∗ ) =

x∈ / B(0, 1), x ∈ B(0, 1),

+∞, 0,

is the indicator function of the closed unit ball B(0, 1) at the origin. (ii) The Fenchel transform of d∗A is given by d∗∗ A = dco A = dco A .

(8.3)

In particular, dco A is the convex envelope of dA . The function dA and the indicator function IA are both zero on A and the Fenchel transform of IA , ∗ = σA = σA , IA

coincides with d∗A on B(0, 1). Proof. (i) From the deﬁnition, d∗A (x∗ ) = sup x∗ · x − dA (x) = sup x∈RN

x∈RN

.

x∗ · x − inf |x − p|

/

p∈A

∗

= sup sup x · x − |x − p| = sup sup x∗ · x − |x − p| x∈RN p∈A

p∈A x∈RN

∗

∗

= sup sup x · p + x · (x − p) − |x − p| p∈A x∈RN ∗

= sup x · p + sup [ x∗ · (x − p) − |x − p| ] x∈RN

p∈A ∗

= sup x · p + sup [ x∗ · x − |x| ] = σA (x∗ ) + IB(0,1) (x∗ ). p∈A

x∈RN

8. Characterization of Convex Sets

323

(ii) We now use the property that σA = σco A : ∗∗ ∗∗ · x∗ − d∗A (x∗ ) = sup x∗∗ · x∗ − σA (x∗ ) − IB(0,1) (x∗ ) d∗∗ A (x ) = sup x x∗ ∈RN x∗ ∈RN . / ∗∗ ∗ ∗ = sup x · x − σA (x ) = sup x∗∗ · x∗ − sup x∗ · x |x∗ |≤1

= sup

|x∗ |≤1

inf (x

∗∗

|x∗ |≤1 x∈co A

x∈co A

∗

− x) · x

≤ sup (x∗∗ − p(x∗∗ )) · x∗ ≤ |x∗∗ − p(x∗∗ )| = dco A (x∗∗ ). |x∗ |≤1

But in fact we have a saddle point. If x∗∗ ∈ co A, pick x∗ = 0 and ∗∗ d∗∗ A (x ) = sup

inf (x∗∗ − x) · x∗ ≥ 0 = dco A (x∗∗ ).

|x∗ |≤1 x∈co A

For x∗∗ ∈ / co A rewrite the function as (x∗∗ − x) · x∗ = (x∗∗ − p(x∗∗ )) · x∗ + (p(x∗∗ ) − x) · x∗ . Since p(x∗∗ ) is the minimizing point of the function |x−x∗∗ |2 over the closed convex set co A it is completely characterized by the variational inequality ∀x ∈ co A,

(p(x∗∗ ) − x∗∗ ) · (x − p(x∗∗ )) ≥ 0.

(8.4)

By choosing the following special x∗ , x∗ =

x∗∗ − p(x∗∗ ) , |x∗∗ − p(x∗∗ )|

we get ∗∗ d∗∗ A (x ) = sup

inf (x∗∗ − x) · x∗

|x∗ |≤1 x∈co A

≥ |x∗∗ − p(x∗∗ )| + (p(x∗∗ ) − x) ·

x∗∗ − p(x∗∗ ) |x∗∗ − p(x∗∗ )|

≥ |x∗∗ − p(x∗∗ )| = dco A (x∗∗ ) in view of (8.4). Therefore sup

inf (x∗∗ − x) · x∗ = dco A (x∗∗ ) = inf

|x∗ |≤1 x∈co A

sup (x∗∗ − x) · x∗ .

x∈co A |x∗ |≤1

Finally from the previous theorem dco A is a convex function that coincides with the convex envelope since it is the bidual of dA .

8.4

Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and c (E; D) Cd,loc

Theorem 8.6. Let D be a nonempty open subset of RN . The subfamily def

Cd (D) = {dA ∈ Cd (D) : A = ∅ convex}

(8.5)

324

Chapter 6. Metrics via Distance Functions

of Cd (D) is closed in Cloc (D). It is compact in C(D) when D is bounded. The subfamily def

Cdc (D) = {dΩ ∈ Cdc (D) : Ω open and convex}

(8.6)

and for E open, ∅ = E ⊂ D, the subfamilies def

Cdc (E; D) = {dΩ : E ⊂ Ω open and convex ⊂ D} , def

c (E; D) = Cd,loc

∃x ∈ R , ∃A ∈ O(N) such that x + AE ⊂ Ω open and convex ⊂ D N

dΩ :

&

(8.7) (8.8)

of Cdc (D) are closed in Cloc (D). If D is bounded, they are compact in C(D). Proof. The set of all convex functions in Cloc (D) is a closed convex cone with vertex at 0. Hence its intersection with Cd (D) is closed. Any Cauchy sequence in Cd (D) converges to some convex dA . From part (i) A is convex. But dA = dA and the nonempty convex set A can be chosen as the limit set. For the complementary distance function, it is suﬃcient to prove the compactness for D bounded. Given any sequence {dΩn }, there exist an open subset Ω of D and a subsequence, still indexed by n, such that dΩn → dΩ in C(D). If Ω is empty, there is nothing to prove. If Ω is not empty, consider two points x and y in Ω and λ ∈ [0, 1]. There exists r > 0 such that B(x, r) ⊂ Ω and B(y, r) ⊂ Ω

⇒ dΩ (x) ≥ r and dΩ (y) ≥ r.

There exists N > 0 such that for all n > N , dΩn − dΩ C(D) < r/2 and dΩn (x) ≥ dΩ (x) − r/2 = r/2 ⇒ B(x, r/2) ⊂ Ωn , dΩn (y) ≥ dΩ (y) − r/2 = r/2 ⇒ B(y, r/2) ⊂ Ωn . By convexity of Ωn , xλ = λx + (1 − λ)y ∈ Ωn , B(xλ , r/2) ⊂ λ B(x, r/2) + (1 − λ)B(y, r/2) ⊂ Ωn ⇒ ∀n > N, dΩn (xλ ) ≥ r/2 ⇒ dΩ (xλ ) ≥ r/2 ⇒ B(xλ , r/2) ⊂ Ω. Therefore xλ ∈ Ω, Ω is convex, and Cdc (D) is compact. Finally, Cdc (E; D) = c c (E; D) = Cd,loc (E; D) ∩ Cdc (D) are closed since Cdc (D) is Cdc (E; D) ∩ Cdc (D) and Cd,loc c c closed and Cd (E; D) and Cd,loc (E; D) are closed by Theorem 2.6.

9

Compactness Theorems for Sets of Bounded Curvature

We have seen two compactness conditions in Theorems 6.6 and 6.7 of section 6.3 for sets of positive reach from a condition in a tubular neighborhood. In this section we give compactness theorems for families of sets of global and local bounded curvature.

9. Compactness Theorems for Sets of Bounded Curvature

325

For the family of sets with bounded curvature, the key result is the compactness of the embeddings

(9.1) BCd (D) = dA ∈ Cd (D) : ∇dA ∈ BV(D)N → W 1,p (D),

BCcd (D) = dΩ ∈ Cdc (D) : ∇dΩ ∈ BV(D)N → W 1,p (D) (9.2) for bounded open Lipschitzian subsets D of RN and p, 1 ≤ p < ∞. It is the analogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets BX(D) = {χ ∈ X(D) : χ ∈ BV(D)} → Lp (D),

(9.3)

which is a consequence of the compactness of the embedding BV(D) → L1 (D)

(9.4)

for bounded open Lipschitzian subsets D of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]). As for characteristic functions in Chapter 5, we give a ﬁrst version involving global conditions on a ﬁxed bounded open Lipschitzian holdall D. In the second version, the sets are contained in a bounded open holdall D with local conditions in their tubular neighborhood or the tubular neighborhood of their boundary.

9.1

Global Conditions in D

Theorem 9.1. Let D be a nonempty bounded open Lipschitzian holdall in RN . The embedding (9.1) is compact. Thus for any sequence {An }, ∅ = An , of subsets of D such that (9.5) ∃c > 0, ∀n ≥ 1, D2 dAn M 1 (D) ≤ c, there exist a subsequence {Ank } and A = ∅, such that ∇dA ∈ BV(D)N and dAnk → dA in W 1,p (D)-strong for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D), lim ∂ij dAnk , ϕ = ∂ij dA , ϕ, 1 ≤ i, j ≤ N,

n→∞

and

D2 dA M 1 (D) ≤ c.

(9.6)

Proof. Given c > 0 consider the set def

Sc = dA ∈ Cd (D) : D 2 dA M 1 (D) ≤ c . By compactness of the embedding (9.4), given any sequence {dAn }, there exist a subsequence, still denoted by {dAn }, and f ∈ BV(D)N such that ∇dAn → f in L1 (D)N . But by Theorem 2.2 (ii), Cd (D) is compact in C(D) for bounded D and there exist another subsequence {dAnk } and dA ∈ Cd (D) such that dAnk → dA in C(D) and, a fortiori, in L1 (D). Therefore, dAnk converges in W 1,1 (D) and also in

326

Chapter 6. Metrics via Distance Functions

L1 (D). By uniqueness of the limit, f = ∇dA and dAnk converges in W 1,1 (D) to dA . For Φ ∈ D1 (D)N ×N as k goes to inﬁnity −→ −→ ∇dAnk · div Φ dx → ∇dA · div Φ dx D D −→ −→ ∇dA · div Φ dx = lim ∇dAnk · div Φ dx ≤ cΦC(D) , ⇒ k→∞

D

D

D dA M 1 (D) ≤ c, and ∇dA ∈ BV(D) . This proves the compactness of the embedding for p = 1 and properties (9.6). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cd (D) in Theorem 4.2 (i). 2

N

When D is bounded open, Cdc (D) is compact in C(D) and closed in W01,p (D), 1 ≤ p < ∞, and we have the analogue of the previous compactness theorem. Theorem 9.2. Let D, ∅ = D ⊂ RN , be bounded open Lipschitzian. The embedding (9.2) is compact. Thus for any sequence {Ωn } of open subsets of D such that ∃h > 0, ∃c > 0, ∀n,

D2 dΩn M 1 (D) ≤ c,

(9.7)

there exist a subsequence {Ωnk } and an open subset Ω of D such that ∇dΩ ∈ BV(D)N and dΩn → dΩ in W01,p (D)

(9.8)

k

for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D)N ×N , D2 dΩn , ϕ → D2 dΩ , ϕ,

D 2 dΩ M 1 (D) ≤ c,

and

χΩ ∈ BV(D). (9.9)

Proof. Given c > 0 consider the set def

Scc = dΩ ∈ Cdc (D) : D 2 dΩ M 1 (D) ≤ c . By compactness of the embedding (9.4), given any sequence {dΩn } there exist a subsequence, still denoted by {dΩn }, and f ∈ BV(D)N such that ∇dΩn → f in L1 (D)N . But by Theorem 2.4 (ii), Cdc (D) is compact in C0 (D) for bounded D and there exist another subsequence {dΩn } and dΩ ∈ Cdc (D) such that dΩn → dΩ k

k

in C0 (D) and, a fortiori, in L1 (D). Therefore, dΩn converges in W01,1 (D) and also k

in L1 (D). By uniqueness of the limit, f = ∇dΩ and dΩn converges in W01,1 (D) k to dΩ . For all Φ ∈ D1 (D)N ×N as k goes to inﬁnity −→ −→ ∇dΩn · divΦ dx → ∇dΩ · divΦ dx k D D −→ −→ ∇dΩ · divΦ dx = lim ∇dΩn · divΦ dx ≤ cΦC(D) , ⇒ D

k→∞

D

k

D 2 dΩ M 1 (D) ≤ c, ∇dΩ ∈ BV(D)N , and χΩ ∈ BV(D). This proves the compactness of the embedding for p = 1 and properties (9.9). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cdc (D) in Theorem 4.2 (i).

9. Compactness Theorems for Sets of Bounded Curvature

9.2

327

Local Conditions in Tubular Neighborhoods

The global conditions (9.5) and (9.7) can be weakened to local ones in a neighborhood of each set of the sequence, and the Lipschitzian condition on D can be removed since only the uniform boundedness of the sets of the sequence is required. Theorem 9.3. Let D, ∅ = D ⊂ RN , be a bounded open holdall and {An }, ∅ = An , be a sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that ∀n,

D2 dAn M 1 (Uh (∂An )) ≤ c.

(9.10)

Then there exist a subsequence {Ank } and a subset A, ∅ = A, of D such that ∇dA ∈ BVloc (RN )N , and for all p, 1 ≤ p < ∞, dAnk → dA in W 1,p (Uh (D))-strong.

(9.11)

Moreover, for all ϕ ∈ D0 (Uh (A)) lim ∂ij dAnk , ϕ = ∂ij dA , ϕ, 1 ≤ i, j ≤ N,

k→∞

D2 dA M 1 (Uh (A)) ≤ c,

(9.12)

and χA belongs to BVloc (RN ). Proof. First notice that, since An is bounded and nonempty, ∂An = ∅. Further Uh (∂An ) can be replaced by Uh (An ) in condition (9.10) since dAn = 0 in An . So it is suﬃcient to prove the theorem for that case. Lemma 9.1. For any A, ∂A = ∅ and h > 0, D2 dA M 1 (Uh (A)) = D2 dA M 1 (Uh (∂A)) . The proof of the lemma will be given after the proof of the theorem. (i) The assumption An ⊂ D implies Uh (An ) ⊂ Uh (D). Since Uh (D) is bounded, there exist a subsequence, still indexed by n, and a set A, ∅ = A ⊂ D, such that dAn → dA in C(Uh (D)) and another subsequence, still denoted by {dAn }, such that dAn → dA

in H 1 (Uh (D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and x in Uh (D) dAn (x) ≤ dA (x) + ε,

dA (x) ≤ dAn (x) + ε.

Therefore, An ⊂ Uh−2 ε (An ) ⊂ Uh−ε (A) ⊂ Uh (An ),

(9.13)

Uh−ε (A) ⊂ Uh−2 ε (An ) ⊂ An .

(9.14)

From (9.10) and (9.13) ∀n ≥ N,

D2 dAn M 1 (Uh−ε (A)) ≤ c.

328

Chapter 6. Metrics via Distance Functions

In order to use the compactness of the embedding (9.4) as in the proof of Theorem 9.1, we would need Uh−ε (A) to be Lipschitzian. To get around this, we construct a bounded Lipschitzian set between Uh−2ε (A) and Uh−ε (A). Indeed, by deﬁnition, Uh−ε (A) = ∪x∈A B(x, h − ε)

and Uh−2ε (A) ⊂ Uh−ε (A),

and by compactness, there exists a ﬁnite sequence of points {xi }ni=1 in A such that |xi − xj | < (h − ε)/2 for all i = j so that no two balls are tangent, and def

Uh−2ε (A) ⊂ UB = ∪ni=1 B(xi , h − ε) ⊂ Uh (D). Since UB is Lipschitzian as the union of a ﬁnite number of nontangent balls, it now follows by compactness of the embedding (9.4) for UB that there exist a subsequence, still denoted by {dAn }, and f ∈ BV(UB )N such that ∇dAn → f in L1 (UB )N . Since Uh (D) is bounded, Cd (Uh (D)) is compact in C(Uh (D)) and there exist another subsequence, still denoted by {dAn }, and ∅ = A ⊂ D such that dAn → dA in C(Uh (D)) and, a fortiori, in L1 (Uh (D)). Therefore, dAn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇dA on UB and dAn converges to dA in W 1,1 (UB ). By Deﬁnition 3.3 and Theorem 5.1, ∇dA and ∇dAn all belong to BVloc (RN )N since they are BV in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), χA ∈ BVloc (RN ). The above conclusions also hold for the subset Uh−2ε (A) of UB . (ii) Convergence in W 1,p (Uh (D)). Consider the integral |∇dAn − ∇dA |2 dx Uh (D) 2 = |∇dAn − ∇dA | dx + |∇dAn − ∇dA |2 dx. Uh−2ε (A)

Uh (D)\Uh−2ε (A)

From part (i) the ﬁrst integral on the right-hand side converges to zero as n goes to inﬁnity. The second integral is on a subset of Uh−2ε (A). From (9.14) for all n ≥ N, |∇dAn (x)| = 1 a.e. in An ⊃ Uh−3ε (An ) ⊃ Uh−2ε (A), |∇dA (x)| = 1 a.e. in A ⊃ Uh−2ε (A). The second integral reduces to |∇dAn − ∇dA |2 dx = Uh (D)\Uh−2ε (A)

2 (1 − ∇dAn · ∇dA ) dx,

Uh (D)\Uh−2ε (A)

which converges to zero since ∇dAn ∇dA in L2 (Uh (D))N -weak in part (i) and the fact that |∇dA | = 1 almost everywhere in Uh (D)\Uh−2ε (A). Therefore, since dAn → dA in C(Uh (D)), dAn → dA in H 1 (Uh (D))-strong, and by Theorem 4.2 (i) the convergence is true in W 1,p (Uh (D)) for all p ≥ 1.

9. Compactness Theorems for Sets of Bounded Curvature

329

(iii) Properties (9.12). Consider the initial subsequence {dAn } which converges to dA in H 1 (Uh (D))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of dAn to dA in H 1 (Uh (D))-weak for each Φ ∈ D1 (Uh (A))N ×N , −→ −→ ∇dAn · div Φ dx = ∇dA · div Φ dx. lim n→∞

Uh (A)

Uh (A)

Each such Φ has compact support in Uh (A), and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (A). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (An ) ⊂ Uh−ε (A) ⊂ Uh (An ).

For n ≥ N (ε) consider the integral −→ ∇dAn · div Φ dx = Uh (A)

⇒

Uh (A)

−→

Uh−2ε (A)

∇dAn

∇dAn · div Φ dx =

−→

∇dAn · div Φ dx U (A )

h n · div Φ dx ≤ D2 dAn M 1 (Uh (An )) ΦC(Uh (An )) ≤ c ΦC(Uh−2ε (A)) = c ΦC(Uh (A)) .

−→

By convergence of ∇dAn to ∇dA in L2 (D∪Uh (A))-weak, for all Φ ∈ D 1 (Uh (A))N ×N −→ ∇dA · div Φ dx ≤ c ΦC(Uh (A)) ⇒ D2 dA M 1 (Uh (A)) ≤ c. Uh (A)

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof. Proof of Lemma 9.1. First check that Uh (A) = Uh (∂A) ∪x∈A−h B(x, h),

def

A−h = x ∈ RN : B(x, h) ⊂ A .

It is suﬃcient to show that all points x of int A are contained in the right-hand side of the above expression. If d∂A (x) < h, then x ∈ Uh (∂A); if d∂A (x) ≥ h, then B(x, h) ⊂ A and x ∈ A−h . For all Φ ∈ D1 (Uh (A))N ×N , Φ has compact support and there exists ε > 0, 0 < 2ε < h, such that supp Φ ⊂ Uh−2ε (A). But Uh−ε (A) ⊂ Uh (A) and there exists {xj ∈ A−h : 1 ≤ j ≤ m} such that Uh−ε (A) ⊂ Uh (∂A)∪m j=1 B(xj , h). Let ψ0 ∈ D(Uh (∂A)) and let ψj ∈ D(B(xj , h)) be a partition of unity such that m ψj = 1 on Uh−2ε (A). 0 ≤ ψj ≤ 1, j=0

330

Chapter 6. Metrics via Distance Functions

Consider the integral −→ ∇dA · div Φ dx = Uh (A)

∇dA · div Φ dx = −→

Uh−2ε (A)

=

−→ ∇dA · div ψ0 Φ dx + Uh−2ε (A)

m ∇dA · div ψj Φ dx −→

Uh−2ε (A)

m

j=1

j=0

−→ ∇dA · div ψj Φ dx. Uh−2ε (A)

By construction for ΦC(Uh (A)) ≤ 1 ψ0 Φ ∈ D1 (Uh−2ε (A) ∩ Uh (∂A))N ×N , ψj Φ ∈ D1 (Uh−2ε (A) ∩ B(xj , h))N ×N ,

ψ0 ΦC(Uh (∂A)) ≤ 1,

ψj ΦC(B(xj ,h)) ≤ 1,

1 ≤ j ≤ m.

But for j, 1 ≤ j ≤ m, B(xj , h) ⊂ int A, where both dA and ∇dA are identically zero. As a result the above integral reduces to −→ −→ −→ ∇dA · div Φ dx = ∇dA · div (ψ0 Φ) dx = ∇dA· div (ψ0 Φ) dx Uh (A)

⇒

Uh−2ε (A)

U (∂A)

h ∇dA · div Φ dx ≤ D2 dA M 1 (Uh (∂A)) ψ0 ΦC(Uh (∂A))

−→

Uh (A)

⇒ D2 dA M 1 (Uh (A)) ≤ D 2 dA M 1 (Uh (∂A)) , and this completes the proof. Theorem 9.4. Let D, ∅ = D ⊂ RN , be a bounded open holdall. Let {Ωn }, ∅ = Ωn ⊂ D, be a sequence of open subsets and assume that ∃h > 0 and ∃c > 0 such that

∀n,

D2 dΩn M 1 (Uh (∂Ωn )) ≤ c.

(9.15)

Then there exist a subsequence {Ωnk } and an open subset Ω of D such that ∇dΩ ∈ BVloc (RN )N and for all p, 1 ≤ p < ∞, dΩn → dΩ in W01,p (D)-strong. k

(9.16)

Moreover for all ϕ ∈ D0 (Uh (Ω)) lim ∂ij dΩn , ϕ = ∂ij dΩ , ϕ, 1 ≤ i, j ≤ N,

k→∞

k

D2 dΩ M 1 (Uh (Ω)) ≤ c, (9.17)

and χΩ belongs to BVloc (RN ). Proof. First note that ∂Ωn = ∅ since Ωn is bounded and nonempty. By Lemma 9.1, Uh (∂Ωn ) can be replaced by Uh (Ωn ) in condition (9.15). Moreover since Ωn can be unbounded, we shall work with the bounded neighborhoods Uh (D Ωn ) of D Ωn rather than with Uh (Ωn ). Lemma 9.2. Let h > 0 and let ∅ = Ω ⊂ D ⊂ RN be bounded open. Then D2 dΩ M 1 (Uh (

D

Ω))

= D2 dΩ M 1 (Uh (Ω)) .

If ∂Ω = ∅, D 2 dΩ M 1 (Uh (∂Ω)) = D2 dΩ M 1 (Uh (Ω)) .

9. Compactness Theorems for Sets of Bounded Curvature

331

The proof of this lemma will be given after the proof of the theorem. (i) By Theorem 2.4 (ii) since D is bounded, there exist a subsequence, still denoted by {dΩn }, and an open subset Ω of D such that dΩn → dΩ

in C0 (D) and Cc (D ∪ Uh (Ω))

and another subsequence, still denoted by {dΩn }, such that dΩn → dΩ

in H01 (D)-weak and H01 (D ∪ Uh (Ω))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N dΩn (x) ≤ dΩ (x) + ε,

dΩ (x) ≤ dΩn (x) + ε.

Clearly, since Ω ⊂ D and Ωn ⊂ D and D is open, D Ωn = ∅ and D Ω = ∅. Furthermore, D Ωn ⊂ Uh−2 ε (D Ωn ) ⊂ Uh−ε (D Ω) ⊂ Uh (D Ωn ), D Uh−ε (D Ω) ⊂ D Uh−2 ε (D Ωn ) ⊂ Ωn .

(9.18) (9.19)

From (9.15) and (9.18) ∀n ≥ N,

D 2 dΩn M 1 (Uh−ε (

D

Ω))

≤ c.

(9.20)

As in part (i) of the proof of Theorem 9.3, we can construct a bounded open Lipschitzian set UB between Uh−2ε (D Ω) and Uh−ε (D Ω) such that Uh−2ε (D Ω) ⊂ UB ⊂ Uh−ε (D Ω). Since UB is bounded and Lipschitzian, it now follows by compactness of the embedding (9.4) for UB that there exist a subsequence, still denoted by {dΩn }, and f ∈ BV(UB )N such that ∇dΩn → f in L1 (UB )N . Since D ∪ UB is bounded, Cdc (D ∪ UB ) is compact in C0 (D ∪ UB ), and there exist another subsequence, still denoted by {dΩn }, and an open subset Ω of D such that dΩn → dΩ in C0 (D ∪ UB ) and, a fortiori, in L1 (D ∪ UB ). Therefore dΩn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇dΩ on BD and dΩn converges to dΩ in W 1,1 (UB ). By Deﬁnition 3.3 and Theorem 5.1, ∇dΩ and ∇dΩn all belong to BVloc (RN )N since they are BV in tubular neighborhoods Uh (D Ω) and Uh (D Ωn ) of their respective boundaries ∂Ω and ∂Ωn . Moreover, by Theorem 5.2 (ii), χΩ ∈ BVloc (RN ) and, a fortiori, χΩ since Ω is open. The above conclusions also hold for the subset Uh−2ε (D Ω) of UB . (ii) Convergence in W01,p (D). Consider the integral |∇dΩn − ∇dΩ |2 dx D |∇dΩn − ∇dΩ |2 dx + |∇dΩn − ∇dΩ |2 dx. = D∩Uh−2ε (D Ω)

D\Uh−2ε (D Ω)

332

Chapter 6. Metrics via Distance Functions

From part (i) the ﬁrst integral on the right-hand side converges to zero as n goes to inﬁnity. The second integral is on a subset of D Uh−2ε (D Ω). From the relations (9.19) for all n ≥ N , |∇dΩn (x)| = 1 a.e. in Ωn ⊃ D Uh−3ε (D Ωn ) ⊃ D Uh−2ε (D Ω), |∇dΩ (x)| = 1 a.e. in Ω ⊃ D Uh−2ε (D Ω). The second integral reduces to |∇dΩn − ∇dΩ |2 dx = D\Uh−2ε (D Ω)

2 (1 − ∇dΩn · ∇dΩ ) dx,

D\Uh−2ε (D Ω)

which converges to zero by weak convergence of ∇dΩn to ∇dΩ in L2 (D)N and the fact that |∇dΩ | = 1 almost everywhere in D\Uh−2ε (D Ω). Therefore, since dΩn → dΩ in C0 (D) dΩn → dΩ in H01 (D)-strong, and by Theorem 4.2 (i) the convergence is true in W01,p (D) for all p ≥ 1. (iii) Properties (9.17). Consider the initial subsequence {dΩn }, which converges to dΩ in H01 (D ∪ Uh (Ω))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of dΩn to dΩ in H01 (D ∪ Uh (Ω))-weak for each Φ ∈ D 1 (Uh (Ω))N ×N , −→ −→ ∇dΩn · div Φ dx = ∇dΩ · div Φ dx. lim n→∞

Uh (Ω)

Uh (Ω)

Now each Φ ∈ D1 (Uh (Ω))N ×N has compact support in Uh (Ω) and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (Ω). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (Ωn ) ⊂ Uh−ε (Ω) ⊂ Uh (Ωn ).

For n ≥ N (ε) consider the integral −→ ∇dΩn · div Φ dx = Uh (Ω)

⇒

Uh (Ω)

∇dΩn

Uh−2ε (Ω)

∇dΩn

· div Φ dx = −→

−→

∇dΩn · div Φ dx

U (Ω )

h n · div Φ dx ≤ D 2 dΩn M 1 (Uh (D Ωn )) ΦC(Uh (D Ωn )) ≤ c ΦC(Uh−2ε (D Ω)) = c ΦC(Uh (Ω)) .

−→

By convergence of ∇dΩn to ∇dΩ in L2 (D∪Uh (Ω))-weak, for all Φ ∈ D1 (Uh (Ω))N ×N −→ ∇dΩ · div Φ dx ≤ c ΦC(Uh (Ω)) ⇒ D2 dΩ M 1 (Uh (Ω)) ≤ c. Uh (Ω)

Finally the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

9. Compactness Theorems for Sets of Bounded Curvature Proof of Lemma 9.2. The proof is similar to that of Lemma 9.1 with Uh (Ω) = Uh (D Ω) ∪ int D = Uh (D Ω)∪x∈(D)−h B(x, h), def

(D)−h = x ∈ RN : B(x, h) ⊂ D , since both dΩ and ∇dΩ are identically zero on D.

333

Chapter 7

Metrics via Oriented Distance Functions 1

Introduction

In this chapter we study oriented distance functions and their role in the description of the geometric properties and smoothness of domains and their boundary. They constitute a special family of the so-called algebraic or signed distance functions. Signed distance functions make sense only when some appropriate orientation of the underlying set is available to assign a sign, while oriented distance functions are always well-deﬁned. Our choice of terminology emphasizes the fact that, for a smooth open domain, the associated oriented distance function speciﬁes the orientation of the normal to the boundary of the set. They enjoy many interesting properties. For instance, they retain the nice properties of the distance functions but also generate the classical geometric properties associated with sets and their boundary. The smoothness of the oriented distance function in a neighborhood of the boundary of the set is equivalent to the local smoothness of its boundary. Similarly, the convexity of the function is equivalent to the convexity of the closure of the set. In addition, their respective gradient and Hessian matrix respectively coincide with the unit outward normal and the second fundamental form on the boundary of the set. Finally, they provide a framework for the classiﬁcation of domains and sets according to their degree of smoothness, much like Sobolev spaces and spaces of continuous and H¨ olderian functions do for functions. The ﬁrst part of the chapter deals with the basic deﬁnitions and constructions and the main results. The second part specializes to speciﬁc subfamilies of oriented distance functions. The last part concentrates on compact families of subsets of oriented distance functions. In the ﬁrst part, section 2 presents the basic properties, introduces the uniform metric topology, and shows its connection with the Hausdorﬀ and complementary Hausdorﬀ topologies of Chapter 6. Section 3 is devoted to the diﬀerentiability properties and the associated set of projections onto the boundary, and it completes the treatment of skeletons and cracks. Section 8 gives the equivalence of the smoothness of a set and the smoothness of its oriented distance function in a neighborhood of its boundary for sets of class C 1,1 or better. 335

336

Chapter 7. Metrics via Oriented Distance Functions

When the domain is suﬃciently smooth the trace of the Hessian matrix of secondorder partial derivatives on the boundary is the classical second fundamental form of geometry. Section 4 deals with the W 1,p -topology on the set of oriented distance functions and the closed subfamily of sets for which the volume of the boundary is zero. In the second part of the chapter, section 5 studies the subfamily of sets for which the gradient of the oriented distance function is a vector of functions of bounded variation: the sets with global or local bounded curvature. There are large classes for which general compactness theorems will be proved in section 11. Examples are given to illustrate the behavior of the norms in tubular neighborhoods as the thickness of the neighborhood goes to zero in section 5.1. Section 9 introduces Sobolev or W s,p -domains which provide a framework for the classiﬁcation of sets according to their degree of smoothness. For smooth sets this classiﬁcation intertwines with the classical classiﬁcation of C k - and H¨ olderian domains. Section 10 extends the characterization of closed convex sets by the distance function to the oriented distance function and introduces the notion of semiconvex sets. The set of equivalence classes of convex subsets of a compact holdall D is again compact, as it was for all the topologies considered in Chapters 5 and 6. Section 7 shows that sets of positive reach introduced in Chapter 6 are sets of locally bounded curvature and that the boundary of their closure has zero volume. A compactness theorem is also given. Section 12 introduces the notion of γ-density perimeter, which is a relaxation of the (N − 1)-dimensional upper Minkowski content, and a compactness theorem for a family of subsets of a bounded holdall that have a uniformly bounded γ-density perimeter. In the last part of the chapter, section 11 gives compactness theorems for sets of global and local bounded curvatures from a uniform bound in tubular neighborhoods of their boundary. They are the analogues of the theorems of Chapter 6. Section 13 gives the compactness theorem for the family of subsets of a bounded holdall that have the uniform fat segment property. Equivalent conditions are given on the local graph functions of the family in section 13.2. The results are specialized to the compactness under the uniform cusp/cone property. In particular we recover the compactness of Theorem 6.11 of section 6.4 in Chapter 5 for the associated family of characteristic functions in Lp (D). Section 14 combines the compactness under the uniform fat segment property with a bound on the De Giorgi perimeter of Caccioppoli sets in section 14.1 and the γ-density perimeter in section 14.2. Section 15 introduces the families of cracked sets and uses them in section 16 to provide an original solution to a variation of the image segmentation problem of D. Mumford and J. Shah [2]. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks, and sets that can be made up of components of diﬀerent codimensions. The Hausdorﬀ (N − 1) measure of their boundary is not necessarily ﬁnite. Yet compact families (in the W 1,p -topology) of such sets can be constructed.

2. Uniform Metric Topology

2

337

Uniform Metric Topology

2.1

The Family of Oriented Distance Functions Cb (D)

The distance function dA provides a good description of a domain A from the “outside.” However, its gradient undergoes a jump discontinuity at the boundary of A which prevents the extraction of information about the smoothness of A from the smoothness of dA in a neighborhood of ∂A. To get around this, it is natural to take into account the negative of the distance function of its complement A that cancels the jump discontinuity at the boundary of smooth sets. Deﬁnition 2.1. Given A ⊂ RN , the oriented distance function from x ∈ RN to A is deﬁned as def

bA (x) = dA (x) − dA (x).

(2.1)

This function gives a level set description of a set whose boundary coincides with the zero level set. From Chapter 6 the function bA is ﬁnite in RN if and only if ∅ = A = RN , since bA = dA = +∞ when A = ∅ and bA = −dA = −∞ when A = ∅. This condition is completely equivalent to ∂A = ∅, in which case bA coincides with the algebraic distance function to the boundary of A: x ∈ int A, dA (x) = d∂A (x), bA (x) = 0, x ∈ ∂A, −dA (x) = −d∂A (x), x ∈ int A.

(2.2)

Noting that bA = −bA , it means that we have implicitly chosen the negative sign for the interior of A and the positive sign for the interior of its complement. As bA is an increasing function from its interior to its exterior, we shall see later that for sets with a smooth boundary, the restriction of the gradient of bA to ∂A coincides with the outward unit normal to ∂A. Changing the sign of bA gives the inward orientation to the gradient and the normal. Associate with a nonempty subset D of RN the family def

Cb (D) = bA : A ⊂ D and ∂A = ∅ .

(2.3)

Theorem 2.1. Let A be a subset of RN . Then the following hold: (i) A = ∅ and A = ∅ ⇐⇒ ∂A = ∅. (ii) Given bA and bB in Cb (D), A ⊃ B ⇒ b A ≤ bB

and

A = B ⇒ b A = bB ,

bA ≤ bB in D ⇐⇒ B ⊂ A and A ⊂ B, bA = bB in D ⇐⇒ B = A and A = B ⇐⇒ B = A and ∂A = ∂B.

338

Chapter 7. Metrics via Oriented Distance Functions In particular, bint A ≤ bA ≤ bA and bA = bA ⇐⇒ ∂A = ∂A

and

bint A = bA ⇐⇒ ∂ int A = ∂A.

(2.4)

(iii) |bA | = dA + dA = max{dA , dA } = d∂A and ∂A = {x ∈ RN : bA (x) = 0}. (iv) bA ≥ 0 ⇐⇒ A ⊃ ∂A ⊃ A ⇐⇒ ∂A = A. (v) bA = 0 ⇐⇒ A = ∂A = A ⇐⇒ ∂A = RN . (vi) If ∂A = ∅, the function bA is uniformly Lipschitz continuous in RN and |bA (y) − bA (x)| ≤ |y − x|.

∀x, y ∈ RN ,

(2.5)

Moreover, bA is (Fr´echet) diﬀerentiable almost everywhere and |∇bA (x)| ≤ 1

a.e. in RN .

(2.6)

Proof. (i) The proof is obvious. (ii) By assumption + − − dA = b+ A ≤ bB = dB and dA = bA ≥ bB = dB in D.

Since A ⊂ D and B ⊂ D, then A ⊃ D and B ⊃ D and dA = dB = 0 in D

⇒ dA ≥ dB in RN

⇒ A ⊂ B.

Also since B ⊂ D for all x ∈ B, dA (x) ≤ dB (x) = 0 and x ∈ A. Therefore, A ⊂ B and B ⊂ A. Conversely, B⊂A

⇒ dA ≤ dB in RN

and

A ⊂ B

⇒ dB ≤ dA in RN

and, a fortiori, in D. The equality case follows from the fact that bA = bB if and only if bA ≥ bB and bA ≤ bB . By deﬁnition int A = A ⊂ A ⊂ A and bint A ≤ bA ≤ bA . (iii) For x in A bA (x) = −dA (x) =⇒ |bA (x)| = dA (x) ≤ d∂A (x) since A ⊃ ∂A and the inf over A is smaller than the inf over its subset ∂A. Similarly, for x in A bA (x) = dA (x) =⇒ |bA (x)| = dA (x) ≤ d∂A (x), and ﬁnally |bA (x)| ≤ max{dA (x), dA (x)} ≤ d∂A (x). Conversely, for each x in A, the set of projections ΠA (x) ⊂ A ∩ A = ∂A is not empty. Hence |bA (x)| = dA (x) =

min |x − y| ≥ inf |x − y| = d∂A (x),

y∈ΠA (x)

y∈∂A

2. Uniform Metric Topology

339

and similarly for all x in A, |bA (x)| = dA (x) ≥ d∂A (x). Therefore |bA (x)| ≥ d∂A (x). (iv) If bA = dA −dA ≥ 0, then dA ≥ dA and A ⊂ A, and necessarily A ⊂ ∂A and A = ∂A. Conversely, if x ∈ ∂A, then by deﬁnition bA (x) = 0. If x ∈ / ∂A, then x ∈ ∂A = A = int A and bA (x) = dA (x) ≥ 0. (v) bA = 0 is equivalent to bA ≥ 0 and bA = −bA ≥ 0. Then we apply (v) twice. But A = ∂A = A ⇐⇒ int A = ∅ = int A ⇐⇒ ∂A = RN . (vi) Clearly, ∀x, y ∈ A, ∀x, y ∈ A,

|bA (y) − bA (x)| = |dA (y) − dA (x)| ≤ |y − x|, |bA (y) − bA (x)| = |dA (y) − dA (x)| ≤ |y − x|.

For x ∈ A and y ∈ int A = A, dA (y) > 0 and bA (y) − bA (x) = dA (y) + dA (x) > 0

⇒ |bA (y) − bA (x)| = dA (y) + dA (x).

By assumption B(y, dA (y)) ⊂ int A. Deﬁne the point dA (y) x=y+ (x − y) ∈ A |x − y| dA (y) ⇒ dA (x) ≤ |x − x| = 1 − (y − x) = |x − y| − dA (y) |x − y| ⇒ |bA (y) − bA (x)| = dA (y) + dA (x) ≤ |x − y|. The argument is similar for x ∈ int A and y ∈ A. The diﬀerentiability follows from Theorem 2.1 (vii) in Chapter 6.

2.2

Uniform Metric Topology

From Theorem 2.1 (i) the function bA is ﬁnite at each point when ∂A = ∅. This excludes A = ∅ and A = RN . The zero function bA (x) = 0 for all x in RN corresponds to the equivalence class of sets A such that A = ∂A = A or ∂A = RN . This class of sets is not empty. For instance, choose the subset of points of RN with rational coordinates or the set of all lines parallel to one of the coordinate axes with rational coordinates. Let D be a nonempty subset of RN and associate with each subset A of D, ∂A = ∅, the equivalence class def

[A]b = B : ∀B, B ⊂ D, B = A and ∂A = ∂B

340

Chapter 7. Metrics via Oriented Distance Functions

and the family of equivalence classes def

Fb (D) = [A]b : ∀A, A ⊂ D and ∂A = ∅ . The equivalence classes induced by bA are ﬁner than those induced by dA since both the closures and the boundaries of the respective sets must coincide. For Cd (D), we have an invariant closed representative A in the equivalence class [A]d and for Cdc (D) an invariant open representative int A for [A]d . For Cb (D), the deﬁciencies of Cd (D) and Cdc (D) combine dA = dA ≤ dint A

and dint A = dA = dA ≤ dA ,

and, in general, there is neither a closed nor an open representative, since bA ≤ bA ≤ bint A . One invariant in the class is ∂A, but it does not completely describes the class without one of the other two A or A. We shall see that for sets verifying a uniform segment property bA = bA = bint A and for convex sets that bA = bA . As in the case of dA we identify Fb (D) with the family Cb (D) through the embedding [A]b → bA : Fb (D) → Cb (D) ⊂ C(D). When D is bounded, the space C(D) endowed with the norm f C(D) is a Banach space. Moreover, for each A ⊂ D, bA is bounded, uniformly continuous on D, and bA ∈ C(D). This will induce the following complete metric: def

ρ([A]b , [B]b ) = bA − bB C(D)

(2.7)

on Fb (D). When D is open but not necessarily bounded, we use the space Cloc (D) deﬁned in section 2.2 of Chapter 6 endowed with the complete metric ρδ deﬁned in (2.9) for the family of seminorms {qK } deﬁned in (2.8). It will be shown below that Cb (D) is a closed subset of Cloc (D) and that this induces the following complete metric on Fb (D): def

ρδ ([A]b , [B]b ) =

∞ 1 qKk (bA − bB ) . 2k 1 + qKk (bA − bB )

(2.8)

k=1

The subfamilies def

Fb0 (D) = {[A]b : ∀A, A ⊂ D, ∂A = ∅, m(∂A) = 0} , def

Cb0 (D) = {bA ∈ Cb (D) : m(∂A) = 0} of Fb (D) and Cb (D) will be important. We now have the equivalent of Theorem 2.2 in Chapter 6.

2. Uniform Metric Topology

341

Theorem 2.2. Let D = ∅ be an open (resp., bounded open) holdall in RN . (i) The set Cb (D) is closed in Cloc (D) (resp., C(D)) and ρδ (resp., ρ) deﬁnes a complete metric topology on Fb (D). (ii) For a bounded open subset D of RN , the set Cb (D) is compact in C(D). (iii) For a bounded open subset D of RN , the map − 3 bA → (b+ A , bA , |bA |) = (dA , dA , d∂A ) : Cb (D) ⊂ C(D) → C(D)

is continuous: for all bA and bB in Cb (D),

max dB − dA C(D) , dB − dA C(D) , d∂B − d∂A C(D) ≤ bB − bA C(D) .

(2.9)

Proof. (i) It is suﬃcient to consider the case for which D is bounded. Consider a sequence {An } ⊂ D, ∂An = ∅, such that bAn → f in C(D) for some f in C(D). Associate with each g in C(D) its positive and negative parts g + (x) = max{g(x), 0},

g − (x) = max{−g(x), 0}.

Then by continuity of this operation + − and dAn = b− in C(D), d A n = b+ An → f An → f

d∂An = |bAn | → |f | in C(D). By Theorem 2.2 (i) of Chapter 6, there exists a closed subset F , ∅ = F ⊂ D, such that f + = dF in RN

and f + > 0 in D.

Moreover F = RN since D is bounded. By Theorem 2.4 and the remark at the beginning of section 2.3 of Chapter 6, there exists an open subset G ⊂ D, G = RN , such that f − = dG in RN and dG ∈ C0 (D). Therefore, f = f + − f − = dF − dG , f = f + − f − = dF in D and f > 0 in D. Deﬁne the sets

def

A+ = x ∈ RN : f (x) > 0 = x ∈ D : f (x) > 0 ∪ D,

def

A− = x ∈ RN : f (x) < 0 = x ∈ D : f (x) < 0 ,

def

A0 = x ∈ RN : f (x) = 0 = x ∈ D : f (x) = 0 .

342

Chapter 7. Metrics via Oriented Distance Functions

They form a partition of RN , RN = A− ∪ A0 ∪ A+ , and RN = A0 ∪ A− = F = ∅,

A0 ∪ A+ = G = ∅

⇒ A0 = F ∩ G = ∅,

D ⊂ A+ and F = A0 ∪ A− ⊂ D, f = dA0 ∪A− − dA0 ∪A+ . If A0 were empty, RN could be partitioned into two disjoint nonempty closed subsets. Since A0 is closed ∂A0 = A0 ∩ A0 = A0 ∩ A+ ∪ A− = A0 ∩ [A+ ∪ A− ] = A0 ∩ [A+ ∪ A− ∪ ∂A− ∪ ∂A+ ] = A0 ∩ [∂A− ∪ ∂A+ ]. Moreover since A− and A+ are open, A− ⊂ A− ∪ A0 , A+ ⊂ A+ ∪ A0 , ∂A− = A− ∩ A− ⊂ [A0 ∪ A+ ] ∩ [A0 ∪ A− ] = A0 , ∂A+ = A+ ∩ A+ ⊂ [A0 ∪ A− ] ∩ [A0 ∪ A+ ] = A0 ⇒ ∂A− ∪ ∂A+ = [∂A− ∪ ∂A+ ] ∩ A0 = ∂A0 ⇒ ∂A− ∪ ∂A+ = ∂A0

⇒ ∂A0 = ∂A− ∪ (∂A+ \∂A− ).

Let Q be the subset of points in RN with rational coordinates. Deﬁne def

B 0 = int A0 ,

def

0 B+ = B 0 ∩ Q,

def

0 B− = B 0 ∩ Q,

and notice that by density of Q and Q in RN , 0 = B0 = B0 . B+ −

Consider the following new partition of RN : 0 0 RN = A+ ∪ (∂A+ \∂A− ) ∪ A− ∪ ∂A− ∪ B+ ∪ B− .

Deﬁne 0 A = A− ∪ (∂A+ \∂A− ) ∪ B+ def

0 ⇒ A = A+ ∪ ∂A− ∪ B−

and 0 = A− ∪ ∂A− ∪ (∂A+ \∂A− ) ∪ B 0 , A = A− ∪ (∂A+ \∂A− ) ∪ B+ 0 = A+ ∪ ∂A+ ∪ ∂A− ∪ B 0 . A = A+ ∪ ∂A− ∪ B−

But ∂A− ∪ ∂A+ = ∂A− ∪ (∂A+ \∂A− ) ⊂ ∂A− ∪ (∂A+ \∂A− ) ⊂ ∂A− ∪ ∂A+ ,

2. Uniform Metric Topology

343

and since ∂A− ∪ ∂A+ = ∂A0

A = A− ∪ ∂A0 ∪ int A0 = A− ∪ A0 A = A+ ∪ ∂A0 ∪ int A0 = A+ ∪ A0

⇒ ∂A = A0 = ∅

⇒ bA = dA − dA = bA − dA = dF − dG = f. (ii) For the compactness, consider any sequence {bAn } ⊂ Cb (D) ⊂ C 0,1 (D). Since D is bounded, bAn and ∇bAn are both pointwise uniformly bounded in D. From Theorem 2.5 in Chapter 2, the injection of C 0,1 (D) into C(D) is compact and there exist f ∈ C(D) and a subsequence {bAnk } such that bAnk → f in C(D). From the proof of the closure in part (i), there exists A ⊂ D, ∂A = ∅, such that f = bA . (iii) For all bA and bB in Cb (D) and x in D, |bB (x)| ≤ |bA (x)| + |bB (x) − bA (x)| ⇒ | |bB (x)| − |bA (x)| | ≤ |bB (x) − bA (x)| ⇒ d∂B − d∂A C(D) = |bB | − |bA | C(D) ≤ bB − bA C(D) . − Moreover, dA = b+ A = (|bA | + bA )/2 and dA = bA = (|bA | − bA )/2, and necessarily ± b± B − bA C(D) ≤ bB − bA C(D) .

By combining the above three inequalities we get (2.9). We have the analogue of Theorem 2.3 of Chapter 6 and its corollary for A, A, and ∂A. In the last case it takes the following form (to be compared with T. J. Richardson [1, Lem. 3.2, p. 44] and S. R. Kulkarni, S. K. Mitter, and T. J. Richardson [1] for an application to image segmentation). Corollary 1. Let D be a nonempty open (resp., bounded open) holdall in RN . Deﬁne for a subset S of RN the sets def

Hb (S) = {bA ∈ Cb (D) : S ⊂ ∂A} , def

Ib (S) = {bA ∈ Cb (D) : ∂A ⊂ S} , def

Jb (S) = {bA ∈ Cb (D) : ∂A ∩ S = ∅} . (i) Let S be a subset of RN . Then Hb (S) is closed in Cloc (D) (resp., C(D)). (ii) Let S be a closed subset of RN . Then Ib (S) is closed in Cloc (D) (resp., C(D)). If, in addition, S∩D is compact, then Jb (S) is closed in Cloc (D) (resp., C(D)). (iii) Let S be an open subset of RN . Then Jb (S) is open in Cloc (D) (resp., C(D)). If, in addition, S ∩D is compact, then Ib (S) is open in Cloc (D) (resp., C(D)). (iv) For D bounded, associate with an equivalent class [A]b the number # ∂A = number of connected components of ∂A.

344

Chapter 7. Metrics via Oriented Distance Functions Then the map def

[A]b → # ∂A : F#b (D) = [A]b : ∀A ⊂ D, ∂A = ∅, and # ∂A < ∞ → R is lower semicontinuous. For a ﬁxed number c > 0, the subset {bA ∈ Cb (D) : # ∂A ≤ c} of Cb (D) is compact in C(D). In particular, the subset {bA ∈ Cb (D) : ∂A is connected} of Cb (D) is compact in C(D).

Proof. The proof is the same as for Theorem 2.3 of Chapter 6 using the Lipschitz continuity of the map bA → d∂A = |bA | from C(D) to C(D).

3

Projection, Skeleton, Crack, and Diﬀerentiability

This section is the analogue of section 3 in Chapter 6: connection between the gradient of bA and the projection onto ∂A and the characteristic functions associated with ∂A, singularities of the gradients, and the notions of skeleton1 and cracks. Let A ⊂ RN , ∅ = ∂A, and recall the notation def

(3.1) Sing (∇bA ) = x ∈ RN : ∇bA (x) for the set of singularities of the gradient of bA , def

Π∂A (x) = {z ∈ ∂A : |z − x| = d∂A (x)}

(3.2)

for the set of projections of x onto ∂A (when Π∂A (x) is a singleton, the unique element is denoted by p∂A (x)), and

def

Sk (∂A) = x ∈ RN : Π∂A (x) is not a singleton = x ∈ RN : ∇d2∂A (x) (3.3) for the skeleton of ∂A. Since Π∂A (x) is a singleton for all x ∈ ∂A we have Sk (∂A) ⊂ RN \∂A. In general, the functions bA and d∂A are diﬀerent and Sing ∇bA is usually smaller than Sing (∇d∂A ) = Sk (∂A) ∪ Ck (∂A). As a result, using the oriented distance function bA rather than d∂A requires a new deﬁnition of the set of cracks with respect to bA that will be justiﬁed in Theorem 3.1 (iv). Deﬁnition 3.1. Given A ⊂ RN , ∅ = ∂A, the set of b-cracks of A is deﬁned as def

Ck b (A) = Sing ∇bA \Sk (∂A).

(3.4)

Sk (∂A), Ck b (A), and Sing (∇bA ) have zero N -dimensional Lebesgue measure, since bA is Lipschitz continuous and hence diﬀerentiable almost everywhere. 1 Our deﬁnition of a skeleton does not exactly coincide with the one used in morphological `re [1]). mathematics (cf., for instance, G. Matheron [1] or A. Rivie

3. Projection, Skeleton, Crack, and Diﬀerentiability

345

Remark 3.1. (1) In general Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A), but Ck b (A) can be strictly smaller than Ck (∂A). Consider the open ball of unit radius A = B1 (0): x 1 x i xj bA (x) = |x| − 1, ∇bA (x) = , D2 bA (x)ij = δij − , |x| |x| |x| |x| Sing (∇bA ) = {0},

Sk (∂A) = {0},

and

Ck b (A) = ∅.

It is easy to check that for all x ∈ ∂A, ∇bA (x) is the outward unit normal, and D 2 bA (x) is the second fundamental form with eigenvalues 0 for the space spanned by the normal x and 1 for the tangent space to ∂A in x orthogonal to x. But d∂A (x) = ||x| − 1| , Sing (∇d∂A ) = {0} ∪ ∂B1 (0),

Sk (∂A) = {0},

and

Ck (∂A) = ∂B1 (0).

Using bA rather than d∂A or dA removes artiﬁcial singularities. (2) In general Sk (∂A) ⊂ RN \∂A, but if ∂A has positive reach h > 0 in the sense of Deﬁnition 6.1 in section 6 of Chapter 6 (there exists h > 0 such that b2A ∈ C 1,1 (Uh (∂A)), the skeleton will remain at a distance h from ∂A. The function bA enjoys properties similar to the ones of dA and dA since |bA | = d∂A = max{dA , dA } and we have the analogues of Theorems 3.2 and 3.3 in Chapter 6. Recall the deﬁnition of def

f∂A (x) =

1 2 |x| − d2∂A (x) . 2

Theorem 3.1. Let A be a subset of RN such that ∅ = ∂A, and let x ∈ RN . (i) For all x ∈ RN , d2∂A (x) = b2A (x), f∂A (x) =

1 2 |x| − b2A (x) 2

is convex and continuous, the function x →

|x|2 k∂A,h (x) = − |bA |(x) : RN \Uh (∂A) → R 2h 2h

is locally convex (and continuous) in RN \Uh (∂A), ∇f∂A and ∇b2A belong to N BVloc (RN )N , and ∇d∂A ∈ BVloc RN \∂(∂A) . (ii) The set Π∂A (x) is nonempty and compact, and ∀x ∈ / ∂A

Π∂A (x) ⊂ ∂(∂A)

and

∀x ∈ ∂A

Π∂A (x) = {x}.

/ A, Π∂A (x) = ΠA (x) ⊂ ∂A. For x ∈ / A, Π∂A (x) = ΠA (x) ⊂ ∂A and for x ∈ Moreover, ∂(∂A) = ∂A ∪ ∂A.

346

Chapter 7. Metrics via Oriented Distance Functions

(iii) For all x and v in RN , the Hadamard semiderivative2 of d2A always exists, b2A (x + tv) − b2A (x) = min 2(x − z) · v, t0 t z∈Π∂A (x) f∂A (x + tv) − f∂A (x) dH f∂A (x; v) = lim = σΠ∂A (x) (v) = σco Π∂A (x) (v), t0 t dH b2A (x; v) = lim

where σB is the support function of the set B, σB (v) = sup z · v, z∈B

and co B is the convex hull of B. In particular,

Sk (∂A) = x ∈ RN : ∇b2A (x) ⊂ RN \∂A.

(3.5)

Given v ∈ RN , for all x ∈ RN \∂(∂A) the Hadamard semiderivative of bA exists, dH bA (x; v) =

1 min (x − p) · v, bA (x) p∈ΠA (x)

(3.6)

and for x ∈ ∂(∂A), dH bA (x; v) exists if and only if lim

t0

bA (x + tv) t

exists.

(3.7)

(iv) The following statements are equivalent: (a) b2A (x) is (Fr´echet) diﬀerentiable at x. (b) b2A (x) is Gateaux diﬀerentiable at x. (c) Π∂A (x) is a singleton. (v) ∇f∂A (x) exists if and only if Π∂A (x) = {p∂A (x)} is a singleton. In that case 1 p∂A (x) = ∇f∂A (x) = x − ∇b2A (x). (3.8) 2 For all x and y in RN , 1 2 1 2 |y| − b2A (y) ≥ |x| − b2A (x) + p(x) · (y − x), (3.9) ∀p(x) ∈ Π∂A (x), 2 2 or equivalently, ∀p(x) ∈ Π∂A (x),

b2A (y) − b2A (x) − 2 (x − p(x)) · (y − x) ≤ |x − y|2 . (3.10)

For all x and y in RN , ∀p(x) ∈ Π∂A (x), ∀p(y) ∈ Π∂A (y), 2A

(p(y) − p(x)) · (y − x) ≥ 0.

function f : RN → R has a Hadamard semiderivative in x in the direction v if def

dH f (x; v) = lim

t0 w→v

(cf. Chapter 9, Deﬁnition 2.1 (ii)).

f (x + tw) − f (x) exists t

(3.11)

3. Projection, Skeleton, Crack, and Diﬀerentiability

347

(vi) The functions p∂A : RN \Sk (∂A) → RN

and

∇b2A : RN \Sk (∂A) → RN

are continuous. For all x ∈ RN \Sk (∂A) 1 p∂A (x) = x − ∇b2A (x) = ∇f∂A (x). 2 In particular, for all x ∈ ∂A, Π∂A (x) = {x}, ∇b2A (x) = 0,

Sk (∂A) = x ∈ RN : ∇b2A (x) and ∇bA (x) ⊂ RN \∂A,

Ck b (A) = x ∈ RN : ∇b2A (x) ∃ and ∇bA (x) ⊂ ∂(∂A),

(3.12)

(3.13)

and Sing (∇bA ) = Sk (∂A) ∪ Ck b (A). (vii) The function ∇bA : RN \ (Sk (∂A) ∪ ∂(∂A)) → RN is continuous. For all x ∈ int ∂A, ∇bA (x) = 0; for all x ∈ RN \ (Sk (∂A) ∪ ∂A), x − p∂A (x) N \A, , x ∈ R ∇b2A (x) |x − p∂A (x)| = , |∇bA (x)| = 1. ∇bA (x) = 2 bA (x) x − p∂A (x) − , x ∈ RN \A, |x − p∂A (x)| (3.14) If ∇d∂A (x) exists for some x ∈ ∂A, then ∇bA (x) = ∇d∂A (x) = 0. In particular, ∇bA (x) = 0 almost everywhere in ∂A. (viii) The functions b2A and bA are diﬀerentiable almost everywhere and m(Sk (∂A)) = m(Ck b (A)) = m(Sing (∇bA )) = 0. If ∂A = ∅, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A).

(3.15)

If ∂A = ∅, then either A = RN or A = ∅. (ix) Given A ⊂ RN , ∂A = ∅, χ∂A (x) = 1 − |∇bA (x)| in RN \Sing (∇bA ),

(3.16)

the identity holds for almost all x in RN , and the set def

∂b A = {x ∈ ∂A : ∇bA (x) exists and ∇bA (x) = 0} ⊂ ∂(∂A) has zero N -dimensional Lebesgue measure.

(3.17)

348

Chapter 7. Metrics via Oriented Distance Functions

(x) Let µ be a measure in the sense of L. C. Evans and R. F. Gariepy [1, Chap. 1] and D ⊂ RN be bounded open such that µ(D) < ∞. The map bA → µ(∂A) : Cb (D) → R

(3.18)

is upper semicontinuous with respect to the topology of uniform convergence on Cb (D). Moreover, if bAn → bA in C(D) for bAn and bA in Cb (D), then µ(int A) ≤ lim inf µ(int An ) ≤ lim sup µ(An ) ≤ µ(A).

(3.19)

If µ(∂A) = 0, the map bA → µ(A) : Cb (D) → R is continuous in bA . def

(xi) Given x ∈ RN , α ∈ [0, 1], p ∈ Π∂A (x), and xα = p + α (x − p), then bA (xα ) = α bA (x)

∀α ∈ [0, 1] ,

and

Π∂A (xα ) ⊂ Π∂A (x).

In particular, if Π∂A (x) is a singleton, then Π∂A (xα ) is a singleton and ∂A, then for ∇b2A (xα ) exists for all α, 0 ≤ α ≤ 1. If, in addition, x = all 0 < α ≤ 1 ∇bA (xα ) exists and ∇bA (xα ) = ∇bA (x). Proof. (i) to (v) follow from Theorems 3.2 and 3.3 of Chapter 6 using the identity b2A = d2∂A . (v) To complete the proof of identities (3.13), recall that the projection of all points of ∂A onto ∂A being a singleton, Sk (∂A ⊂ RN \∂A. Therefore, for any x ∈ Sk (∂A), we have bA (x) = 0 and if ∇b2A (x) does not exist, then ∇bA (x) cannot exist. This sharpens the characterization (3.5) of Sk (∂A). From this we readily get the characterization of Ck b (A) = Sing (∇bA )\Sk (∂A) and Ck b (A) ⊂ ∂A. This last property can be improved to Ck b (A) ⊂ ∂(∂A) by noticing that for x ∈ int ∂A, bA (x) = 0 and ∇b2A (x) = ∇bA (x) = 0. (vi) From Theorem 3.3 (iv) of Chapter 6. (vii) In RN \A, bA = dA and ∇bA is continuous in RN \(Sk (A) ∪ A); in N R \A, bA = −dA and ∇bA is continuous in RN \(Sk (A) ∪ A); in int ∂A, bA = dA = dA = 0 and ∇bA = 0. So, in view of part (v) it is continuous in RN \(Sk (A) ∪ A) ∪ RN \(Sk (A) ∪ A) ∪ int ∂A = RN \ (Sk (∂A) ∪ ∂(∂A)) . If x ∈ RN \A, then bA = −dA and repeat the proof of Theorem 3.3 (vi) in Chapter 6 to obtain −∇bA (x) = ∇dA (x) =

x − p∂A (x) x − p∂A (x) = |x − p∂A (x)| −bA (x)

and p∂A (x) ∈ ∂A is unique. Similarly, for x ∈ RN \A, bA = dA , ∇bA (x) = ∇dA (x) =

x − p∂A (x) x − p∂A (x) = |x − p∂A (x)| bA (x)

and p∂A (x) ∈ ∂A is unique. In both cases |∇bA (x) = 1.

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

349

Finally, from Theorem 3.3 (i) in Chapter 6, whenever d∂A is diﬀerentiable at a point x ∈ ∂A, ∇d∂A (x) = 0. But for all v in RN and t > 0 bA (x + tv) − bA (x) |bA (x + tv)| d∂A (x + tv) − d∂A (x) = = t t t since |bA | = d∂A . Therefore, if ∇d∂A (x) exists, ∇bA (x) exists and ∇bA (x) = ∇d∂A (x) = 0. Finally, since ∇d∂A (x) exists almost everywhere in RN , then for almost all x ∈ ∂A, ∇bA (x) = 0. (viii) From Theorem 3.3 (vii) in Chapter 6 and the fact that Ck b (A) ⊂ C(∂A). Indeed, for x ∈ ∂A, if ∇dA (x) and ∇dA (x) exist, they are 0 and for all t > 0 and v ∈ RN bA (x + tv) − bA (x) dA (x + tv) − dA (x) dA (x + tv) − dA (x) = →0 − t t t as t 0. Hence ∇bA (x) = 0. As a consequence, if ∇bA (x) does not exist, then either ∇dA (x) or ∇dA (x) does not exist and Ck b (A) ⊂ C(A) ∪ C(A) = C(∂A) ⊂ ∂(∂A) from Theorem 3.3 (vii) in Chapter 6. (ix) Since bA is a Lipschitzian function, it is diﬀerentiable almost everywhere in RN , and in view of the previous considerations, when it is diﬀerentiable |∇bA (x)| =

1, x ∈ / ∂A, 0 a.e. in ∂A

⇒ χ∂A (x) = 1 − |∇bA (x)|

a.e. in RN .

Therefore, the set ∂b A has at most a zero measure. (x) From Theorem 4.1 in Chapter 6 applied to A, A, and ∂A. (xi) Same proof as the one of Theorem 3.3 (ix) in Chapter 6. Remark 3.2. In general ∇dA (x) and ∇dA (x) do not exist for x ∈ ∂A. This is readily seen by constructing the directional derivatives for the half-space A = {(x1 , x2 ) ∈ R2 : x1 ≤ 0}

(3.20)

at the point (0, 0). Nevertheless ∇bA (0, 0) exists and is equal to (1, 0), which is the outward unit normal at (0, 0) ∈ ∂A to A. Note also that for all x ∈ ∂A, |∇bA (x)| = 1. This is possible since mN (∂A) = 0. If ∇bA (x) exists, then it is easy to check that |∇bA (x)| ≤ 1. For suﬃciently smooth domains, the subset ∂b A of the boundary ∂A coincides with the reduced boundary of ﬁnite perimeter sets.

4 4.1

W 1,p (D)-Metric Topology and the Family Cb0 (D) Motivations and Main Properties

1,p From Theorem 2.1 (vi) bA is locally Lipschitzian and belongs to Wloc (RN ) for all p ≥ 1. So the previous constructions for dA and dA can be repeated with the space

350

Chapter 7. Metrics via Oriented Distance Functions

1,p (RN ) in place of Cloc (D) to generate new W 1,p metric topologies on the family Wloc Cb (D). Moreover the other distance functions can be recovered from the map − bA → (b+ A , bA , |bA |) = (dA , dA , d∂A )

and the characteristic functions from the maps bA → b− A = dA → χint A = |∇dA |,

(4.1)

bA → = dA → χint A = |∇dA |, bA → χ∂A = 1 − |∇bA |.

(4.2) (4.3)

b+ A

One of the advantages of the function bA is that the W 1,p -convergence of sequences will imply the Lp -convergence of the corresponding characteristic functions of int A, int A, and ∂A, that is, continuity of the volume of these sets. It will be useful to introduce the following set and terminology. Deﬁnition 4.1. (i) A set A ⊂ RN , ∂A = ∅, is said to have a thin 3 boundary if mN (∂A) = 0. (ii) Denote by def

Cb0 (D) = {bA : ∀A, A ⊂ D, ∂A = ∅ and m(∂A) = 0}

(4.4)

the subset of oriented distance functions with thin boundaries. In this section we give the analogues of Theorems 4.2 and 4.3 in Chapter 6. Theorem 4.1. Let D be an open (resp., bounded open) subset of RN . 1,p (i) The topologies induced by Wloc (D) (resp., W 1,p (D)) on Cb (D) are all equivalent for p, 1 ≤ p < ∞. 1,p (D) (resp., W 1,p (D)) for p, 1 ≤ p < ∞, and (ii) Cb (D) is closed in Wloc def

ρD ([A2 ]b , [A1 ]b ) =

∞ bA2 − bA1 W 1.p (B(0,n)) 1 n 2 1 + bA2 − bA1 W 1.p (B(0,n)) n=1 def

(resp., ρD ([A2 ]b , [A1 ]b ) = bA2 − bA1 W 1,p (D) ) deﬁnes a complete metric structure on Fb (D). (iii) For p, 1 ≤ p < ∞, the map 1,p bA → χ∂A = 1 − |∇bA | : Cb (D) ⊂ Wloc (D) → Lploc (D)

is “Lipschitz continuous”: for all bounded open subsets K of D and bA1 and bA2 in Cb (D), χ∂A2 − χ∂A1 Lp (K) ≤ ∇bA2 − ∇bA1 Lp (K) ≤ bA2 − bA1 W 1,p (K) . In particular, Cb0 (D) is a closed subset of Cb (D) for the W 1,p -topology. 3 This

terminology should not be confused with the notion of thin set in the capacity sense.

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

351

(iv) Let D be a bounded open subset of RN . For each bA ∈ Cb (D), bA W 1,p (D) = |bA | W 1,p (D) = d∂A W 1,p (D) , dA W 1,p (D) ≤ bA W 1,p (D) ,

dA W 1,p (D) ≤ bA W 1,p (D) .

(4.5)

The map − 1.p (D) → W 1.p (D)3 (4.6) bA → (b+ A , bA , |bA |) = (dA , dA , d∂A ) : Cb (D) ⊂ W

is continuous. (v) Let D be an open (resp., bounded open) subset of RN . For all p, 1 ≤ p < ∞, the map bA → (χ∂A , χint A , χint A ) 1,p (D) → Lploc (D)3 (resp.,W 1,p (D) → Lp (D)3 ) : Wloc

(4.7)

is continuous. Proof. The proofs of (i) and (ii) are essentially the same as the proof of Theorem 4.2 of Chapter 6 using the properties established for Cb (D) in C(D) of Theorem 2.2. (iii) From Theorem 3.1 (ix) for any two subsets A1 and A2 of D with nonempty boundaries and for any open U in D |∇bA2 | ≤ |∇bA1 | + |∇bA2 − ∇bA1 | + |∇bA2 − ∇bA1 | ⇒ |χ∂A1 − χ∂A2 | ≤ |∇bA2 − ∇bA1 |

⇒ χ∂A1 ≤ χ∂A2 |χ∂A2 − χ∂A1 |p dx ≤ ∇bA2 − ∇bA1 pLp (U ) ≤ bA2 − bA1 pW 1,p (U ) ⇒ U

for p, 1 ≤ p < ∞, and with the ess sup norm for p = ∞. The closure of Cb0 (D) follows from the continuity of the maps bA → m(∂A ∩ U ) = χ∂A dx : W 1,p (U ) → R ∂A∩U

for all bounded open subsets U of D. (iv) First observe that |bA (x)| = dA (x) + dA (x) and ∇|bA (x)| = ∇dA (x) + ∇dA (x), bA (x) = dA (x) − dA (x) and ∇bA (x) = ∇dA (x) − ∇dA (x) and since ∇bA = ∇dA = ∇dA = 0 almost everywhere on ∂A = A ∩ A, then |∇bA (x)| = |∇dA (x)| + |∇dA (x)| = |∇|bA (x)| | for almost all x in RN . From this we readily get (4.5). However, this is not suﬃcient to prove the continuity since the map (4.6) is not linear. To get around this consider a sequence {bAn } ⊂ Cb (D) converging to bA in W 1,p (D). From Theorem 2.2 (iii), |bAn | = d∂An , b+ An = + − dAn , and b− = d converge to |b | = d , b = d , and b = d in C(D) A ∂A A An A An A A

352

Chapter 7. Metrics via Oriented Distance Functions

and hence in W 1,p (D)-weak. To prove that the convergence is strong, consider the L2 -norm |∇dAn − ∇dA |2 + |∇dAn − ∇dA |2 dx D = |∇dAn |2 + |∇dA |2 + |∇dAn |2 + |∇dA |2 D − 2 ∇dAn · ∇dA − 2 ∇dAn · ∇dA dx |∇bAn |2 + |∇bA |2 − 2 ∇dAn · ∇dA − 2 ∇dAn · ∇dA dx = D 2 2 2 → 2 |∇bA | − 2 |∇dA | − 2 |∇dA | dx = 2 |∇bA |2 − 2 |∇bA |2 dx = 0 D

D

by weak L -convergence of ∇dAn and ∇dAn to ∇dA and ∇dA . So both sequences dAn and dAn converge in W 1,2 (D)-strong, and hence from part (i) in W 1,p (D)-strong for all p, 1 ≤ p < ∞. (v) It is suﬃcient to prove the result for D bounded open. From part (iii) it is true for χ∂A , and from part (iv) and Theorem 4.2 (ii) and (iii) of Chapter 6 for the other two. 2

4.2

Weak W 1,p -Topology

Theorem 4.2. Let D be a bounded open domain in RN . (i) If {bAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. (ii) If {bAn } converges in C(D), then it weakly converges in W 1,p (D) for all p, 1 ≤ p < ∞. Conversely, if {bAn } weakly converges in W 1,p (D) for some p, 1 ≤ p < ∞, it converges in C(D). (iii) Cb (D) is compact in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Proof. (i) Recall that for D bounded there exists a constant c > 0 such that for all bA ∈ Cb (D) |bA (x)| ≤ c and |∇bA (x)| ≤ 1 a.e. in D. If {bAn } weakly converges in W 1,p (D) for some p ≥ 1, then {bAn } weakly converges in Lp (D), {∇bAn } weakly converges in Lp (D)N . By Lemma 3.1 (i) in Chapter 5 both sequences weakly converge for all p ≥ 1 and hence {bAn } weakly converges in W 1,p (D) for all p ≥ 1. (ii) If {bAn } converges in C(D), then by Theorem 2.2 (ii) there exists bA ∈ Cb (D) such that bAn → bA in C(D) and hence in Lp (D). So for all ϕ ∈ D(D)N ∇bAn · ϕ dx = − bAn div ϕ dx → − bA div ϕ dx = ∇bA · ϕ dx. D

D

D

D

4. W 1,p (D)-Metric Topology and the Family Cb0 (D)

353

By density of D(D) in L2 (D), ∇bAn → ∇bA in L2 (D)N -weak and hence bAn → bA in W 1,2 (D)-weak. From part (i) it converges in W 1,p (D)-weak for all p, 1 ≤ p < ∞. Conversely, the weakly convergent sequence converges to some f in W 1,p (D). By compactness of Cb (D), there exist a subsequence, still indexed by n, and bA such that bAn → bA in C(D) and hence in W 1,p (D)-weak. By uniqueness of the limit, bA = f . Therefore, all convergent subsequences in C(D) converge to the same limit. So the whole sequence converges in C(D). This concludes the proof. (iii) Consider an arbitrary sequence {bAn } in Cd (D). From Theorem 2.2 (ii) Cb (D) is compact and there exist a subsequence {bAnk } and bA ∈ Cb (D) such that bAnk → bA in C(D). From part (ii) the subsequence weakly converges in W 1,p (D) and hence Cb (D) is compact in W 1,p (D)-weak. For sets with thin boundaries the strong and weak W 1,p (D)-convergences of elements of Cb0 (D) to an element of Cb0 (D) are equivalent as in the case of Theorem 3.2 in section 3.2 of Chapter 5 for characteristic functions in Lp (D)-strong. Theorem 4.3. Let ∅ = D ⊂ RN be bounded open. (i) Let {An } be a sequence of subsets of D such that ∂An = ∅ and m(∂An ) = 0 ¯ be such that ∂A = ∅ and m(∂A) = 0. Then and A ⊂ D ⇒ bAn → bA in W 1,2 (D)-strong,

bAn bA in W 1,2 (D)-weak

and hence in W 1,p (D)-strong for all p, 1 ≤ p < ∞. (ii) The map bA → mN (int A) = mN (A) = mN (A) : Cb0 (D) → R

(4.8)

is continuous with respect to the W 1,p -topology on Cb0 (D) for all p, 1 ≤ p < ∞. Proof. (i) From Theorem 4.1 (i) on the equivalence of the W 1,p -topologies, it is suﬃcient to prove the result for p = 2. By Theorem 4.2 (ii), the weak L2 -convergence of {bAn } to bA implies the strong convergence in C(D) and hence in L2 (D)-strong. Since, for all n ≥ 1, m(∂An ) = 0 = m(∂A), |∇bA | = 1 = |∇bAn | almost everywhere in D by Theorem 3.1 (vii). As a result |∇bAn − ∇bA |2 dx = |∇bAn |2 + |∇bA |2 − 2∇bAn · ∇bA dx D D 2 χ∂A dx = 0. = 2 (1 − ∇bAn · ∇bA ) dx → 2 (1 − |∇bA | ) dx = 2 D

D

D

Therefore ∇bAn → ∇bA in L2 (D)N -strong and bAn → bA in W 1,2 (D)-strong, since the convergence bAn → bA in L2 (D)-strong follows from the weak convergence in W 1,2 (D). The convergence in W 1,p (D)-strong follows from the equivalence of the topologies on Cb (D) (cf. Theorem 4.1 (i)). (ii) From Theorem 3.1 (x) since mN (∂A) = 0 for bA ∈ Cb0 (D) and Cb0 (D) is closed in the W 1,p -topology.

354

5

Chapter 7. Metrics via Oriented Distance Functions

Boundary of Bounded and Locally Bounded Curvature

We introduce the family of sets with bounded or locally bounded curvature which are the analogues for bA of the sets of exterior and interior bounded curvature associated with dA and dA in section 5 of Chapter 6. They include C 1,1 -domains, convex sets, and the sets of positive reach of H. Federer [3]. They lead to compactness theorems for subfamilies of Cb (D) in W 1,p (D). Deﬁnition 5.1. (i) Given a bounded open nonempty holdall D in RN and a subset A of D, ∂A = ∅, its boundary ∂A is said to be of bounded curvature with respect to D if ∇bA ∈ BV(D)N .

(5.1)

This family of sets will be denoted as follows: def

BCb (D) = bA ∈ Cb (D) : ∇bA ∈ BV(D)N . (ii) Given a subset A of RN , ∂A = ∅, its boundary ∂A is said to be of locally bounded curvature if ∇bA belongs to BVloc (RN )N . The family of sets whose boundary is of locally bounded curvature will be denoted by def

BCb = bA ∈ Cb (RN ) : ∇bA ∈ BVloc (RN )N . As in Theorem 6.2 of Chapter 5 for Caccioppoli sets, and in Theorem 5.1 of Chapter 6 for sets with a locally bounded curvature, it is suﬃcient to satisfy the BV property in a neighborhood of each point of the boundary ∂A. Theorem 5.1. Let A, ∂A = ∅, be a subset of RN . Then ∂A is of locally bounded curvature if and only ∀x ∈ ∂(∂A), ∃ρ > 0 such that ∇bA ∈ BV(B(x, ρ))N ,

(5.2)

where B(x, ρ) is the open ball of radius ρ > 0 in x. Proof. From Deﬁnition 5.1 and Theorem 3.1 (i). Theorem 5.2. (i) Let D be a nonempty bounded open Lipschitzian holdall in RN and let A ⊂ D, ∂A = ∅. If ∇bA ∈ BV(D)N , then ∇χ∂A M 1 (D) ≤ 2 D 2 bA M 1 (D) ,

χ∂A ∈ BV(D),

and ∂A has ﬁnite perimeter with respect to D. (ii) For any subset A of RN , ∂A = ∅, ∇bA ∈ BVloc (RN )N and ∂A has locally ﬁnite perimeter.

⇒

χ∂A ∈ BVloc (RN ),

5. Boundary of Bounded and Locally Bounded Curvature

355

Proof. Given ∇bA in BV(D)N , there exists a sequence {uk } in C ∞ (D)N such that uk → ∇bA in L1 (D)N

and Duk M 1 (D) → D2 bA M 1 (D)

as k goes to inﬁnity, and since |∇bA (x)| ≤ 1, this sequence can be chosen in such a way that ∀k ≥ 1, |uk (x)| ≤ 1. This follows from the use of molliﬁers (cf. E. Giusti [1, Thm. 1.17, p. 15]). For all V in D(D)N

−

(|∇bA | − 1) div V dx =

|∇bA |2 div V dx.

2

χ∂A div V dx = D

D

D

For each uk |uk |2 div V dx = −2 [ ∗Duk ]uk · V dx = −2 uk · [Duk ]V dx, D

D

D

where ∗Duk is the transpose of the Jacobian matrix Duk and |uk |2 div V dx ≤ 2 |uk | |Duk | |V | dx D

D

≤ 2 Duk L1 V C(D) ≤ 2 Duk M 1 V C(D) since for W 1,1 (D)-functions ∇f L1 (D)N = ∇f M 1 (D)N . Therefore, as k goes to inﬁnity, χ∂A div V dx = |∇bA |2 div V dx ≤ 2D2 bA M 1 V C(D) . D

D

Therefore, ∇χ∂A ∈ M 1 (D)N .

5.1

Examples and Limit of Tubular Norms as h Goes to Zero

It is informative to compute the Laplacian of bA for a few examples. The ﬁrst three examples are illustrated in Figure 7.1. Example 5.1 (Half-plane in R2 ). (cf. Example 5.1 in Chapter 6). Consider the domain A = {(x1 , x2 ) : x1 ≤ 0},

∂A = {(x1 , x2 ) : x1 = 0}.

It is readily seen that bA (x1 , x2 ) = x1 ,

∇bA (x1 , x2 ) = (1, 0),

∆bA (x) = 0,

356

Chapter 7. Metrics via Oriented Distance Functions

and

b∂A (x1 , x2 ) = |x1 |,

∇b∂A (x1 , x2 ) = ∆b∂A , ϕ = 2 ϕ dx.

x1 ,0 , |x1 |

∂A

Figure 7.1. ∇bA for Examples 5.1, 5.2, and 5.3. Example 5.2 (Ball of radius R > 0 in R2 ). (cf. Example 5.2 in Chapter 6). Consider the domain A = {x ∈ R2 : |x| ≤ R},

∂A = {x ∈ R2 : |x| = R}.

Clearly, x bA (x) = |x| − R, ∇bA (x) = , |x| 1 ∆bA , ϕ = ϕ dx. 2 |x| R Also b∂A (x) = |x| − R ,

∇b∂A (x) =

ϕ ds −

∆b∂A , ϕ = 2 ∂A

A

|x| > R,

x , |x| x − |x| ,

1 ϕ dx + |x|

|x| < R,

A

1 ϕ dx. |x|

Again ∆b∂A contains twice the boundary measure on ∂A. Example 5.3 (Unit square in R2 ). (cf. Example 5.3 in Chapter 6). Consider the domain A = {x = (x1 , x2 ) : |x1 | ≤ 1, |x2 | ≤ 1} . Since A is symmetrical with respect to both axes, it is suﬃcient to specify bA in the ﬁrst quadrant. We use the notation Q1 , Q2 , Q3 , and Q4 for the four quadrants in

5. Boundary of Bounded and Locally Bounded Curvature

357

the counterclockwise order and c1 , c2 , c3 , and c4 for the four corners of the square in the same order. We also divide the plane into three regions: D1 = {(x1 , x2 ) : |x2 | ≤ min{1, |x1 |}} , D2 = {(x1 , x2 ) : |x1 | ≤ min{1, |x2 |}} , D3 = {(x1 , x2 ) : |x1 | ≥ 1 and |x2 | ≥ 1} . Hence for 1 ≤ i ≤ 4 |x2 | − 1, bA (x) = |x − ci |, |x1 | − 1,

x ∈ D2 ∩ Qi , x ∈ D3 ∩ Qi , x ∈ D1 ∩ Qi ,

and for the whole plane ∆bA , ϕ =

4 i=1

D3 ∩Qi

(0, 1), x−ci , ∇bA (x) = |x−c i| (1, 0),

√ 1 ϕ dx + 2 |x − ci |

x ∈ D2 ∩ Qi , x ∈ D3 ∩ Qi , x ∈ D1 ∩ Qi ,

ϕ dx, D1 ∩D2

where D1 ∩ D2 is made up of the two diagonals of the square where ∇bA has a singularity (that is, the skeleton). Moreover, for 1 ≤ i ≤ 4 x ∈ D2 ∩ Qi , ||x2 | − 1| , b∂A (x) = |x − ci |, x ∈ D3 ∩ Qi , ||x1 | − 1| , x ∈ D1 ∩ Qi , and ∆b∂A , ϕ =

4 i=1

D3 ∩Qi

√ 1 ϕ dx − 2 |x − ci |

ϕ dx + 2 D1 ∩D2

ϕ dx. ∂A

Notice that the structures of the Laplacian are similar to the ones observed in the previous examples except for the presence of a singular term along the two diagonals of the square. All C 2 -domains with a compact boundary belong to all the categories of Deﬁnition 5.1 and of Deﬁnition 5.1 of Chapter 6. The h-dependent norms D2 bA M 1 (Uh (∂A)) , D 2 dA M 1 (Uh (A)) , and D2 dA M 1 (Uh (A)) are all decreasing as h goes to zero. The limit is particularly interesting since it singles out the behavior of the singular part of the Hessian matrix in a shrinking neighborhood of the boundary ∂A. Example 5.4. If A ⊂ RN is of class C 2 with compact boundary, then lim D2 bA M 1 (Uh (∂A)) = 0.

h0

Example 5.5. Let A = {xi }Ii=1 be I distinct points in RN . Then ∂A = A, and lim D 2 bA M 1 (Uh (∂A)) =

h0

2 I − 1, N = 1, 0, N ≥ 2.

358

Chapter 7. Metrics via Oriented Distance Functions

Example 5.6. Let A be a closed line in RN of length L > 0. Then ∂A = A, and lim D2 bA M 1 (Uh (∂A)) =

h0

2 L, N = 2, 0, N ≥ 3.

Example 5.7. Let N = 2. For the ﬁnite square and the ball of ﬁnite radius lim ∆dA M 1 (Uh (∂A)) = H1 (∂A),

h0

where H1 is the one-dimensional Hausdorﬀ measure (cf. Examples 5.2 and 5.3 in Chapter 6). Also, looking at ∆bA in Uh (A) as h goes to zero provides information about Sk (A). Example 5.8. Let A be the unit square in R2 . Then lim ∆bA M 1 (Uh (A))

h0

√ 2 = H1 (Sk (A)), 2

where H1 is the one-dimensional Hausdorﬀ measure and Sk(A) = Skint (A) is the skeleton of A made up of the two interior diagonals (cf. Example 5.3). It would seem that in general lim ∆bA M 1 (Uh (A)) = | [∇bA ] · n| dH1 , h0

Sk(A)

where [∇bA ] is the jump in ∇bA and n is the unit normal to Sk(A) (if it exists!).

6

Approximation by Dilated Sets/Tubular Neighborhoods

We have seen in Theorem 7.1 of Chapter 6 that dA can be approximated by the distance function dAr of the dilated set Ar = {x ∈ RN : dA (x) ≤ r}, r > 0, of A in the W 1,p -topology. The approximation can also be achieved with the oriented distance function: bAr (x) → bA (x) uniformly in RN . But it requires that the measure of the boundary ∂A be zero to get the approximation in the W 1,p -topology. Lemma 6.1. Given A ⊂ RN , ∂ A¯ = ∅ if and only if A = ∅ and there exists h > 0 such that Ah = RN . Proof. If ∂ A¯ = ∅, then A = ∅ and A = ∅, A = ∅ and A = ∅, and RN = A = ∩h>0 Ah

⇒ ∃h > 0 such that Ah = RN .

Conversely, A ⊃ A = ∅ and, for some h > 0, A = ∩k>0 Ak ⊂ Ah = RN . Therefore A = ∅ and A = ∅ imply that ∂A = ∅.

6. Approximation by Dilated Sets/Tubular Neighborhoods

359

Theorem 6.1. Given A ⊂ RN such that ∂A = ∅, there exists h > 0 such that for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN . (i) For all r, 0 < r ≤ h, A ⊃ Ar = ∅, ∂Ar = ∅, bUr (A) = bAr , and Ur (A) and Ar are invariant open and closed representatives in the equivalence class [Ar ]b . Moreover, dUr (A) (x) = dAr (x) = 0,

if dA (x) ≥ r,

dUr (A) (x) = dAr (x) ≥ r − dA (x),

if 0 < dA (x) < r,

(6.1)

dUr (A) (x) = dAr (x) ≥ dA (x) + r,

if dA (x) = 0,

(6.2)

∀x ∈ R , N

0 ≤ bA (x) − bAr (x) ≤ r,

(6.3)

and, as r → 0, bUr (A) (x) = bAr (x) → bA (x) uniformly in RN , χAr → χA in Lploc (RN ),

∀p, 1 ≤ p < ∞.

(ii) The following conditions are equivalent: (a) m(∂A) = 0. 1,p (RN ) for some p, 1 ≤ p < ∞. (b) As r → 0, bUr (A) = bAr → bA in Wloc 1,p (RN ) for some p, 1 ≤ p < ∞. (c) As r → 0, dUr (A) = dAr → dA in Wloc

If (a), (b), or (c) is veriﬁed, then (b) and (c) are veriﬁed for all p, 1 ≤ p < ∞. In particular, (a) is veriﬁed for all bounded sets A = ∅ in RN such that m(∂A) = 0. Proof. The existence of the h > 0 such that Ah = RN and A = ∅ is a consequence of Lemma 6.1. From this for all r, 0 < r ≤ h, ∅ = A ⊂ A ⊂ Ar ⊂ Ah = RN . (i) From the initial remarks, RN = A ⊃ A¯ ⊃ Ar ⊃ Ah = ∅ and ∂Ar = ∅ and ∂A = ∅, and the functions dA , dAr , bA , and bAr are well-deﬁned. From Theorem 3.1 of Chapter 6, Ar = Ur (A), and Ar = Ur (A), and ∂Ar = d−1 A {r}. Hence, dAr = dUr (A) . Hence, dAr = dUr (A) , dAr = dUr (A) , and bAr = bUr (A) . For x ∈ RN such that dA (x) ≥ r, x ∈ Ur (A) and dUr (A) (x) = 0. Inequality (6.1). For all y ∈ Ur (A) and all p ∈ A, r < dA (y) ≤ |y − p| ≤ |y − x| + |x − p|, r ≤ inf |y − x| + inf |x − p| = dUr (A) (x) + dA (x). y∈Ur (A)

p∈A

Inequality (6.2). Since A ∩ Ur (A) = ∅, for x ∈ A, there exists pr ∈ ∂Ur (A) = d−1 A {r} such that dUr (A) (x) = |pr − x| > 0. Therefore dA (pr ) = r. Either dA (x) = 0 or dA (x) > 0. If dA (x) = 0, then dUr (A) (x) = |pr − x| ≥ dA (pr ) = r = dA (x) + r.

360

Chapter 7. Metrics via Oriented Distance Functions

def If dA (x) > 0, then x ∈ / A and x ∈ int A¯ = A\∂A, Bx = B(x, dA (x)) ⊂ int A, and Bx ⊂ A. Deﬁne pr − x def p = x + dA (x) ∈ Bx ⇒ |p − x| = dA (x) and p ∈ Bx ⊂ A. |pr − x|

By construction dUr (A) (x) = |x − pr | = |x − p| + |p − pr | = dA (x) + |p − pr |, |p − pr | ≥ inf |p − pr | ≥ inf |p − pr | = dA (pr ) = r p ∈Bx

p ∈A

and dUr (A) (x) ≥ dA (x) + r. Inequality (6.3). From Theorem 2.1 (ii), A¯ ⊂ Ar implies that bA¯ (x) ≥ bAr (x). As for the second part of inequality (6.3), we use identity (7.2) from Theorem 7.1 in Chapter 6 for dUr (A) = dAr combined with (6.2) for x ∈ A and with (6.1) for x ∈ RN \A: r ≤ dA (x)

⇒ x ∈ Ur (A) ⊂ A

0 < dA (x) < r

⇒ bUr (A) (x) = dUr (A) (x) = dA (x) − r = bA (x) − r, ⇒ dUr (A) (x) = 0 and dA (x) = 0 ⇒ bUr (A) (x) = −dUr (A) (x) ≤ dA (x) − r = bA (x) − r,

⇒ dUr (A) (x) = dA (x) = 0

⇒ x ∈ A and dUr (A) (x) = 0 ⇒ bUr (A) (x) = −dUr (A) (x) ≤ −dA (x) − r = bA (x) − r.

dA (x) = 0

(ii) If m(∂A) = 0, from part (i), dUr (A) (x) → dA (x) implies that ∇dUr (A) ∇dA in L2 (D)N -weak for all bounded open sets. Consider the estimate |∇dUr (A) − ∇dA |2 dx = |∇dUr (A) |2 + |∇dA |2 − 2 ∇dUr (A) · ∇dA dx D D 1 − χUr (A) + 1 − χ − 2 ∇dUr (A) · ∇dA dx = A

D

from the identities of Theorem 3.3 (viii) in Chapter 6. As r → 0, {Ur (A)} is a family of increasing closed sets and Ur (A) ∪r>0 Ur (A) = ∩r>0 Ur (A) = A

⇒ m(D ∩ Ur (A)) → m(D ∩ A).

Going to the limit 1 − χUr (A) + 1 − χ − 2 ∇dUr (A) · ∇dA dx A D 2 − χA − χ − 2 |∇dA |2 dx = χ − χA dx = → χ∂A dx = 0, D

A

since |∇dA | = 1 − χ

A

D

A

D

, ∂A = A\A, and m(∂A) = 0. Therefore dUr (A) → dA

1,p in W (D) and hence in Wloc (RN ) for all p, 1 ≤ p < ∞. So, from Theorem 7.1 1,p in Chapter 6, bUr (A) = dUr (A) − dUr (A) → dA − dA = bA in Wloc (RN ) for all p, 1 ≤ p < ∞. 1,2

7. Federer’s Sets of Positive Reach

7

361

Federer’s Sets of Positive Reach

7.1

Approximation by Dilated Sets/Tubular Neighborhoods

We have seen in Theorem 6.1 that the oriented distance function bA can be approximated uniformly in RN by the oriented distance function bAr of its dilated set Ar , r > 0, and that the W 1,p -convergence can be achieved if and only if m(∂A) = 0. It turns out that the W 1,p -convergence can be obtained for sets with positive reach such that ∂A = ∅ without assuming a priori that m(∂A) = 0. Theorem 7.1. Let A ⊂ RN such that ∂A = ∅ and reach (A) > 0. (i) A = ∅ and there exists h such that 0 < h ≤ reach (A) and Ah = RN . For all r, 0 < r < h, the sets Ur (A) are of class C 1,1 , bUr (A) = bAr , and bAr (x) = bA (x) − r in RN .

(7.1)

As r goes to zero, bAr (x) → bA (x) uniformly in RN , 1,p bAr → bA in Wloc (RN ),

dAr → d A ,

dAr → dA ,

d∂Ar → d∂A in

(7.2) 1,p Wloc (RN ),

(7.3)

m(∂A) = 0, and χAr → χA = χint A in Lploc (RN ) for all p, 1 ≤ p < ∞. (ii) Furthermore, ∂A is of locally bounded curvature and of locally ﬁnite perimeter: ∇bA ∈ BVloc (RN )N

and

χ∂A ∈ BVloc (RN ).

Moreover, for all r, 0 < r < h, / 1 1. 2 bA (x) = f∂Ar (x) − f∂A (x) + r = bA (x) − b2Ar (x) + r, r r ∀x ∈ RN , ∀v ∈ RN , dH bA (x; v) exists.

(7.4)

(7.5) (7.6)

Remark 7.1. Recall from Theorem 3.1 (iii) that the Hadamard semiderivative of bA always exists at points oﬀ the boundary. The last statement in part (ii) says that for a closed set with nonempty boundary and positive reach, the Hadamard semiderivative exists in all points of its boundary and there is an explicit expression for it. In particular, this is true for all closed convex sets. Proof. (i) The uniform convergence of bAr to bA in RN was proved in Theorem 6.1 (i), and it remains to prove the W 1,p -convergence without the a priori assumption that m(∂A) = 0 that will come as a consequence rather than as a hypothesis. To do that 1,p (RN ). we ﬁrst prove the convergence dUr (A) → dA in Wloc From Theorem 6.1 (i), it suﬃcient to show that inequalities (6.1) and (6.2) become equalities for x ∈ Uh (A): for 0 < r < h dUr (A) (x) = 0,

if dA (x) ≥ r,

dUr (A) (x) ≥ r − dA (x),

if 0 < dA (x) < r,

dUr (A) (x) ≥ dA (x) + r, if dA (x) = 0.

(7.7) (7.8)

362

Chapter 7. Metrics via Oriented Distance Functions

For x ∈ RN such that dA (x) ≥ r, dA (x) = dUr (A) (x) = 0 and, from identity (7.2) of Chapter 6 in Theorem 6.1, bAr (x) = dAr (x) = dA (x) − r = bA (x) − r. For x ∈ RN such that 0 < dA (x) < r, dA (x) = 0 = dAr (x). From Theorem 6.2 (iii) of Chapter 6 for all r, 0 < dA (x) < r < h ≤ reach (A), def

x(r) = pA (x) + r

x − pA (x) dA (x)

⇒ dA (x(r)) = r and pA (x(r)) = pA (x)

/ Ur (A), dUr (A) (x) = inf p∈∂Ar |x − p| and and x(r) ∈ ∂Ar = ∂Ur (A). Since x ∈ dUr (A) (x) = dAr (x) = d∂Ar (x) ≤ |x(r) − x| = r − dA (x). Combining this with inequality (7.7), bAr (x) = bA (x) − r. Finally, for x ∈ RN such that 0 = dA (x), x ∈ A, and since ∂A = ∅, there exists p¯ ∈ ∂A such that p − x|. There exists a sequence {yn } ⊂ Ur (A)\A such that yn → p and dA (x) = |¯ cr such that p)| ≤ cr |yn − p¯| |pA (yn ) − pA (¯

⇒ pA (yn ) → pA (¯ p) = p¯.

Associate with each yn the points def

qn = pA (yn ) + r ∇dA (yn ). Since 0 < dA (qn ) ≤ r < h ≤ reach (A), by Theorem 6.2 (iii) of Chapter 6 pA (qn ) = pA (yn ),

dA (qn ) = r,

qn ∈ ∂Ar .

The sequence {qn } is bounded since p)| |qn − p¯| ≤ |qn − pA (qn )| + |pA (qn ) − p¯| = |qn − pA (qn )| + |pA (yn ) − pA (¯ ≤ r + cr |yn − p¯|. There exist q and a subsequence, still indexed by n, such that qn → q ⇒ r = dA (qn ) → dA (q) ⇒ pA (yn ) = pA (qn ) → pA (q) ⇒ dA (q) = r and pA (q) = p¯, r = dA (q) = |q − pA (q)| = |q − p¯|. Finally, since x ∈ A, A ∩ Ar = ∅, and ∂Ar = ∂Ar = d−1 A {r} dAr (x) =

inf dA (pr )=r

|x − pr | ≤ |x − q| ≤ |x − p¯| + |¯ p − q| = dA (x) + r.

Hence, from inequality (7.8), dAr (x) = dA (x) + r, and since dA (x) = 0 = dAr (x), bAr (x) = bA (x) − r. Since ∇bAr = ∇bA , for all bounded open subsets D of RN as r → 0, bAr − bA W 1,p (D) = bAr − bA Lp (D) ≤ r m(D)1/p → 0.

7. Federer’s Sets of Positive Reach

363

From Theorem 7.1 of Chapter 6 1,p (RN ) dAr → dA in Wloc

and χAr → χA in Lploc (RN )

for all p, 1 ≤ p < ∞. For all 0 < r < h ≤ reach (A), dAr = dAr − bAr → dA − bA = 1,p 1,p (RN ) and d∂Ar = dAr + dAr → dA + dA = d∂A in Wloc (RN ). Finally, dA in Wloc 1,p N since bAr → bA in Wloc (R ), 0 = χ∂Ar = 1 − |∇bAr | → 1 − |∇bA | = χ∂A in Lploc (RN ) and χ∂A = 0 and m(∂A) = 0. (ii) For 0 < r < h < reach (A), the open set Ur (A) is deﬁned via the r-level N set d−1 A {r} = {x ∈ R : dA (x) = r} of the function dA . From Theorem 6.2 (vi) in Chapter 6, for each x ∈ d−1 A {r} = ∂Ur (A) there exists a neighborhood Bρ (x), ρ = min{r, h − r}/2 > 0, of x in Uh (A)\A such that dA ∈ C 1,1 (Bρ (x)). From part (i) for all 0 < r < h and y ∈ Bρ (x) ⊂ Uh (A)\A, dA (y) = 0, bAr (y) = bA (y)−r = dA (y)−r. Therefore, bAr ∈ C 1,1 (Bρ (x)), and, a fortiori, for all x ∈ ∂Ar , there exists Bρ (x) such that ∇bAr = ∇dA ∈ BV(Bρ (x)). By Theorem 5.1, ∂Ar is of locally bounded curvature. By Deﬁnition 5.1, ∇bAr ∈ BVloc (RN )N and, since bAr = bA − r in RN , we also have ∇bA = ∇bAr ∈ BVloc (RN )N . By Theorem 5.2 (ii), χ∂A ∈ BVloc (RN ). Finally, since bAr (x) = bA (x) − r in RN from part (i) |x|2 − b2Ar (x) = |x|2 − b2A (x) − r2 + 2r bA (x), f∂Ar (x) − f∂A (x) + r2 = r bA (x) and bA is the diﬀerence of two convex continuous functions for which the Hadamard semiderivative exists in all points x ∈ RN and for all directions v ∈ RN .

7.2

Boundaries with Positive Reach

The next theorem is a specialization of Theorem 6.2 of Chapter 6 to the oriented distance function. Theorem 7.2. Let A ⊂ RN , ∂A = ∅. (i) Associate with a ∈ ∂A the sets def

P (a) = v ∈ RN : Π∂A (a + v) = {a} ,

def

Q(a) = v ∈ RN : d∂A (a + v) = |v| .

Then P (a) and Q(a) are convex and P (a) ⊂ Q(a) ⊂ (−Ta ∂A)∗ . (ii) Given a ∈ ∂A and v ∈ RN , assume that def

0 < r(a, v) = sup {t > 0 : Π∂A (a + tv) = {a}} . Then for all t, 0 ≤ t < r(a, v), Π∂A (a + tv) = {a} and d∂A (a + tv) = t |v|. Moreover, if r(a, v) < +∞, then a + r(a, v) v ∈ Sk (∂A). (iii) If there exists h such that 0 < h ≤ reach (∂A), then for all x ∈ Uh (∂A)\∂A and 0 < |t| < h, Π∂A (x(t)) = {p∂A (x)} and bA (x(t)) = t, where def

x(t) = p∂A (x) + t ∇bA (x) = p∂A (x) + t

x − p∂A (x) . bA (x)

(7.9)

364

Chapter 7. Metrics via Oriented Distance Functions

(iv) Given b ∈ ∂A, x ∈ RN \Sk (∂A), a = p∂A (x), such that reach (∂A, a) > 0, then (x − a) · (a − b) ≥ −

|a − b|2 |x − a| . 2 reach (∂A, a)

(7.10)

(v) Given 0 < r < q < ∞ and x ∈ RN \Sk (∂A) such that |bA (x)| ≤ r and reach (∂A, p∂A (x)) ≥ q, y ∈ RN \Sk (∂A) such that |bA (y)| ≤ r and reach (∂A, p∂A (y)) ≥ q, then |p∂A (y) − p∂A (x)| ≤

q |y − x|. q−r

(7.11)

(vi) If 0 < reach (∂A) < +∞, then for all h, 0 < h < reach (∂A), and all x, y ∈ (∂A)h = {x ∈ RN : |bA (x)| ≤ h} reach (∂A) |p∂A (y) − p∂A (x)| ≤ |y − x|, reach (∂A) − h 2 ∇bA (y) − ∇b2A (x) ≤ 2 1 + reach (∂A) |y − x|; (7.12) reach (∂A) − h

for all s, 0 < s < h < reach (∂A), x, y ∈ ∂As,h = x ∈ RN : s ≤ |bA (x)| ≤ h , reach (∂A) 1 2+ |y − x| . |∇bA (y) − ∇bA (x)| ≤ s reach (∂A) − h If reach (∂A) = +∞, ∀x, y ∈ RN , |p∂A (y) − p∂A (x)| ≤ |y − x|, 2 ∇bA (y) − ∇b2A (x) ≤ 4 |y − x|;

(7.13)

for all 0 < s and all x, y ∈ {x ∈ RN : s ≤ d∂A (x)} |∇bA (y) − ∇bA (x)| ≤

3 |y − x| . s

(vii) If reach (∂A) > 0, then, for all 0 < r < reach (∂A), Ur (∂A) is a set of class C 1,1 , Ur (∂A) = (∂A)r , ∂Ur (∂A) = {x ∈ RN : |bA (x)| = r}, and (∂A)r = Ur (∂A) = {x ∈ RN : |bA (x)| ≥ r}. (viii) For all a ∈ ∂A,

Ta ∂A =

|bA (a + th)| =0 . h ∈ R : lim inf t0 t N

In particular, if a ∈ int ∂A, Ta ∂A = RN .

8. Boundary Smoothness and Smoothness of bA

365

The next theorem is a specialization of Theorem 6.3 of Chapter 6 to the oriented distance function. Theorem 7.3. Given A ⊂ RN such that ∂A = ∅, the following conditions are equivalent: (i) ∂A has positive reach, that is, reach (∂A) > 0. (ii) There exists h > 0 such that b2A ∈ C 1,1 (Uh (∂A)). (iii) There exists h > 0 such that for all s, 0 < s < h, bA ∈ C 1,1 (Us,h (∂A)). (iv) There exists h > 0 such that bA ∈ C 1 (Uh (∂A)\∂A). (v) There exists h > 0 such that for all x ∈ Uh (∂A), Π∂A (x) is a singleton. Recall from Theorem 3.1 (viii) and (3.15) the following properties. If ∂A = ∅, Sk (∂A) = Sk (A) ∪ Sk (A) ⊂ RN \∂A, Ck b (A) ⊂ Ck (∂A) = Ck (A) ∪ Ck (A) ⊂ ∂(∂A). If ∂A has positive reach, then both A and A have positive reach.

8

Boundary Smoothness and Smoothness of bA

In this section the smoothness of the boundary ∂A is related to the smoothness of the function bA in a neighborhood of ∂A. The next two theorems give a complete characterization of sets A such that m(∂A) = 0 and bA ∈ C k, in a neighborhood of ∂A. We ﬁrst present two equivalence theorems that will follow from the more detailed Theorems 8.3 and 8.4. It is useful to recall that for a set that is locally of class C k, in a point of ∂A, ∂A is locally a C k, -submanifold of dimension N − 1 in that point, but the converse is not true (cf. Deﬁnitions 3.1 and 3.2 in Chapter 2)4 . Theorem 8.1 (Local version). Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤ ≤ 1 be a real number, and let x ∈ ∂A. 4 It is important to note that Theorem 8.1 is not true when b is replaced by d A ∂A since its gradient ∇d∂A is discontinuous across ∂A. Part (iii) of Theorem 8.1 in the direction (⇒) was asserted by J. Serrin [1] in 1969 for N = 3 and proved in 1977 by D. Gilbarg and N. S. Trudinger [1] for k ≥ 2 (provided that d∂A is replaced by bA in Lemma 1, p. 382 in [1], and Lemma 14.16, p. 355 in [2]) with a diﬀerent proof by S. G. Krantz and H. R. Parks [1] in 1981. Another proof with the ´sio [17], who extended the result function bA was given in 1994 by M. C. Delfour and J.-P. Zole from (iii) to (ii) in the direction (⇒) down to the C 1,1 case and established the equivalence (⇔) in the whole range from C ∞ to C 1,1 . The counterexample of Example 6.2 for domains of class C 1,1− is the same as the one provided earlier in S. G. Krantz and H. R. Parks [1], where they observe only that the domain is C 2−ε , leaving the reader under the misleading impression that part (ii) of Theorem 8.1 would not be true for domains ranging from class C 1,1 to C 2 . Part (i) of Theorem 8.1 in the direction (⇒) was proved in the C 1 case by S. G. Krantz and H. R. Parks [1] in 1981. The equivalence (⇔) here for C 1,λ , 0 ≤ λ < 1, was given in 2004 by M. C. Delfour and ´sio [40]. J.-P. Zole

366

Chapter 7. Metrics via Oriented Distance Functions

(i) (k = 1, 0 ≤ < 1). A is a set of class C 1, in a bounded open neighborhood V (x) of x and Sk (∂A) ∩ V (x) = ∅ if and only if ∃ρ > 0 such that bA ∈ C 1, (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.1)

(ii) (k = 1, = 1). A is locally a set of class C 1,1 at x if and only if ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.2)

(iii) (k ≥ 2, 0 ≤ ≤ 1). A is locally a set of class C k, at x if and only if ∃ρ > 0 such that bA ∈ C k, (Bρ (x))

and

m(∂A ∩ Bρ (x)) = 0.

(8.3)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A, and ∂A is locally a C k, -submanifold of dimension N − 1 at x. Remark 8.1. The condition Sk (∂A) ∩ V (x) = ∅ in part (i) is really necessary. Example 6.2 in Chapter 6 gives a family of two-dimensional sets of class C 1,λ , 0 ≤ λ < 1, for which bA ∈ C 1,λ and d2∂A = b2A ∈ / C 1 in any neighborhood of the point (0, 0) of its boundary ∂A. From Theorem 8.1 we get the global version that has had and still has an interesting history of inaccurate statements or incomplete proofs Theorem 8.2 (Global version). Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, and let 0 ≤ ≤ 1 be a real number. (i) (k = 1, 0 ≤ < 1). A is a set of class C 1, and ∂A ∩ Sk (∂A) = ∅ if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C 1, (Bρ (x)).

(8.4)

(ii) (k = 1, = 1). A is a set of class C 1,1 if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x)).

(8.5)

(iii) (k ≥ 2, 0 ≤ ≤ 1). A is a set of class C k, if and only if m(∂A) = 0

and

∀x ∈ ∂A, ∃ρ > 0 such that bA ∈ C k, (Bρ (x)).

(8.6)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A, and ∂A is a C k, -submanifold of dimension N − 1. Remark 8.2. Theorems 8.1 and 8.2 deal only with sets of class C 1, whose boundary is a C 1, submanifold of dimension (N − 1). For arbitrary closed submanifolds M of RN of codimension greater than or equal to 1, ∇bM generally does not exist on M . In that case the smoothness of M is related to the existence and the smoothness of

8. Boundary Smoothness and Smoothness of bA

367

∇d2M in a neighborhood of M that implies that M is locally of positive reach. The reader is referred to Theorem 6.5 in section 6.2 of Chapter 6. The above two equivalence theorems follow from the following two local theorems in each direction. Theorem 8.3. Let A ⊂ RN be such that ∂A = ∅. Given x ∈ ∂A, assume that ∃ a bounded open neighborhood V (x) of x such that bA ∈ C k, (V (x))

(8.7)

for some integer k ≥ 1 and some real , 0 ≤ ≤ 1, and that m(∂A ∩ V (x)) = 0. (i) Then A is locally a set of class C k, in V (x) and bA = bint A = bA in V (x). (ii) Π∂A (y) = {p∂A (y)} is a singleton for all y ∈ V (x) and 1,1 k, (V (x)) ∩ Cloc (V (x)) b2A ∈ Cloc

and

0,1 k−1, p∂A ∈ Cloc (V (x))N ∩ Cloc (V (x))N .

Remark 8.3. If condition (8.7) is veriﬁed, the condition m(∂A) = 0 is equivalent to ∂A = RN . The condition bA ∈ Cb0 (RN ) is necessary to rule out the case ∅ = ∂A = RN that yields bA = 0 and A = A = RN that trivially satisﬁes condition (8.7). Proof. (i) Since ∇bA exists and is continuous everywhere in V (x), |∇bA | is continuous in V (x). But |∇bA (x)| = 1 in V (x)\∂A and |∇bA (x)| = 0 almost everywhere in V (x) ∩ ∂A. By continuity, either |∇bA | = 1 or |∇bA | = 0 in V (x). If ∇bA = 0 in V (x), then bA is constant in V (x) and equal to bA (x) = 0. Therefore, V (x) ⊂ ∂A and this yields the contradiction 0 < m(V (x)) ≤ m(∂A ∩ V (x)) = 0. Consider def the set Ω = {x ∈ RN : −bA (x) > 0}. By Theorem 4.2 in Chapter 2, Ω is a set of class C k, since bA ∈ C k, (V (x)) and ∇bA = 0 on b−1 A {0} ∩ V (x). In particular, ∂Ω = b−1 {0} = ∂A, int A = Ω, and int A = int Ω in V (x). Finally A bA = bΩ = bint A = bA . k, k, (ii) By assumption, bA ∈ Cloc (V (x)) and hence b2A belongs to Cloc (V (x)). By Theorem 3.1 (iv), Π∂A (y) = {p∂A (y)} is a singleton in V (x). By Theorem 7.3, 1,1 k, 0,1 (V (x)) and, a fortiori, in Cloc (V (x)). This yields p∂A ∈ Cloc (V (x)) ∩ b2A ∈ Cloc k−1, Cloc (V (x)). In the other direction the smoothness of the domain implies the smoothness of bA in a neighborhood of ∂A when the domain is at least of class C 1,1 . A counterexample will be given in dimension 2. Theorem 8.4. Let A ⊂ RN , ∂A = ∅, let k ≥ 1 be an integer, let 0 ≤ ≤ 1 be a real number, and let x ∈ ∂A if it is not. (i) (k = 1, 0 ≤ < 1). If A is locally a set of class C 1, in a bounded open neighborhood V (x) of x and Sk (∂A) ∩ V (x) = ∅, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C 1, (Bρ (x))

and

p∂A ∈ C 0,1 (Bρ (x)).

(8.8)

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Chapter 7. Metrics via Oriented Distance Functions

(ii) (k = 1, = 1). If A is locally a set of class C 1,1 in a bounded open neighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C 1,1 (Bρ (x))

and

p∂A ∈ C 0,1 (Bρ (x)).

(8.9)

(iii) (k ≥ 2, 0 ≤ ≤ 1). If A is locally a set of class C k, in a bounded open neighborhood V (x) of x, then m(∂A ∩ V (x)) = 0 and ∃ρ > 0 such that bA ∈ C k, (Bρ (x))

and

p∂A ∈ C k−1, (Bρ (x)).

(8.10)

Moreover, in all cases, ∇bA = n ◦ p∂A in Bρ (x), where n is the unit exterior normal to A on ∂A. Proof. (i) From Deﬁnition 3.2 of Chapter 2 of a locally a C 1, -submanifold of dimension N − 1, m(∂A ∩ V (x)) = 0 in a bounded open neighborhood V (x) and the set A can be locally described by the level sets of the C 1, -function def

f (y) = gx (y) · eN ,

(8.11)

since by deﬁnition int A ∩ V (x) = {y ∈ V (x) : f (y) > 0} , ∂A ∩ V (x) = {y ∈ V (x) : f (y) = 0} . The boundary ∂A in V (x) is the zero level set of f and the gradient ∇f (y) = Dgx (y)∗ eN = 0 in V (x) is normal to that level set. Thus the outward unit normal n to A on V (x) ∩ ∂A is given by n(y) = −

∇f (y) Dgx (y)∗ eN Dhx (gx (y))−∗ eN =− = − . |∇f (y)| |Dgx (y)∗ eN | |Dhx (gx (y))−∗ eN |

(8.12)

Since V (x) ∩ Sk (∂A) = ∅, for all y ∈ V (x), ∃ unique p∂A (y) ∈ ∂A,

d∂A (y) = inf |y − p| = |y − p∂A (y)|, p∈∂A

1 1 p∂A (y) = y − ∇d2∂A (y) = y − ∇b2A (y). 2 2 Choose ρ > 0 such that B3ρ (x) ⊂ V (x). Then for all y ∈ Bρ (x) |p∂A (y) − x| ≤ |p∂A (y) − y| + |y − x| ≤ 2 |y − x| ≤ 2r ⇒ p∂A (y) ∈ B2ρ (x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f −1 {0}.

(8.13)

8. Boundary Smoothness and Smoothness of bA

369

Therefore, for each y ∈ Bρ (x), there exists a unique p∂A (y) ∈ V (x) ∩ f −1 {0} such that d2∂A (y) =

inf p∈V (x), f (p)=0

|p − y|2 = |y − p∂A (y)|2 .

By using the Lagrange multiplier theorem for the Lagrangian |z − y|2 + λ f (z) ˆ ∈ R such that there exists a λ ˆ ∇f (p∂A (y)) = 0 and f (p∂A (y)) = 0 2(p∂A (y) − y) + λ ˆ |∇f (p∂A (y))| ⇒ 2d∂A (y) = |λ| ⇒ p∂A (y) = y + bA (y)

∇f (p∂A (y)) = y − bA (y) n(p∂A (y)) |∇f (p∂A (y))|

since ∇f (p∂A (y)) = 0 on ∂A ∩ V (x). Combining this with (8.13) 1 y − ∇b2A (y) = p∂A (y) = y − bA (y) n(p∂A (y)) 2 ⇒ ∀y ∈ Br (x)\∂A, ∇bA (y) = n(p∂A (y))

∀y ∈ Br (x),

(8.14)

since V (x)∩Sk (∂A) = ∅ implies that ∇bA (y) exists in V (x)\∂A by Theorem 3.1 (vii). Consider the normal n(z) = −∇f (z)/|∇f (z)| in V (x)\∂A. For z2 and z1 in ∂A ∩ B2ρ (x) the diﬀerence ∇f (z2 ) ∇f (z1 ) − |n(z2 ) − n(z1 )| = |∇f (z2 )| |∇f (z1 )| 1 1 1 = (∇f (z2 ) − ∇f (z1 )) + − ∇f (z1 ) |∇f (z2 )| |∇f (z2 )| |∇f (z1 )| 1 ∇f (z2 ) − ∇f (z1 ) + (|∇f (z1 )| − |∇f (z2 )|) ∇f (z1 ) = |∇f (z2 )| |∇f (z1 )| 1 1 |∇f (z2 ) − ∇f (z1 )| ≤ 2 |∇f (z2 ) − ∇f (z1 )| . ≤2 |∇f (z2 )| inf z∈∂A∩B2ρ (x) |∇f (z)| But f ∈ C 1, (B2ρ (x)) and since |∇f | = 0 is continuous on V (x), |∇f (z2 ) − ∇f (z1 )| ≤ cx |z2 − z1 |

def

and α =

inf

|∇f (z)| > 0

z∈∂A∩B2ρ (x)

and the outward unit normal is -H¨olderian, ∀z2 , z1 ∈ B2ρ (x) ∩ ∂A,

|n(z2 ) − n(z1 )| ≤

2 cx |z2 − z1 | , α

where cx is the Lipschitz constant of f in B2ρ (x). By construction, B3ρ (x) ∩ Sk (∂A) ⊂ V (x) ∩ Sk (∂A) = ∅ and for all z ∈ Bρ (x), B2ρ (z) ∩ Sk (∂A) = ∅

370

Chapter 7. Metrics via Oriented Distance Functions

implies that reach (∂A, z) ≥ 2r > 0. By Theorem 6.2 (v) in Chapter 6 applied to ∂A with 0 < r = ρ < q = 2ρ, ∀z1 , z2 ∈ Bρ (x),

|p∂A (z1 ) − p∂A (z2 )| ≤ 2 |z1 − z2 |

(8.15)

and hence p∂A ∈ C 0,1 (Bρ (x))N . So the composition n◦p∂A belongs to C 0, (Bρ (x))N since we have already shown that n ∈ C 0, (B2ρ (x) ∩ ∂A)N . Finally, ∇bA ∈ L∞ (Bρ (x))N , n(p∂A ) ∈ C 0, (Bρ (x))N , m(∂A) = 0, and from (8.14) ∇bA (y) = n(p∂A (y))

a.e. in Bρ (x)

imply that ∇bA ∈ C 0, (Bρ (x))N , bA ∈ C 1, (Bρ (x)), and ∇bA = n ◦ p∂A in Bρ (x). (ii) The ﬁrst steps are the same as in part (i) up to (8.12) with C 1,1 in place 1, of C . For each y ∈ V (x), Π∂A (y) is compact and nonempty and ∀p ∈ Π∂A (y),

d2∂A (y) = inf |a − y|2 = |p − y|2 . a∈∂A

Choose r > 0 such that B3r (x) ⊂ V (x). Then for all y ∈ Br (x) and p ∈ Π∂A (y) |p − x| ≤ |p − y| + |y − x| ≤ 2 |y − x| ≤ 2r ⇒ p ∈ B2r (x) ∩ ∂A ⊂ V (x) ∩ ∂A = V (x) ∩ f −1 {0}. Therefore, for each y ∈ Br (x) and p ∈ Π∂A (y), p ∈ V (x) ∩ f −1 {0} such that d2∂A (y) =

inf p∈V (x), f (p)=0

|p − y|2 = |y − p∂A (y)|2 .

By using the Lagrange multiplier theorem for the Lagrangian |a − y|2 + λ f (a) ˆ ∈ R and p ∈ Π∂A (y) such that there exist λ ˆ ∇f (p) = 0 and f (p) = 0 2(p − y) + λ ⇒ p = y + bA (y)

ˆ |∇f (p)| ⇒ 2d∂A (y) = |λ|

∇f (p) = y − bA (y) n(p) |∇f (p)|

⇒ ∀y ∈ Br (x), ∀p ∈ Π∂A (y),

p = y − bA (y) n(p),

(8.16)

since ∇f (p) = 0 on ∂A ∩ V (x). By the same proof as in part (i) with = 1 2 ∀z2 , z1 ∈ B2r (x) ∩ ∂A, |n(z2 ) − n(z1 )| ≤ cx |z2 − z1 |, α where cx is the Lipschitz constant associated with f in B2r (x). Therefore, n ∈ C 0,1 (B2r (x) ∩ ∂A). From (8.16) for y ∈ Br (x) and for all p1 , p2 ∈ Π∂A (y), |p2 − p1 | ≤ |bA (y)| |n(p2 ) − n(p1 )| ≤ |bA (y)|

2 cx |p2 − p1 |. α

8. Boundary Smoothness and Smoothness of bA

371

By choosing h such that 0 < h < min{r, α/(2 cx )}, h 2 cx /α < 1, for y ∈ Bh (x) ⊂ Br (x) ∩ {z ∈ RN : |bA (z)| ≤ h} and all p1 , p2 ∈ Π∂A (y), p2 = p1 , Π∂A (y) is a singleton, and p∂A (y) = y − bA (y) n(p∂A (y)). By construction, Bh (x)∩Sk (∂A) = ∅ and for all z ∈ Bh/3 (x), B2h/3 (z)∩Sk (∂A) = ∅ implies that reach (∂A, z) ≥ 2h/3 > 0. By Theorem 6.2 (v) in Chapter 6 applied to ∂A with 0 < r = h/3 < q = 2h/3 ∀z1 , z2 ∈ Bh/3 (x),

|p∂A (z1 ) − p∂A (z2 )| ≤ 2 |z1 − z2 |

(8.17)

and hence p∂A ∈ C 0,1 (Bh/3 (x))N . Thus n ◦ p∂A belongs to C 0,1 (Bh/3 (x))N since we have already shown that n ∈ C 0,1 (B2r (x) ∩ ∂A)N . The remainder and the conclusion of the proof is the same as in part (i) with ρ = h/3. (iii) When A is of class C k, in V (x) for an integer k, k ≥ 2, and a real number , 0 ≤ ≤ 1, it is C 1,1 and the previous constructions and conclusions of (ii) are veriﬁed. There exists ρ > 0 such that for each y ∈ Bρ (x) there exists a unique p∂A (y) ∈ B2ρ (x) such that p∂A (y) = y − bA (y)

∇f (p∂A (y)) . |∇f (p∂A (y))|

Since ∇f is continuous and ∇f = 0 on ∂A ∩ V (x), there exists r, 0 < 2r < ρ, such that inf

|∇f (z)| > 0

z∈B2r (x)

and deﬁning N (z) = ∇f (z)/|∇f (z)|, ∇f (z) 1 D2 f (z) − ∇N (z) = |∇f (z)| |∇f (z)|

∗

∇f (z) D f (z) |∇f (z)|

2

belongs to C k−1, (B2r (x)), where ∗v denotes the transpose of a vector v ∈ RN , so that (u ∗v)ij = ui vj is the tensor product. Consider the map def

(y, z) → G(y, z) = z − y − bA (y)

∇f (z) : Br (x) × B2r (x) → RN . |∇f (z)|

From part (ii), there exists a unique p(y) = p∂A (y) ∈ B2r (x) ∩ ∂A such that p∂A ∈ C 0,1 (B2r (x))N , bA ∈ C 1,1 (Br (x)), and G(y, p(y)) = 0 where DGz (y, z) = I −

⇒ DGy (y, p(y)) + DGz (y, p(y)) Dp(y) = 0,

bA (y) ∇f (z) D2 f (z) − |∇f (z)| |∇f (z)|

∗

D2 f (z)

DGy (y, z)) = −I − N (z) ∗∇bA (y).

∇f (z) |∇f (z)|

,

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Chapter 7. Metrics via Oriented Distance Functions

DGz (y, z) is at least C 1,1 in (y, z) since bA ∈ C 1,1 and N is at least C 1,1 for k ≥ 2. Since f is at least C 2 in V (x), f and its ﬁrst and second derivatives are bounded in some smaller neighborhood Br (x) of x. By reducing the upper bound on |bA (y)|, there exists a new smaller neighborhood Br (x) of x such that Dz G(y, z) is invertible for all (y, z) ∈ Br (x) × B2r (x). So the conditions of the implicit function theorem are met. There exist a neighborhood W (x) ⊂ Br (x) of x and a unique C 1 -mapping p : W (x) → RN such that ∀y ∈ W (x),

G(y, p(y)) = 0

that coincides with p∂A (cf., for instance, L. Schwartz [3, Thm. 26, p. 283]). Therefore Dp∂A (y) = (DGz (y, p∂A (y)))−1 [I + N (p∂A (y)) ∗∇bA (y)] and Dp∂A is C 0 and p∂A is C 1 . Thus ∇bA = N ◦ p∂A ∈ C 1 and bA is now C 2 . Since N ∈ C k−1, and D2 f ∈ C k−2, , we can repeat this argument until we get Dp∂A in C k−2, , and ﬁnally p∂A ∈ C k−1, , ∇bA = N ◦ p∂A ∈ C k−1, , and bA ∈ C k, . From Theorem 8.1 (ii) additional information is available on the ﬂow of ∇bΩ . def

Theorem 8.5. Let Ω ⊂ RN be of class C 1,1 . Denote its boundary by Γ = ∂Ω. (i) For each x ∈ Γ there exists a bounded open neighborhood W (x) of x such that bΩ ∈ C 1,1 (W (x)),

∇bΩ = n ◦ pΓ = ∇bΩ ◦ pΓ ,

|∇bΩ | = 1 in W (x), (8.18)

where n is the unit exterior normal to Ω on Γ. (ii) Associate with x ∈ Γ and y in W (x) the ﬂow Tt (y) of ∇bΩ in W (x), that is, dx (t) = ∇bΩ (x(t)), dt

x(0) = y,

def

Tt (y) = x(t).

(8.19)

Then for t in a neighborhood of 0 we have the following properties: pΓ ◦ Tt = pΓ ,

∇bΩ ◦ Tt = ∇bΩ ,

Tt (y) = y + t ∇bΩ (y).

(8.20)

(iii) Almost everywhere in W (x) d DTt = D 2 bΩ dt

−1 D2 bΩ ◦ Tt = D2 bΩ I + tD2 bΩ ,

DTt = I + t D 2 bΩ , ⇒ D2 bΩ ◦ Tt DTt = D 2 bΩ ,

(DTt )−1 ∇bΩ = ∇bΩ = ∗(DTt )−1 ∇bΩ .

(8.21) (8.22) (8.23)

Proof. For simplicity, we use the notation b = bΩ and p = pΓ . (i) The result follows from Theorem 8.4 (i). (ii) By assumption from part (i) the function b belongs to C 1,1 (W (x)) and hence p ∈ C 0,1 (W (x); RN ). Thus almost everywhere in W (x), Dp = I − ∇b ∗∇b − bD2 b

⇒ Dp∇b = 0.

9. Sobolev or W m,p Domains

373

Now dTt d (p ◦ Tt ) = Dp ◦ Tt = Dp ◦ Tt ∇b ◦ Tt = (Dp ∇b) ◦ Tt = 0 dt dt ⇒ p ◦ Tt = p in W (x). But since p and Tt belong to C 0,1 (W (x); RN ), the identity necessarily holds everywhere in W (x). Moreover, ∇b ◦ Tt = n ◦ p ◦ Tt = n ◦ p = ∇b in W (x). Finally dTt = ∇b ◦ Tt = ∇b dt

⇒ Tt (y) = y + t∇b(y) in W (x).

In particular, almost everywhere in W (x), d DTt = D2 b, dt ⇒ (DTt )−1 ∇b = ∇b = ∗(DTt )−1 ∇b.

DTt = I + t D2 b DTt ∇b = ∇b

9

⇒

Sobolev or W m,p Domains

The N -dimensional Lebesgue measure of the boundary of a Lipschitzian subset of RN is zero. However, this is generally not true for sets of locally bounded curvature, as can be seen from the following example. Example 9.1. Let B be the open unit ball centered in 0 of R2 and deﬁne A = {x ∈ B : x with rational coordinates}. Then ∂A = B, bA = dB , and for all h > 0 ∇bA ∈ BV(Uh (∂A))2 , 1 ∆bA , ϕ = ϕ dx. ϕ dx + |x| ∂B B The next natural question that comes to mind is whether the boundary or the skeleton has locally ﬁnite (N − 1)-Hausdorﬀ measure. Then come questions about the mean curvature of the boundary. For the function bA , ∆bA = tr D2 bA is proportional to the mean curvature of the boundary, which is only a measure in RN for sets of bounded local curvature. This calls for the introduction of some classiﬁcation of sets that would complement the classical H¨olderian terminology introduced in Chapter 2, would ﬁll the gap in between, and would possibly say something about sets whose boundary is not even continuous. The following deﬁnition seems to have ´sio [32]. been ﬁrst introduced in M. C. Delfour and J.-P. Zole

374

Chapter 7. Metrics via Oriented Distance Functions

Deﬁnition 9.1 (Sobolev domains). Given m > 1 and p ≥ 1, a subset A of RN is said to be an (m, p)-Sobolev domain or simply a W m,p -domain if ∂A = ∅ and there exists h > 0 such that m,p bA ∈ Wloc (Uh (∂A)).

This classiﬁcation gives an analytical description of the smoothness of boundaries in terms of the derivatives of bA . The deﬁnition is obviously vacuous for m = 1 1,∞ since bA ∈ Wloc (RN ) for any set A such that ∂A = ∅. For 1 < m < 2 we shall see that it encompasses sets that are less smooth than sets of locally bounded curvature. For m = 2 the W 2,p -domains are of class C 1,1−N/p for p > N . For m > 2 they are intertwined with sets of class C k, . Theorem 9.1. Given any subset A of RN , ∂A = ∅, ∇bA ∈ BVloc (RN )N 1 1+η,p ⇒ ∀p, 1 ≤ p < ∞, ∀η, 0 ≤ η < , bA ∈ Wloc (RN ). p N N Proof. Since |∇bA (x)| ≤ 1 almost everywhere, ∇bA ∈ BVloc (RN )N ∩ L∞ loc (R ) and the theorem follows directly from Theorem 6.9 in Chapter 5.

It is quite interesting that a domain of locally bounded curvature is a W 2−ε,1 domain for any arbitrary small ε > 0. So it is almost a W 2,1 -domain, and domains of class W 2−ε,1 seem to be a larger class than domains of locally bounded curvature. In addition, their boundary does not generally have zero measure. Now consider the “threshold case” m = 2. Theorem 9.2. Given an integer N ≥ 1, let A be a subset of RN such that ∂A = ∅. (i) If m(∂A) = 0 and there exist p > N and h > 0 such that 2,p bA ∈ Wloc (Uh (∂A)),

(9.1)

then reach (∂A) ≥ h > 0,

1,1−N/p

bA ∈ Cloc

(Uh (∂A)),

and A is a H¨ olderian set of class C 1,1−N/p . 2,p (Uh (∂A)) is equivalent to ∆bA ∈ (ii) In dimension N = 2 the condition bA ∈ Wloc p Lloc (Uh (∂A)).

Proof. (i) Consider the function |∇bA |2 . Since |∇bA | ≤ 1, then 2,p 1,∞ (Uh (∂A)) ∩ Wloc (Uh (∂A)) bA ∈ Wloc

2,p 1,∞ ⇒ b2A ∈ Wloc (Uh (∂A)) ∩ Wloc (Uh (∂A)).

In particular, for all x ∈ ∂A, bA , b2A ∈ W 2,p (B(x, h)) ∩ W 1,∞ (B(x, h)).

10. Characterization of Convex and Semiconvex Sets

375

For p > N , from R. A. Adams [1, Thm. 5.4, Part III, and Rem. 5.5 (3), p. 98], bA , b2A ∈ C 1,λ (B(x, h)), ⇒

bA , b2A

∈

1,λ Cloc (Uh (A)),

0 < λ ≤ 1 − N/p 0 < λ ≤ 1 − N/p.

From Theorem 8.2 (i), if m(∂A) = 0, the set A is a set of class C 1,1−N/p , ∂A ∩ Sk (∂A) = ∅, and reach (∂A) ≥ h. (ii) From part (i), |∇bA (x)|2 = 1 in Uh (∂A), and D2 bA (x) ∇bA (x) = 0 almost everywhere. Hence in dimension N = 2, 2 2 bA (x) = ∂21 bA (x) = −∂1 bA (x) ∂2 bA (x) ∆bA (x), ∂12 2 ∂11 bA (x) = (∂2 bA (x))2 ∆bA (x), 2 ∂22 bA (x) = (∂1 bA (x))2 ∆bA (x).

As can be seen, a certain amount of work would be necessary to characterize all the Sobolev domains and answer the many associated open questions.

10 10.1

Characterization of Convex and Semiconvex Sets Convex Sets and Convexity of bA

We have seen in Chapter 6 that the convexity of dA is equivalent to the convexity of A. This characterization remains true with bA in place of dA . However, if in both cases the convexity of dA or bA is suﬃcient to get the convexity of A, it is not the case for A as can be seen from the following example. Example 10.1. Let B be the open unit ball in RN and A be the set B minus all the points in B with rational coordinates. By deﬁnition, A¯ = B, A = RN , dA = dB = dA ,

∂A = B,

∂A = ∂B = ∂A,

dA = dA = dRN = 0

⇒ bA = dB = bB = bA . By Theorem 8.1 of Chapter 6 the functions dB and, a fortiori, bA are convex, but A is not convex and ∂A = ∂A. Theorem 10.1. (i) Let A be a convex subset of RN . Then bA = bA

and

[A]b = [A]b .

(10.1)

Hence, the equivalence class [A]b can be identiﬁed with the unique closed set A in the class.

376

Chapter 7. Metrics via Oriented Distance Functions

(ii) Let A be a convex subset of RN such that int A = ∅. Then bint A = bA = bA

and

[int A]b = [A]b = [A]b .

(10.2)

Hence, the equivalence class [A]b can be identiﬁed with either the unique closed set A or the unique open set int A = A = ∅ in the class. If, in addition, A = RN , then ∂A = ∅ and int A = ∅. (iii) Let A be a subset of RN such that ∂A = ∅.5 Then bA is convex

⇐⇒

A is convex.

(10.3)

Proof. (i) If A = ∅, then A = ∅, A = RN , and A = RN . Hence bA = bA . If A = RN , we use the identity bA = −bA . If ∂A = ∅, then dA = dA and, by Theorem 3.5 (ii) in Chapter 5, A = A, and bA = bA . (ii) From Theorem 3.5 (ii) in Chapter 5, int A = A and int A = A = A. Hence bA = bint A . From part (i), bA = bA . If, in addition, A = RN , always from Theorem 3.5 (ii) in Chapter 5, ∂A = ∅ and int A = ∅. (iii) (⇒) Denote by xλ the convex combination λx + (1 − λ)y of two points x and y in A for some λ ∈ [0, 1]. By convexity of bA , bA (xλ ) ≤ λbA (x) + (1 − λ)bA (y) = − [λdA (x) + (1 − λ)dA (y)] ≤ 0 ⇒ dA (xλ ) = b+ A (xλ ) = max{bA (xλ ), 0} = 0

⇒ xλ ∈ A

and A is convex. (⇐) We ﬁrst prove that for A closed the function bA is convex. From this we conclude that if A is convex, then bA is convex and from part (i) bA = bA is convex. So for A closed, consider three cases. The ﬁrst one deserves a lemma. Lemma 10.1. Let A be a subset of RN such that ∂A = ∅ and let A be convex. Then bA = −dA is convex in A. Proof. Again we can assume that A is closed. Associate with x and y in A the radii rx = dA (x), ry = dA (y), and rλ = λrx + (1 − λ)ry and the closed balls B x of center x and radius rx , B y of center y and radius ry , and B λ of center xλ and radius rλ . By the deﬁnition of dA , B x ⊂ A and B y ⊂ A since A is closed. Associate with 5 In 1985 D. H. Armitage and U. ¨ Kura [1] showed that a proper closed domain Ω is convex if and only if −bA is superharmonic and that, for N = 2, the result still holds if −bA is superharmonic only on A. They also showed that, for A compact, d∂A is subharmonic on A if and only if A is convex. This work was pursued in 1987 by M. J. Parker [1] and in 1988 by M. J. Parker [2]. ´sio [17] that the property that A is convex In 1994 it was shown in M. C. Delfour and J.-P. Zole if and only if dA is convex remains true with bA in place of dA .

10. Characterization of Convex and Semiconvex Sets

377

each z ∈ B λ the points zx = x and zy = y if rλ = 0, and if rλ > 0 the points rx (z − xλ ) ⇒ zx ∈ B x , rλ ry def zy = y + (z − xλ ) ⇒ zy ∈ B y rλ λrx + (1 − λ)ry ⇒ λzx + (1 − λ)zy = xλ + (z − xλ ) = z. rλ def

zx = x +

Obviously z = xλ if rλ = 0. Therefore, B λ ⊂ λB x + (1 − λ)B y ⊂ A, since A is closed and convex. In particular dA (xλ ) ≥ dB λ (xλ ) ≥ rλ = λdA (x) + (1 − λ)dA (y), and this proves the lemma. For the second case consider x and y in A. By deﬁnition bA (x) = dA (x)

and bA (y) = dA (y).

By Theorem 8.1 of Chapter 6, dA is convex when A is convex and bA (xλ ) ≤ dA (xλ ) ≤ λdA (x) + (1 − λ)dA (y) = λbA (x) + (1 − λ)bA (y). The third and last case is the mixed one for x ∈ A and y ∈ A. Deﬁne def

xλ = λx + (1 − λ)y,

def

pλ = p∂A (xλ ).

Since A is convex, denote by H the tangent hyperplane to A through pλ , by H + the closed half-space containing A, and by H − the other closed subspace associated with H. By the deﬁnition of H, d∂A (xλ ) = dH (xλ ),

H ⊂ H ±,

and by convexity of A, A ⊂ H + and H − ⊂ A. The projection onto H is a linear operator and pλ = pH (xλ ) = λpH (x) + (1 − λ)pH (y), pλ − xλ = λ(pH (x) − x) + (1 − λ)(pH (y) − y). If y ∈ H + and x ∈ H + , dH (xλ ) = λdH (x) + (1 − λ)dH (y),

(10.4)

378

Chapter 7. Metrics via Oriented Distance Functions

and if y ∈ H + and x ∈ H − ,

dH (xλ ) =

λdH (x) − (1 − λ)dH (y), (1 − λ)dH (y) − λdH (x),

if xλ ∈ H − , if xλ ∈ H + .

(10.5)

By convexity any y ∈ A belongs to H + and since H − ⊂ A, we readily have dH (y) = dH − (y) ≥ dA (y) = −bA (y). First consider the case xλ ∈ A. Then dA (xλ ) > 0, xλ ∈ H − , and necessarily x ∈ H − . From (10.5) bA (xλ ) = dA (xλ ) = dH (xλ ) = λdH (x) − (1 − λ)dH (y). But since x ∈ H − and A ⊂ H + dH (x) = dH + (x) ≤ dA (x) and bA (xλ ) = dH (xλ ) ≤ λdA (x) − (1 − λ)dA (y) = λbA (x) + (1 − λ)bA (y). Next consider the case xλ ∈ A for which xλ ∈ H + and −bA (xλ ) = dA (xλ ) = dH (xλ ). If x ∈ H − , then since A ⊂ H + , dH (x) = dH + (x) ≤ dA (x), and from (10.5) −bA (xλ ) = dH (xλ ) = (1 − λ)dH (y) − λdH (x) ≥ (1 − λ)dA (y) − λdA (x) = −[(1 − λ)bA (y) + λbA (x)]. If, on the other hand, x ∈ H + , then x, y ∈ H + and xλ ∈ A ⊂ H + . So from (10.4) −bA (xλ ) = dH (xλ ) = (1 − λ)dH (y) + λdH (x) ≥ (1 − λ)dA (y) − λdA (x) = −[(1 − λ)bA (y) + λbA (x)]. This covers all cases and concludes the proof.

10. Characterization of Convex and Semiconvex Sets

10.2

379

Families of Convex Sets Cb (D), Cb (E; D), and Cb,loc (E; D)

We have seen in Theorem 10.1 (i) that for a convex subset A of D, bA = bA and that A is the unique closed representative in the equivalence class. To work with open convex subsets is more delicate. Always from Theorem 10.1 (ii), int A becomes the unique open representative in the equivalence class when the convex set A has a nonempty interior. In order to work with families of open rather than closed convex subsets, it will be necessary to add a constraint on the interior of the subsets to prevent the occurrence of subsets with an empty interior. Theorem 10.2. Let D be a nonempty open (resp., bounded open) subset of RN . (i) The family def

Cb (D) = bA : A ⊂ D, ∂A = ∅, A convex

(10.6)

is closed in Cloc (D) (resp., compact in C(D)). It coincides with the family bA : A ⊂ D, ∂A = ∅, A closed and convex .

(ii) Let E be a nonempty open subset of D. Then the families def

Cb (E; D) = {bΩ : E ⊂ Ω ⊂ D, ∂Ω = ∅, Ω open and convex} , def

Cb,loc (E; D) =

bΩ :

∃x ∈ RN , ∃A ∈ O(N) such that x + AE ⊂ Ω and Ω is open and convex ⊂ D

&

(10.7) (10.8)

are closed in Cloc (D) (resp., compact in C(D)). 1,p (D)- (resp., (iii) The conclusions of parts (i) and (ii) remain true with Wloc 1,p W (D)-) strong in place of Cloc (D) (resp., C(D)).

Proof. It is suﬃcient to prove the result for D bounded. Furthermore, by compactness of Cb (D) in C(D), it is suﬃcient to prove that Cb (D), Cb (E; D), and Cb,loc (E; D) are closed in C(D). (i) Let {bAn } be a Cauchy sequence in Cb (D). It converges to some bA ∈ Cb (D). The function bA is convex as the pointwise limit of a family of convex functions. By Theorem 10.1 (iii), A is convex, and by Theorem 10.1 (i) bA = bA is convex. Therefore the Cauchy sequence converges to bA in Cb (D) and Cb (D) is closed in C(D). (ii) From part (i), given a Cauchy sequence {bΩn } in Cb (E; D), there exists an open convex subset Ω of D such that bΩn → bΩ . Moreover, since dΩn → dΩ , E ⊂ Ωn ⇒ E ⊃ Ωn ⇒ dE ≤ dΩn ⇒ dE ≤ dΩ ⇒ E ⊃ Ω ⇒ E ⊂ Ω. c The proof for Cb,loc (E; D) is similar to the proof for Cd,loc (E; D) in Theorem 2.6 of Chapter 6. (iii) Let {bAn } ⊂ Cb (D) be a Cauchy sequence in W 1,p (D). From Theorem 4.2 (ii), it is also Cauchy in C(D). Hence from part (i) there exists a convex set

380

Chapter 7. Metrics via Oriented Distance Functions

A, ∂A = ∅, such that bAn → bA in C(D) and, a fortiori, in W 1,p (D). This shows that Cb (D) is closed in W 1,p (D). To show that it is compact consider a sequence {bAn } ⊂ Cb (D). From part (i) there exist a subsequence, still indexed by n, and a convex subset A, ∂A = ∅, of D such that bAn → bA in C(D). Since D is bounded, it also converges in Lp (D). Since |∇bAn | is pointwise bounded by one, there exists another subsequence of {bAn }, still indexed by n, such that ∇bAn converges to ∇bA in W 1.p (D)N -weak. But An and A are convex. Thus m(∂A) = 0 = m(∂An ) and |∇bAn | = 1 = |∇bA | almost everywhere in RN . As a result for p = 2 2 |∇bAn − ∇bA | dx = |∇bAn |2 + |∇bA |2 dx − 2∇bAn · ∇bA dx D D 2 − 2∇bAn · ∇bA dx → 0. = D

Hence we get the compactness in W 1,2 (D)-strong and by Theorem 4.1 (i) in W 1,p (D)strong, 1 ≤ p < ∞.

10.3

BV Character of bA and Semiconvex Sets

The next theorem is a consequence of the fact that the gradient of a convex function is locally of bounded variations. Theorem 10.3. For all convex sets A such that ∂A = ∅, (i) ∇bA ∈ BVloc (RN )N , ∇χ∂A ∈ BVloc (RN ), ∇dA ∈ BVloc (RN )N , ∇χA ∈ BVloc (RN ), and for all x ∈ RN and v ∈ RN , dH bA (x, v) exists. (ii) The Hessian matrix D2 bA of second-order derivatives is a matrix of signed Radon measures which are nonnegative on the diagonal. (iii) bA has a second-order derivative almost everywhere, and for almost all x and y in RN , bA (y) − bA (x) − ∇bA (x) · (y − x) − 1 (y − x) · D 2 bA (x) (y − x) = o(|y − x|2 ) 2 as y → x. Proof. Same argument as in the proof of Theorem 8.2 of Chapter 6. When A is convex and of class C 2 , then for any x0 ∈ ∂A, there exists a strictly convex neighborhood N (x0 ) of x0 such that bA ∈ C 2 (N (x0 )) and ∀x ∈ N (x0 ), ∀ξ ∈ RN ,

D 2 bA (x)ξ · ξ ≥ 0,

(10.9)

or, since D 2 bA (x) ∇bA (x) = 0 in N (x0 ), then for all x ∈ N (x0 ), ∀ξ ∈ RN such that ξ · ∇bA (x) = 0,

D2 bA (x)ξ · ξ ≥ 0.

(10.10)

This is related to the notion of strong elliptic midsurface in the theory of shells: there exists c > 0 ∀x ∈ ∂A, ∀ξ ∈ RN such that ξ · ∇bA (x) = 0,

D 2 bA (x) ξ · ξ ≥ c|ξ|2 .

All this motivates the introduction of the following notions.

11. Compactness and Sets of Bounded Curvature

381

Deﬁnition 10.1. Let A be a closed subset of RN such that ∂A = ∅. (i) The set A is locally convex (resp., locally strictly convex ) if for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 such that bA is convex (resp., strictly convex) in N (x0 ). (ii) The set A is semiconvex if ∃α ≥ 0,

bA (x) + α|x|2 is convex in RN .

(iii) The set A is locally semiconvex if for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 and ∃α ≥ 0,

bA (x) + α|x|2 is convex in N (x0 ).

Remark 10.1. When A is a C 2 domain with a compact boundary, D 2 bA is bounded in a bounded neighborhood of ∂A and A is necessarily locally semiconvex. When A is a locally semiconvex set, for each x0 ∈ ∂A there exists a convex neighborhood N (x0 ) of x0 such that ∇bA ∈ BV(N (x0 ))N . If, in addition, ∂A is compact, then there exists h > 0 such that ∇bA ∈ BV(Uh (∂A))N . Remark 10.2. Given a ﬁxed constant β > 0, consider all the subsets of D that are semiconvex with constant 0 ≤ α ≤ β. Then this set is closed for the uniform and the W 1,p -topologies, 1 ≤ p < ∞.

11

Compactness and Sets of Bounded Curvature

For the family of sets with bounded curvature, the key result is the compactness of the embedding

(11.1) BCb (D) = bA ∈ Cb (D) : ∇bA ∈ BV(D)N → W 1,p (D) for bounded open Lipschitzian subsets of RN and p, 1 ≤ p < ∞. It is the analogue of the compactness Theorem 6.3 of Chapter 5 for Caccioppoli sets BX(D) = {χ ∈ X(D) : χ ∈ BV(D)} → Lp (D),

(11.2)

which is a consequence of the compactness of the embedding BV(D) → L1 (D)

(11.3)

for bounded open Lipschitzian subsets of RN (cf. C. B. Morrey, Jr. [1, Def. 3.4.1, p. 72, Thm. 3.4.4, p. 75] and L. C. Evans and R. F. Gariepy [1, Thm. 4, p. 176]). As in the case of characteristic functions in Chapter 5, we give a ﬁrst version involving global conditions on a ﬁxed bounded open Lipschitzian holdall D. In the second version the sets are contained in a bounded open holdall D with local conditions in the tubular neighborhood of their boundary.

382

11.1

Chapter 7. Metrics via Oriented Distance Functions

Global Conditions on D

Theorem 11.1. Let D be a nonempty bounded open Lipschitzian subset of RN . The embedding (11.1) is compact. Thus for any sequence {An }, ∂An = ∅, of subsets of D such that ∃c > 0, ∀n ≥ 1, D2 bAn M 1 (D) ≤ c, (11.4) there exist a subsequence {Ank } and a set A, ∂A = ∅, such that ∇bA ∈ BV(D)N and bAnk → bA in W 1,p (D)-strong for all p, 1 ≤ p < ∞. Moreover, for all ϕ ∈ D0 (D), lim ∂ij bAnk , ϕ = ∂ij bA , ϕ,

n→∞

1 ≤ i, j ≤ N,

D2 bA M 1 (D) ≤ c.

(11.5)

def

Proof. Given c > 0 consider the set Sc = bA ∈ Cb (D) : D2 bA M 1 (D) ≤ c . By compactness of the embedding (11.3), given any sequence {bAn } there exist a subsequence, still denoted by {bAn }, and f ∈ BV(D)N such that ∇bAn → f in L1 (D)N . But by Theorem 2.2 (ii), Cb (D) is compact in C(D) for bounded D and there exist another subsequence {bAnk } and bA ∈ Cb (D) such that bAnk → bA in C(D) and, a fortiori, in L1 (D). Therefore, bAnk converges in W 1,1 (D) and also in L1 (D). By uniqueness of the limit, f = ∇bA and bAnk converges in W 1,1 (D) to bA . For Φ ∈ D1 (D)N ×N as k goes to inﬁnity −→ −→ ∇bAnk · div Φ dx → ∇bA · div Φ dx D D −→ −→ ∇bA · div Φ dx = lim ∇bAnk · div Φ dx ≤ cΦC(D) , ⇒ k→∞

D

D

D bA M 1 (D) ≤ c, and ∇bA ∈ BV(D) . This proves the sequential compactness of the embedding (11.1) for p = 1 and properties (11.5). The conclusions remain true for p ≥ 1 by the equivalence of the W 1,p -topologies on Cb (D) in Theorem 4.1 (i). 2

11.2

N

Local Conditions in Tubular Neighborhoods

The global condition (11.4) is now weakened to a local one in a neighborhood of each set of the sequence. Simultaneously, the Lipschitzian condition on D is removed since only the uniform boundedness of the sets of the sequence is required. ∅, be a Theorem 11.2. Let ∅ = D ⊂ RN be bounded open and {An }, ∂An = sequence of subsets of D. Assume that there exist h > 0 and c > 0 such that ∀n,

D2 bAn M 1 (Uh (∂An )) ≤ c.

(11.6)

Then, there exist a subsequence {Ank } and A ⊂ D, ∅ = ∂A, such that ∇bA ∈ BVloc (RN )N and for all p, 1 ≤ p < ∞, bAnk → bA in W 1,p (Uh (D))-strong.

(11.7)

11. Compactness and Sets of Bounded Curvature

383

Moreover, for all ϕ ∈ D0 (Uh (∂A)), lim ∂ij bAnk , ϕ = ∂ij bA , ϕ,

1 ≤ i, j ≤ N,

k→∞

D2 bA M 1 (Uh (∂A)) ≤ c,

(11.8)

and χ∂A belongs to BVloc (RN ). Proof. (i) By assumption, An ⊂ D implies that Uh (An ) ⊂ Uh (D). Since Uh (D) is bounded, there exist a subsequence, still indexed by n, and a subset A of D, ∂A = ∅, such that bAn → bA

in C(Uh (D))-strong

and another subsequence, still indexed by n, such that bAn → bA

in H 1 (Uh (D))-weak.

For all ε > 0, 0 < 3ε < h, there exists N > 0 such that for all n ≥ N and all x ∈ Uh (D), d∂An (x) ≤ d∂A (x) + ε,

d∂A (x) ≤ d∂An (x) + ε.

(11.9)

Therefore, ∂An ⊂ Uh−2 ε (∂An ) ⊂ Uh−ε (∂A) ⊂ Uh (∂An ), Uh−ε (∂A) ⊂ Uh−2 ε (∂An ) ⊂ ∂An .

(11.10) (11.11)

From (11.6) and (11.10) ∀n ≥ N,

D 2 bAn M 1 (Uh−ε (∂A)) ≤ c.

In order to use the compactness of the embedding (11.3) as in the proof of Theorem 11.1, we would need Uh−ε (∂A) to be Lipschitzian. To get around this we construct a bounded Lipschitzian set between Uh−2ε (∂A) and Uh−ε (∂A). Indeed by deﬁnition, Uh−ε (∂A) = ∪x∈∂A B(x, h − ε)

and

Uh−2ε (∂A) ⊂ Uh−ε (∂A),

and by compactness there exists a ﬁnite sequence of points {xi }ni=1 in ∂A such that def

Uh−2ε (∂A) ⊂ UB = ∪ni=1 B(xi , h − ε) ⊂ Uh (D). Since UB is Lipschitzian as the union of a ﬁnite number of balls, it now follows by compactness of the embedding (11.3) for UB that there exist a subsequence, still denoted by {bAn }, and f ∈ BV(UB )N such that ∇bAn → f in L1 (UB )N . Since Uh (D) is bounded, Cb (Uh (D)) is compact in C(Uh (D)) and there exist another subsequence, still denoted by {bAn }, and A ⊂ D, ∂A = ∅, such that bAn → bA in C(Uh (D)) and, a fortiori, in L1 (Uh (D)). Therefore, bAn converges in W 1,1 (UB ) and also in L1 (UB ). By uniqueness of the limit, f = ∇bA on UB and bAn converges to bA in W 1,1 (UB ). By Deﬁnition 5.1 and Theorem 3.1 (i), ∇bA and ∇bAn all belong

384

Chapter 7. Metrics via Oriented Distance Functions

to BVloc (RN )N since they are of bounded variation in tubular neighborhoods of their respective boundaries. Moreover, by Theorem 5.2 (ii), χ∂A ∈ BVloc (RN ). The above conclusions also hold for the subset Uh−2ε (∂A) of UB . (ii) Convergence in W 1,p (Uh (D)). Consider the integral |∇bAn − ∇bA |2 dx Uh (D)

|∇bAn − ∇bA |2 dx +

=

|∇bAn − ∇bA |2 dx.

Uh−2ε (∂A)

Uh (D)\Uh−2ε (∂A)

From part (i), the ﬁrst integral on the right-hand side converges to zero as n goes to inﬁnity. The second integral is on a subset of Uh−2ε (∂A). From (11.11) for all n ≥ N, |∇bAn (x)| = 1 a.e. in ∂An ⊃ Uh−3ε (∂An ) ⊃ Uh−2ε (∂A), |∇bA (x)| = 1 a.e. in ∂A ⊃ Uh−2ε (∂A). The second integral reduces to

|∇bAn − ∇bA |2 dx =

Uh (D)\Uh−2ε (∂A)

2 (1 − ∇bAn · ∇bA ) dx,

Uh (D)\Uh−2ε (∂A)

which converges to zero by weak convergence of ∇bAn to ∇bA in the space L2 (Uh (D))N in part (i) and the fact that |∇bA | = 1 almost everywhere in Uh (D)\ Uh−2ε (∂A). Therefore, since bAn → bA in C(Uh (D)) bAn → bA in H 1 (Uh (D))-strong, and by Theorem 4.1 (i) the convergence is true in W 1,p (Uh (D)) for all p ≥ 1. (iii) Properties (11.8). Consider the initial subsequence {bAn } which converges to bA in H 1 (Uh (D))-weak constructed at the beginning of part (i). This sequence is independent of ε and the subsequent constructions of other subsequences. By convergence of bAn to bA in H 1 (Uh (D))-weak for each Φ ∈ D1 (Uh (∂A))N ×N , lim

n→∞

Uh (∂A)

−→

∇bAn · div Φ dx =

−→

∇bA · div Φ dx. Uh (∂A)

Each such Φ has compact support in Uh (∂A), and there exists ε = ε(Φ) > 0, 0 < 3ε < h, such that supp Φ ⊂ Uh−2ε (∂A). From part (ii) there exists N (ε) > 0 such that ∀n ≥ N (ε),

Uh−2 ε (∂An ) ⊂ Uh−ε (∂A) ⊂ Uh (∂An ).

12. Finite Density Perimeter and Compactness For n ≥ N (ε) consider the integral −→ ∇bAn · div Φ dx = Uh (∂A)

∇bAn

385

· div Φ dx = −→

Uh−2ε (∂A)

⇒

∇bAn

Uh (∂A)

−→

∇bAn · div Φ dx

U (∂A )

n h · div Φ dx ≤ D2 bAn M 1 (Uh (∂An )) ΦC(Uh (∂An )) ≤ c ΦC(Uh−2ε (∂A)) = c ΦC(Uh (∂A)) .

−→

By convergence of ∇bAn to ∇bA in the space L2 (Uh (D))-weak, then for all Φ ∈ D1 (Uh (∂A))N ×N −→ ∇bA · div Φ dx ≤ c ΦC(Uh (∂A)) ⇒ D2 bA M 1 (Uh (∂A)) ≤ c. Uh (∂A)

Finally, the convergence remains true for all subsequences constructed in parts (i) and (ii). This completes the proof.

12

Finite Density Perimeter and Compactness

´sio [14] is a reThe density perimeter introduced by D. Bucur and J.-P. Zole laxation of the (N − 1)-dimensional upper Minkowski content which leads to the compactness Theorem 12.1. Deﬁnition 12.1. Let h > 0 be a ﬁxed real number and let ∅ = A ⊂ RN , ∂A = ∅. Consider the quotient mN (Uk (∂A)) def Ph (∂A) = sup , (12.1) 2k 0 0 such that ∀n,

Ph (∂An ) ≤ c.

(12.2)

Then there exist a subsequence {Ank } and A ⊂ D, ∂A = ∅, such that Ph (∂A) ≤ lim inf Ph (∂An ) ≤ c, n→∞

∀p, 1 ≤ p < ∞,

bAnk → bA in W

1,p

(Uh (D))-strong.

(12.3) (12.4)

386

Chapter 7. Metrics via Oriented Distance Functions

Proof. The proof essentially rests on Theorem 4.3 since Ph (∂A) ≤ c implies mN (∂A) = 0. Since D is bounded, the family of oriented distance functions Cb (D) is compact in C(D) and W 1,p (D)-weak for all p, 1 ≤ p < ∞ (cf. Theorems 2.2 (ii) and 4.2 (iii)). So there exist bA ∈ Cb (D) and a subsequence, still indexed by n, such that bAn → bA in the above topologies. Moreover, for all k, 0 < k < h, and all ε, 0 < ε < h − k, ∃N (ε) > 0 such that ∀n ≥ N (ε),

Uk−ε (∂An ) ⊂ Uk (∂A) ⊂ Uk+ε (∂An )

(cf. proof of part (i) of Theorem 11.2). As a result for all n ≥ N (ε), mN (Uk−ε (∂An )) k − ε mN (Uk (∂A)) mN (Uk+ε (∂An )) k + ε k+ε ≤ ≤ ≤c 2(k − ε) k 2k 2(k + ε) k k mN (Uk+ε (∂An )) k + ε mN (Uk (∂A)) k+ε ≤ ≤ Ph (∂An ) ⇒ 2k 2(k + ε) k k mN (Uk (∂A)) k+ε ≤ lim inf Ph (∂An ) . (12.5) ⇒ n→∞ 2k k Going to the limit as ε goes to zero in the second and fourth terms mN (Uk (∂A)) ≤ lim inf Ph (∂An ) ≤ c n→∞ 2k ⇒ Ph (∂A) ≤ lim inf Ph (∂An ) ≤ c ⇒ mN (∂A) = 0.

∀k, 0 < k < h,

n→∞

The theorem now follows from the fact that mN (∂A) = 0 and Theorem 4.3. Corollary 1. Let D = ∅ be a bounded open subset of RN and A, ∂A = ∅, be a subset of D such that ∃h > 0 and c > 0 such that Ph (∂A) ≤ c.

(12.6)

Then the mapping bA → Ph (∂A ) : Cb (D) → R ∪{+∞} is lower semicontinuous in A for the W 1,p (D)-topology. Proof. Since we have a metric topology, it is suﬃcient to prove the property for W 1,p (D)-converging sequences {bAn } to bA . From that point on the argument is the same as the one used to get (12.5) in the proof of Theorem 12.1 after the extraction of the subsequence. Remark 12.1. It is important to notice that even if {An } is a W 1,p -convergent sequence of bounded open subsets of RN with a uniformly bounded perimeter, the limit set A need not be an open set or have a nonempty interior int A such that bA = bint A . It would be tempting to say that bAn → bA implies dAn → dA and use the open set int A = A for which dA = dint A to conclude that bA = bint A . This is incorrect as can be seen in the following example shown in Figure 7.2. Consider a family {An } of open rectangles in R2 of width equal to 1 and height 1/n, n ≥ 1, an integer. Their density perimeter is bounded by 4, and the bAn ’s converge to bA for A equal to the line of length 1 which has an empty interior.

13. Compactness and Uniform Fat Segment Property

An

1 n

1

387

A

1

Figure 7.2. W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R with uniformly bounded density perimeter to a set with empty interior. 2

13 13.1

Compactness and Uniform Fat Segment Property Main Theorem

In Theorem 6.11 of section 6.4 of Chapter 5, we have seen a compactness theorem for the family of subsets of a bounded holdall D satisfying the uniform cone property. This compactness theorem is no longer true when the uniform cone property is replaced by a uniform segment property. This is readily seen by considering the following example. Example 13.1. Given an integer n ≥ 1, consider the following sequence of open domains in R2 : def Ωn = (x, y) ∈ R2 : |x| < 1 and |x|1/n < y < 2 . They satisfy the uniform segment property of Deﬁnition 6.1 (ii) of Chapter 2 by choosing λ = r = 1/4. The sequence {Ωn } converges to the closed set def

A = (x, y) ∈ R2 : |x| ≤ 1 and 1 ≤ y ≤ 2 ∪ L, def

L = (0, y) ∈ R2 : 0 ≤ y ≤ 1 in the uniform topologies associated with dΩn and bΩn or in the Lp -topologies associated with χΩn and χΩn . However, the segment property is not satisﬁed along the line L and the corresponding family of subsets of the holdall D = B(0, 4) satisfying the uniform segment property with r = λ = 1/4 is not closed and, a fortiori, not compact. This example shows that a uniform segment property is too meager to make the corresponding family compact. Looking back at the proof of Theorem 6.11 of section 6.4 in Chapter 5, the fact that the cone is an open set around the segment is used critically. That was the motivation to introduce the more general uniform fat segment property (Deﬁnition 6.1 (iii) in section 6 of Chapter 2) that includes as special cases both the uniform cone and the uniform cusp properties (cf. Deﬁnitions 6.3 and 6.4 in section 6.4 of Chapter 2). However, the proof of Theorem 6.11 in Chapter 5 was based on the fact that the perimeter of all the sets in the family

388

Chapter 7. Metrics via Oriented Distance Functions

were uniformly bounded. There is no hope of getting that property for general H¨ olderian domains, and a completely diﬀerent proof is necessary. Given a bounded open subset D of RN consider the family & Ω satisﬁes the uniform fat def L(D, r, O, λ) = Ω ⊂ D : . (13.1) segment property for (r, O, λ) We ﬁrst give a general proof of the compactness of L(D, r, O, λ) in the C(D)- and W 1,p (D)-topologies associated with the oriented distance function bΩ . As a consequence, we get the compactness for the C(D)- and W 1,p (D)-topologies associated with dΩ and dΩ and the Lp (D)-topologies associated with χΩ and χΩ . We recover the compactness Theorem 6.11 of Chapter 5 under the uniform cone property for χΩ in Lp (D) as a special case. In the last part of this section, we give several other ways to specify the compact families by using the equivalent conditions of Chapter 2 of the uniform fat segment property in terms of conditions on the local epigraphs via the dominating function and equicontinuity of the local graph functions. Theorem 13.1. Let D be a nonempty bounded open subset of RN and let 1 ≤ p < ∞. Assume that L(D, r, O, λ) is not empty for r > 0, λ > 0, and an open / O. Then the family subset O of RN such that (0, eN ) ⊂ O and 0 ∈ def

B(D, r, O, λ) = {bΩ : ∀Ω ∈ L(D, r, O, λ)} is compact in C(D) and W 1,p (D). As a consequence the families def

Bd (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)}, def Bdc (D, r, O, λ) = {dΩ : ∀Ω ∈ L(D, r, O, λ)}, def

Bd∂ (D, r, O, λ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ)} are compact in C(D) and W 1,p (D), and the families def

X(D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)}, def X c (D, r, O, λ) = {χΩ : ∀Ω ∈ L(D, r, O, λ)} are compact in Lp (D). From Lemma 6.1 of Chapter 2, Theorem 13.1 can be specialized to structured open sets O. Corollary 1. The conclusions of Theorem 13.1 hold when O is specialized to open regions of the form

(13.2) O(h, ρ, λ) = ξ ∈ RN : ξ ∈ BH (0, ρ) and h(|ξ |) < ξN < λ for dominating functions h ∈ H. Theorem 13.1 can be further specialized to Lipschitzian and H¨ olderian sets.

13. Compactness and Uniform Fat Segment Property

389

Corollary 2. The conclusions of Theorem 13.1 hold under the uniform cone property of Deﬁnition 6.3 (ii) in section 6.4 of Chapter 2 with O = O(h, ρ, λ) with h(θ) = θ/ tan ω and ρ = λ tan ω,

0 < ω < π/2,

(13.3)

and the uniform cusp property of Deﬁnition 6.4 (ii) of Chapter 2 with

O = O(h , ρ, λ) with h (θ) = λ (θ/ρ) ,

0 < < 1.

The proof of Theorem 13.1 will require Theorem 4.3 and the following lemma. Lemma 13.1. Given a sequence {bΩn } ⊂ Cb (D) such that bΩn → bΩ in C(D) for some bΩ ∈ Cb (D), we have the following properties: ∀x ∈ Ω, ∀R > 0,

∃N (x, R) > 0, ∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅,

and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅.

(13.4)

Moreover, ∀x ∈ ∂Ω, ∀R > 0, ∃N (x, R) > 0, ∀n ≥ N (x, R), B(x, R) ∩ Ωn = ∅ and B(x, R) ∩ Ωn ) = ∅, and B(x, R) ∩ ∂Ωn = ∅. Proof. We proceed by contradiction. Assume that ∃x ∈ Ω,

∃R > 0,

∀N > 0,

∃n ≥ N,

B(x, R) ∩ Ωn = ∅.

So there exists a subsequence {Ωnk }, nk → ∞, such that bΩnk (x) = dΩnk (x) ≥ R = 0 = dΩ (x) ≥ bΩ (x), which contradicts the fact that bΩnk → bΩ . Of course, the same assertion is true

for the complements and for all x ∈ Ω, ∀R > 0,

∃N (x, R) > 0,

∀n ≥ N (x, R),

B(x, R) ∩ Ωn = ∅.

When x ∈ ∂Ω, x ∈ Ω ∩ Ω and we combine the two results. For the last result use the fact that the open ball cannot be partitioned into two nonempty disjoint open subsets. Proof of Theorem 13.1. (i) Compactness in C(D). Since for all Ω in L(D, r, O, λ), Ω is locally a C 0 -epigraph, ∂Ω = ∅, bΩ ∈ Cb (D), and mN (∂Ω) = 0. Consider an arbitrary sequence {Ωn } in L(D, r, O, λ). For D compact, Cb (D) is compact in C(D) and there exist Ω ⊂ D, ∂Ω = ∅, and a subsequence {Ωnk } such that bΩnk → bΩ in C(D). It remains to prove that Ω ∈ L(D, r, O, λ), that is, ∀x ∈ ∂Ω, ∃A ∈ O(N) such that ∀y ∈ Ω ∩ B(x, r),

y + λAO ⊂ int Ω.

390

Chapter 7. Metrics via Oriented Distance Functions

From Lemma 13.1, for each x ∈ ∂Ω, for all k ≥ 1, there exists nk ≥ k such that B x, r/2k ∩ ∂Ωnk = ∅. Denote by xk an element of that intersection: ∀k ≥ 1, xk ∈ B x, r/2k ∩ ∂Ωnk . By construction xk → x. Next consider y ∈ B(x, r) ∩ Ω. From the ﬁrst part of the lemma, there exists a subsequence of {Ωnk }, still denoted by {Ωnk }, such that for all k ≥ 1, B(y, r/2k ) ∩ Ωnk = ∅. For each k ≥ 1 denote by yk a point of that intersection. By construction yk ∈ Ωnk → y ∈ Ω ∩ B(x, r). There exists K > 0 large enough such that for all k ≥ K, yk ∈ B(xk , r). To see this, note that y ∈ B(x, r) and that ∃ρ > 0,

B(y, ρ) ⊂ B(x, r)

and |y − x| + ρ/2 < r.

Now |yk − xk | ≤ |yk − y| + |y − x| + |x − xk | . r ρ ρ/ r r ≤ k + r − + k ≤ r + k−1 − < r. 2 2 2 2 2 Since r/ρ > 1 the result is true for r 2k−1

−

ρ 2 + log

r . ρ

So we have constructed a subsequence {Ωnk } such that for k ≥ K xk ∈ ∂Ωnk → x ∈ ∂Ω

and yk ∈ Ωnk ∩ B(xk , r) → y ∈ Ω ∩ B(x, r).

For all k, there exists Ak ∈ O(N), Ak ∗Ak = ∗Ak Ak = I, such that yk + λAk O ⊂ int Ωnk . Pick another subsequence of {Ωnk }, still denoted by {Ωnk }, such that ∃A ∈ O(N), A ∗A = ∗A A = I,

Ak → A.

Now consider z ∈ y + λAO. Since y + λAO is open there exists ρ > 0 such that B(z, ρ) ⊂ y + λAO, and there exists K ≥ K such that ∀k ≥ K ,

B(z, ρ/2) ⊂ yk + λAk O ⊂ int Ωnk = Ωnk

⇒ B(z, ρ/2) ⊃ Ωnk ⇒ 0 < ρ/2 = dB(z,ρ/2) (z) ≤ dΩn (z) → dΩ (z) k

⇒ 0 < ρ/2 ≤ dΩ (z)

⇒ z ∈ Ω = int Ω

⇒ y + λAO ⊂ int Ω.

Hence Ω ⊂ D satisﬁes the uniform fat segment property.

13. Compactness and Uniform Fat Segment Property

391

(ii) W 1,p (D)-compactness. From Theorem 4.1 (i), it is suﬃcient to prove the result for p = 2. Consider the subsequence {Ωnk } ⊂ L(D, λ, O, r) and let Ω ∈ L(D, λ, O, r) be the set previously constructed such as bΩnk → bΩ in C(D). Hence bΩnk → bΩ in L2 (D). Since D is compact |bΩnk |2 dx ≤ diam (D)2 dx ≤ diam (D)2 mN (D), ∀Ω ⊂ D, D D 2 |∇bΩnk | dx ≤ dx = mN (D), D

D

and there exists a subsequence, still denoted by {bΩnk }, which converges weakly to bΩ . Since all the sets in L(D, r, O, λ) verify a uniform segment property, they are locally C 0 -epigraphs, mN (∂Ωnk ) = 0 = mN (∂Ω) and, by Theorem 4.3, bΩnk → bΩ in W 1,p (D)-strong, 1 ≤ p < ∞. (iii) The other compactness follows by continuity of the maps (4.6) and (4.7) in Theorem 4.1 and the fact that mN (∂Ω) = 0 implies χint Ω = χΩ and χint Ω = χΩ almost everywhere for Ω ∈ L(D, λ, O, r).

13.2

Equivalent Conditions on the Local Graph Functions

We have shown in Theorem 6.2 of Chapter 2 that the uniform fat segment property is equivalent to the equi-C 0 epigraph property. This means that we can equivalently characterize the compact family L(D, r, O, λ) in terms of conditions on the local graph functions such as the ones introduced in Theorem 5.1 of Chapter 2. Recall the following equivalent conditions: (i) Ω veriﬁes the fat segment property. (ii) Ω is an equi-C 0 -epigraph; that is, Ω is a C 0 -epigraph and the local graph functions are uniformly bounded and equicontinuous. (iii) Ω is a C 0 -epigraph and there exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ζ , ξ ∈ BH (0, ρ) such that |ξ − ζ | < ρ,

|ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |). (13.5)

(iv) Ω is a C 0 -epigraph and there exist ρ > 0 and h ∈ H such that BH (0, ρ) ⊂ V and for all x ∈ ∂Ω ∀ξ ∈ BH (0, ρ),

ax (ξ ) ≤ h(|ξ |).

(13.6)

Recall also from Deﬁnition 5.1 (ii) in Chapter 2 that Ω is said to be a C 0 epigraph if it is locally a C 0 -epigraph and the neighborhoods U(x) and Vx can be chosen in such a way that Vx and A−1 x (U(x) − x) are independent of x: there exist bounded open neighborhoods V of 0 in H and U of 0 in RN such that PH (U ) ⊂ V and ∀x ∈ ∂Ω,

Vx = V

and ∃Ax ∈ O(N) such that U(x) = x + Ax U.

392

Chapter 7. Metrics via Oriented Distance Functions

By applying each of the last three characterizations to all members of a family of subsets of a bounded holdall D, we get the analogue of the compactness Theorem 13.1 under the uniform fat segment property. Theorem 13.2. Let D be a bounded open nonempty subset of RN and consider the family L(D, ρ, U ) of all subsets Ω of D that verify the C 0 -epigraph property: there exist ρ > 0 and a bounded open neighborhood U of 0 such that

def (13.7) U ⊂ ζ ∈ RN : PH (ζ) ∈ BH (0, ρ) , V = BH (0, ρ), and for each Ω ∈ L(D, ρ, U ) and each x ∈ ∂Ω, there exists AΩ x ∈ O(N) such that def

U Ω (x) = x + AΩ xU

(13.8)

Ω and a C 0 -graph function aΩ x : V → R such that ax (0) = 0 and

U Ω (x) ∩ ∂Ω = U Ω (x) ∩ x + Ax (ζ + ζN eN ) : ζ ∈ V, ζN = aΩ x (ζ ) ,

U Ω (x) ∩ int Ω = U Ω (x) ∩ x + Ax (ζ + ζN eN ) : ζ ∈ V, ζN > aΩ x (ζ ) .

Then, for all h ∈ H, the subfamilies {aΩ x : Ω ∈ L(D, ρ, U ), x ∈ ∂Ω} def L0 (D, ρ, U ) = Ω ∈ L(D, ρ, U ) : is uniformly bounded , and equicontinuous def

L1 (D, ρ, U, h) = Ω ∈ L(D, ρ, U ) : ∀x ∈ ∂Ω, ∀ζ ∈ V, aΩ x (ζ ) ≤ h(|ζ |) , ∀x ∈ ∂Ω, ∀ζ , ξ ∈ V def L2 (D, ρ, U, h) = Ω ∈ L(D, ρ, U ) : such that |ξ − ζ | < ρ |ax (ξ ) − ax (ζ )| ≤ h(|ξ − ζ |)

(13.9)

(13.10)

(13.11) (13.12)

are compact (possibly empty) in C(D) and W 1,p (D), 1 ≤ p < ∞, and the conclusions of Theorem 13.1 hold. Remark 13.1. The geometric fat segment property of Corollary 1 was introduced in the form (13.2) ´sio [37] under the name uniform in the book of M. C. Delfour and J.-P. Zole cusp property in 2001, where the condition was specialized to (13.3) for Lipschitzian and H¨ olderian domains as stated in Corollary 2. Condition (13.10) is a streamlined version of the suﬃcient condition given in an unpublished report by D. Tiba [2], announced without proof in the note ¨ ki, and D. Tiba [1] in 2000, and later documented by W. Liu, P. Neittaanma in D. Tiba [3] in 2003. He gets a compactness result by introducing the following conditions on the local C 0 -epigraphs: the boundaries of the domains belong (after a rotation and a translation) to a family F of uniformly equicontinuous functions deﬁned in a ﬁxed neighborhood V of 0 in RN −1 . This means that there exists a modulus of continuity µ(ε) > 0 such that ∀ε > 0, ∀f ∈ F, ∀x, y ∈ V,

|x − y| < µ(ε)

⇒ |f (y) − f (x)| < ε.

Then the compactness follows by Ascoli–Arzel`a Theorem 2.4 of Chapter 2.

14. Compactness under Uniform Fat Segment Property and Bound on a Perimeter 393 The connection between that condition and our geometric condition was ´sio [1, 2] in ﬁrst established in M. C. Delfour, N. Doyon, and J.-P. Zole 2005. Condition (13.11) that relaxes the equicontinuity condition (13.10) to the simpler dominating function condition and condition (13.12) were both introduced ´sio [43] in 2007 along with the general form in M. C. Delfour and J.-P. Zole of the uniform fat segment property that involves only the unparametrized open set O.

14

Compactness under the Uniform Fat Segment Property and a Bound on a Perimeter

Domains that are locally Lipschitzian epigraphs or equivalently satisfy the local uniform cone property enjoy the additional property that for any bounded open set D the (N − 1)-Hausdorﬀ measure of D ∩ ∂Ω is ﬁnite. In general this is no longer true for domains verifying a uniform cusp property for some function h(θ) = |θ/ρ|α , 0 < α < 1 (cf., for instance, Example 6.2 given6 in section 6.5 of Chapter 2 of a bounded domain Ω in RN for which HN −1 (∂Ω) = +∞ and the Hausdorﬀ dimension of ∂Ω is exactly N − α). Yet, there are many domains with cusps whose boundary has ﬁnite HN −1 measure.

14.1

De Giorgi Perimeter of Caccioppoli Sets

One of the classical notions of perimeter is the one introduced in Deﬁnition 6.2 in section 6 of Chapter 5 in the context of the problem of minimal surfaces for Caccioppoli sets. Theorem 13.1 extends to the following subfamily of L(D, r, O, λ) for a bounded open subset D of RN and a constant c : Ω satisﬁes the uniform fat def L(D, r, O, λ, c ) = Ω ⊂ D : segment property for (r, O, λ) . (14.1) and PD (Ω) ≤ c Theorem 14.1. Let D be a nonempty bounded open subset of RN and let 1 ≤ p < ∞. Assume that L(D, r, O, λ, c ) is not empty for r > 0, λ > 0, c > 0, and a / O. Then the family bounded open subset O of RN such that (0, eN ) ⊂ O and 0 ∈ B(D, r, O, λ, c ) = {bΩ : ∀Ω ∈ L(D, r, O, λ, c )} def

is compact in C(D) and W 1,p (D). As a consequence the families Bd (D, r, O, λ, c ) = {dΩ : ∀Ω ∈ L(D, r, O, λ, c )}, def Bdc (D, r, O, λ, c ) = {dΩ : ∀Ω ∈ L(D, r, O, λ, c )}, def Bd∂ (D, r, O, λ, c ) = {d∂Ω : ∀Ω ∈ L(D, r, O, λ, c )} def

6 This

´sio [1]. was originally in M. C. Delfour, N. Doyon, and J.-P. Zole

394

Chapter 7. Metrics via Oriented Distance Functions

are compact in C(D) and W 1,p (D), and the families X(D, r, O, λ, c ) = {χΩ : ∀Ω ∈ L(D, r, O, λ, c )} , def X c (D, r, O, λ, c ) = {χΩ : ∀Ω ∈ L(D, r, O, λ, c )} def

are compact in Lp (D).

14.2

Finite Density Perimeter

It is also possible to use the density perimeter introduced in Deﬁnition 12.1 in place of the De Giorgi perimeter. The proof of the next result combines Theorem 14.1, which says that the family L(D, r, O, λ, c ) is compact, with Theorem 12.1, which says that the family of sets verifying (12.2) is compact in W 1,p (D). The intersection of the two families of oriented distance functions is compact in W 1,p (D). Theorem 14.2. For ﬁxed h > 0 and an open subset Ω of RN , Theorem 14.1 remains true when PD (Ω) is replaced by the density perimeter Ph (∂Ω).

15

The Families of Cracked Sets

In this section we introduce new families of thin sets, that is, sets A such that mN (∂A) = 0, by introducing conditions on the semiderivative of the distance function at points of the boundary. They have been used in M. C. Delfour and ´sio [38] in the context of the image segmentation problem of D. MumJ.-P. Zole ford and J. Shah [2]. Cracked sets are more general than sets which are locally the epigraph of a continuous function in the sense that they include domains with cracks and sets that can be made up of components of diﬀerent codimensions. The Hausdorﬀ (N − 1) measure of their boundary is not necessarily ﬁnite. Yet, compact families (in the W 1,p -topology) of such sets can be constructed. First recall lower and upper semiderivatives7 of Dini type for the diﬀerential quotient of a function f : V (x) ⊂ RN → R deﬁned in a neighborhood V (x) of a point x ∈ RN in the direction d ∈ RN : f (x + td) − f (x) def f (x + td) − f (x) , = lim inf δ0 0 xk,j−1 + 2δk .

406

Chapter 7. Metrics via Oriented Distance Functions

Ω2 f = 0

frame D

Ω1 f = 1

Figure 7.5. The two open components Ω1 and Ω2 of the open domain Ω for N = 2. Note that gk,j (xk,j−1 + δk ) = (δk )α is independent of j and is the maximum of the function gk,j . The uniform cusp property is veriﬁed for ρ = 1/8, λ = h(ρ), and h(θ) = θα . The boundary of Ω1 is made up of straight lines of total length 9 plus the length of the curve def

C = {(x, f (x)) : 0 ≤ x < 1} ,

C = ∪∞ k=0 Ck ,

η /2

k Ck = ∪j=1 Ck,j ,

def

Ck,j = {(x, f (x)) : xk,j−1 ≤ x < xk,j−1 + δk } . The length of the curve Ck,j is bounded below by HN −1 (Ck,j ) ≥ 2

'

ηk /2

(δk )2 + (δk )2α ≥ 2 (δk )α

ηk 2 (δk )α = ηk (δk )α = ηk HN −1 (Ck ) ≥ 2 HN −1 (Ck ) ≥ ηk 1−α

⇒ HN −1 (C) =

α

1 2k+1

∞

⇒ HN −1 (Ck ) ≥

1 ηk 2k+1

HN −1 (Ck,j ),

j=1

α

= ηk

1−α

1 2k+1

α ,

α 1−α α 1 = 2[(2k+1 ) 1−α ] 2k+1 α 1 ≥ 21−α (2k+1 )α = 21−α 2k+1

HN −1 (Ck ) ≥ +∞ 21−α = +∞.

k=0

When at least one solution has a bounded perimeter, it is possible to show that there is one that minimizes the h-density perimeter.

16. A Variation of the Image Segmentation Problem of Mumford and Shah

407

Theorem 16.3. Assume that the assumptions of Theorem 16.1 are veriﬁed. There exists an Ω∗ in F ∗ (D, h, α) which minimizes the h-density perimeter. Proof. If for all Ω in F ∗ (D, h, α) the h-density perimeter is +∞, the theorem is true. If for some Ω ∈ F ∗ (D, h, α), Ph (Γ) ≤ c, then there exists a sequence {Ωn } in F ∗ (D, h, α) such that Ph (Γn ) →

inf

Ω∈F ∗ (D,h,α)

Ph (Γ).

By the compactness Theorems 12.1 and 16.2, there exist a subsequence and Ω∗ , Γ∗ = ∅, such that bΩn → bΩ∗ in W 1,p (D), Ω∗ ∈ F (D, h, α), mN (Ω) = mN (D), and Ph (Γ∗ ) ≤ lim inf Ph (Γn ) ≤ c. n→∞

Finally by going back to the proof of Theorem 16.1 and using the fact that all the Ωn ’s are already minimizers in F ∗ (D, h, α), it can be shown that Ω∗ is indeed one of the minimizers in the set F ∗ (D, h, α).

16.4

Uniform Bound or Penalization Term in the Objective Function on the Density Perimeter

To complete the results on the segmentation problem, we turn to the existence of a segmentation for a family of sets with a uniform bound or with a penalization term in the objective function on the h-density perimeter. Theorem 16.4. Given a bounded open frame ∅ = D ⊂ RN with a Lipschitzian boundary and real numbers h > 0 and c > 0,10 there exists an open subset Ω∗ of D, Γ∗ = ∅, with ﬁnite h-density perimeter such that mN (Ω∗ ) = mN (D) (Ph (Γ∗ ) ≤ c for (16.8)), and y ∈ H 1 (Ω∗ ) solutions of the respective problems inf inf1 ε |∇ϕ|2 + |ϕ − f |2 dx, (16.8) Ω open ⊂D, Ph (Γ)≤c ϕ∈H (Ω) mN (Ω)=mN (D)

Ω

inf

ε |∇ϕ|2 + |ϕ − f |2 dx + c Ph (Γ).

inf

Ω open ⊂D ϕ∈H 1 (Ω) mN (Ω)=mN (D)

(16.9)

Ω

Proof. The proof for the objective function (16.8) is exactly the same as the one of Theorem 16.1. It uses Lemma 16.2 to show that the minimizing set has a thin boundary and the compactness Theorem 12.1. The proof for the objective function (16.9) uses the fact that there is a minimizing sequence for which the h-density perimeter is uniformly bounded and the lower semicontinuity of the density perimeter in the W 1,p -topology is given by Corollary 1. ´sio [8]. Problem (16.9) was originally considered in D. Bucur and J.-P. Zole The above two identiﬁcation problems can be further specialized to the family of cracked sets F(D, h, α). 10 Note that the constant c must be large enough to take into account the contribution of the boundary of D.

408

Chapter 7. Metrics via Oriented Distance Functions

Corollary 1. Given a bounded open frame D ⊂ RN with a Lipschitzian boundary and real numbers h > 0, α > 0, and c > 0, there exists an open subset Ω∗ of D in F(D, h, α) such that mN (Ω∗ ) = mN (D) (Ph (Γ∗ ) ≤ c for (16.10)), and y ∈ H 1 (Ω∗ ) solutions of the problem inf

inf1

ε |∇ϕ|2 + |ϕ − f |2 dx,

(16.10)

ε |∇ϕ|2 + |ϕ − f |2 dx + c Ph (Γ).

(16.11)

Ω open ⊂D, Ω∈F (D,h,α) ϕ∈H (Ω) Ph (Γ)≤c, mN (Ω)=mN (D)

Ω

inf

inf1

Ω open ⊂D, Ω∈F (D,h,α) ϕ∈H (Ω) mN (Ω)=mN (D)

Ω

Proof. Since the minimizing sequence {bΩn } constructed in the proof of Theorem 16.1 strongly converges to bΩ∗ in W 1,p (D) for all p, 1 ≤ p < ∞, from property (12.3) in Theorem 12.1, we have Ph (Γ∗ ) ≤ lim inf Ph (Γn ) ≤ c n→∞

and the optimal Ω∗ constructed in the proof of the theorem satisﬁes the additional constraint on the density perimeter.

Chapter 8

Shape Continuity and Optimization 1

Introduction and Generic Examples

The underlying philosophy behind Chapters 5, 6, and 7 was to introduce metric topologies on sets associated with families of functions parametrized by sets rather than parametrize sets by functions. In the case of the characteristic function we have seen several examples where it appears explicitly in the modeling of the problem. The distance functions provide other topologies and constructions. For instance the oriented distance function gives a direct analytic access to geometric entities such as the normal and the fundamental forms on the boundary of a geometric domain without ad hoc local bases or Christoﬀel symbols. Many of the advantages of working in a truly intrinsic framework for shape derivatives will be illustrated in Chapters 9 and 10. Now that we have meaningful topologies on sets, we can consider the continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc. In this chapter we concentrate on continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the objective function is deﬁned as an inﬁmum over a family of functions over a ﬁxed domain or set such as eigenvalue and compliance problems. What is nice about them is that no adjoint system is necessary to characterize the minimizing function. Other problems have the structure of optimal control theory, that is, a state equation depending on the control that describes the evolution of the state and an objective functional that depends on the state and the control. In that case, the characterization of the optimal control involves an adjoint state equation coupled with the state equation. In the context of shape and geometric optimization the control will be the underlying geometry. The state equation will be a static or dynamical partial differential equation on the domain, and the objective functional will depend on the domain and the state that itself depends on the domain. As in control theory, we need continuity of the objective functional and the state with respect to the geometry and compactness of the family of domains over which the optimization takes 409

410

Chapter 8. Shape Continuity and Optimization

place. Since it is not possible to give a complete general theory of shape optimization problems within this book, we choose to concentrate on a few simple generic examples that are illustrative of the underlying technicalities involved. Several examples of optimization problems involving the Lp -topology on measurable sets via the characteristic function were given in Chapter 5. In this chapter, we ﬁrst characterize the continuity of the transmission problem and the upper semicontinuity of the ﬁrst eigenvalue of the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value problems with respect to their underlying domain of deﬁnition since they require diﬀerent constructions and topologies that are generic of the two types of boundary conditions even for more complex nonlinear partial diﬀerential equations. In the above problems the strong continuity of solutions of the elliptic equation or the eigenvector equation with respect to the underlying domain is the key element in the proof of the existence of optimal domains. To get that continuity, some extra conditions have to be imposed on the family of open domains. Those issues have received a lot of attention for the Laplace equation with homogeneous Dirichlet boundary conditions. With a sequence of domains in a ﬁxed holdall D associate a sequence of extensions by zero of the solutions in the ﬁxed space H01 (D). The Poincar´e inequality is uniform for that sequence, as the ﬁrst eigenvalue of the Laplace equation in each domain is dominated by the one associated with the larger holdall D. By a classical compactness argument, the sequence of extensions converges to some limiting element y in H01 (D). To complete the proof of the continuity, two more fundamental questions remain. Is y the solution of the Laplace equation for the limit domain? Does y satisfy the Dirichlet boundary condition on the boundary of the limit domain? The ﬁrst issue can be resolved by assuming that the Sobolev spaces associated with the moving domains converge in the Kuratowski sense, i.e., with the property that any element in the Sobolev space associated with the limit domain can be approached by a sequence of elements in the moving Sobolev spaces. That property is obtained in examples where the compactivorous property 1 follows from the choice of the deﬁnition of convergence of the domains. If the domains are open subsets of D, then the complementary Hausdorﬀ topology has that property. The family of subsets of D satisfying a uniform fat segment property has that property. For the second issue it is necessary to impose some constraints at least on the limit domain. The stability condition introduced by J. Rauch and M. Taylor [1] and used by E. N. Dancer [1] and D. Daners [1] precisely assumes that the limit domain is suﬃciently smooth, just enough to have the solution in H01 (Ω). Of course, assuming such a regularity on the limit domain makes things easier but is a little bit artiﬁcial. The more fundamental issue is to identify the families of domains for which this stability property is preserved in the limit. The uniform cone property was used in the context of shape optimization by D. Chenais [1, 2] in ´sio [3, 5, 6, 8, 9] showed that this is true 1973. In 1994 D. Bucur and J.-P. Zole under general capacitary conditions and have constructed new compact subfamilies 1 A sequence of open sets converging to a limit open set in some topology has the compactivorous property if any compact subset of the limit set is contained in all sets of the sequence after a certain rank: the sequence eats up the compact set after a certain rank (cf. Theorem 2.4 (iii) in Chapter 6).

1. Introduction and Generic Examples

411

of domains with respect to the Hausdorﬀ complementary topology. Furthermore ˇ ´ k [1, 2] in D. Bucur [1] proved that the condition given in 1993 by V. Sver a dimension 2 involving a bound on the number of connected components of the complement of the domain can be recovered from the more general capacity conditions that are suﬃcient in the case of the Laplacian with homogeneous Dirichlet boundary conditions. In a more recent paper D. Bucur [5] proved that they are almost necessary. Intuitively, those capacity conditions are such that, locally, the complement of the domains in the family under consideration has enough capacity to preserve and retain the homogeneous Dirichlet boundary condition in the limit.

1.1

First Generic Example

The ﬁrst example is the optimization of the ﬁrst eigenvalue of an associated elliptic diﬀerential operator: sup λA (Ω) A∇ϕ · ∇ϕ dx Ω∈A(D) def A Ω , (1.1) inf 1 λ (Ω) = A |ϕ|2 dx 0 = ϕ∈H (Ω) inf λ (Ω) 0 Ω Ω∈A(D) where A(D) is a family of admissible open subsets of D.

1.2

Second Generic Example

As a second generic example, consider the transmission problem over the ﬁxed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D: ﬁnd y = y(χΩ ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D), (k1 χΩ + k2 (1 − χΩ )) ∇y · ∇ϕ dx = f, ϕ (1.2) D

for some strictly positive constants k1 and k2 , α = max{k1 , k2 } > 0, f ∈ H −1 (D), H −1 (D) the topological dual of H01 (D), and ·, · denotes the duality pairing between H −1 (D) and H01 (D). This problem was studied in section 4 of Chapter 5 in the context of the compliance problem with two materials.

1.3

Third Generic Example

As a third example, consider the minimization of the objective function def 1 |uΩ − g|2 dx J(Ω) = 2 Ω

(1.3)

over the family of open subsets Ω of a bounded open holdall D of RN , where g in L2 (D), uΩ ∈ H01 (Ω) is the solution of the variational problem ∀ϕ ∈ H01 (Ω), A∇uΩ · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω) , (1.4) Ω

A ∈ L∞ (D; L(RN , RN )) is a matrix function on D such that ∗A = A and αI ≤ A ≤ βI for some coercivity and continuity constants 0 < α ≤ β, and f |Ω denotes the restriction of f in H −1 (D) to H −1 (Ω).

412

1.4

Chapter 8. Shape Continuity and Optimization

Fourth Generic Example

As a fourth generic example, consider the previous example but with the homogeneous Dirichlet boundary value problem replaced by the homogeneous Neumann boundary value problem of the mathematician for the Laplacian ∂y = 0 on ∂Ω. ∂n

−∆y + y = f in Ω,

(1.5)

The classical (physical) homogeneous Neumann boundary value problem ∂y = 0 on ∂Ω ∂n

−∆y = f in Ω, has a unique solution up to a constant if

2

D

(1.6)

f dx = 0.

Upper Semicontinuity and Maximization of the First Eigenvalue

Given a bounded open nonempty domain D ⊂ RN , consider the minimization of the Raleigh quotient for an open nonempty subset Ω of D: A∇ϕ · ∇ϕ dx def D inf1 A∇ϕ · ∇ϕ dx, (2.1) λA (Ω) = = inf |ϕ|2 dx 0=ϕ∈H0 (Ω;D) ϕ∈H01 (Ω;D) D D ϕL2 =1

where A ∈ L∞ (D; L(RN , RN )) is a matrix function on D such that ∗

and αI ≤ A(x) ≤ βI

A(x) = A(x)

(2.2)

for some coercivity and continuity constants 0 < α ≤ β. We set λA (∅) = +∞. The minimizers are characterized as follows. Theorem 2.1. (i) Given a bounded open nonempty domain Ω in RN , there exist uΩ ∈ H01 (Ω; D), uΩ = 0, and λA (Ω) > 0 such that A∇uΩ · ∇ϕ − λA (Ω) uΩ ϕ dx = 0 (2.3) ∀ϕ ∈ H01 (Ω; D), Ω

and ∀ϕ ∈

H01 (Ω; D),

A

λ (Ω) =

D

A∇uΩ · ∇uΩ dx ≤ |uΩ |2 dx D

D

A∇ϕ · ∇ϕ dx . |ϕ|2 dx D

Conversely, if there exist u ∈ H01 (Ω; D), u = 0, and λ such that A∇u · ∇ϕ − λ u ϕ dx = 0 ∀ϕ ∈ H01 (Ω; D), Ω

(2.4)

(2.5)

2. Upper Semicontinuity and Maximization of the First Eigenvalue and

∀ϕ ∈ H01 (Ω; D),

then

λ=

D

λ≤

D

A∇ϕ · ∇ϕ dx , |ϕ|2 dx D

A∇u · ∇u dx = inf |u|2 dx ϕ∈H01 (Ω;D) D

D

413

(2.6)

A∇ϕ · ∇ϕ dx , |ϕ|2 dx D

u is a nonzero solution of the minimization problem (2.1), and λ is equal to the ﬁrst eigenvalue λA (Ω) associated with Ω. (ii) Given a bounded open nonempty domain D, there exists λA D > 0 (that depends only on the diameter of D and A) such that for all open subsets Ω of D, λA (Ω) ≥ λA (D) ≥ λA D > 0, where λD (Ω) is the ﬁrst eigenvalue of the symmetrical linear operator ϕ → −div (A ∇ϕ) on D. Proof. (i) For any bounded open Ω, the inﬁmum is bounded below by 0 and hence ﬁnite. Let {ϕn } be a minimizing sequence such that ϕn L2 (Ω) = 1. By coercivity, α ∇ϕn 2L2 (Ω) is bounded. Hence {ϕn } is a bounded sequence in H01 (Ω) and there exist ϕ ∈ H01 (Ω) and a subsequence, still indexed by n, such that ϕn ϕ in H 1 (Ω)-weak. By the Rellich–Kondrachov compactness theorem, the subsequence strongly converges in L2 (Ω). Then ϕ2n dx → ϕ2 dx and A∇ϕ · ∇ϕ dx ≤ lim inf A∇ϕn · ∇ϕn dx 1= n→∞ Ω Ω Ω Ω A∇ϕ · ∇ϕ dx A∇ϕn · ∇ϕn dx ⇒ Ω 2 ≤ lim inf Ω 2 = λA (Ω). n→∞ ϕ dx ϕ dx Ω Ω n By deﬁnition of λA (Ω), 0 = ϕ ∈ H01 (Ω) is a minimizing element and inequality (2.4) is veriﬁed. Since ϕ = 0, the Rayleigh quotient is diﬀerentiable in ϕ and its directional derivative in the direction ψ is given by A∇ϕ · ∇ψ dx ϕ ψ dx Ω 2 −2 A∇ϕ · ∇ϕ dx Ω 4 . 2 ϕL2 (Ω) ϕL2 (Ω) Ω A nonzero solution ϕ of the minimization problem on Ω is necessarily a stationary point, and for all ψ ∈ H01 (Ω), A∇ϕ · ∇ϕ dx A Ω A∇ϕ · ∇ψ dx = ϕ ψ dx = λ (Ω) ϕ ψ dx ϕ2L2 (Ω) Ω Ω Ω and we get (2.3). Conversely, any λ and nonzero u ∈ H01 (Ω; D) that verify (2.5) yield λ ≥ λA (Ω). But, from inequality (2.6), λ ≤ λA (Ω) and λ = λA (Ω). Hence, u is a minimizer of the Rayleigh quotient over H01 (Ω; D).

414

Chapter 8. Shape Continuity and Optimization

(ii) Let diam (D) be the diameter of D and x ∈ RN be a point such that D ⊂ Bx = B(x, diam (D)). Therefore, from Theorem 2.3 in Chapter 2, H01 (Ω; Bx ) ⊂ H01 (D; Bx ) ⊂ H01 (Bx ),

H01 (Ω; D) ⊂ H01 (D)

with the associated isometries. Hence, by deﬁnition of the minimum of a functional over an increasing sequence of sets, λA (Ω) ≥ λA (D) ≥ λA (Bx ). If λA (Bx ) = 0, we repeat the above construction with H01 (Bx ) in place of H01 (Ω; Bx ) and end up with an element ϕ ∈ H01 (Bx ) such that ϕ2 dx = 1 and α |∇ϕ|2 dx ≤ A∇ϕ · ϕ dx = 0, Bx

Bx

Bx

which is impossible in H01 (Bx ). Finally λA (Bx ) is independent of the choice of x since the eigenvalue is invariant under a translation of the domain: it depends only on the diameter of D. The case λA (∅) = +∞ is compatible with the results. Remark 2.1. For A equal to the identity operator I, denote by λ(Ω), λ(D), and λD the quantities 1 λA (Ω), λA (D), and λA D of Theorem 2.1. For any open Ω ⊂ D and any ϕ ∈ H0 (Ω; D) 1 ϕL2 (D) ≤ √ ∇ϕL2 (D) λD 4 1 ⇒ ∀Ω open ⊂ D, ϕH 1 (Ω) ≤ 1 + ∇ϕL2 (Ω) λD

(2.7) (2.8)

and the spaces H01 (Ω) and H01 (D) will be endowed with the respective equivalent norms ∇ϕL2 (Ω) and ∇ϕL2 (D) . Before considering the existence of minimizers/maximizers, we investigate the question of the continuity of the ﬁrst eigenvalue λA (Ω) with respect to Ω for the uniform complementary Hausdorﬀ topology on def

Cdc (D) = dΩ : ∀Ω open subset in D and Ω = RN (cf. (2.11) of Deﬁnition 2.2 in Chapter 6). Theorem 2.2. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). The mapping dΩ → λA (Ω) : Cdc (D) ⊂ C0 (D) → R ∪{+∞}

(2.9)

is upper semicontinuous. Proof. Since λA (Ω) is ﬁnite in Ω = ∅ and +∞ in Ω = ∅, it is upper semicontinuous in ∅. For Ω = ∅, consider a sequence {Ωn } of open subsets of D such that dΩn → dΩ in C(D) and the associated sequence {λA (Ωn )}. We want to show that lim sup λA (Ωn ) ≥ λA (Ω).

2. Upper Semicontinuity and Maximization of the First Eigenvalue

415

Consider the new sequence {Ωn ∩Ω}. Since Ω ⊂ D, for all n, (Ωn ∩Ω) = Ωn ∪Ω ⊃ D = ∅ and d(Ωn ∩Ω) = dΩn ∪Ω = min{dΩn , dΩ } → dΩ in C(D). Let un ∈ H01 (Ωn ∩ Ω), un L2 (D) = 1, be an eigenvector for the eigenvalue λn = λA (Ωn ∩ Ω). From Theorem 2.1 un is solution of the variational equation (2.3): A∇un · ∇ϕn dx = λn un ϕn dx. (2.10) ∀ϕn ∈ H01 (Ωn ; D), D

D

Since Ω = ∅, there exists a compact ball Kρ = B(x, ρ) ⊂ Ω for some x ∈ Ω and ρ > 0. By the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6, there exists an integer N (Kρ ) such that for all n ≥ N (Kρ ), Kρ ⊂ Ωn ∩ Ω and by Theorem 2.1 (ii), 0 < λn ≤ λA (B(x, ρ)). Therefore, the sequence of eigenvalues {λn } is bounded. By coercivity α∇un }2L2 (D) ≤ λn ∇un }2L2 (D) = λn and {un } is bounded in H01 (Ω; D) since Ωn ∩ Ω ⊂ Ω. Letting λ∗ = lim sup λn , there exist subsequences, still indexed by n, and u ∈ H01 (Ω; D) such that λn → λ ∗ ,

un L2 (D) = 1,

un u in H01 (Ω; D)-weak

⇒ uL2 (D) = 1.

Since for all n, λA (Ωn ) ≤ λA (Ωn ∩ Ω), it is readily seen that lim sup λA (Ωn ) ≤ lim sup λA (Ωn ∩ Ω) = λ∗ and that it is suﬃcient to show that λ∗ = λA (Ω) to get the upper semicontinuity. We cannot directly go to the limit in (2.10), since the test functions ϕn depend on Ωn . To get around this diﬃculty, recall the density of D(Ω) in H01 (Ω; D) and use the compactivorous property a second time. For any ϕ ∈ D(Ω), K = supp ϕ ⊂ Ω is compact and there exists N (K) such that for all n ≥ N (K), supp ϕ ⊂ Ωn ∩ Ω. Therefore, for all n ≥ N (K) 1 1 un ∈ H0 (Ωn ∩ Ω; D) ⊂ H0 (Ω; D), A∇un · ∇ϕ dx = λn un ϕ dx. D

By letting n → ∞

A∇u · ∇ϕ dx = λ∗

∀ϕ ∈ D(Ω), D

D

u ϕ dx, D

and, by density of D(Ω) in H01 (Ω; D), the variational equation is veriﬁed in H01 (Ω; D). Hence, u ∈ H01 (Ω; D), uL2 (Ω) = 1, satisﬁes the variational equation A∇u · ∇ϕ dx = λ∗ u ϕ dx. ∀ϕ ∈ H01 (Ω; D), D

D

416

Chapter 8. Shape Continuity and Optimization

Therefore, since λA (Ω) is the smallest eigenvalue with respect to H01 (Ω; D), λA (Ω) ≤ λ∗ . To show that λ∗ = λA (Ω), it remains to prove that λ∗ ≤ λA (Ω). For each n A∇ϕn · ∇ϕn dx 1 A . ∀ϕn ∈ H0 (Ωn ∩ Ω; D), λ (Ωn ∩ Ω) ≤ D ϕn 2L2 (D) By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N such that supp ϕ ⊂ Ω ∩ Ωn for all n ≥ N . Therefore for all n ≥ N A∇ϕ · ∇ϕ dx A λ (Ωn ∩ Ω) ≤ D ϕ2L2 (D) and by taking the lim sup

∀ϕ ∈ D(Ω),

∗

λ ≤

D

A∇ϕ · ∇ϕ dx . ϕ2L2 (D

By density this is true for all ϕ ∈ H01 (Ω; D), ϕ = 0, and λ∗ ≤ λA (Ω). As a result λ∗ = λA (Ω). Finally, λA (Ωn ) ≤ λA (Ωn ∩ Ω)

⇒ lim sup λA (Ωn ) ≤ lim sup λA (Ωn ∩ Ω) = λ∗ = λA (Ω),

and we have the upper semicontinuity of the map dΩ → λA (Ω). In view of the previous theorem, we now deal with the maximization of the ﬁrst eigenvalue λA (Ω) associated with the operator A with respect to a family of open subsets Ω of D. Theorem 2.3. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). (i) Then λA (Ω) = λA (∅) = +∞.

sup

(2.11)

Ω∈Cdc (D)

The conclusion remains true when Cdc (D) is replaced by Cdc (D) = {dΩ : Ω convex and open ⊂ D}

(2.12)

(cf. Theorem 8.6 in Chapter 6). (ii) Given an open set E, ∅ = E ⊂ RN , consider the compact family c Cd,loc (E; D)

=

dΩ

∃x ∈ RN , ∃A ∈ O(N) such that : x + AE ⊂ Ω open ⊂ D

&

of Theorem 2.6 in Chapter 6. Then, there exists an open Ω∗ such that dΩ∗ ∈ c Cd,loc (E; D) and λA (Ω∗ ) =

sup c Ω∈Cd,loc (E;D)

λA (Ω) ≤ β λI (E).

(2.13)

3. Continuity of the Transmission Problem

417

c (E; D) is replaced by Cdc (E; D) or The conclusion remains true when Cd,loc c c the subfamilies of convex subsets Cd (E; D) and Cd,loc (E; D) of Theorem 8.6 in Chapter 6.

(iii) There exists Ω∗ ∈ L(D, r, O, λ) such that λA (Ω∗ ) =

sup

λ(Ω) ≤ βλI (O).

(2.14)

Ω∈L(D,r,O,λ)

Proof. (i) From Theorem 2.2 and the compactness of Cdc (D) in C(D). (ii) All the families are compact by Theorems 2.6 and 8.6 in Chapter 6. So there is a maximizer dΩ∗ of λA (Ω). The upper bound is obvious. (iii) By compactness of L(D, r, O, λ) in W 1,p (D) and hence in Cdc (D) for the C(D)-topology.

3

Continuity of the Transmission Problem

As a second generic example, consider the transmission problem over the ﬁxed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D: ﬁnd y = y(χΩ ) ∈ H01 (D) such that (k1 χΩ + k2 (1 − χΩ )) ∇y · ∇ϕ dx = f, ϕ (3.1) ∀ϕ ∈ H01 (D), D

for some strictly positive constants k1 and k2 , α = max{k1 , k2 } > 0, f ∈ H −1 (D), H −1 (D) the topological dual of H01 (D), and ·, · denotes the duality pairing between H −1 (D) and H01 (D). This problem was studied in section 4 of Chapter 5 in the context of the compliance problem with two materials. Theorem 3.1. Let D be a bounded open domain in RN and k1 and k2 be some strictly positive constants such that α = max{k1 , k2 } > 0 and f ∈ H −1 (D). Denote by y(χ) the solution of the transmission problem (3.1) associated with χ ∈ X(D). The mapping χ → y(χ) : X(D) → H01 (D) is continuous when X(D) is endowed with the strong Lp (D)-topology, 1 ≤ p < ∞. The same result is true with co X(D) in place of X(D). Remark 3.1. This theorem applies to the compact families of sets of Theorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat segment property and for the conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 of Chapter 2) since the convergence of bΩn → bΩ in W 1,p (D) implies the convergence of χΩn → χΩ in Lp (D). It also applies to the compact family of convex sets C(D) = {χΩ : Ω convex subset of D} introduced in (3.24) in section 3.4 of Chapter 5 (cf. Theorem 3.5 (iii) and Corollary 1 to Theorem 6.3 in section 6.1 of Chapter 5).

418

Chapter 8. Shape Continuity and Optimization

´sio [2], [24, Proof of Theorem 3.1. The proof follows the arguments of J.-P. Zole sect. 4.2.1, pp. 446–448]. It is suﬃcient to prove that for any convergent sequence χn → χ in L2 (D)-strong, the corresponding solutions yn = y(χn ) ∈ H01 (D) converge to y = y(χ) in H01 (D)-strong. First, the sequence {yn } is bounded in H01 (D): α∇yn 2L2 (D) ≤ (k1 χn + k2 (1 − χn )) ∇yn · ∇yn dx D

⇒ α∇yn 2L2 (D)

= f, yn ≤ f H −1 (D) yn H01 (D) 4 1 ≤ f H −1 (D) yn H01 (D) ≤ f H −1 (D) 1 + ∇yn L2 (Ω) λD

since we have the equivalence of norms from (2.7) in Remark 2.1. So there is a subsequence, still denoted by {yn }, that weakly converges to some y ∈ H01 (D). To show that y is solution of (3.1) corresponding to χ, we go to the limit in ∀ϕ ∈ H01 (D), (k1 χn + k2 (1 − χn )) ∇ϕ · ∇yn dx = f, ϕ. D

Since (k1 χn +k2 1−χn ))∇ϕ → (k1 χ+k2 (1−χ))∇ϕ in L2 (D)-strong and ∇yn → ∇y in L2 (D)-weak, we get that y ∈ H01 (D) is the unique solution of ∀ϕ ∈ H01 (D), (k1 χ + k2 (1 − χ)) ∇y · ∇ϕ dx = f, ϕ. D

Finally, we prove the strong convergence. Indeed (k1 χn + k2 (1 − χn )) ∇(yn − y) · ∇(yn − y) dx α ∇(yn − y)2L2 (D) ≤ D ≤ (k1 χn + k2 (1 − χn )) ∇yn · ∇yn dx D (k1 χn + k2 (1 − χn )) ∇yn · ∇y dx −2 D (k1 χn + k2 (1 − χn )) ∇y · ∇y dx + D (k1 χn + k2 (1 − χn )) ∇y · ∇y dx =f, yn − 2 f, y + D (k1 χ + k2 (1 − χ)) ∇y · ∇y dx = 0. → −f, y + D

Since every strongly convergent subsequence converges to the same limit, the whole sequence strongly converges to y in H01 (D).

4 4.1

Continuity of the Homogeneous Dirichlet Boundary Value Problem Classical, Relaxed, and Overrelaxed Problems

We have seen in section 3.5 of Chapter 5 two relaxations of the homogeneous Dirichlet boundary value problem for the Laplacian to measurable subsets Ω of a bounded

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

419

open holdall D by introducing the following closed subspaces of H01 (D): def

H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D ,

def H•1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )∇ϕ = 0 a.e. in D

(4.1) (4.2)

(cf. (3.27) and (3.28) in Theorem 3.6 of Chapter 5). It is readily seen that H1 (Ω; D) ⊂ H•1 (Ω; D). For Ω open, H01 (Ω; D) ⊂ H1 (Ω; D) ⊂ H•1 (Ω; D) and the three closed subspaces of H01 (D) are generally not equal as can be seen from Examples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5. The choice of a relaxation is very much problem dependent and should be guided by the proper modeling of the underlying physical or technological phenomenon at hand. In general, there is a price to pay for the above relaxations. Both H1 (Ω; D) and H•1 (Ω; D) depend on the equivalence class of measurable sets [Ω] and not speciﬁcally on Ω. In general, there is not even an open representative of Ω in the class. From section 3.3 in Chapter 5, [Ω] = [I], where I is the measure theoretic interior of Ω, m(B(x, r) ∩ Ω) N I = x ∈ R : lim =1 , r0 m(B(x, r)) int I = Ω1 , where

Ω1 = x ∈ RN : ∃ρ > 0 such that m Ω ∩ B(x, ρ) = m B(x, ρ) is open, and int Ω ⊂ int I and ∂I ⊂ ∂Ω (cf. Deﬁnition 3.3 and Theorem 3.3 in Chapter 5). In view of the cleaning operation in the deﬁnition of I, all cracks of zero measure will be deleted or not seen by the functions of the spaces H1 (Ω; D) and H•1 (Ω; D). When mN (∂Ω) = 0, [int I] = [Ω] and int I = Ω1 is an open representative in the class [Ω]. Let D be a bounded open subset of RN and let f ∈ H −1 (D), where H −1 (D) is the topological dual of H01 (D). For all open subsets Ω of D ϕ → f, ϕH −1 (D)×H01 (D) : D(Ω) → R

(4.3)

is well-deﬁned and continuous with respect to the H 1 (Ω)-topology and extends to an element f |Ω ∈ H −1 (Ω). The map ϕ → f, ϕH −1 (D)×H01 (D) : H1 (Ω; D) → R is also well-deﬁned and continuous with respect to the closed subspace H1 (Ω; D) of H01 (D) and deﬁnes an element f |Ω of the dual H1 (Ω; D) of H1 (Ω; D). Similarly, the map ϕ → f, ϕH −1 (D)×H01 (D) : H•1 (Ω; D) → R is also well-deﬁned and continuous with respect to the closed subspace H•1 (Ω; D) of H01 (D) and deﬁnes an element f |•Ω of the dual H•1 (Ω; D) of H•1 (Ω; D). Let A ∈ L∞ (D; L(RN , RN )) be a symmetrical matrix function on D such that ∗

A=A

and αI ≤ A ≤ βI

(4.4)

420

Chapter 8. Shape Continuity and Optimization

for some coercivity and continuity constants 0 < α ≤ β. Consider the classical homogeneous Dirichlet boundary value problem for the generalized Laplacian ∃y(Ω) ∈ H01 (Ω; D) such that, A∇y(Ω) · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω) , ∀ϕ ∈ H01 (Ω; D),

(4.5)

D

the relaxed homogeneous Dirichlet boundary value problem ∃y(χΩ ) ∈ H1 (Ω; D) such that, 1 A∇y(χΩ ) · ∇ϕ dx = f |Ω , ϕH1 (Ω;D) ×H1 (Ω;D) , ∀ϕ ∈ H (Ω; D),

(4.6)

D

and the overrelaxed homogeneous Dirichlet boundary value problem ∃y(χΩ ) ∈ H•1 (Ω; D) such that, A∇y(χΩ ) · ∇ϕ dx = f |•Ω , ϕH•1 (Ω;D) ×H•1 (Ω;D) . ∀ϕ ∈ H•1 (Ω; D),

(4.7)

D

By convention, set y(∅) = 0 and y(χ∅ ) = 0 as elements of H01 (D). In addition we shall keep the terminology homogeneous Dirichlet boundary value problem for what we just called the classical homogeneous Dirichlet boundary value problem for the generalized Laplacian. Since, in general, H01 (Ω; D) is a distinct closed subspace of H1 (Ω; D), a solution of (4.6) is not necessarily a solution of (4.5). In the other direction, a solution y = y(Ω) ∈ H01 (Ω; D) of (4.5) belongs to H1 (Ω; D) and is a solution of ∃y ∈ H1 (Ω; D) such that, A∇y · ∇ϕ dx = f |Ω , ϕH1 (Ω;D) ×H1 (Ω;D) , ∀ϕ ∈ H01 (Ω; D),

(4.8)

D

but y is not necessarily unique in H1 (Ω; D) since this variational equation is veriﬁed only on the closed subspace H01 (Ω; D) of H1 (Ω; D). However, the space H1 (Ω; D) is very close to the space H01 (Ω; D) and the two spaces will coincide on smooth sets and more generally on crack-free sets that will be introduced in section 7 (cf. Deﬁnition 7.1 (ii) and Theorem 7.3 (ii)). The following terminology has been introduced by J. Rauch and M. Taylor [1]. Deﬁnition 4.1. An open subset Ω of D is said to be stable with respect to D if H01 (Ω; D) = H1 (Ω; D). The same considerations apply to the overrelaxed problem (4.7) in H•1 (Ω; D) with the additional observation that even for sets with a very smooth boundary, the solution does not coincide with the one of the (classical) Dirichlet problem when the Ω has more than one connected component as seen from Examples 3.3, 3.4, and 3.5 in section 3.5 of Chapter 5. The problem in H•1 (Ω; D) is associated with diﬀerent physical phenomena.

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

4.2

421

Classical Dirichlet Boundary Value Problem

We have the following general results. Theorem 4.1. Let D be a bounded open nonempty subset in RN . Associate with each open subset Ω of D the solution y(Ω) ∈ H01 (Ω; D) of the classical Dirichlet problem (4.5) on Ω. (i) Let Ω and the sequence {Ωn } be open subsets of D such that dΩn → dΩ in C(D). Then y(Ωn ∩ Ω) → y(Ω) in H01 (D)-strong

(4.9)

and there exist a subsequence {Ωnk } and y0 ∈ H1 (Ω; D) such that y(Ωnk ) y0 in H01 (D)-weak

(4.10)

and y0 is a (not necessarily unique) solution of (4.8). (ii) The function dΩ → y(Ω) : Cdc (D) → H01 (D)-strong

(4.11)

is continuous for each Ω such that H01 (Ω; D) = H1 (Ω; D). Remark 4.1. We have the additional information that y0 ∈ H1 (Ω; D) ⊂ H01 (int Ω; D) in part (i) of the theorem (cf. Theorem 7.3 (i)). Remark 4.2. The projection operator PΩ : H01 (D) → H01 (Ω; D)

(4.12)

is deﬁned for u ∈ H01 (D) as the solution of the variational equation ∃PΩ u ∈ H01 (Ω; D), ∀ϕ ∈ H01 (Ω; D), ∇(PΩ u) · ∇ϕ dx = ∇u · ∇ϕ dx. D

D

Applying the theorem to a sequence of open subsets {Ωn } of D converging to Ω in the complementary Hausdorﬀ topology with def f, ϕ = ∇u · ∇ϕ dx, D

there exist u0 ∈ H1 (Ω; D) and a subsequence {Ωnk } such that dΩn → dΩ in C(D),

PΩnk u u0 in H01 (D)-weak.

The projection operator PΩ is continuous at Ω if H01 (Ω; D) = H1 (Ω; D), but this is only a suﬃcient condition.

422

Chapter 8. Shape Continuity and Optimization

Proof. (i) The proof of the continuity of (4.9) follows the same steps as the proof of the upper semicontinuity of the ﬁrst eigenvalue of the Laplacian in Theorem 2.2. As for the convergence (4.10), it is readily seen that the sequence {yn = y(Ωn )} of elements of H01 (Ωn ; D) is bounded in the bigger space H01 (D): by coercivity 4 1 2 α ∇yn L2 (D) ≤ f H −1 (D) yn H01 (D) ≤ 1 + f H −1 (D) ∇yn L2 (D) λD from Remark 2.1, where λD > 0 is the ﬁrst eigenvalue of the Laplacian on D. There exist a subsequence, still indexed by n, and some y0 in H01 (D) such that yn y0 in H01 (D)-weak and hence yn → y0 in L2 (D)-strong. From Corollary 1 to Theorem 4.4 in Chapter 6 ∀x ∈ RN , ⇒ ∀ a.a. x ∈ RN ,

lim inf χΩn (x) ≥ χΩ (x) n→∞

0 = lim inf (1 − χΩn (x))|yn (x)|2 ≥ (1 − χΩ (x))|y0 (x)|2 n→∞

since (1 − χΩn (x)) ≥ 0, |yn | ≥ 0, and |yn |2 → 0 a.e. in D. By deﬁnition, y0 ∈ H1 (Ω; D). Finally, by the compactivorous property of Theorem 2.4 in section 2.3 of Chapter 6, for each ϕ ∈ D(Ω) there exists N such that for all n > N , supp ϕ ⊂ Ωn . Therefore, by substituting ϕ in (4.5) on Ωn , for all n > N A∇yn · ∇ϕ dx = f |Ωn , ϕH −1 (Ωn )×H01 (Ωn ) = f |Ω , ϕH −1 (Ω)×H01 (Ω) (4.13) 2

D

and by going to the limit

∀ϕ ∈ D(Ω), D

A∇y0 · ∇ϕ dx = f |Ω , ϕH −1 (Ω)×H01 (Ω)

(4.14)

and, by density of D(Ω) in H01 (Ω; D), y0 ∈ H1 (Ω; D) is a solution of (4.8). (ii) If, in addition, H1 (Ω; D) = H01 (Ω; D), then y0 ∈ H01 (Ω; D) is the unique solution of (4.5). Hence y0 = y(Ω), solution of the classic problem (4.5), is independent of the choice of the converging subsequence. Therefore the whole sequence {yn } converges to y0 = y(Ω) in H01 (D)-weak. The strong continuity in H01 (D) now follows from the convergence of un → u0 2 in L (D)-strong: α∇(un − u0 )2L2 (D) ≤ A∇(un − u0 ) · ∇(un − u0 ) dx D = A∇un · ∇un dx + A∇u0 · ∇u0 dx − 2 A∇un · ∇u0 dx D D D A∇un · ∇u0 dx = f, un + f, u0 − 2 D A∇u0 · ∇u0 dx = 0 → 2 f, u0 − 2 D

and the whole sequence {yn } converges to y0 = y(Ω) in H01 (D)-strong. Therefore, the function (4.11) is continuous in Ω.

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

4.3

423

Overrelaxed Dirichlet Boundary Value Problem

In this section we concentrate on the overrelaxation (4.7) of the homogeneous Dirichlet boundary value problem. To show the continuity with respect to the characteristic function of domains, we ﬁrst approximate its solution by transmission problems. 4.3.1

Approximation by Transmission Problems

The overrelaxed Dirichlet problem (4.7) corresponding to χ = χΩ ∈ X(D) is ﬁrst approximated by a transmission problem over D indexed by ε > 0 going to zero: ﬁnd yε = yε (χ) ∈ H01 (D) such that ∀ϕ ∈ H01 (D) 1 χ + (1 − χ) ∇yε · ∇ϕ dx = χf ϕ dx. ε D D

(4.15)

They are special cases of the general transmission problem (3.1) over the ﬁxed bounded open nonempty domain D associated with the Laplace equation and a measurable subset Ω of D for k1 = 1 and k2 = 1/ε and χ = χΩ . Theorem 4.2. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H•1 (Ω; D) be the solution of the overrelaxed Dirichlet problem (4.7). For ε, 0 < ε < 1, let yε be the solution of problem (4.15) in H01 (D). Then y(χΩ ) = lim yε in H01 (D)-strong. ε→0

Proof. For ε, 0 < ε < 1, substitute ϕ = yε in (4.15) to get the bounds √ ∇yε L2 (D) ≤ c(D) f L2 (D) , (1 − χΩ ) ∇yε L2 (D) ≤ ε c(D) f L2 (D) . There exist a sequence εn 0 and y0 ∈ H01 (D) such that yn = yεn y0 in H01 (D)weak and (1 − χΩ ) ∇yn → 0 in L2 (D)-strong. Therefore, (1 − χΩ ) ∇y0 = 0 and y0 ∈ H•1 (Ω; D). For ϕ ∈ H•1 (Ω; D), (4.15) reduces to ∇yε · ∇ϕ dx = χΩ f ϕ dx ∀ϕ ∈ H•1 (Ω; D), D

and as n → ∞

D

∀ϕ ∈

H•1 (Ω; D),

∇y0 · ∇ϕ dx =

D

χΩ f ϕ dx. D

Hence y0 ∈ H•1 (Ω; D) is the solution of (4.7). Since the limit is the same for all convergent sequences, yε → y0 in H01 (D)-weak. By subtracting the last two equations ∀ϕ ∈ H•1 (Ω; D), [∇yε − ∇y0 ] · ∇ϕ dx = 0 ⇒ [∇yε − ∇y0 ] · ∇y0 dx = 0. D

D

424

Chapter 8. Shape Continuity and Optimization

Finally, we estimate the following term: 2 |∇(yε − y0 )| dx = [∇yε − ∇y0 ] · ∇yε dx − [∇yε − ∇y0 ] · ∇y0 dx D D D [∇yε − ∇y0 ] · ∇yε dx = D ∇yε · ∇yε dx − ∇y0 · ∇yε dx = D D = χΩ f yε dx − ∇y0 · ∇yε dx D D χΩ f y0 dx − ∇y0 · ∇y0 dx = 0. → D

As a result yε → y0 in 4.3.2

D

H01 (D)-strong.

Continuity with Respect to X(D) in the Lp (D)-Topology

Theorem 4.3. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H•1 (Ω; D) be the solution of the overrelaxed Dirichlet problem (4.7). Then the function χΩ → y(χΩ ) : X(D) → H01 (D)-strong

(4.16)

is continuous for all Lp (D)-topologies, 1 ≤ p < ∞, on X(D). Proof. Given χ = χΩ ∈ X(D), let {χΩn } in X(D) be a converging sequence to χ = χΩ . In view of Theorem 4.2, let εn > 0 be such that the approximating solution yn ∈ H01 (D) of (4.15), 1 1 χΩ n + ∀ϕ ∈ H0 (D), (1 − χΩn ) ∇yn · ∇ϕ dx = χΩn f ϕ dx, εn D D is such that yn − y(χΩn )H 1 (D) < 1/n and recall the bounds ∇yn L2 (D) ≤ c(D) f L2 (D) , √ (1 − χΩn ) ∇yn L2 (D) ≤ εn c(D) f L2 (D) ≤ c(D) f L2 (D) .

(4.17) (4.18)

Therefore the sequence {y(χΩn } is also bounded in H01 (D) and there exist subsequences, still indexed by n, and y0 ∈ H01 (D) such that yn y0 and y(χΩn ) y0 in H01 (D)-weak. Moreover, (1 − χΩn ) ∇y(χΩn ) = 0. Hence (1 − χΩ ) ∇y0 = 0 and y0 ∈ H•1 (Ω; D). For ϕ ∈ H•1 (Ω; D) 1 χΩ n + ∀ϕ ∈ H01 (D), (1 − χΩn ) ∇yn · (χΩ ∇ϕ) dx = χΩn f ϕ dx. εn D D

4. Continuity of the Homogeneous Dirichlet Boundary Value Problem

425

χΩ f ϕ dx. The left-hand side can be rewritten as 1 χΩ χΩn ∇yn · ∇ϕ + χΩ (1 − χΩn ) (1 − χΩn ) ∇yn · ∇ϕ dx εn D χΩ ∇y0 · ∇ϕ dx = ∇y0 · ∇ϕ dx →

The right-hand side converges to

D

D

D

since χΩ χΩn → χΩ and χΩ (1 − χΩn ) → 0 in L2 (D)-strong and the bound (4.18). Finally, y0 ∈ H•1 (Ω; D) is solution of 1 ∇y0 · ∇ϕ dx = χΩ f ϕ dx. ∀ϕ ∈ H• (Ω; D), D

D

Since the limit point y0 = y(χΩ ) is independent of the choice of the sequence, y(χΩn ) y(χΩ ) in H01 (D)-weak. Finally, 2 |∇ (y(χΩn ) − y(χΩ ))| dx D 2 2 = |∇y(χΩn )| dx + |∇y(χΩ )| dx − 2 ∇y(χΩn ) · ∇y(χΩ ) dx D D D χΩn f y(χΩn ) dx + χΩ f y(χΩ ) dx − 2 ∇y(χΩn ) · ∇y(χΩ ) dx = D D D χΩ f y(χΩ ) dx − 2 ∇y(χΩ ) · ∇y(χΩ ) dx = 0 →2 D

D

and y(χΩn ) → y(χΩ ) in

H01 (D)-strong.

Remark 4.3. The penalization technique used in this section can be applied to higher-order elliptic, and even parabolic, problems and the Navier–Stokes equation (cf. R. Dziri ´sio [6]) provided that the relaxation is acceptable in the modeling and J.-P. Zole of the physical phenomenon. The continuity of the classical homogeneous Dirichlet problem for the Laplacian will be given in section 6 under capacity conditions.

4.4

Relaxed Dirichlet Boundary Value Problem

In this section we consider the relaxation (4.6) of the homogeneous Dirichlet boundary value problem for the Laplacian to measurable subsets of a bounded open holdall D for the space H1 (Ω; D). Theorem 4.4. Let D be a bounded open nonempty subset in RN and Ω be a measurable subset of D. Let y(χΩ ) ∈ H1 (Ω; D) be the solution of the relaxed Dirichlet problem (4.1). Then the function χΩ → y(χΩ ) : X(D) → H01 (D)-strong is continuous for all Lp (D)-topology, 1 ≤ p < ∞, on X(D).

(4.19)

426

Chapter 8. Shape Continuity and Optimization

Proof. Given χΩ ∈ X(D), let {χΩn } in X(D) be a converging sequence to χΩ and let y(χΩ ) ∈ H1 (Ω; D) and y(χΩn ) ∈ H1 (Ωn ; D) be the corresponding solutions of (4.1). Since H1 (Ω; D) ⊂ H•1 (Ω; D), we have the continuity χΩ → y(χΩ ) from Theorem 4.3. To complete the proof we need to show that the limiting element y(χΩ ) in the proof of Theorem 4.3 is solution of the variational equation (4.1) in the smaller space H1 (Ω; D). Indeed, for each n, (1 − χΩn )y(χΩn ) = 0 implies (1 − χΩ )y(χΩ ) = 0 and y(χΩ ) ∈ H1 (Ω; D). Finally, since y(χΩ ) is solution of the variational equation (4.7) for all ϕ ∈ H•1 (Ω; D), it still is solution for ϕ in the smaller subspace H1 (Ω; D). Hence y(χΩ ) ∈ H1 (Ω; D) is the solution of (4.1).

5

Continuity of the Homogeneous Neumann Boundary Value Problem

As a fourth generic example, consider the homogeneous Neumann boundary value problem of mathematicians for the Laplacian ∂y = 0 on ∂Ω, −∆y + y = f in Ω, ∂n ∇y · ∇ϕ + y ϕ dx = f ϕ dx. ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), Ω

(5.1) (5.2)

Ω

The classical homogeneous Neumann boundary value problem −∆y = f in Ω, ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω),

∂y = 0 on ∂Ω, ∂n ∇y · ∇ϕ dx = f ϕ dx Ω

(5.3) (5.4)

Ω

has a unique solution up to a constant if D f dx = 0. One way to get around the nonuniqueness is to introduce the closed subspace def Ha1 (Ω) = ϕ ∈ H 1 (Ω) : ϕ dx = 0 Ω

of functions with zero average. Then there exists a unique y ∈ Ha1 (Ω) such that ∀ϕ ∈ Ha1 (Ω), ∇y · ∇ϕ dx = f ϕ dx. (5.5) Ω

Ω

Moreover, the solution of (5.5) can be approximated by the ε-problems ∇yε · ∇ϕ + ε yε ϕ dx = f ϕ dx ∃yε ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), Ω

(5.6)

Ω

and yε → y in Ha1 (Ω)-strong. Note that, by assumption on f , yε ∈ Ha1 (Ω). One of the key conditions to get the continuity with respect to the domain Ω is the following density property.

5. Continuity of the Homogeneous Neumann Boundary Value Problem

427

Deﬁnition 5.1. An open set Ω is said to have the H 1 -density property if the set {ϕ|Ω : ϕ ∈ C01 (RN )}

(5.7)

is dense in H 1 (Ω). This property is not veriﬁed for a domain Ω with a crack in R2 , since elements of H 1 (Ω) have diﬀerent traces on each side of the crack. Recall from section 6.1 in Chapter 2 that this condition is veriﬁed for a large class of sets Ω that can be characterized by the geometric segment property that is equivalent to the property that the set be locally a C 0 -epigraph (cf. section 6.2 of Chapter 2). We recall Theorem 6.3 of Chapter 5. Theorem 5.1. If the open set Ω has the segment property, then the set

f |Ω : ∀f ∈ C0∞ (RN ) of restrictions of functions of C0∞ (RN ) to Ω is dense in W m,p (Ω) for 1 ≤ p < ∞ and m ≥ 1. In particular C k (Ω) is dense in W m,p (Ω) for any m ≥ 1 and k ≥ m. Lemma 5.1. Let D be a bounded open subset of RN , let f ∈ L2 (D), and let Ω and the sequence {Ωn } be open subsets of D. Denote by yn ∈ H 1 (Ωn ) the solution of (5.1) in Ωn and by y ∈ H 1 (Ω) the solution of (5.1) in Ω. Assume the following: (i) χΩn → χΩ in L2 (D); (ii) dΩn → dΩ in C(D); (iii) Ω has the H 1 -density property. Then 9n ) → (˜ : in (L2 (D) × L2 (D))-weak, y , ∇y) (˜ yn , ∇y where z˜ denotes the extension by zero of a function z from Ω or Ωn to D. Proof. The proof of the lemma follows the approach of W. Liu and J. E. Rubio [1] in 1992. Since f ∈ L2 (D), we readily have the following estimates: 2 2 2 9n 2 2 ˜ yn 2L2 (D) + ∇y L (D) = yn L2 (Ωn ) + ∇yn L2 (Ωn ) ≤ f L2 (D) .

There exist subsequences, still indexed by n, z ∈ L2 (D), and Z ∈ L2 (D)N such that y˜n z weakly in L2 (RN )

9n Z weakly in L2 (RN )N . and ∇y

By the H 1 -density property of Ω, the variational equation (5.2) on Ωn can be rewritten in the form / . 9n · ∇ϕ + y˜n ϕ dx = χΩn ∇y χΩn f ϕ dx (5.8) ∀ϕ ∈ C01 (RN ), RN

RN

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Chapter 8. Shape Continuity and Optimization

and we can go to the limit as n → ∞: ∀ϕ ∈ C01 (RN ), χΩ [Z · ∇ϕ + z ϕ] dx = RN

χΩ f ϕ dx.

(5.9)

RN

By deﬁnition of the distributional derivative, for all n ∂i yn ϕn dx = − yn ∂i ϕn dx, ∀ϕn ∈ D(Ωn ). Ωn

Ωn

By the compactivorous property (cf. Theorem 2.4 (iii) in Chapter 6) and the convergence dΩn → dΩ in C(D), for each ϕ ∈ D(Ω), there exists N such that for all n > N , supp ϕ ⊂ Ωn . Hence for all n > N y ϕ dx = − y˜n ∂i ϕ dx, ∀ϕ ∈ D(Ω) ∂9 i n Ω

Ω

and by going to the limit Zi ϕ dx = − z ∂i ϕ dx, Ω

∀ϕ ∈ D(Ω),

⇒ ∇z = Z ∈ L2 (Ω)

Ω

and from (5.9), z ∈ H 1 (Ω) is solution of the variational equation 1 N ∀ϕ ∈ C0 (R ), ∇z · ∇ϕ + z ϕ dx = f ϕ dx. Ω

Ω

By the H 1 -density property of Ω, this extends to all ϕ ∈ H 1 (Ω), which implies that z = y. Since all weakly convergent subsequences converge to the same limit, the whole sequence weakly converges and this completes the proof. Remark 5.1. We know that the H 1 -density condition holds for the family of domains satisfying the segment property of Deﬁnition 6.1 in Chapter 2 and not only for H 1 -spaces but also W m,p -spaces (cf. Theorem 6.3 in Chapter 2), thus extending the property and the results to a broader class of partial diﬀerential equations. We have shown in Theorem 6.5 of Chapter 2 that a domain satisfying the segment property is locally a C 0 -epigraph in the sense of Deﬁnition 5.1 of Chapter 2. Theorem 5.2. Let D be a bounded open subset of RN , let f ∈ L2 (D), and let L(D, r, O, λ) be the compact family of sets in W 1,p (D), 1 ≤ p < ∞, verifying the uniform fat segment property. Then the function : L(D, r, O, λ) → (L2 (D) × L2 (D))-weak, Ω → (˜ y (Ω), ∇y(Ω)) where z˜ denotes the extension by zero of a function z from Ω to D, is continuous. It is also true when L(D, r, O, λ) is replaced by the family of convex sets def

Cb (D) = bΩ : Ω ⊂ D, ∂Ω = ∅, Ω convex (cf. Theorem 10.2 in Chapter 7).

(5.10)

6. Elements of Capacity Theory

429

Proof. In L(D, r, O, λ) endowed with the W 1,p (D)-topology, 1 ≤ p < ∞, the three assumptions of the lemma are veriﬁed. Remark 5.2. Therefore the previous arguments are veriﬁed for the compact families of sets of Theorems 13.1, 14.1, and 14.2 in section 13 of Chapter 7 under the uniform fat segment property and for the conditions on the local graph functions of Theorem 13.2 (cf. Theorem 5.1 of Chapter 2). However, the subsets of a bounded holdall verifying a uniform segment property do not include the very important domains with cracks ´sio for which some speciﬁc results have been obtained by D. Bucur and J.-P. Zole [2, 4] (and D. Bucur and N. Varchon [1] in dimension 2). An issue that was not tackled is the nonhomogeneous boundary condition that involves a boundary integral in the variational formulation: ∂y = g on ∂Ω, (5.11) −∆y + y = f in Ω, ∂n ∃y ∈ H 1 (Ω), ∀ϕ ∈ H 1 (Ω), ∇y · ∇ϕ + y ϕ dx = f ϕ dx + g ϕ dΓ. (5.12) Ω

6

Ω

Γ

Elements of Capacity Theory

Recall the relaxation of the homogeneous Dirichlet boundary value problem for the Laplacian (4.6) in section 4.4 by using the space def

H1 (Ω; D) = ϕ ∈ H01 (D) : (1 − χΩ )ϕ = 0 a.e. in D .

(6.1)

The classical solution lies in the closed subspace H01 (Ω; D) of H1 (Ω; D), and we know that they are not equal when Ω is not stable with respect to D. The diﬀerence between the two spaces is that a function in H01 (Ω; D) is zero in D\Ω as an H01 (D)function and not just almost everywhere as an L2 (D)-function. Some information is lost in the relaxation, and this requires the ﬁner notion of quasi-everywhere that will characterize the set over which an H01 (D)-function is zero. This section introduces the basic elements and results from capacity theory that will play an essential role in establishing the continuity of the classical homogeneous Dirichlet problem (4.5) with respect to its underlying domain. The characterization of the space H01 (Ω; D) in terms of capacity will be given in Theorem 6.2. By introducing capacity constraints on sets, it will be possible to construct larger compact families that include domains with cracks.

6.1

Deﬁnition and Basic Properties

For p > N the elements of W 1,p (RN ) can be redeﬁned on sets of measure zero to be continuous functions (cf. R. A. Adams [1, Thm. 5.4, p. 97]): W 1,p (RN ) → C 1,λ (RN ),

0 < λ ≤ 1 − N/p.

(6.2)

430

Chapter 8. Shape Continuity and Optimization

But for p ≤ N this is no longer the case. We ﬁrst recall the deﬁnition of (1, p)capacity with respect to RN and with respect to an open subset G of RN along with a number of basic technical results. Deﬁnition 6.1. Let p, 1 < p < ∞, be a real number and N ≥ 1 be an integer. (i) (L. I. Hedberg [2, p. 239]) The capacity is deﬁned as follows: for a compact subset K ⊂ RN def Cap1,p (K) = inf |∇ϕ|p dx : ϕ ∈ C0∞ (RN ), ϕ > 1 on K ; RN

for an open subset G ⊂ RN

def Cap1,p (G) = sup Cap1,p (K) : ∀K ⊂ G, K compact ; for an arbitrary subset E ⊂ D

def Cap1,p (E) = inf Cap1,p (G) : ∀G ⊃ E, G open . In what follows, we shall use the simpler notation Capp as in L. C. Evans and R. F. Gariepy [1, pp. 147] instead of Cap1,p . (ii) (L. C. Evans and R. F. Gariepy [1, pp. 160] and J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 87]) A function f is said to be quasi-continuous on D ⊂ RN if, for each ε > 0, there exists an open set Gε such that Cap1,p (Gε ) < ε and f is continuous on D\Gε . (iii) A set E in RN is said to be quasi-open if, for all ε > 0, there exists an open set Gε such that Cap1,p (Gε ) < ε and E ∪ Gε is open. In the above deﬁnitions the capacity is deﬁned with respect to RN . In J. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 2, p. 27], the capacity is deﬁned relative to an open subset D of RN instead of the whole space RN . Deﬁnition 6.2. Let N ≥ 1 be an integer, let p, 1 < p < ∞, and let D be a ﬁxed open subset of RN . (i) The (1, p)-capacity is deﬁned as follows: for a compact subset K ⊂ D def p ∞ |∇ϕ| dx : ϕ ∈ C0 (D), ϕ > 1 on K ; Cap1,p (K, D) = inf D

for an open subset G ⊂ D

def Cap1,p (G, D) = sup Cap1,p (K, D) : ∀K ⊂ G, K compact ; for an arbitrary subset E ⊂ D

def Cap1,p (E, D) = inf Cap1,p (G, D) : ∀G open such that E ⊂ G ⊂ D .

6. Elements of Capacity Theory

431

(ii) A function f is said to be quasi-continuous in D if, for all ε > 0, there exists an open set Gε ⊂ D such that capD (Gε ) < ε and f is continuous on D\Gε . (iii) A set E in D is said to be quasi-open if, for all ε > 0, there exists an open set Gε ⊂ D such that capD (Gε ) < ε and E ∪ Gε is open. Clearly, by deﬁnition, ∀E ⊂ D,

Cap1,p (E) ≤ Cap1,p (E, D).

(6.3)

The number Cap1,p (E, D) ∈ [0, ∞] is called the (variational) (p, mN )-capacity of the condenser (E, G), where mN is the N -dimensional Lebesgue measure. It can easily be shown that for a quasi-open set there exists a decreasing sequence {Ωn } of open sets such that Ωn ⊃ E and capD (Ωn \E) goes to zero as n goes to inﬁnity. We say that a property holds quasi-everywhere (q.e.) in D if it holds in the complement D\E of a set E of zero capacity. A set of zero capacity has zero measure, but the converse is not true. The capacity is a countably subadditive set function, but it is not additive even for disjoint sets. Hence the union of a countable number of sets of zero capacity has zero capacity.

6.2

Quasi-continuous Representative and H 1 -Functions

The following theorem is from L. I. Hedberg [2] (sect. 2, p. 241 and Thm. 1.1, p. 237, and footnote, p. 238 referring to T. Wolﬀ and the paper by L. I. Hedberg and Th. H. Wolff [1]). See also J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.4, p. 89] Theorem 6.1. Let p, 1 < p < ∞, be a real number, N ≥ 1 be an integer, and D ⊂ RN be open. (i) Every f in W 1,p (D) has a quasi-continuous representative: there exists a quasi-continuous function f1 deﬁned on D such that f1 = f almost everywhere in D (hence f1 is a representative of f in W 1,p (D) ). Any two quasicontinuous representatives f1 and f2 of f ∈ W 1,p (D) such that f1 = f2 almost everywhere are equal quasi-everywhere in D. (ii) Let f ∈ W 1,p (RN ). Let K ⊂ RN be closed, and suppose that f |K = 0 (trace deﬁned quasi-everywhere). Then f |K ∈ W01,p (K). The following key lemma completes the picture (cf. J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.5, p. 90]). See also L. I. Hedberg [2, p. 241]. Lemma 6.1. (i) Let Ω be an open subset of RN , let p, 1 ≤ p < ∞, and consider an element u of W 1.p (Ω). Then u ∈ W01.p (Ω) if and only if there exists a quasicontinuous representative u1 of u in RN such that u1 = 0 quasi-everywhere in Ω and u1 = u almost everywhere in Ω.

432

Chapter 8. Shape Continuity and Optimization

(ii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consider an element u of W01.p (D). Then u|Ω ∈ W01.p (Ω) if and only if there exists a quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere in D\Ω and u1 = u almost everywhere in Ω. (iii) Let Ω and D be two bounded open subsets of RN such that Ω ⊂ D and consider an element u of W 1.p (D). Then u|Ω ∈ W01.p (Ω) if and only if there exists a quasi-continuous representative u1 of u such that u1 = 0 quasi-everywhere in D\Ω and u1 = u almost everywhere in Ω. A function ϕ in W 1,p (D) is said to be zero quasi-everywhere in a subset E of D if there exists a quasi-continuous representative of ϕ which is zero quasi-everywhere in E. This makes sense since any two quasi-continuous representatives of an element ϕ of W 1,p (D) are equal quasi-everywhere. For any ϕ ∈ W01,p (D) and t ∈ R, the set {x ∈ D : ϕ(x) > t} is quasi-open. Moreover, the subspace W01,p (Ω; D) introduced in section 2.5.3 in Chapter 2 can now be characterized by the capacity. Theorem 6.2. Given two bounded open subsets Ω and D of RN , W01,p (Ω; D) = ϕ ∈ W01,p (D) : (1 − χΩ )ϕ = 0 q.e. in D , if Ω ⊂ D,

W01,p (Ω; D) = ϕ ∈ W 1,p (D) : (1 − χΩ )ϕ = 0 q.e. in D , if Ω ⊂ D.

(6.4) (6.5)

The characterization of W01,p (Ω; D) requires the notion of capacity in a very essential way. It cannot be obtained by saying that the function and its derivatives are zero almost everywhere in D\Ω. Recall the deﬁnition of the other extension H1 (Ω; D) of H 1 (Ω) to a measurable set D containing Ω (cf. Chapter 5, section 3.5, Theorem 3.6, identity (3.27)). Its deﬁnition readily extends from H 1 to W 1,p as follows: def W1,p (Ω; D) = ψ ∈ W01,p (D) : (1 − χΩ )ψ = 0 a.e. in D . By deﬁnition W01,p (Ω; D) ⊂ W1,p (Ω; D), but the two spaces are generally not equal, as can be seen from the following example. Denote by Br , r > 0, the open ball of radius r > 0 in R2 . Deﬁne Ω = B2 \∂B1 and D = B3 . The circular crack ∂B1 in Ω has zero measure but nonzero capacity. Since ∂B1 has zero measure, W1,p (Ω; D) contains functions ψ ∈ W01,p (B2 ) whose restrictions to B2 are not zero on the circle ∂B1 and hence do not belong to W01,p (Ω). Yet for Lipschitzian domains and, more generally, for domains Ω that are stable with respect to D in the sense of Deﬁnition 4.1, the two spaces are indeed equal.

6.3

Transport of Sets of Zero Capacity

Any Lipschitz continuous transformation of RN which has a Lipschitz continuous inverse transports sets of zero capacity onto sets of zero capacity. Lemma 6.2. Let D be an open subset of RN and let T be an invertible transformation of D such that both T and T −1 are Lipschitz continuous. For any E ⊂ D Cap1,2 (E, D) = 0 ⇐⇒ Cap1,2 (T (E), D) = 0.

6. Elements of Capacity Theory

433

Proof. (a) E = K, K compact in D. Observe that, in the deﬁnition of the capacity, it is not necessary to choose the functions ϕ in C0∞ (D). They can be chosen in a larger space as long as |∇ϕ|2 dx < ∞ and ϕ > 1 on K. D

So the capacity is also given by def Cap1,2 (K, D) = inf |∇ϕ|2 dx : ϕ ∈ H01 (D) ∩ C(D), ϕ > 1 on K . D

But the space

H01 (D)

∩ C(D) is stable under the action of T :

ϕ ∈ H01 (D) ∩ C(D) ⇐⇒ T ◦ ϕ ∈ H01 (D) ∩ C(D). Consider the matrix A(x) = | det(DT (x))| ∗DT (x)( ∗DT (x))−1 . def

By assumption on T , the elements of the matrix A belong to L∞ (D) and αI ≤ A(x) ≤ α−1 I a.e. in D.

∃α > 0,

Deﬁne E(K) = {ϕ ∈ H01 (D) ∩ C(D) : ϕ > 1 on K}. For any ϕ ∈ E(T (K)), ϕ ◦ T ∈ E(K) and |∇ϕ|2 dx = A ∇(ϕ ◦ T ) · ∇(ϕ ◦ T ) dx ≥ α |∇(ϕ ◦ T )|2 dx. D

D

D

Let K be a compact subset such that Cap1,2 (K, D) = 0. For each ε > 0 there exists ϕ ∈ E(K) such that 2 |∇ϕ| dx ≤ ε ⇒ ∀ε, |∇(ϕ ◦ T )|2 dx ≤ ε/α or ϕ ◦ T ∈ E(T −1 (K)) D

D

⇒ ∀ε,

Cap1,2 (T −1 (K), D) ≤ ε/α

⇒ Cap1,2 (T −1 (K), D) = 0

and we can repeat the proof with T −1 in place of T . (b) E = G, G open. Let

Cap1,2 (G, D) = sup Cap1,2 (K, D) : ∀K ⊂ G, K compact = 0. Therefore capD (K) = 0, which implies that Cap1,2 (T (K), D) = 0 and hence

Cap1,2 (T (G), D) = sup Cap1,2 (K , D) : ∀K ⊂ T (G), K compact

= sup Cap1,2 (T (K), D) : ∀K ⊂ G, K compact = 0. (c) General case. Let def

Cap1,2 (E, D) = inf {capD (G) : ∀G ⊃ E, G open} .

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Chapter 8. Shape Continuity and Optimization

Therefore, for all ε > 0, there exists G ⊃ E open such that Cap1,2 (E, D) ≤ Cap1,2 (G, D) ≤ ε

⇒ Cap1,2 (T (E), D) ≤ Cap1,2 (T (G), D)

since T (E) ⊂ T (G) is open. By deﬁnition of Cap1,2 (G, D) we have ∀K ⊂ G,

Cap1,2 (K, D) ≤ ε

⇒ Cap1,2 (T (K), D) ≤ αε

⇒ Cap1,2 (T (G), D) = sup{Cap1,2 (T (K), D) : K ⊂ G} ≤ αε. Finally, for all ε > 0, Cap1,2 (T (E), D) ≤ αε and hence Cap1,2 (T (E), D) = 0.

7

Crack-Free Sets and Some Applications

In this section we show that the result for the second question raised in the introduction to this chapter is true for measurable sets in the following large family.

7.1

Deﬁnitions and Properties

Deﬁnition 7.1. (i) The interior boundary of a set Ω is deﬁned as def

∂i Ω = ∂Ω\∂Ω.

(7.1)

The exterior boundary of a set Ω is deﬁned as def

∂e Ω = ∂Ω\∂Ω.

(7.2)

(ii) A subset Ω of RN is said to be crack-free if ∂i Ω = ∅; its complement Ω is said to be crack-free if ∂e Ω = ∅. The intuitive terminology crack-free arises from the following identities: ∂i Ω = ∂Ω\∂Ω = int Ω\int Ω,

∂e Ω = ∂Ω\∂Ω = int Ω\int Ω.

(7.3)

A crack-free set Ω does not have internal cracks that would disappear after the cleaning operation with respect to the closure of the set2 . It is veriﬁed for sets that 2 Since ∂Ω = Ω ∩ Ω ⊂ Ω ∩ Ω = ∂Ω, the deﬁnition of a crack-free set Ω is equivalent to the property ∂Ω = ∂Ω. In 1994 A. Henrot [2] introduced the terminology Carath´ eodory set for such ¨ki, J. Sprekels, and D. Tiba [1], a set that was later adopted by D. Tiba [3], P. Neittaanma and A. Henrot and M. Pierre [4]. However, this terminology does not seem to be standard. For instance, in the literature on polynomial approximations in the complex plane C, a Carath´ eodory set is deﬁned as follows.

Deﬁnition 7.2 (O. Dovgoshey [1] or D. Gaier [1]). A bounded subset Ω of C is said to be a Carath´ eodory set if the boundary of Ω coincides with the boundary of the unbounded component of the complement of Ω. A Carath´ eodory domain is a Carath´ eodory set if, in addition, Ω is simply connected. In V. A. Martirosian and S. E. Mkrtchyan [1], Ω is further assumed to be measurable. This deﬁnition excludes not only interior cracks but also bounded holes inside the set Ω as can be seen from the example of the annulus Ω = {x ∈ R2 : 1 < |x| < 2} in R2 . So, it is more restrictive than ∂Ω = ∂Ω. In order to avoid any ambiguity, we choose to keep the intuitive terminology crack-free.

7. Crack-Free Sets and Some Applications

435

are locally the epigraph of a C 0 -function. Yet, it is also veriﬁed for sets that are not locally the epigraph of a C 0 -function. The following equivalent characterization of a crack-free set in terms of the oriented distance function was used as its deﬁnition in M. C. Delfour and J.´sio [43]. P. Zole Theorem 7.1. A subset Ω of RN is crack-free if and only if bΩ = bΩ . Its complement Ω is crack-free if and only if bΩ = bΩ (or, equivalently, bΩ = bint Ω ). Proof. If ∂Ω = ∂Ω, then |bΩ | = d∂Ω = d∂Ω = |bΩ |. Hence, dΩ + dΩ = |bΩ | = |bΩ | = dΩ + dΩ implies dΩ = dΩ and bΩ = bΩ . Conversely, if bΩ = bΩ , then d∂Ω = |bΩ | = |bΩ | = d∂Ω and ∂Ω = ∂Ω since both sets are closed. We also have the following equivalences: bΩ = bint Ω ⇐⇒ bΩ = bΩ ⇐⇒ Ω = Ω ⇐⇒ ∂Ω = ∂Ω ⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int Ω = int Ω

(7.4)

and bΩ = bint Ω ⇐⇒ bΩ = bΩ ⇐⇒ Ω = Ω ⇐⇒ ∂Ω = ∂Ω ⇐⇒ int Ω = Ω ⇐⇒ ∂int Ω = ∂Ω ⇐⇒ int Ω = int Ω.

(7.5)

Remark 7.1. An open set Ω is crack-free if and only if Ω = int Ω. Indeed, by deﬁnition, Ω is crackfree if and only if Ω = Ω. But since Ω is open Ω = Ω and Ω = Ω = int Ω. We give the following general technical theorem and its corollary. ´sio [43]). Let K be a compact Theorem 7.2 (M. C. Delfour and J.-P. Zole subset of RN and let v ∈ W 1,p (RN ). Then v = 0 almost everywhere on K implies that v has a quasi-continuous representative v ∗ such that v ∗ |int K ∈ W01,p (int K). Corollary 1. Let Ω be a bounded set in RN and let v ∈ W 1,p (RN ). Then v = 0 almost everywhere on Ω implies that v has a quasi-continuous representative v ∗ such that v ∗ |int Ω ∈ W01,p (int Ω). Proof. Choose the quasi-continuous representative 1 def v dy v ∗ (x) = lim δ0 |B(x, δ)| B(x,δ)

(7.6)

of v and let {Gε } be the associated family of open subsets of (1, p)-capacity less than ε of Deﬁnition 6.1 (ii). (a) If there exists ε > 0 such that Gε ⊂ int K, then K ⊂ Gε and v ∗ is continuous in K. But, by assumption, v = 0 almost everywhere in the open set K. Hence, by deﬁnition of v ∗ , v ∗ = 0 everywhere in the open subset K ⊂ K, and, by continuity, v ∗ = 0 everywhere in K.

436

Chapter 8. Shape Continuity and Optimization

(b) If for all ε > 0, Gε ∩ K = ∅, then either x ∈ ∪ε>0 Gε ∩ K or x ∈ ∩ε>0 Gε ∩ ∂K, since K = int K ∪ ∂K = K ∪ ∂K. In the ﬁrst case, v ∗ (x) = 0 since x ∈ K. In the second case, Capp (∩ε>0 Gε ∩ ∂K) ≤ Capp (Gε ) < ε and ∩ε>0 Gε ∩ ∂K has zero (1, p)-capacity. From (a) and (b), v ∗ = 0 quasi-everywhere in K. Therefore by Theorem 5 in L. I. Hedberg and Th. H. Wolff [1], which generalizes Theorem 1.1 in L. I. Hedberg [2], v ∗ |int K ∈ W01,p (int K). Remark 7.2. In L. I. Hedberg [2, 3] a compact set K is said to be (1, p)-stable if v = 0 quasieverywhere on K implies that v|int K ∈ W01,p (int K). A suﬃcient condition (cf. Theorem 6.4 in L. I. Hedberg [2]) for the (1, p)-stability of K is that K be (1, p)-thick (1, p) quasi-everywhere on ∂K. In the proof of Theorem 7.2 we have shown that, under the weaker suﬃcient condition v = 0 almost everywhere on K, v = 0 quasi-everywhere on K. However, this condition would not be suﬃcient for the (2, p)-stability as shown in the example of L. I. Hedberg [3] (cf. Thm. 3.14, p. 100). Remark 7.3. There is also a result in dimension N = 2 in Theorem 7 of L. I. Hedberg and Th. H. Wolff [1]: for p, 2 ≤ p < ∞, and q, q −1 + p−1 = 1, K is (1, q)stable if and only if Capq (∂K ∩ e1,q (K)) = 0, where e1,q (K) = {x ∈ RN : K is (1, q)-thin at x}. It says that the part of ∂K where K is thin has zero (1, q)capacity. However, in the equivalence of parts (b) and (d) of Theorem 7, there seems to be a typo: the (1, q)-stability is deﬁned as follows: “K is (1, q)-stable if v = 0 q.e. on K implies that v ∈ W01,q (K).” It should probably read “v ∈ W01,q (int K).” We now turn to our speciﬁc problem. Given a measurable subset Ω of a bounded open holdall D ⊂ RN , deﬁne the closed subspace def W1,p (Ω; D) = ψ ∈ W01,p (D) : (1 − χΩ )ψ = 0 a.e. (7.7) of W01,p (D). This space provides a relaxation of the Dirichlet boundary value problem to domains that are only measurable. The next theorem will show that in fact W1,p (Ω; D) can be embedded in W01,p (int Ω). Furthermore, for crack-free measurable subsets Ω of D, W1,p (Ω; D) coincides with W01,p (int Ω). This will yield the continuity of the homogeneous Dirichlet problem with respect to the open domains Ω in the family L(D, r, O, λ) of subsets of D verifying a uniform fat segment property. Theorem 7.3. Let D be bounded open in RN . (i) For any measurable subset Ω ⊂ D such that int Ω = ∅, ∃ a quasi-continuous representative v ∗ 1,p v ∈ W (Ω; D) =⇒ 1,p such that v ∗ | int Ω ∈ W0 (int Ω) and W01,p (int Ω; D) ⊂ W1,p (Ω; D) ⊂ W01,p (int Ω; D).

(7.8)

7. Crack-Free Sets and Some Applications (ii) For any measurable subset Ω ⊂ D such that int Ω = ∅, ∃ a quasi-continuous representative v ∗ v ∈ W 1,p (Ω; D) =⇒ such that v ∗ |int Ω ∈ W01,p (int Ω) and Ω crack-free

437

(7.9)

and W01,p (int Ω; D) = W1,p (Ω; D) = W01,p (int Ω; D). Proof. (i) From Theorem 7.2 with K = Ω and v˜, the extension by zero of v from D to RN . Given ϕ ∈ W01,p (int Ω), the zero extension e0 ϕ ∈ W01,p (int Ω; D) and (1 − χint Ω )ϕ = 0 almost everywhere in D. This implies that (1 − χΩ )ϕ = 0 almost everywhere in D and ϕ ∈ W1,p (Ω; D). (ii) From (i) and the equivalences (7.4): int Ω = int Ω. Remark 7.4. The condition that v ∈ W1,p (Ω; D) is weaker than the capacitary condition of ´sio [12] and hence its conclusion that v ∗ ∈ W01,p (int Ω) D. Bucur and J.-P. Zole is also weaker. It gives information only about the function on ∂Ω rather than on the whole ∂Ω. The “crack-free” property of Deﬁnition 7.1 is purely geometric and even set theoretic. It is diﬀerent from and probably stronger than the capacitary ´sio [12]. Instead of forcing Ω to be condition of D. Bucur and J.-P. Zole “uniformly thick” at quasi all points of the boundary ∂Ω in order to preserve the zero Dirichlet condition on the whole boundary, it erases the “interior boundary” ∂Ω\∂Ω. Its advantage is that it does not require an a priori knowledge of the diﬀerential operator and does not necessitate the Maximum Principle or harmonic functions; its disadvantage is that it cannot handle problems where the trace of the function on ∂Ω\∂Ω is an important feature of the problem at hand. Yet, it should readily be extendable to Sobolev spaces on C 1,1 -submanifolds of codimension 1 to deal with the Laplace–Beltrami or shell equations.

7.2 7.2.1

Continuity and Optimization over L(D, r, O, λ) Continuity of the Classical Homogeneous Dirichlet Boundary Condition

In light of Theorem 4.1, it remains to ﬁnd families of sets with properties that would enable us to sharpen part (i) or that would verify the stability condition H01 (Ω; D) = H1 (Ω; D). In the second case, the family of subsets of a bounded holdall D that satisfy the uniform fat segment property meets all the requirements. Theorem 7.4. Let D be a bounded open nonempty subset in RN and Ω be open subsets of D. Denote by y(Ω) ∈ H01 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map Ω → y(Ω) : L(D, r, O, λ) → H01 (D)-strong

(7.10)

is continuous with respect to any of the topologies on L(D, r, O, λ) of Theorem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p (D), dΩ in C(D) and W 1,p (D), bΩ , C(D) in C(D) and W 1,p (D), χΩ in Lp (D), and χΩ in Lp (D).

438

Chapter 8. Shape Continuity and Optimization

Proof. Since all topologies are equivalent on L(D, r, O, λ), consider a domain Ω and a sequence {Ωn } such that dΩn → dΩ in W 1,p (D) for some p, 1 ≤ p < ∞. By Theorem 6.1 (ii) of Chapter 2, all sets in L(D, r, O, λ) are crack-free since int Ω = ∅,

Ω = int Ω = Ω

⇒ b Ω = bΩ .

By Theorem 7.3 (ii), H1 (Ω; D) = H01 (Ω; D). Finally, from part (ii) of Theorem 4.1, we get the continuity of the solutions of the classical Dirichlet problem. 7.2.2

Minimization/Maximization of the First Eigenvalue

Consider the ﬁrst eigenvalue λA (Ω) introduced in (2.1) A∇ϕ · ∇ϕ dx def A D λ (Ω) = inf1 |ϕ|2 dx 0=ϕ∈H0 (Ω;D) D

(7.11)

for the Laplace equation on the open Ω of D. It was shown in Theorem 2.2 that λA (Ω) is an upper semicontinuous function of Ω and that there is a solution to the maximization problem with respect to Ω. We now turn to the minimization of λA (Ω) with respect to Ω that requires the lower semicontinuity and hence the continuity of λA (Ω) with respect to Ω. To do that we choose the smaller family L(D, r, O, λ) of sets verifying a uniform fat segment property. Theorem 7.5. Let D be a bounded open domain in RN and A be a matrix function satisfying assumption (2.2). (i) The mapping dΩ → λA (Ω) : L(D, r, O, λ) → R ∪{+∞}

(7.12)

is continuous with respect to any of the topologies on L(D, r, O, λ) of Theorem 13.1 in Chapter 7, that is, dΩ in C(D) and W 1,p (D), dΩ in C(D) and W 1,p (D), bΩ , C(D) in C(D) and W 1,p (D), χΩ in Lp (D), and χΩ in Lp (D). (ii) The minimization and maximization problems inf

λ(Ω) and

Ω∈L(D,r,O,λ)

sup

λ(Ω)

(7.13)

Ω∈L(D,r,O,λ)

have solutions in L(D, r, O, λ). Moreover, 0 < λA (D) ≤

inf Ω∈L(D,r,O,λ)

λ(Ω) ≤

sup

λ(Ω) ≤ β λI (O).

(7.14)

Ω∈L(D,r,O,λ)

Proof. (i) Continuity. Go back to the proof of Theorem 2.2. Given Ω ∈ L(D, r, O, λ), consider a sequence {Ωn } in L(D, r, O, λ) such that dΩn → dΩ in C(D). Let λA (Ωn ) and un ∈ H01 (Ωn ; D), un L2 (D) = 1, be the eigenvalue and an eigenvector associated with Ωn . By deﬁnition of L(D, r, O, λ), for all x ∈ ∂Ω, there exists Ax ∈ O(N) such that for all y ∈ B(x, r) ∩ Ω, y + Ax O ⊂ Ω. Fix x0 ∈ ∂Ω. There exist z ∈ B(x0 , r) ∩ Ω

7. Crack-Free Sets and Some Applications

439

and ε > 0 such that B(z, ε) ⊂ B(x0 , r) ∩ Ω. As a result B(z, ε) + Ax O ⊂ Ω and z + Ax O ⊂ Ω. By the compactivorous property, there exists N such that, for all n > N , z + Ax O ⊂ Ωn and ∀n > N,

λn = λA (Ωn ) ≤ λA (z + Ax O) ≤ βλI (z + Ax O) = βλI (O)

since the eigenvalue is independent of a translation and a rotation of the set. This means that the sequence {λn } of eigenvalues is bounded. By coercivity α∇un }2L2 (D) ≤ λn un 2L2 (D) = λn and {un } is bounded in H01 (D). There exist λ0 and u0 ∈ H1 (Ω; D) and subsequences, still indexed by n, such that λn → λ 0 ,

un L2 (D) = 1,

⇒ u0 L2 (D) = 1.

un u0 in H01 (D)-weak

But for all Ω ∈ L(D, r, O, λ) we have H1 (Ω; D) = H01 (Ω; D) and u0 ∈ H01 (Ω; D). For any ϕ ∈ D(Ω), K = supp ϕ ⊂ Ω is compact and there exists N (K) such that for all n ≥ N (K), supp ϕ ⊂ Ωn . Therefore for all n ≥ N (K) A∇un · ∇ϕ dx = λn un ϕ dx. un ∈ H01 (Ωn ; D), D

By letting n → ∞

D

∀ϕ ∈ D(Ω),

A∇u · ∇ϕ dx = λ0 D

u ϕ dx, D

and, by density of D(Ω) in H01 (Ω; D), the variational equation is veriﬁed in H01 (Ω; D). Hence, u0 ∈ H01 (Ω; D), uL2 (Ω) = 1, satisﬁes the variational equation A∇u0 · ∇ϕ dx = λ0 u ϕ dx. ∀ϕ ∈ H01 (Ω; D), D

D

Since λA (Ω) is the smallest eigenvalue with respect to H01 (Ω; D), λA (Ω) ≤ λ0 . To show that λ0 = λA (Ω), it remains to prove that λ0 ≤ λA (Ω). For each n A∇ϕn · ∇ϕn dx 1 A . ∀ϕn ∈ H0 (Ωn ; D), λ (Ωn ) ≤ D ϕn 2L2 (D) By the compactivorous property for each ϕ ∈ D(Ω), ϕ = 0, there exists N such that supp ϕ ⊂ Ωn and ϕ ∈ D(Ωn ). Therefore for all n > N A∇ϕ · ∇ϕ dx A λ (Ωn ) ≤ D ϕ2L2 (D) and by going to the limit ∀ϕ ∈ D(Ω),

λ0 ≤

D

A∇ϕ · ∇ϕ dx . ϕ2L2 (D

440

Chapter 8. Shape Continuity and Optimization

By density this is true for all ϕ ∈ H01 (Ω; D), ϕ = 0, and λ0 ≤ λA (Ω). As a result λ0 = λA (Ω). Therefore the whole sequence {λA (Ωn )} converges to λA (Ω) and we have the continuity of the map dΩ → λA (Ω). The strong continuity in H01 (D) now follows from the convergence of un → u0 in L2 (D)-strong: α∇(un − u0 )2L2 (D) ≤ A∇(un − u0 ) · ∇(un − u0 ) dx D A∇un · ∇un dx + A∇u0 · ∇u0 dx − 2 A∇un · ∇u0 dx = D D D un · un dx + λ0 u0 · u0 dx − 2 A∇un · ∇u0 dx = λn D D D u0 · u0 dx − 2 A∇u0 · ∇u0 dx = 0. → 2λ0 D

D

(ii) This follows from the continuity and the compactness of L(D, r, O, λ) in all the mentioned topologies. The inequalities have been proven in part (i). Corollary 1. Under the assumptions of Theorem 7.5, the maximization and minimization problems (7.13) also have solutions in L(D, r, O, λ) under an equality or an inequality constraint of the form m(Ω) = α, m(Ω) ≤ α, or m(Ω) ≥ α provided that there exists Ω ∈ L(D, r, O, λ) such that m(Ω ) = α, m(Ω ) ≤ α, or m(Ω ) ≥ α. Proof. This follows from Theorem 13.1 and its Corollaries 1 and 2 of section 13 in Chapter 7, where it is shown that the convergence is strong not only in the Hausdorﬀ topologies but also for the associated characteristic functions in L1 (D), thus making the volume function continuous with respect to open domains in L(D, r, ω, λ). For the spaces of convex subsets recall from Theorem 8.6 of Chapter 6 and Theorem 10.2 of Chapter 7 that the compactness remains true in the W 1,1 (D)-topology.

8

Continuity under Capacity Constraints

We have established the continuity of the homogeneous Dirichlet problem, that is, the continuity of the function dΩ → u(Ω) : Cdc (D) → H01 (D)-strong for the family of domains verifying a uniform fat segment property. Domains with cracks lie between families of L(D, r, O, λ) type and Cdc (D). Topologies using the characteristic function χΩ are excluded since they do not see interior boundaries of zero measure. Families that contains sets with cracks must be chosen in such a way that the homogeneous Dirichlet boundary condition on the sequence of solutions u(Ωn ) in the converging sets Ωn is preserved by the solution u(Ω) on the boundary of the limit domain Ω in section 4. We have to be careful since the relative capacity of the

8. Continuity under Capacity Constraints

441

complement of the domains near the boundary must not vanish. In order to handle this point, we introduce the following concepts and terminology related to the local capacity of the complement near the boundary points. The following deﬁnition is due to J. Heinonen, T. Kilpelainen, and O. Martio [1, pp. 114–115]. Deﬁnition 8.1. (i) Given r > 0, a set E ⊂ RN , and a point x, the function def

capx,r (E) =

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

(8.1)

is called the capacity density function. (ii) We say that a set E is thick at a point x if

1

0

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

1/(p−1)

dr = ∞. r

(8.2)

This is called the Wiener criterion. (iii) We say that an open set Ω satisﬁes the Wiener density condition if ∀x ∈ ∂Ω, 0

1

Capp (Ω ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

1/(p−1)

dr = ∞. r

(8.3)

This means that the set Ω is thick at every boundary point x ∈ ∂Ω (that is, Ω has “suﬃcient” capacity around each boundary point). Remark 8.1. If Ω satisﬁes the (r, c)-capacity density condition, the complement is thick in any point of the boundary. The Wiener criterion can be strengthened by introducing the following condition (J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 127]). Deﬁnition 8.2. (i) Given r > 0, c > 0 , and an open set Ω, Ω is said to satisfy the (r, c)-capacity density condition if ∀x ∈ ∂Ω, ∀r , 0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

≥ c.

(8.4)

We also say that Ω is uniformly (r, c)-thick. (ii) Given r > 0, c > 0 , and an open set Ω, Ω is said to satisfy the strong (r, c)-capacity density condition if ∀x ∈ ∂Ω, ∀r , 0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) ≥ c. Capp (B(x, r ), B(x, 2r ))

We also say that Ω is strongly uniformly (r, c)-thick.

(8.5)

442

Chapter 8. Shape Continuity and Optimization

(iii) For r, 0 < r < 1, and c > 0 deﬁne the following family of open subsets of D: & Ω satisﬁes the strong def . (8.6) Oc,r (D) = Ω ⊂ D : (r, c)-capacity density condition

Example 8.1. Only a few capacities can be computed explicitly. The computation of the spherical condensers Capp (B(x, r), B(x, R)) and Capp (B(x, r), B(x, R)) leads to some simpliﬁcations in the denominator of the above expression (cf. J. Heinonen, T. Kilpelainen, and O. Martio [1, p. 35]). For 0 < r < R < ∞ Capp (B(x, r), B(x, R)) = Capp (B(x, r), B(x, R)

(8.7)

and for p > 1 p−1 p−N 1−p |N − p| p−N N −1 , R p−1 − r p−1 ω p−1 Capp (B(x, r), B(x, R)) = 1−N R N −1 ω , log r

p = N, p = N,

where ω N −1 is the volume (Lebesgue measure) of the ball in RN −1 . In particular Capp (B(x, r), B(x, 2r)) = c1 (N, p) rN −p , N

Capp (B(x, r)) = Capp (B(x, r), R ) = c2 (N, p) r Capp ({x0 }, R ) = c3 (N, p) r N

N −p

1 < p, N −p

,

1 < p < N,

,

(8.8)

p < N,

for constants ci (N, p) that depend only on p and N . Remark 8.2. Deﬁnition 8.2 (i) uses a slightly diﬀerent deﬁnition of the previous capacity density function of Deﬁnition 8.1: def

capx,r (E) = instead of def

capx,r (E) =

Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r)) Capp (E ∩ B(x, r), B(x, 2r)) Capp (B(x, r), B(x, 2r))

.

Note that Capp (B(x, r), B(x, 2r)) = Capp (B(x, r), B(x, 2r)) = c1 (N, p) rN −p ,

1 < p,

and Capp (E ∩ B(x, r), B(x, 2r)) ≤ Capp (E ∩ B(x, r), B(x, 2r)) imply that ∀r > 0, ∀x,

capx,r (E) ≥ capx,r (E)

8. Continuity under Capacity Constraints

443

and the strong (r, c)-capacity density condition implies the (r, c)-capacity density condition. But for an open subset Ω of the bounded open subset D of RN , the following two conditions are equivalent: ∀x ∈ ∂Ω, ∀ 0 < r < r, and ∀x ∈ ∂Ω, ∀0 < r < r,

Capp (Ω ∩ B(x, r ), B(x, 2r )) ≥c Capp (B(x, r ), B(x, 2r ))

Capp (Ω ∩ B(x, r ), B(x, 2r )) Capp (B(x, r ), B(x, 2r ))

≥ c,

(8.9)

(8.10)

and also from the Oc,r (D) family viewpoint the two deﬁnitions are equivalent. It is clear that (8.9) implies (8.10), but the converse is not obvious. Recall Theorem 4.1 and the notation un for the solution of (4.5) in H01 (Ωn ; D) and u ∈ H1 (Ω; D) for the weak limit in H01 (D) which satisﬁes (4.8). The following theorem gives the main continuity result. Theorem 8.1. Let D be a bounded open nonempty subset of RN of class C 2 for N ≥ 3 and Ω be an open subset of D. Assume that A is a matrix function that satisﬁes the conditions (4.4) and that the elements of A belong to C 1 (D). Denote by u(Ω) ∈ H01 (Ω; D) the solution of the Dirichlet problem (4.5) on Ω. The map Ω → u(Ω) : Oc,r (D) → H01 (D)-strong

(8.11)

is continuous with respect to the uniform topology C(D) on Cdc (D). Proof. When the dimension N of the space is equal to 1, H 1 (D) ⊂ C(D) and no approximation of f is necessary. When N ≥ 2, we ﬁrst prove the result for f ∈ H s (D), s > N/2 − 2 (since for D of class C 2 the corresponding solution will belong to H s+2 (D) ⊂ C(D)), and then, by approximation of f , we prove it for f ∈ H −1 (D). (i) First consider f ∈ H s (D), s > N/2−2, for N ≥ 2 (s = −1 for N = 1). The proof will make use of the following results from J. Heinonen, T. Kilpelainen, and O. Martio [1, Chap. 3, Thm. 3.70, p. 80]. Lemma 8.1. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisﬁed. Let v be an A-harmonic function; that is, v ∈ 1,p Wloc (Ω) is a weak solution of equation A∇v(Ω) · ∇ϕ dx = 0 (8.12) ∀ϕ ∈ D(Ω), D

in the open set Ω. Then there exists a continuous function u on Ω such that v(Ω) = u almost everywhere. Note that the continuous representative from Lemma 8.1 is in fact a quasicontinuous H01 (Ω) representative. Indeed, let v1 = v almost everywhere and let v1 be continuous on Ω. We want to show that v1 is a quasi-continuous representative of v. There exists a quasi-continuous representative v2 of v, which is equal to v almost

444

Chapter 8. Shape Continuity and Optimization

everywhere. So v1 is continuous, v2 is quasi-continuous, and v1 = v2 almost everywhere. Using J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 4.12], we get v1 = v2 quasi-everywhere. Always from J. Heinonen, T. Kilpelainen, and O. Martio [1, Thm. 6.27, p. 122] we get the following lemma. Lemma 8.2. Let the assumptions of Theorem 8.1 on the open domain D and the matrix function A be satisﬁed. Let Ω belong to Oc,r (D). If θ ∈ W 1,p (Ω) ∩ C(Ω) and if h is an A-harmonic function in Ω such that h − θ ∈ W01,p (Ω), then ∀x0 ∈ ∂Ω,

lim h(x) = θ(x0 ).

x→x0

Note that the fact that Ω belongs to Oc,r (D) involves the notion of thickness in any poi