Springer Monographs in Mathematics
Kung-Ching Chang
Methods in Nonlinear Analysis
ABC
Kung-Ching Chang School of ...

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Kung-Ching Chang

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Springer Monographs in Mathematics

Kung-Ching Chang

Methods in Nonlinear Analysis

ABC

Kung-Ching Chang School of Mathematical Sciences Peking University 100871 Beijing People’s Republic of China E-mail: [email protected]

Library of Congress Control Number: 2005931137 Mathematics Subject Classiﬁcation (2000): 47H00, 47J05, 47J07, 47J25, 47J30, 58-01, 58C15, 58E05, 49-01, 49J15, 49J35, 49J45, 49J53, 35-01 ISSN 1439-7382 ISBN-10 3-540-24133-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24133-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

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Preface

Nonlinear analysis is a new area that was born and has matured from abundant research developed in studying nonlinear problems. In the past thirty years, nonlinear analysis has undergone rapid growth; it has become part of the mainstream research ﬁelds in contemporary mathematical analysis. Many nonlinear analysis problems have their roots in geometry, astronomy, ﬂuid and elastic mechanics, physics, chemistry, biology, control theory, image processing and economics. The theories and methods in nonlinear analysis stem from many areas of mathematics: Ordinary diﬀerential equations, partial diﬀerential equations, the calculus of variations, dynamical systems, diﬀerential geometry, Lie groups, algebraic topology, linear and nonlinear functional analysis, measure theory, harmonic analysis, convex analysis, game theory, optimization theory, etc. Amidst solving these problems, many branches are intertwined, thereby advancing each other. The author has been oﬀering a course on nonlinear analysis to graduate students at Peking University and other universities every two or three years over the past two decades. Facing an enormous amount of material, vast numbers of references, diversities of disciplines, and tremendously diﬀerent backgrounds of students in the audience, the author is always concerned with how much an individual can truly learn, internalize and beneﬁt from a mere semester course in this subject. The author’s approach is to emphasize and to demonstrate the most fundamental principles and methods through important and interesting examples from various problems in diﬀerent branches of mathematics. However, there are technical diﬃculties: Not only do most interesting problems require background knowledge in other branches of mathematics, but also, in order to solve these problems, many details in argument and in computation should be included. In this case, we have to get around the real problem, and deal with a simpler one, such that the application of the method is understandable. The author does not always pursue each theory in its broadest generality; instead, he stresses the motivation, the success in applications and its limitations.

VI

Preface

The book is the result of many years of revision of the author’s lecture notes. Some of the more involved sections were originally used in seminars as introductory parts of some new subjects. However, due to their importance, the materials have been reorganized and supplemented, so that they may be more valuable to the readers. In addition, there are notes, remarks, and comments at the end of this book, where important references, recent progress and further reading are presented. The author is indebted to Prof. Wang Zhiqiang at Utah State University, Prof. Zhang Kewei at Sussex University and Prof. Zhou Shulin at Peking University for their careful reading and valuable comments on Chaps. 3, 4 and 5. Peking University September, 2003

Kung Ching Chang

Contents

1

Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Diﬀerential Calculus in Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1.1.1 Frechet Derivatives and Gateaux Derivatives . . . . . . . . . . 1.1.2 Nemytscki Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 High-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Implicit Function Theorem and Continuity Method . . . . . . . . . . 1.2.1 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Continuity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lyapunov–Schmidt Reduction and Bifurcation . . . . . . . . . . . . . . 1.3.1 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Lyapunov–Schmidt Reduction . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 A Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Hard Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Small Divisor Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Nash–Moser Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 7 9 12 12 17 23 30 30 33 43 47 49 54 55 62

2

Fixed-Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1 Order Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2 Convex Function and Its Subdiﬀerentials . . . . . . . . . . . . . . . . . . . 80 2.2.1 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2.2 Subdiﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.3 Convexity and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.4 Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.5 Monotone Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.6 Maximal Monotone Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

VIII

Contents

3

Degree Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.1 The Notion of Topological Degree . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.2 Fundamental Properties and Calculations of Brouwer Degrees . 137 3.3 Applications of Brouwer Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.3.1 Brouwer Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . 148 3.3.2 The Borsuk-Ulam Theorem and Its Consequences . . . . . 148 3.3.3 Degrees for S 1 Equivariant Mappings . . . . . . . . . . . . . . . . 151 3.3.4 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.4 Leray–Schauder Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.5 The Global Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.6.1 Degree Theory on Closed Convex Sets . . . . . . . . . . . . . . . 175 3.6.2 Positive Solutions and the Scaling Method . . . . . . . . . . . . 180 3.6.3 Krein–Rutman Theory for Positive Linear Operators . . . 185 3.6.4 Multiple Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.6.5 A Free Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3.6.6 Bridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 3.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.7.1 Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.7.2 Strict Set Contraction Mappings and Condensing Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.7.3 Fredholm Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4

Minimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.1 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.1.1 Constraint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.1.2 Euler–Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.1.3 Dual Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.2 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.2.1 Fundamental Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement . . . . . . . . . . 223 4.3 Quasi-Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.3.1 Weak Continuity and Quasi-Convexity . . . . . . . . . . . . . . . 232 4.3.2 Morrey Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 4.3.3 Nonlinear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 4.4 Relaxation and Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.4.1 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.4.2 Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4.5 Other Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.5.1 BV Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.5.2 Hardy Space and BMO Space . . . . . . . . . . . . . . . . . . . . . . . 266 4.5.3 Compensation Compactness . . . . . . . . . . . . . . . . . . . . . . . . 271 4.5.4 Applications to the Calculus of Variations . . . . . . . . . . . . 274

Contents

IX

4.6 Free Discontinuous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 4.6.1 Γ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 4.6.2 A Phase Transition Problem . . . . . . . . . . . . . . . . . . . . . . . . 280 4.6.3 Segmentation and Mumford–Shah Problem . . . . . . . . . . . 284 4.7 Concentration Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.7.1 Concentration Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.7.2 The Critical Sobolev Exponent and the Best Constants 295 4.8 Minimax Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 4.8.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 301 4.8.2 Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5

Topological and Variational Methods . . . . . . . . . . . . . . . . . . . . . . 315 5.1 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 5.1.2 Deformation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 5.1.3 Critical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 5.1.4 Global Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 5.1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 5.2 Minimax Principles (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5.2.1 A Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5.2.2 Category and Ljusternik–Schnirelmann Multiplicity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 5.2.3 Cap Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 5.2.4 Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 5.2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 5.3.1 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 5.3.2 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 5.3.3 Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 5.4 Prescribing Gaussian Curvature Problem on S 2 . . . . . . . . . . . . . 380 5.4.1 The Conformal Group and the Best Constant . . . . . . . . . 380 5.4.2 The Palais–Smale Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 387 5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 5.5 Conley Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 5.5.1 Isolated Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 5.5.2 Index Pair and Conley Index . . . . . . . . . . . . . . . . . . . . . . . . 397 5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

1 Linearization

The ﬁrst and the easiest step in studying a nonlinear problem is to linearize it. That is, to approximate the initial nonlinear problem by a linear one. Nonlinear diﬀerential equations and nonlinear integral equations can be seen as nonlinear equations on certain function spaces. In dealing with their linearizations, we turn to the diﬀerential calculus in inﬁnite-dimensional spaces. The implicit function theorem for ﬁnite-dimensional space has been proved very useful in all diﬀerential theories: Ordinary diﬀerential equations, diﬀerential geometry, diﬀerential topology, Lie groups etc. In this chapter we shall see that its inﬁnite-dimensional version will also be useful in partial diﬀerential equations and other ﬁelds; in particular, in the local existence, in the stability, in the bifurcation, in the perturbation problem, and in the gluing technique etc. This is the contents of Sects. 1.2 and 1.3. Based on Newton iterations and the smoothing operators, the Nash–Moser iteration, which is motivated by the isometric embeddings of Riemannian manifolds into Euclidean spaces and the KAM theory, is now a very important tool in analysis. Limited in space and time, we restrict ourselves to introducing only the spirit of the method in Sect. 1.4.

1.1 Diﬀerential Calculus in Banach Spaces There are two kinds of derivatives in the diﬀerential calculus of several variables, the gradients and the directional derivatives. We shall extend these two to inﬁnite-dimensional spaces. Let X, Y and Z be Banach spaces, with norms · X , · Y , · Z , respectively. If there is no ambiguity, we omit the subscripts. Let U ⊂ X be an open set, and let f : U → Y be a map.

2

1 Linearization

1.1.1 Frechet Derivatives and Gateaux Derivatives Deﬁnition 1.1.1 (Fr´echet derivative) Let x0 ∈ U ; we say that f is Fr´echet diﬀerentiable (or F-diﬀerentiable) at x0 , if ∃A ⊂ L(X, Y ) such that f (x) − f (x0 ) − A(x − x0 ) Y = ◦( x − x0 X ) . Let f (x0 ) = A, and call it the Fr´echet (or F-) derivative of f at x0 . If f is F-diﬀerentiable at every point in U , and if x → f (x), as a mapping from U to L(X, Y ), is continuous at x0 , then we say that f is continuously diﬀerentiable at x0 . If f is continuously diﬀerentiable at each point in U , then we say that f is continuously diﬀerentiable on U , and denote it by f ∈ C 1 (U, Y ). Parallel to the diﬀerential calculus of several variables, by deﬁnition, we may prove the following: 1. If f is F-diﬀerentiable at x0 , then f (x0 ) is uniquely determined. 2. If f is F-diﬀerentiable at x0 , then f must be continuous at x0 . 3. (Chain rule) Assume that U ⊂ X, V ⊂ Y are open sets, and that f is F-diﬀerentiable at x0 , and g is F-diﬀerentiable at f (x0 ), where f

g

U −−−−→ V −−−−→ Z Then

(g ◦ f ) (x0 ) = g ◦ f (x0 ) · f (x0 ) .

Deﬁnition 1.1.2 (Gateaux derivative) Let x0 ∈ U ; we say that f is Gateaux diﬀerentiable (or G-diﬀerentiable) at x0 , if ∀h ∈ X, ∃ df (x0 , h) ⊂ Y , such that f (x0 + th) − f (x0 ) − tdf (x0 , h)Y = ◦(t) as t → 0 for all x0 + th ⊂ U . We call df (x0 , h) the Gateaux derivative (or G-derivative) of f at x0 . We have

d f (x0 + th) |t=0 = df (x0 , h) , dt if f is G-diﬀerentiable at x0 . By deﬁnition, we have the following properties: 1. If f is G-diﬀerentiable at x0 , then df (x0 , h) is uniquely determined. 2. df (x0 , th) = tdf (x0 , h) ∀t ∈ R1 . 3. If f is G-diﬀerentiable at x0 , then ∀h ∈ X, ∀y ∗ ∈ Y ∗ , the function ϕ(t) = y ∗ , f (x0 + th) is diﬀerentiable at t = 0, and ϕ (t) = y ∗ , df (x0 , h) . 4. Assume that f : U → Y is G-diﬀerentiable at each point in U , and that the segment {x0 + th | t ∈ [0, 1]} ⊂ U , then f (x0 + h) − f (x0 ) Y sup df (x0 + th, h) Y 0 0, ∃δ = δ(M, ε) > 0 such that | ϕu (x, ξ) − ϕu (x, ξ ) |< ε, ∀x ∈ Ω , as |ξ|, |ξ | M and |ξ − ξ | δ. We choose M = u0 + h , then for |t| < δ < 1, |ϕu (x, u0 (x) + tθ(x)h(x)) − ϕu (x, u0 (x))| < ε . It follows that df (u0 , h)(x) = ϕu (x, u0 (x))h(x). Noticing that the multiplication operator h → A(u)h = ϕu (x, u(x)) · h(x) is linear and continuous, and the mapping u → A(u) from C(Ω) into L(C(Ω), C(Ω)) is continuous, from Theorem 1.1.3, f is F-diﬀerentiable, and (f (u0 ) · v)(x) = ϕu (x, u0 (x)) · v(x) ∀v ∈ C(Ω) .

1.1 Diﬀerential Calculus in Banach Spaces

5

We investigate nonlinear diﬀerential operators on more general spaces. Let Ω ⊂ Rn be a bounded open set, and let m be a nonnegative integer, older space C m,γ (Ω)) is deﬁned to be the function γ ∈ (0, 1). C m (Ω) (and the H¨ m space consisting of C functions (with γ-H¨older continuous m-order partial derivatives). The norms are deﬁned as follows: |∂ α u(x)| , u C m = max x∈Ω

|α|≤m

and u C m,γ = u C m +

|∂ α u(x) − ∂ α u(y)| , |x − y|γ |α|=m

max x,y∈Ω

where α = (α1 , α2 , . . . , αn ) is a multi-index, |α| = α1 + α2 + · · · + αn , ∂ α = ∂xα11 ∂xα22 · · · ∂xαnn . We always denote by m∗ the number of the index set {α = (α1 , α2 , . . . , αn ) | |α| m}, and Dm u the set {∂ α u | |α| m}. ∗ Suppose that r is a nonnegative integer, and that ϕ ∈ C ∞ (Ω×Rr ). Deﬁne a diﬀerentiable operator of order r: f (u)(x) = ϕ(x, Dr u(x)) . Suppose m r, then f : C m (Ω) → C m−r (Ω) (and also C m,γ (Ω) → (Ω)) is F-diﬀerentiable. Furthermore C ϕα (x, Dr u0 (x)) · ∂ α h(x), ∀h ∈ C m (Ω) , (f (u0 )h)(x) = m−r,γ

|α|≤r

where ϕα is the partial derivative of ϕ with respect to the variable index α. The proof is similar to Example 3. ∗

Example 4. Suppose ϕ ∈ C ∞ (Ω × Rr ). Deﬁne

f (u) = ϕ(x, Dr u(x))dx ∀u ∈ C r (Ω) . Ω

Then f : C r (Ω) → R1 is F-diﬀerentiable. Furthermore

f (u0 ), h = ϕα (x, Dr u0 (x))∂ α h(x)dx

∀h ∈ C r (Ω) .

Ω |α|≤r

Proof. Use the chain rule: ϕ(·,D r u(·))

C r (Ω) −−−−−−−→ C(Ω) −−−Ω−→ R1 , and combine the results of Examples 1 and 3.

6

1 Linearization

In particular, the following functional occurs frequently in the calculus of variations (r = 1, r∗ = n + 1). Assume that ϕ(x, u, p) is a function of the form: ϕ(x, u, p) =

n 1 2 |p| + ai (x)pi + a0 (x)u , 2 i=1

where p = (p1 , p2 , . . . , pn ), and ai (x), i = 0, 1, . . . , n, are in C(Ω). Set

n 1 2 |∇u(x)| + f (u) = ai (x)∂xi u + a0 (x)u(x) dx , Ω 2 i=1 we have

∇u(x) · ∇h(x) +

f (u), h = Ω

n

ai (x)∂xi h(x) + a0 (x)h(x) dx

i=1

∀h ∈ C 1 (Ω) . Example 5. Let X be a Hilbert space, with inner product (, ). Find the Fderivative of the norm f (x) = x , as x = θ. Let F (x) = x 2 . Since t−1 ( x + th 2 − x 2 ) = 2(x, h) + t h 2 , we have dF (x, h) = 2(x, h). It is continuous for all x, therefore F is Fdiﬀerentiable, and F (x)h = 2(x, h) . 1

Since f = F 2 , by the chain rule F (x) = 2 x ·f (x) . As x = θ,

x ,h . f (x)h = x In the applications to PDE as well as to the calculus of variations, Sobolev spaces are frequently used. We should extend the above studies to nonlinear operators deﬁned on Sobolev spaces. ∀p 1, ∀ nonnegative integer m, let

W m,p (Ω) = {u ∈ Lp (Ω) | ∂ α u ∈ Lp (Ω) | |α| m} , where ∂ α u stands for the α-order generalized derivative of u, i.e., the derivative in the distribution sense. Deﬁne the norm ⎞ p1 ⎛ ∂ α u pLp (Ω) ⎠ . u W m,p = ⎝ |α≤m

The Banach space is called the Sobolev space of index {m, p}. W m,2 (Ω) is denoted by H m (Ω), and the closure of C0∞ (Ω) under this norm is denoted by H0m (Ω).

1.1 Diﬀerential Calculus in Banach Spaces

7

1.1.2 Nemytscki Operator On Sobolev spaces, we extend the composition operator u → ϕ(x, u(x)) such that ϕ may not be continuous in x. The class of operators is sometimes called Nemytski operators. Deﬁnition 1.1.4 Let (Ω, B , µ) be a measure space. We say that ϕ : Ω × RN → R1 is a Caratheodory function, if 1. ∀a.e.x ∈ Ω, ξ → ϕ(x, ξ) is continuous. 2. ∀ξ ∈ RN , x → ϕ(x, ξ) is µ-measurable. The motivation in introducing the Caratheodory function is to make the composition function measurable if u(x) is only measurable. Indeed, there exists a sequence of simple functions {un (x)}∞ , such that un (x) → u(x) a.e., ϕ(x, un (x)) is measurable according to (2). And from (1), ϕ(x, un (x)) → ϕ(x, u(x)) a.e., therefore ϕ(x, u(x)) is measurable. p

Theorem 1.1.5 Assume p1 , p2 1, a > 0 and b ∈ Ldµ2 (Ω). Suppose that ϕ is a Caratheodory function satisfying p1

|ϕ(x, ξ)| b(x) + a|ξ| p2 . Then f : u(x) → ϕ(x, u(x)) is a bounded and continuous mapping from p p Ldµ1 (Ω, RN ) to Ldµ2 (Ω, RN ). Proof. The boundedness follows from the Minkowski inequality: p1 p

f (u) p2 b p2 + a u p 2 , 1

where · p is the Lpdµ (Ω, RN ) norm. We turn to proving the continuity. It is p1 suﬃcient to prove that ∀{un }∞ 1 if un → u in L , then there is a subsequence p2 {uni } such that f (uni ) → f (u) in L . Indeed one can ﬁnd a subsequence {uni } of {un } which converges a.e. to u, along which uni − uni−1 p1 < 21i , i = 2, 3, . . .; therefore |uni (x)| Φ(x) := |un1 (x)| +

∞

|uni (x) − uni−1 (x)| .

i=2

Since Φ is measurable, and

|Φ(x)|p1 dµ

p1

1

Ω

un1 p1 +

∞ i=2

p

we conclude that Φ ∈ Ldµ1 (Ω). Noticing

uni − uni−1 p1 < +∞ ,

8

1 Linearization

f (uni ) = ϕ(x, uni (x)) → ϕ(x, u(x)) and

p1

a.e. ,

p

|f (uni )(x)| b(x) + a(Φ(x)) p2 ∈ Ldµ2 (Ω, RN ) , we have f (uni ) − f (u) p2 → 0, according to Lebesgue dominance theorem. This proves the continuity of f . Corollary 1.1.6 Let Ω ⊂ Rn be a smooth bounded domain, and let 1 ≤ ∗ p1 , p2 ≤ ∞. Suppose that ϕ : Ω × Rm → R is a Caratheodory function satisfying m αj |ξj | p2 , |ϕ(x, ξ0 , . . . , ξm )| b(x) + a j=0

where ξj is a #{α = (α1 , . . . , αn )| |α| = j}-vector, αj ( p1 − 1

m−j −1 , n )

a>

0, and b ∈ L (Ω). Then f (u)(x) = ϕ(x, D u(x)) deﬁnes a bounded and continuous map from W m,p1 (Ω) into Lp2 (Ω). p2

m

Corollary 1.1.7 Suppose that Ω ⊂ Rn and that ϕ : Ω×R1 → R1 and ϕξ (x, ξ) 2n are Caratheodory functions. If |ϕξ (x, ξ)| b(x) + a|ξ|r , where b ∈ L n+2 (Ω), n+2 (if n ≤ 2, then the restriction is not necessary), then the a > 0, and r = n−2 functional

ϕ(x, u(x))dx

f (u) = Ω

is F-diﬀerentiable on H 1 (Ω), with F-derivative

f (u), v = ϕξ (x, u(x)) · v(x)dx , Ω

where , is the inner product on H 1 (Ω). Proof. The Sobolev embedding theorem says that the injection i : H 1 (Ω) → 2n 2n L n−2 (Ω) is continuous, so is the dual map i∗ : L n+2 (Ω) → (H 1 (Ω))∗ . 2n 2n According to Theorem 1.1.5, ϕξ (·, ·) : L n−2 → L n+2 is continuous. Therefore the Gateaux derivative

ϕξ (x, u(x)) · v(x)dx ∀v ∈ H 1 (Ω) df (u, v) = Ω

is continuous from H 1 (Ω) to (H 1 (Ω))∗ . Applying Theorem 1.1.3, we conclude that f is F-diﬀerentiable on H 1 (Ω). The proof is complete. Corollary 1.1.8 In Corollary 1.1.6, the diﬀerential operator f (u(x)) = ϕ(x, Dm u(x)) from C l,γ (Ω) to C l−m,γ (Ω), l m, 0 γ < 1, is F-diﬀerentiable, with ϕα (x, Dm u(x))∂ α h(x) ∀h ∈ C l,γ (Ω) . (f (u0 )h)(x) = |α|≤m

1.1 Diﬀerential Calculus in Banach Spaces

9

1.1.3 High-Order Derivatives The second-order derivative of f at x0 is deﬁned to be the derivative of f (x) at x0 . Since f : U → L(X, Y ), f (x0 ) should be in L(X, L(X, Y )). However, if we identify the space of bounded bilinear mappings with L(X, L(X, Y )), and verify that f (x0 ) as a bilinear mapping is symmetric, see Theorem 1.1.9 below, then we can deﬁne equivalently the second derivative f (x0 ) as follows: For f : U → Y , x0 ∈ U ⊂ X, if there exists a bilinear mapping f (x0 )(·, ·) of X × X → Y satisfying 1 f (x0 +h)−f (x0 )−f (x0 )h− f (x0 )(h, h) = ◦( h 2 ) ∀h ∈ X, as h → 0 , 2 then f (x0 ) is called the second-order derivative of f at x0 . By the same manner, one deﬁnes the mth-order derivatives at x0 successively: f (m) (x0 ) : X × · · · × X → Y is an m-linear mapping satisfying m (j) f (x0 )(h, . . . , h) f (x0 + h) − = ◦( h m ) , j! j=0 as h → 0. Then f is called m diﬀerentiable at x0 . Similar to the ﬁnite-dimensional vector functions, we have: Theorem 1.1.9 Assume that f : U → Y is m diﬀerentiable at x0 ∈ U . Then for any permutation π of (1, . . . , m), we have f (m) (x0 )(h1 , . . . , hm ) = f (m) (x0 )(hπ(1) , . . . , hπ(m) ) . Proof. We only prove this in the case where m = 2, i.e., f (x0 )(ξ, η) = f (x0 )(η, ξ) ∗

∀ξ, η ∈ X .

∗

Indeed ∀y ∈ Y , we consider the function ϕ(t, s) = y ∗ , f (x0 + tξ + sη) . It is twice diﬀerentiable at t = s = 0; so is ∂2 ∂2 ϕ(0, 0) = ϕ(0, 0) . ∂t∂s ∂s∂t Since f (x0 + tξ + sη) is continuous as |t|, |s| small, one has ∂ ϕ(t, ·)|s=0 = y ∗ , f (x0 + tξ)η ; ∂s and then, ∂2 ϕ(t, s)|t=s=0 = y ∗ , f (x0 )(ξ, η) . ∂t∂s Similarly ∂2 ϕ(t, s)|t=s=0 = y ∗ , f (x0 )(η, ξ) . ∂s∂t This proves the conclusion.

10

1 Linearization

Theorem 1.1.10 (Taylor formula) Suppose that f : U → Y is continuously m-diﬀerentiable. Assume the segment {x0 + th | t ∈ [0, 1]} ⊂ U . Then f (x0 + h) =

m 1 (j) f (x0 )(h, . . . , h) j! j=0

1 1 (1 − t)m f (m+1) (x0 + th)(h, . . . , h)dt . + m! 0

Proof. ∀y ∗ ∈ Y ∗ , we consider the function: ϕ(t) = y ∗ , f (x0 + th) . From the Hahn–Banach theorem and the Taylor formula for single-variable functions:

1 m 1 (j) 1 ϕ (0) + (1 − t)m ϕ(m+1) (t)dt , ϕ(1) = j! m! 0 j=0 we obtain the desired Taylor formula for mappings between B-spaces.

Example 1. X = Rn , Y = R1 . If f : X → Y is twice continuously diﬀerentiable, then 2 ∂ f (x) . f (x) = Hf (x) = ∂xi ∂xj i,j=1,...,n Example 2. X = C 1 (Ω, RN ), Y = R1 . Suppose that g ∈ C 2 (Ω × RN , R1 ). Deﬁne

1 |∇u|2 + g(x, u(x)) f (u) = 2 Ω Ω as u ∈ X. By deﬁnition, we have

f (u) · ϕ = [∇u(x)∇ϕ(x) + gu (x, u(x))ϕ(x)]dx , Ω

and f (u)(ϕ, ψ) =

[∇ψ(x)∇ϕ(x) + guu (x, u(x))ϕ(x)ψ(x)]dx .

Ω With some additional growth conditions on guu : 4

(x, u)| ≤ a(1 + |u| n−2 ), a > 0 ∀u ∈ RN , |guu

f is twice diﬀerentiable in H01 (Ω, RN ). As an operator from H01 (Ω, RN ) into itself, f (u) = id + (−)−1 gu (·, u(·)) .

1.1 Diﬀerential Calculus in Banach Spaces

11

is self-adjoint, or equivalently, the operator − + guu (x, u(x))· deﬁned on L2 2 1 N is self-adjoint with domain H ∩ H0 (Ω, R ).

Example 3. Let X = H01 (Ω, R3 ), where Ω is a plane domain. Consider the volume functional

u · (ux ∧ uy ) . Q(u) = Ω

One has Q (u) · ϕ =

ϕ(ux ∧ uy ) + u · [(ϕx ∧ uy ) + (ux ∧ ϕy )] , Ω

and Q (u)(ϕ, ψ) =

ϕ[(ψx ∧ uy ) + (ux ∧ ψy )] + ψ[(ϕx ∧ uy ) + (ux ∧ ϕy )] Ω

+ u[(ϕx ∧ ψy ) + (ψx ∧ ϕy )] , ∀ϕ, ψ ∈ H01 (Ω, R3 ). If further we assume u ∈ C 2 (Ω, R3 ), then from integration by parts and the antisymmetry of the exterior product, we have

u(ϕx ∧ uy ) = − (ϕ ∧ uy ) · ux + (ϕ ∧ uxy ) · u Ω

Ω = (ux ∧ uy )ϕ − (ϕ ∧ uxy ) · u , Ω

and

Ω

u(ux ∧ ϕy ) = − uy (ux ∧ ϕ) + u(uxy ∧ ϕ) Ω Ω

ϕ(ux ∧ uy ) + (ϕ ∧ uxy )u . =

Ω

Therefore, Q (u) = 3

Ω

ϕ · (ux ∧ uy ) . Ω

By the same manner, we obtain

Q (u)(ϕ, ψ) = 3 u[(ϕx ∧ ψy ) + (ϕy ∧ ψx )] . Ω C

2

Geometrically, let u : Ω → R3 be a parametrized surface in R3 ; Q(U ) is the volume of the body enclosed by the surface. As exercises, one computes the ﬁrst- and second-order diﬀerentials of the following functionals:

12

1 Linearization

1. X = W01,p (Ω, R1 ), Y = R1 , 2 < p < ∞,

f (u) = |∇u|p dx . Ω

2. X = C01 (Ω, Rn ), where Ω ⊂ Rn is a domain,

f (u) = det(∇u(x))dx . Ω

3. X = C01 (Ω, R1 ), where Ω ⊂ Rn is a domain,

f (u) = 1 + |∇u|2 dx . Ω

1.2 Implicit Function Theorem and Continuity Method 1.2.1 Inverse Function Theorem It is known that the implicit function theorem for functions of several variables plays important roles in many branches of mathematics (diﬀerential manifold, diﬀerential geometry, diﬀerential topology, etc.). Its extension to inﬁnite-dimensional space is also extremely important in nonlinear analysis, as well as in the study of inﬁnite-dimensional manifolds. Theorem 1.2.1 (Implicit function theorem) Let X, Y , Z be Banach spaces, U ⊂ X × Y be an open set. Suppose that f ∈ C(U, Z) has an F-derivative w.r.t. y, and that fy ∈ C(U, L(Y, Z)). For a point (x0 , y0 ) ∈ U , if we have f (x0 , y0 ) = θ , fy−1 (x0 , y0 ) ∈ L(Z, Y ) ; then ∃r, r1 > 0, ∃|u ∈ C(Br (x0 ), Br1 (y0 )), such that ⎧ ⎨ Br (x0 ) × Br1 (y0 ) ⊂ U , u(x0 ) = y0 , ⎩ f (x, u(x)) = θ ∀x ∈ Br (x0 ) . Furthermore, if f ∈ C 1 (U, Z), then u ∈ C 1 (Br (x0 ), Y ), and u (x) = −fy−1 (x0 , u(x0 )) ◦ fx (x, u(x))

∀x ∈ Br (x0 ) .

(1.1)

1.2 Implicit Function Theorem and Continuity Method

13

Proof. (1) After replacing f by g(x, y) = fy−1 (x0 , y0 ) ◦ f (x + x0 , y + y0 ) , one may assume x0 = y0 = θ, Z = Y and fy (θ, θ) = idY . (2) We shall ﬁnd the solution y = u(x) ∈ Br1 (θ) of the equation f (x, y) = θ

∀x ∈ Br (θ) .

Setting R(x, y) = y − f (x, y) , it is reduced to ﬁnding the ﬁxed point of R(x, ·) ∀x ∈ Br (θ). We shall apply the contraction mapping theorem to the mapping R(x, ·). Firstly, we have a contraction mapping: R(x, y1 ) − R(x, y2 ) = y1 − y2 − [f (x, y1 ) − f (x, y2 )]

1 fy (x, ty1 + (1 − t)y2 )dt · (y1 − y2 ) = y 1 − y2 −

0 1

idY − fy (x, ty1 + (1 − t)y2 ) dt· y1 − y2 . 0

Since fy : U → L(X, Y ) is continuous, ∃r, r1 > 0 such that R(x, y1 ) − R(x, y2 )

0, where f (x, θ)

0 such that Bδ (y0 ) ⊂ V and g : Bδ (y0 ) → g(Bδ (y0 )) is a diﬀermorphism. Furthermore (g −1 ) (x0 ) = g −1 (y0 ), with x0 = g(y0 ) .

(1.5)

1.2 Implicit Function Theorem and Continuity Method

15

Proof. Set f (x, y) = x − g(y),

f ∈ C 1 (X × V, X) .

We use the implicit function theorem (IFT) to f , there exist r > 0 and a unique u ∈ C 1 (Br (x0 ), Br (y0 )) satisfying x = g ◦ u(x) . Since g is continuous, ∃δ ∈ (0, r) such that g(Bδ (y0 )) ⊂ Br (x0 ), therefore g : Bδ (y0 ) → g(Bδ (y0 )) is a diﬀeomorphism. And (1.5) follows from (1.1). In the spirit of the IFT, we have a nonlinear version of the Banach open mapping theorem. Theorem 1.2.4 (Open mapping) Let X, Y be Banach spaces, and let δ > 0 and y0 ∈ Y . Suppose that g ∈ C 1 (Bδ (y0 ), X) and that g (y0 ) : Y → X is an open map, then g is an open map in a neighborhood of y0 . Proof. We want to prove that ∃δ1 ∈ (0, δ) and r > 0, such that Br (g(y0 )) ⊂ g(Bδ1 (y0 )) . With no loss of generality, we may assume y0 = θ and g(y0 ) = θ. Let A = g (θ). Since A is surjective, ∃C > 0 such that inf

z∈ker A

y − z Y C Ay X , ∀ y ∈ Y ,

(1.6)

provided by the Banach inverse theorem. One chooses δ1 ∈ (0, δ) and r > 0, satisfying 1 ∀y ∈ Bδ1 (θ) , g (y) − A 2(C + 1) and r

0, φt ◦ h ◦ ψ−t − id ∈ C 0 (Rn , Rn ) is in a small neighborhood of θ, and the above equation has a unique solution h there; therefore φt ◦ h ◦ ψ−t = h . Then we can extend the procedure step by step to all t, i.e. we have φt ◦ h = h ◦ ψt ∀t ∈ R1 .

20

1 Linearization

Example 2. (Local existence of isothermal coordinates) Given a surface M 2 with a Riemannian metric g, i.e., in local coordinates x = (x1 , x2 ), g = Edx21 + 2F dx1 dx2 + Gdx22 , where E, F and G are functions of local coordinates x = (x1 , x2 ), and ∀(x1 , x2 ), Eξ 2 +2F ξη +Gη 2 is a positive deﬁnite quadratic form. Our problem is to ﬁnd a local coordinate u(x) = (u1 (x1 , x2 ), u2 (x1 , x2 )) in a neighborhood of x0 = (x01 , x02 ) such that there exists a function λ = λ(x1 , x2 ) > 0 satisfying g = λ(x1 , x2 )(du21 + du22 ) . The local coordinate u = (u1 , u2 ) is called an isothermal coordinate. We shall ﬁnd u = (u1 , u2 ) satisfying ⎧ ⎨ λ(x)|∂x1 u(x)|2 = E(x) , λ(x)∂x1 u(x) · ∂x2 u(x) = F (x) , ⎩ λ(x)|∂x2 u(x)|2 = G(x) .

(1.11)

We may assume (x01 , x02 ) = (0, 0) and F (0, 0) = 0 (by translation and rotation of the local coordinates). After eliminating λ, (1.11) is equivalent to: F |∂x1 u|2 = E∂x1 u · ∂x2 u (1.12) F |∂x2 u|2 = G∂x1 u · ∂x2 u . This is a ﬁrst-order nonlinear diﬀerential system. In a neighborhood O of θ = (0, 0), deﬁne F (x) · |∂x1 u(x)|2 − E(x) · ∂x1 u(x) · ∂x2 u(x) ϕ(x, ∇u(x)) = , F (x) · |∂x2 u(x)|2 − G(x) · ∂x1 u(x) · ∂x2 u(x) , ∀x ∈ O, and let f be the map satisfying f (u)(x) = ϕ(x, ∇u(x)) . We want to solve the equation: f (u) = θ .

(1.13)

Let x = εy, where y ∈ D, the unit disk on the coordinate plane, and ε > 0 is a parameter, and let (1.14) εv(y) = u(x) − u(x) . where u = (p1 x1 + p2 x2 , q1 x1 + q2 x2 ), and (p1 , p2 , q1 , q2 ) is chosen such that the following conditions are satisﬁed: ∂(u1 , u2 ) = 0 , ∂(x1 , x2 )

1.2 Implicit Function Theorem and Continuity Method

21

and ϕ(0, ∇u) = θ . In fact, this is an algebraic system with four unknowns, and is trivially solvable. Equation (1.13) is equivalent to ϕ(εy, ∇y v(y) + ∇x u(εy)) = 0 . ¯ denote the space C 1,α (D) ¯ modulo a constant. Set F : R1 × Let C˙ 1,α (D) ˙ 1,α 0 α 2 ∩ C0 (D) C (D)) → C (D) , for some α ∈ (0, 1), where (C 1,α

F (ε, v) = ϕ(εy, ∇y v(y) + ∇x u(εy)) . Thus F (0, θ) = θ . It remains to ﬁnd a pair (ε, v) for small ε > 0, satisfying F (ε, v) = θ . Then by the transform(1.14), the solution u in a small neighborhood of θ εD is obtained. Note that (p1 , q1 , p2 , q2 ) = (u1,x1 , u2,x1 , u1,x2 , u2,x2 ). Let us introduce the notation (E0 , F0 , G0 ) = (E, F, G)|(0,0) ; we have Fv (0, θ)h = −(A1 ∇h1 + A2 ∇h2 ) , where h = (h1 , h2 ) ∈ C 1,α ∩ C00 (D) C˙ 1,α (D), and (E0 p2 − 2F0 p1 ) E0 p1 A1 = , G0 p2 (G0 p1 − 2F0 p2 ) (E0 q2 − 2F0 q1 ) E0 q1 . A2 = G0 q2 (G0 q1 − 2F0 q2 ) This is a constant coeﬃcient ﬁrst-order linear diﬀerential operator with the 2 × 2 matrix symbol: L = (A1 ω, A2 ω) , where ω = (ξ, η). Since det L = −2F0 (E0 ξ 2 − 2F0 ξη + G0 η 2 )

∂(u, v) = 0 ∂(x, y)

∀(ξ, η) ∈ R2 \{θ}, L is elliptic. According to the elliptic theory, Fv (0, ·) has a bounded inverse. The IFT is applied to conclude the existence of ε0 > 0 such that the equation F (ε, v) = θ has a unique solution vε , ∀ε ∈ (−ε0 , ε0 ).

22

1 Linearization

Remark 1.2.7 The boundary condition for the ﬁrst-order system can be deduced from the second-order elliptic theory. In fact, h = (h1 , h2 ) satisﬁes A1 ∇h1 + A2 ∇h2 = f ,

(1.15)

where f is given. Suppose that A2 is invertible (it is available); we have ∇h2 = C∇h1 + A−1 2 f , where

(1.16)

C = −A−1 2 A1 ,

Let E= Then E −1 =

0 1 −1 0

0 1

−1 0

. ,

and E 2 = −I. Since h2,xy = h2,yx ,

(1.17)

we obtain a second-order equation for h1 , in which the principal symbol of the second-order equation reads as 1 E 2 (A2 ω)T E −1 A1 ω det(A2 ) 1 (A2 ω)T E −1 A1 ω = det(A2 ) det(L) . = det(A2 )

−ω T EA−1 2 A1 ω = −

The later is positive (or negative) deﬁnite because the ﬁrst-order system is elliptic. Thus the Dirichlet boundary condition for h1 is well posed, and then h2 follows from (1.16) modulo a constant. Remark 1.2.8 There are many applications of the IFT similar to the above examples, e.g. a necessary and suﬃcient condition for an almost complex structure being a complex structure (Newlander–Nirenberg theorem, see L. Nirenberg [NN]), prescribing Ricci curvature problem (see DeTurck [DT] etc.). For applications to boundary value problems in ordinary and partial diﬀerential equations see S.N. Chow, J. Hale [CH] and J. Mawhin [Maw 3]

1.2 Implicit Function Theorem and Continuity Method

23

1.2.3 Continuity Method We have shown the usefulness of the IFT in the existence of solutions for small perturbations of a given equation which has a known solution. As to large perturbations, the IFT is not enough, we have to add new ingredients. The continuity method is a general principle, which can be applied to prove the existence of solutions for a variety of nonlinear equations. Let X and Y be Banach spaces, and f : X → Y be C 1 . Find the solution of the equation: f (x) = θ . Let us introduce a parameter t ∈ [0, 1] and a map F : [0, 1] × X → Y such that both F and Fx are continuous; in addition, F (1, x) = f (x) . Assume that there exists x0 ∈ X satisfying F (0, x0 ) = θ; we want to extend the solution x0 of the equation F (0, x) = θ to a solution of F (1, x) = θ .

(1.18)

For this purpose, we deﬁne a set S = {t ∈ [0, 1] | such that F (t, x) = θ is solvable} . What we want to do is to prove: (1) S is an open set (relative to [0, 1]). For this purpose, it is suﬃcient to prove that ∀t0 ∈ S, ∃xt0 ∈ X, which solves F (t0 , xt0 ) = θ such that Fx−1 (t0 , xt0 ) ∈ L(Y, X), provided by the IFT. (2) S is a closed set. Usually it depends on the a priori estimates for the solution set {x ∈ X | ∃t ∈ S, such that F (t, x) = θ}. For most PDE problems it requires special knowledge and features of the equations and techniques in hard analysis. We present here two major ideas: (a) If there exist a Banach space X1 , which is compactly embedded in X, and a constant C > 0 such that xt X1 C

∀t ∈ S ,

where xt is a solution of F (t, x) = θ, then S is closed.

24

1 Linearization

In fact, we have {tn }∞ ⊂ S, which implies tn → t∗ , and xtn X1 C . Since the embedding X1 → X is compact, xtn subconverges to some point x∗ ∈ X in X. From the continuity, it follows that F (t∗ , x∗ ) = θ. This proves t∗ ∈ S, i.e., S is closed. (b) If ∀t ∈ S, there exists a unique local solution xt of the equation F (t, ·) = θ, and if there exists C > 0 such that x˙ t X C , where x˙ t is the derivative of xt , then S is a closed set. Proof. Let {tn }∞ 1 be a sequence included in an open interval contained in S, with tn ↑ t∗ . Then

tn xtn − xtm x˙ t dt C(tn − tm ) → 0 , tm

as n m → ∞. Let x∗ be the limit. If the IFT is applicable to (t∗ , x∗ ), then we have t∗ ∈ S, i.e., S is closed. Once (1) and (2) are proved, S is a nonempty (0 ∈ S) open and closed set. Therefore S = [0, 1], and then the equation F (1, ·) = θ is solvable. As an application of the continuity method, we have: Theorem 1.2.9 (Global implicit function theorem) Let X, Y be Banach spaces, and let f ∈ C 1 (X, Y ) with f (x)−1 ∈ L(Y, X) ∀x ∈ X. If ∃ constants A, B > 0 such that f (x)−1 A x +B

∀x ∈ X ,

then f is a diﬀeomorphism. Proof. (1) Surjective. We want to prove that ∀y ∈ Y, ∃x ∈ X satisfying f (x) = y . ∀x0 ∈ X, deﬁne F : [0, 1] × X → Y as follows: F (t, x) = f (x) − [(1 − t)f (x0 ) + ty] . Set S = {t ∈ [0, 1]| F (t, ·) = θ is solvable}. Obviously, 0 ∈ S, and since Fx−1 (t, x) = f (x)−1 ∈ L(Y, X), S is open, from the IFT.

1.2 Implicit Function Theorem and Continuity Method

25

It remains to prove the closeness of S. Indeed, in a component (a, b) of S, there is a branch of solutions xt satisfying F (t, xt ) = θ and then

∀t ∈ (a, b) ,

f (xt )x˙ t = y − f (x0 ) .

Thus, x˙ t f (xt )−1 y − f (x0 ) (A xt +B) y − f (x0 ) . Set c =

a+b 2 ,

(1.19)

so

t

xt xc +

y − f (x0 ) (A xs +B)ds

as t > c .

c

Applying Gronwall’s inequality, ∃ a constant C > 0 such that xt C .

(1.20)

Substituting (1.20) into (1.19), we have another constant C1 > 0 such that x˙ t C1 , ∀t ∈ (a, b) . This proves the closeness of S, therefore f is surjective. (2) Injective. We argue by contradiction. If ∃y ∈ Y and x0 , x1 ∈ X, satisfying f (xi ) = y, i = 0, 1. Let γ : [0, 1] → X be the segment connecting these two points: γ(s) = (1 − s)x0 + sx1 s ∈ [0, 1] . Thus f ◦γ is a loop passing through y. If we could ﬁnd x : [0, 1] → X satisfying x(i) = xi i = 0, 1,

and

f ◦ x(s) = y ∀s ∈ [0, 1] , then this would contradict with the locally homeomorphism of f . Deﬁne I = [0, 1] and T : I × C0 (I, X) → C0 (I, Y ) as follows: (t, u(s)) → f (γ(s) + u(s)) − ty − (1 − t)f (γ(s)) , where C0 (I, X) = {u ∈ C(I, X)| u(0) = u(1) = θ} . We want to solve T (t, u) = θ . Obviously, T (0, θ) = θ; and if we have u ∈ C0 (I, X) satisfying T (1, u(·)) = θ, then x(s) = u(s) + γ(s) is what we need. Now,

26

1 Linearization

f

r

y

x1 x0

r Fig. 1.2.

1.

Tu (t, u) = f (γ(·) + u(·)) ∈ L(C0 (I, X), C0 (I, Y )) , which has a bounded inverse. Therefore S = {t ∈ [0, 1]| T (t, u) = θ is solvable} is open, from the IFT.

2. Let ut (s) be a solution at t ∈ S. Then f (γ(s) + ut (s)) · u˙ t (s) = y − f ◦ γ(s) , where u˙ t denotes the derivative with respect to t. Again we obtain u˙ t C0 (I,X) (A ut C0 (I,X) +B1 ) y − f ◦ γ C0 (I,Y ) , where B1 > 0 is another constant depending on B and x0 , x1 only. As in paragraph (1), ∃C > 0 such that u˙ t C0 (I,X) C

∀t ∈ S .

Again, by the continuity method, 1 ∈ S. This is the contradiction. The injectivity of f is proved. Next we shall give an example showing how a priori estimates give the closeness of S. Theorem 1.2.10 Suppose f ∈ C 1 (Ω × R1 × Rn , R1 ), where Ω ⊂ Rn is a bounded domain of smooth boundary. Assume ∃ constants C > 0, satisfying 1 1 → R+ such that (1) There exists an increasing function c : R+

|f (x, η, ξ)| ≤ c(|η|)(1 + |ξ|2 ), ∀ (x, η, ξ) ∈ Ω × R1 × Rn . (2) ∂f (x, η, ξ) 0 . ∂η

1.2 Implicit Function Theorem and Continuity Method

27

(3) Assume ∃M > 0 such that < 0, if η > M f (x, η, θ) = > 0, if η < −M . Assume φ ∈ C 2,γ , for some γ ∈ (0, 1). Then the equation −u = f (x, u(x), ∇u(x)) x ∈ Ω u|∂Ω = φ

(1.21)

possesses a unique solution in C 2,γ . Lemma 1.2.11 Under the assumption (3), if u ∈ C 2 (Ω) is a solution of (1.21), then u C(Ω) max max |φ(x)|, M . ∂Ω

Proof. Assume that |u(x)| attains its maximum at x0 ∈ Ω. We divide our discussion into two cases. (1) x0 ∈ ∂Ω; the proof is done. ◦

(2) x0 ∈ Ω, then ∇u(x0 ) = 0 and −u(x0 ) = f (x0 , u(x0 ), θ). If u(x0 ) > M , then LHS 0, but RHS < 0. It is impossible. Similarly, if u(x0 ) < −M , then LHS 0, but RHS > 0. Again, it is impossible. We have |u(x0 )| M . Lemma 1.2.12 Assume a ∈ C 0,γ (Ω) and φ ∈ C 2,γ (∂Ω). Then the equation −u + u = a(x)(1 + |∇u|2 ) (1.22) u|∂Ω = φ has a unique solution u ∈ C 2,γ (Ω), and u C 2,γ C( a C 0,γ , φ C 2,γ ) . Proof. We apply the continuity method to study equation (1.22). Deﬁne a map F : X × [0, 1] → Y as follows: (u, τ ) → (−u + u − a(τ + |∇u|2 ), u|∂Ω − τ φ) , where X = C 2,γ (Ω), Y = C 0,γ (Ω) × C 2,γ (∂Ω) and γ ∈ (0, γ). Noticing that Fu (u, τ )v = ((−v + v − a∇u · ∇v), v|∂Ω ) ,

28

1 Linearization

and from the Schauder estimates, ∀(u, τ ) ∈ X × [0, 1], Fu (u, τ ) has a bounded inverse. Deﬁne the set: S = {τ ∈ [0, 1]| ∃uτ solving F (uτ , τ ) = θ} . Equation (1.22) is solvable if 1 ∈ S. Since S is open, and 0 ∈ S, it remains to verify that the set S is closed. To this end, it is suﬃcient to prove that there is a constant C, depending on a and φ such that ∀uτ ∈ S, uτ C 2,γ C .

(1.23)

τ Since uτ satisﬁes F (uτ , τ ) = θ, u˙ τ = du dτ exists, from the IFT, and satisﬁes: −u˙ τ + u˙ τ = a + 2a∇uτ · ∇u˙ τ , (1.24) u˙ τ |∂Ω = φ .

Set g = a + 2a∇uτ · ∇u˙ τ ; we have u˙ τ W 2,p C(p)(1+ g Lp ) C(p, a C )(1+ ∇uτ L2p ∇u˙ τ L2p ) , provided by the Lp estimates. Since u˙ τ satisﬁes (1.24), from Lemma 1.2.11, u˙ τ C(Ω) is bounded by a constant depending on a C(Ω) and φ, and from the Gagliardo–Nirenberg inequality 1

1

∇u˙ τ L2p Cp ∇2 u˙ τ L2 p u˙ τ L2 p +C u˙ τ L∞ , we obtain: u˙ τ W 2,p C(1+ ∇uτ 2L2p ) . Again, by the Gagliardo–Nirenberg inequality, we have 1

1

∇uτ L2p Cp ∇2 uτ L2 p uτ L2 p +C uτ L∞ . Repeating the use of Lemma 1.2.11, uτ C is bounded by a constant depending on φ C , and we obtain u˙ τ W 2,p C( φ C 2,γ , a C , p)(1+ uτ W 2,p ) . From

d uτ W 2,p u˙ τ W 2,p , dτ and the Gronwall inequality, we obtain uτ W 2,p CeCτ , where C depends on p, φC 2,γ and aC .

1.2 Implicit Function Theorem and Continuity Method

As a consequence of the Sobolev embedding theorem, for p > uτ C 1,γ C( φ C 2,γ , a C , γ) .

n 1−γ ,

29

we have (1.25)

Substituting into the equation F (uτ , τ ) = θ, we apply the Schauder estimate: uτ C 2,γ C( φ C 2,γ , a C 0,γ , γ) . The continuity method is applicable; we have a solution u of (1.22). Indeed, the solution is unique. Let u1 , u2 be two solutions and set ω = u1 − u2 , then −ω + ω = a∇(u1 + u2 ) · ∇ω in Ω , ω|∂Ω = 0 . ◦

If max ω > 0, then ∃x0 ∈ Ω such that maxΩ ω = ω(x0 ). Thus, ∇ω(x0 ) = 0, −ω(x0 ) 0 and ω(x0 ) > 0. This is impossible. Similarly, one proves that min ω < 0 is impossible. Therefore ω ≡ 0. Now we come back to the proof of the theorem. Proof. Applying the continuity method, we study the equation: −u = tf (x, u(x), ∇u(x)) t ∈ [0, 1] u|∂Ω = tφ

(1.26)

and turn to considering the operator: F : I × C 2,σ (Ω) → C σ (Ω) × C 2,σ (∂Ω),

I = [0, 1] ,

(t, u) → (−u − tf (x, u(x), ∇u(x)), u|∂Ω − tφ) , where σ ∈ (0, γ). We want to solve F (1, u) = θ. However, ∀u ∈ C 2,σ (Ω), Fu (t, u)v = (−v − tfη (x, u(x), ∇u(x))v − tfξ (x, u(x), ∇u(x))∇v, v|∂Ω ) . By assumption (2) and the maximum principle for linear elliptic equations, ∀g ∈ C σ (Ω) × C 2,σ (∂Ω) Fu (t, u)v = g has a unique solution, i.e., Fu (t, u) : C 2,σ ∩ C0 (Ω) → C σ (Ω) × C 2,σ (∂Ω) has a bounded inverse. Thus the set S = {t ∈ I| F (t, u) = θ is solvable} is open, from the IFT. We prove that S is closed. Noting that if ut satisﬁes F (t, ut ) = θ ,

30

1 Linearization

then we set at (x) =

tf (x, ut (x), ∇ut (x)) + ut (x) . 1 + |∇ut (x)|2

By assumptions (1) and (3) and Lemma 1.2.11, at C ≤ C(ut C ) ≤ C(M, φC ) . Since ut satisﬁes (1.22), in which a is replaced by at , and φ by tφ. According to equation (1.25), ut C 1,γ ≤ C(φC 2,γ , M, γ) , and then at C 0,γ ≤ C(φC 2,γ , M, γ, f ) . Again by Lemma 1.2.12, we have uC 2,γ ≤ C(φC 2,γ , M, γ, f ) . This is the required estimate. Since the embedding C 2,γ (Ω) → C 2,σ (Ω) × C 2,σ (∂Ω) is compact, we conclude that S is closed, as we have seen previously. The existence of the solution for F (1, u) = θ follows from the continuity method. The uniqueness is a consequence of the maximum principle. Once we obtain a solution u in C 2,σ , it follows directly by Schauder esti mates that u ∈ C 2,γ Remark 1.2.13 We shall return to this example by dropping assumption (2) in Chap. 3.

1.3 Lyapunov–Schmidt Reduction and Bifurcation 1.3.1 Bifurcation We often meet an equation with a parameter λ: F (x, λ) = 0 . The following phenomenon has been observed: a branch of solutions x(λ) depending on λ, is either disappeared or split into several branches, as λ attains some critical values. This kind of phenomenon is called bifurcation. For example, a simple algebraic equation: x3 − λx = 0 λ ∈ R1 ,

1.3 Lyapunov–Schmidt Reduction and Bifurcation

31

x

λ

0

Fig. 1.3.

has a solution x = 0 ∀λ ∈ R1 . As λ 0, this is the unique solution; but as λ > 0, we have two more branches of solutions √ x=± λ. See Figure 1.3. Bifurcation phenomena occur extensively in nature. Early in 1744, Euler observed the bending of a rod pressed along the direction of its axis. Let θ be the angle between the real axis and the tangent of the central line of the rod, and let λ be the pressure. The length of the rod is normalized to be π. We obtain the following diﬀerential equation with the two free end point conditions: θ¨ + λ sin θ = 0 , ˙ ˙ θ(0) = θ(π) =0. Obviously, θ ≡ 0 is always a solution of the ODE. Actually the solution is unique, if λ is not large. As λ increasingly passes through a certain value λ0 , it is shown by experiment that there exists a bending solution θ = 0. The same phenomenon occurs in the bending of plates, shells etc. In addition, bifurcation occurs in the study of thermodynamics (B´ernard problem), rotation of ﬂuids, solitary waves, superconductivity and lasers. Mathematically, we describe the bifurcation by the following: Deﬁnition 1.3.1 Let X, Y be Banach spaces, and let ∧ be a topological space. Suppose that F : X × ∧ → Y is a continuous map. ∀λ ∈ ∧, let Sλ = {x ∈ X| F (x, λ) = θ} be the solution set of the equation F (x, λ) = θ, where λ is a parameter (Fig. 1.4). Assume θ ∈ Sλ , ∀λ ∈ ∧. We call (θ, λ0 ) a bifurcation point, if for any neighborhood U of (θ, λ0 ), there exists (x, λ) ∈ U with x ∈ Sλ \{θ}.

32

1 Linearization

θ λ

λ Fig. 1.4.

The following problems are of primary concern: (1) What is the necessary and suﬃcient condition for a bifurcation point (θ, λ0 )? (2) What is the structure of Sλ near λ = λ0 ? (3) How do we compute the solutions near the bifurcation points? (4) How about the global structure of ∪λ∈∧ Sλ ? (5) Let F (x, λ) = θ be the steady equation of the evolution equation: x˙ = F (x, λ) , we study the stability of solutions in Sλ as λ approaches λ0 . In this section, we focus our discussions on problems (1) and (2). (4) will be studied in Chap. 3. For simplicity, we assume that U ⊂ X is an open neighborhood of the origin θ of the Banach space X, and that F : U × ∧ → Y is continuous, and satisﬁes F (θ, λ) = θ ∀λ ∈ ∧ . What is the necessary condition for a bifurcation point λ0 ∈ ∧? (1) Assume that Fx (x, λ) is continuous. If (θ, λ0 ) is a bifurcation point, then Fx (θ, λ0 ) does not have a bounded inverse. Proof. By the IFT directly.

(2) Assume F (x, λ) = Lx − λx + N (x, λ) , where L ∈ L(X, Y ), λ ∈ R1 , and that N : U × R1 → X is continuous with N (x, λ) = ◦( x ) as x → θ uniformly for λ in a neighborhood of λ0 . If (θ, λ0 ) is a bifurcation point, then λ0 ∈ σ(L), i.e., λ0 is a spectrum of L. Proof. If not, λ0 ∈ ρ(L), the resolvent set of L. Since ρ(L) is open, ∃ε > 0 and Cε > 0 such that (L − λI)−1 Cε as |λ − λ0 | < ε . It follows that

1.3 Lyapunov–Schmidt Reduction and Bifurcation

33

x (L − λI)−1 N (x, λ) = ◦( x ) if |λ − λε | < ε, and x → θ, as x ∈ Sλ . Thus ∃δ > 0, such that Bδ × (λ0 − ε, λ0 + ε) ∩ Sλ = {(θ, λ)| |λ − λ0 | < ε} , i.e., (θ, λ0 ) is not a bifurcation point.

(3) The above condition is not suﬃcient. For example, let X = R2 , let u x= , v and let

3 u u −v . F (x, λ) = −λ + u3 v v

Obviously, Fx (θ, λ) = (1 − λ)Id. λ = 1 is in the spectrum, but (θ, λ) is not a bifurcation point, because: F (x, λ) = θ ⇔ u4 + v 4 = 0, i.e., x = θ . In order to study the suﬃcient condition and the local behavior of the solution set near its bifurcation points, we introduce the following: 1.3.2 Lyapunov–Schmidt Reduction Let X, Y be Banach spaces, and let ∧ be a topological space. Assume that F : U × ∧ → Y is continuous, where U ⊂ X is a neighborhood of θ. We assume that Fx (θ, λ0 ) is a Fredholm operator, i.e., (1) Im Fx (θ, λ0 ) is closed in Y , (2) d = dim ker Fx (θ, λ0 ) < ∞, (3) d∗ = codim Im Fx (θ, λ0 ) < ∞. Set X1 = ker Fx (θ, λ0 ), Y1 = Im Fx (θ, λ0 ) . Since both dim X1 , and codim Y1 are ﬁnite, we have the direct sum decompositions: X = X1 ⊕ X2 , Y = Y1 ⊕ Y2 , and the projection operator P : Y → Y1 . ∀x ∈ X, there exists a unique decomposition: x = x1 + x2 , xi ∈ Xi , i = 1, 2 .

Thus F (x, λ) = θ ⇔

P F (x1 + x2 , λ) = θ , (I − P )F (x1 + x2 , λ) = θ .

34

1 Linearization

Now, P Fx (θ, λ0 ) : X2 → Y1 is a surjection as well as an injection. According to the Banach theorem, it has a bounded inverse. If we already have F (θ, λ0 ) = θ, then from the IFT, we have a unique solution u : V1 × V → V2 satisfying P F (x1 + u(x1 , λ), λ) = θ , where Vi is a neighborhood of θ in U ∩ Xi , i = 1, 2, and V is a neighborhood of λ0 . It remains to solve the equation: (I − P )F (x1 + u(x1 , λ), λ) = θ on V1 × V . This is a nonlinear system of d variables and d∗ equations. The above procedure is called the Lyapunov–Schmidt reduction. It reduces an inﬁnite-dimensional problem to a ﬁnite-dimensional system. Many applied mathematicians have been doing the same reduction in their concrete problems in their own language. Before going further, let us study the simple properties of the solution x2 = u(x1 , λ). Lemma 1.3.2 Under the assumptions (1), (2), (3) of the Lyapunov–Schmidt reduction, if F ∈ C p (U × ∧, Y ), p 1, satisﬁes F (θ, λ) = θ, where ∧ is again a Banach space, then we have u(θ, λ) = θ,

u (θ, λ0 ) = θ . If p = 1, then u(x1 , λ) = ◦( x1 +|λ − λ0 |) ; and if p = 2, then u(x1 , λ) = O( x1 2 +|λ − λ0 |2 ) . Proof. According to the IFT, the solution in V1 × V → V2 is unique. Since F (θ, λ) = θ, we have u(θ, λ) = θ. Again, by the IFT, u (θ, λ0 )(x1 , λ) = −(P Fx (θ, λ0 ))−1 (P Fx (θ, λ0 )x1 + P Fλ (θ, λ0 )λ) ∀(x1 , λ) ∈ X1 × ∧. From F (θ, λ) = θ, it follows that Fλ (θ, λ) = 0. Therefore u (θ, λ0 ) = θ , provided x1 ∈ X = ker Fx (θ, λ0 ) . The last two conclusions follow from Taylor’s formula.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

35

Next, we turn to the case d = d∗ = 1. Theorem 1.3.3 (Crandall–Rabinowitz) Suppose that U ⊂ X is an open neighborhood of θ, and that F ∈ C 2 (U × R1 , Y ) satisﬁes F (θ, λ) = θ. If Fx (θ, λ0 ) is a Fredholm operator with d = d∗ = 1, and if Fxλ (θ, λ0 )u0 ∈ / Im Fx (θ, λ0 )

(1.27)

for all u0 ∈ ker Fx (θ, λ0 )\{θ}, then (θ, λ0 ) is a bifurcation point, and there exists a unique C 1 curve (λ, ψ) : (−δ, δ) → R1 × Z satisfying F (su0 + ψ(s), λ(s)) = θ λ(0) = λ0 , ψ(0) = ψ (0) = θ , where δ > 0, and Z is the complement space of span {u0 } in X. Furthermore, there is a neighborhood of (θ, λ0 ), in which F −1 (θ) = {(θ, λ)| λ ∈ R1 } ∪ {(su0 + ψ(s), λ(s))| |s| < δ} . Proof. Decompose the spaces X and Y according to the Lyapunov–Schmidt reduction, and write down the reduction equation. By assumptions, ker Fx (θ, λ0 ) = span{u0 } , and ∃φ∗ ∈ Y ∗ \{θ}, such that ker φ∗ = ImFx (θ, λ0 ), from the Hahn–Banach theorem. The reduction equation reads as: g(s, λ) = φ∗ , F (su0 + u(su0 , λ), λ) = 0 .

(1.28)

We have the trivial solution s = 0, and we look for a nontrivial solution. Noticing gs (0, λ0 ) = φ∗ , Fx (θ, λ0 )(u0 + us (θ, λ0 )) , and u0 ∈ ker Fx (θ, λ0 ), by applying Lemma 1.3.2, we obtain gs (0, λ0 ) = 0. Therefore, it is impossible to get a solution s = s(λ) directly from the IFT. However, we may consider λ as a function of s, in this case, the only diﬃculty is that g(0, λ) = 0 for all λ. Let us introduce a new function: 1 g(s, λ) as s = 0 h(s, λ) = s as s = 0 . gs (0, λ) When s = 0, the solutions of h(s, λ) = 0 are the same as (1.28). Here we deﬁne gs (0, λ) to be the value of h at s = 0, in order to make h ∈ C 1 . We assume this conclusion at this moment, and postpone the veriﬁcation to the end. Since h(0, λ0 ) = gs (0, λ0 ) = 0 , and

36

1 Linearization hλ (0, λ0 ) = gsλ (0, λ0 )

= φ∗ , Fxλ (θ, λ0 )(u0 + us (θ, λ0 )) + Fx (θ, λ0 )usλ (θ, λ0 )

= φ∗ , Fxλ (θ, λ0 )u0

= 0 . Again, the IFT is applied, and we obtain a C 1 -curve λ = λ(s), |s| < δ, satisfying h(s, λ(s)) = 0 , λ(0) = λ0 . Set ψ(s) = u(su0 , λ(s)) ; we have ψ(0) = u(0, λ0 ) = θ , ψ (0) = ∇u(θ, λ0 )(u0 , λ (0)) = θ , and

g(s, λ(s)) = φ∗ , F (su0 + u(su0 , λ(s)), λ(s)) = 0 ;

i.e., F (su0 + ψ(s), λ(s)) = θ . This is what we need. Now we turn to verifying h ∈ C 1 (Bη ) for some η > 0, where Bη = {(s, λ) ∈ 2 R | |s|2 + |λ − λ0 |2 < η 2 }. Indeed, only want to verify that h is C 1 at s = 0. By deﬁnition, 1 lim g(s, λ) = gs (0, λ) , s→0 s which implies the continuity of h. Moreover, 1 hs (0, λ) = lim [h(s, λ) − h(0, λ)] s→0 s 1 = lim 2 [g(s, λ) − g(0, λ) − gs (0, λ)s] s→0 s 1 = gss (0, λ) , 2 therefore hs (s, λ)

−

hs (0, λ)

1 1 2 = 2 gs (s, λ)s − g(s, λ) − gss (0, λ)s s 2 = ◦(1) as |s| → 0 .

Since

hλ (0, λ) = gsλ (0, λ) , 1 (0, λ) → 0 as s → 0 . hλ (s, λ) − hλ (0, λ) = gλ (s, λ) − gsλ s The proof is complete.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

37

Z

( su0 +ψ(s), λ (s) ) λ0

u0

λ

Span{ u0} Fig. 1.5.

We present two simple applications. Example 1. (Euler elastic rod) We study the bending problem raised at the beginning of this section: ϕ¨ + λ sin ϕ = 0 in (0, π) , ϕ(0) ˙ = ϕ(π) ˙ =0. Let ˙ = u(π) ˙ = 0)} , X = {u ∈ C 2 [0, π]| u(0) Y = C[0, π] ; and let F : X × R1 → Y be the map: (u, λ) → u + λ sin u . It is a continuous map satisfying F (θ, λ) = θ. According to the necessary condition, if (θ, λ) is a bifurcation point, then λ is in the spectrum of the d 2 ) + λI, i.e., linearized operator ( dt λ = n2 , for some n = 1, 2, . . . d 2 ) + n2 I) = {s cos nt| s ∈ R1 }, and since the diﬀerential Since ker (( dt d 2 operator ( dt ) under the free end point condition is self-adjoint, we have

coker Fu (θ, n2 ) = {s cos nt| s ∈ R1 } . In addition thus

(θ, n2 ) = cos nu|u=θ = I , Fuλ (θ, n2 ) cos nt = cos nt ∈ / Im Fu (θ, n2 ) . Fuλ

38

1 Linearization

All the assumptions in the Crandall–Rabinowitz theorem are satisﬁed, and we obtain a family of C 1 curves (λn (s), ψn (s)) : (−δ, δ) → R1 × Zn , where Zn is the complement of span{cos nt}, satisfying λn (0) = n2 , d ds ψn (0)

= ψn (0) = θ ,

for n = 1, 2, 3, . . . If we set ϕn (s, t) = s cos nt + (ψn (s))(t)

t ∈ [0, π] ,

then

∂ ∂t

2 ϕn (s, t) + λn (s) sin ϕn (s, t) = 0 0 < t < π, |s| < δ , ∂ ∂ ϕn (s, 0) = ϕn (s, π) = 0 . ∂t ∂t

We obtain the bifurcation diagram of Fig. 1.6.

1

4

9

Fig. 1.6.

Example 2. We consider the following elliptic BVP. Let Ω ⊂ Rn be a bounded open domain with smooth boundary ∂Ω, and p ∈ (1, ∞): −u − λu = |u|p−1 u in Ω , u|∂Ω = 0 . Set X = C 2,γ ∩ C0 (Ω), and Y = C γ (Ω) for some γ ∈ (0, 1) , F : (u, λ) → −u − λu − |u|p−1 u , we have Fu (θ, λ) = − − λI . Thus, for (θ, λ) being a bifurcation point, −λ has to be in the spectrum of the Laplacian under the 0-Dirichlet data.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

39

Let us ﬁrst consider the ﬁrst eigenvalue λ1 , which is simple, i.e., ker − − λI is one dimensional; let ϕ1 be the associate eigenfunction. By the same reasoning, coker Fu (θ, λ1 ) is also one dimensional, and Fuλ (θ, λ1 )ϕ1 = −ϕ1 ∈ / Im Fu (θ, λ1 ) . Again, we apply the Crandall–Rabinowitz theorem, and conclude that (θ, λ1 ) is a bifurcation point, and in its neighborhood, the solution set is the C 1 curve (−δ, δ) → (sϕ1 (x) + ψ(s)(x), λ1 (s)) plus the trivial solution set (θ, λ), where ψ(s) ∈ Z, and Z is the complement of span{ϕ1 } in C 2,γ ∩ C0 (Ω). We have not studied eigenvalues other than λ1 , because we do not know if they are simple, i.e., if the condition d = 1 is satisﬁed. In fact, if λ is a simple eigenvalue, then we have the similar result. For some special cases, in which d = 1, we can extend Theorem 1.3.3 as follows: Theorem 1.3.4 Let ∧, X and Y be Banach spaces. Suppose that F ∈ C 2 (X × ∧, Y ) has the form F (x, λ) = L(λ)x + P (x, λ) , where L(λ) ∈ L(X, Y ) ∀λ ∈ ∧, and P (θ, λ) = θ, Px (θ, λ) = θ, Pxλ (θ, λ0 ) = θ for some λ0 ∈ ∧ . If there exist u0 ∈ ker L(λ0 )\{θ} and a closed linear subspace Z ⊂ X such that (z, λ) → L(λ0 )z + λL (λ0 )u0 : Z × ∧ → Y is a linear homeomorphism, then there exist a neighborhood U of (θ, λ0 ) in (span{u0 } × Z) × ∧, δ > 0, and a C 1 map: (−δ, δ) → Z × ∧, deﬁned by s → (ϕ(s), λ(s)) satisfying F −1 (θ) ∩ U \{(θ, λ)| λ ∈ ∧} = {(s(u0 + ϕ(s)), λ(s))| |s| < δ} , and (ϕ(0), λ(0)) = (θ, λ0 ) .

Proof. Similar to Theorem 1.3.3, we deﬁne 1 F (s(u0 + z), λ) Φ(s, z, λ) = s Fx (θ, λ)(u0 + z)

as s = 0 as s = 0 .

One wants to verify that Φ ∈ C 1 ((R1 × Z) × ∧, Y ). It is suﬃcient to verify the continuous diﬀerentiability at s = 0. Since Φ(s, z, λ) − Φ(0, z, λ) = s−1 [F (s(u0 + z), λ) − F (θ, λ) − Fx (θ, λ)(s(u0 + z))]

1 1 Fxx (rts(u0 + z), λ)tdtdr · (u0 + z)2 , =s 0

0

40

1 Linearization

we have Φs (0, z, λ) =

1 Fxx (θ, λ)(u0 + z)2 . 2

Furthermore as s → 0, Φs (s, z, λ) − Φs (0, z, λ) s2 −2 2 = −s F (s(u0 + z), λ) − sFx (θ, λ)(u0 + z) − Fxx (θ, λ)(u0 + z) 2 +s−1 [Fx (s(u0 + z), λ) − Fx (θ, λ)](u0 + z) = o(1) , and Φλ (s, z, λ) − Φλ (0, z, λ) = s−1 [Fλ (s(u0 + z), λ) − Fλ (θ, λ) − Fxλ (θ, λ)s(u0 , z)] = o(1) . Thus, Φ ∈ C 1 . Now, we have Φ(0, z, λ) = L(λ)(u0 + z) + Px (θ, λ)(u0 + z) , Φ(0, θ, λ0 ) = L(λ0 )u0 + Px (θ, λ0 )u0 = θ , and Φ(z,λ) (0, θ, λ0 )(z, λ) = L(λ0 )z + L (λ0 )u0 · λ + Pλx (θ, λ0 )u0 + Px (θ, λ0 )z , = L(λ0 )z + λL (λ0 )u0 , ∀(z, λ) ∈ Z × ∧. By the assumption, the last linear operator is a homeomorphism; then the IFT is applied. Therefore ∃ a neighborhood U of (θ, λ0 ), and a unique C 1 -curve: s → (ϕ(s), λ(s)) ∈ Z × ∧ ∀ |s| < ε, satisfying (ϕ(0), λ(0)) = (θ, λ0 ) , Φ(s, ϕ(s), λ(s)) = θ, i.e., F (s(u0 + ϕ(s)), λ(s)) = θ .

The proof is complete. The above theorem is applied to Hopf bifurcations in ODE.

Example 3. (Hopf bifurcation) We study the periodic solutions for a linear ordinary diﬀerential system: x˙ = Ax , where x ∈ C 1 ([0, 2π], Rn ) and A ∈ M (n, R), the n × n matrix. Let B 0 A= , 0 C

(1.29)

1.3 Lyapunov–Schmidt Reduction and Bifurcation

where B=

0 1 −1 0

41

, and C ∈ M (n − 2, R) .

We assume that (e2πC − I) is invertible. Obviously the linear system (1.29) admits a family of periodic solutions with two real parameters a and b: ⎞ ⎛ a cos t + b sin t ⎜ −a sin t + b cos t ⎟ ⎟ ⎜ ⎟ ⎜ 0 x(t) = ⎜ ⎟ , ∀a, b ∈ R1 . ⎟ ⎜ .. ⎠ ⎝ . 0 Now, we perturb A by introducing a parameter µ ∈ (−1, 1), and consider the nonlinear diﬀerential system x˙ = A(µ)x + P (x, µ) ,

where A(µ) =

B(µ) 0 0 C(µ)

, B(µ) =

µ −β(µ)

(1.30) β(µ) µ

,

and C(µ) ∈ M (n − 2, R), satisfying e2πC(µ) − I ∈ GL(n − 2, R). We assume that P : Rn × (−1, 1) → Rn satisﬁes P (θ, µ) = Px (θ, 0) = Px,µ (θ, 0) = θ . and that β : (−1, 1) → R1 , satisﬁes β(0) = 1, and β (0) = 0 . Obviously, ∀µ ∈ (−1, 1), x = θ is always a solution. We are interested in nontrivial solutions bifurcating from the branch of trivial solutions. Theorem 1.3.5 Suppose that A(µ) and P (x, µ) are C 2 functions satisfying the above assumptions. Then ∃ positive constants a0 , δ0 and a C 1 map (µ, ω, x) : (−a0 , a0 ) → R1 × R1 × C 1 (R1 , Rn ) , satisfying µ(0) = 0, ω(0) = 1 , where x(a) is a 2πω(a)-periodic function, which satisﬁes (1.30), and is of the form:

42

1 Linearization

⎛

⎞ a sin ω(a)−1 t ⎜ a cos ω(a)−1 t ⎟ ⎜ ⎟ ⎜ ⎟ 0 (x(a))(t) = ⎜ ⎟ + ◦(|a|) . ⎜ ⎟ .. ⎝ ⎠ . 0 Furthermore, every 2πω(a)-periodic solution y(t) of (1.30), satisfying |y(t)| < δ0 ∀t, coincides with x(a)(t) modulating a phase shift when |µ| < α0 and |ω − 1| < δ0 . Proof. Since ω depends on µ, we introduce a new scale τ , and let t = ωτ ; (1.30) is rewritten as dx = ωA(µ)x + ωP (x, µ) . dτ

(1.31)

We ﬁnd the 2π-periodic solution of (1.31). Set ∧ = R2 , λ = (ω, µ) ∈ ∧ , 1 X = C2π (Rn ) = {u ∈ C 1 (R, Rn )| u is 2π periodic} , Y = C2π (Rn ) = {u ∈ C(R, Rn )| u is 2π periodic} ,

and set dx − ωA(µ)x , dτ P(x, λ) = ωP (x, µ) ,

L(λ)x =

λ0 = (1, 0) . Let

⎞ ⎞ ⎛ sin τ cos τ ⎜ cos τ ⎟ ⎜ − sin τ ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 0 ⎟ ⎜ u0 = ⎜ ⎟ , and u1 = ⎜ 0 ⎟ ; ⎜ .. ⎟ ⎜ .. ⎟ ⎝ . ⎠ ⎝ . ⎠ 0 0 ⎛

then we have ker L(λ0 ) = span{u0 , u1 } . Deﬁne Z=

z ∈ X|

2π

z(t)uj (t)dt = 0, j = 0, 1 .

0

It remains to verify the following conditions: ∀y ∈ Y , the linear equation L(λ0 )z + λL (λ0 )u0 dz − A(0)z − ωA(0)u0 − µA (0)u0 = dτ =y has a unique solution (ω, µ, z) ∈ R2 × Z.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

43

X

µ (1,0, θ) ω Fig. 1.7. Nontrivial solutions

According to the Fredholm alternative, this is equivalent to saying that ∀y ∈ Y, ∃|(ω, µ) ∈ R2 such that y + ωA(0)u0 + µA (0)u0 ∈ ker L∗ (λ0 )⊥ . However, ker L∗ (λ0 ) = ker L(λ0 ); this is again equivalent to 2π 2π 0 (A(0)u0 (t))u0 (t)dt 0 (A (0)u0 (t))u0 (t)dt = 0 . det 2π (A(0)u0 (t))u1 (t)dt 2π (A (0)u0 (t))u1 (t)dt 0 0 Computing the determinant directly, it equals −4π 2 β (0). Thus all assumptions in Theorem 1.3.4 are satisﬁed. We obtain our conclusion. This kind of bifurcation phenomenon is shown in Fig. 1.7. 1.3.3 A Perturbation Problem We present here an example of perturbation problems. This is a nonlinear Schr¨ odinger equation (NSE); we look for a nonspreading wave packet solution. Assume that the potential V ∈ C 2 (R1 ) is bounded, and the standing wave ψ = ψ(t, x) satisﬁes the following NSE: ih

h2 ∂ 2 ψ ∂ψ =− + V (x)ψ − ψ 3 . ∂t 2 ∂x2

where h > 0 is a small constant, ψ(t, x) = exp(−iλt/h)ϕ(x) with ϕ(x) → 0 as |x| → ∞, and λ < inf V . It is reduced to the following equation: −

h2 ϕ (x) + V (x)ϕ(x) = λϕ(x) + ϕ(x)3 . 2

If V has a nondegenerate critical point x0 , i.e., V (x0 ) = 0, V (x0 ) = 0, 0 we make a change of variables y = x−x h , and the equation is reduced to

44

1 Linearization

1 − ϕ (y) + (Vh (y) − λ)ϕ(y) = ϕ(y)3 , 2

(1.32)

where Vh (y) = V (x0 + hy) and ϕ(y) → 0, as |y| → ∞. This is a perturbation problem, because h > 0 is small. The special case where h = 0 reads as: 1 (1.33) − ϕ + Eϕ = ϕ3 , 2 where E = V (x0 ) − λ, and ϕ(y) → 0 as |y| → ∞. The equation has a special solution: z(y) = α sech (αy), α = (2E)1/2 . Let τθ ϕ = ϕ(y − θ). Since equation (1.33) is autonomous, it has a onedimensional solution manifold Z = {zθ = τθ z| θ ∈ R1 }. It is easily check: Tz Z = span{e}, where e = z , and then Tzθ Z = span{τθ e}. We write eθ = τθ e. Let us deﬁne X = H 2 (R1 ), Y = L2 (R1 ), and 1 F (h, u) = − u + (Vh − λ)u − u3 . 2 Since V is bounded, F : R1 × X → Y . Thus, F (0, zθ ) = 0, ∀ θ ∈ R1 . Since Lθ := Fu (0, zθ ) = −

1 d2 + (E − 3zθ2 )·, with D(Lθ ) = X , 2 dt2

Lθ is a self-adjoint operator on Y . We verify that Lθ is a Fredholm operator: (1) Im (Lθ ) is closed. (2) Ker (Lθ ) = Coker(Lθ ) = Tzθ Z. In fact, only (1) remains to be veriﬁed; it follows from: Lemma 1.3.6 Assume inf (V ) > λ. Then ∃γ > 0, such that Lθ vY ≥ γvX ∀v ∈ (Tzθ Z)⊥ . Proof. We prove it by contradiction. Suppose that ∃φn ∈ (Tzθ Z)⊥ such that φn X = 1, and Lθ φn Y → 0. Then φn φ in X. We claim that φ = 0. In fact, (1) by weakly convergence, φ ∈ (Tzθ Z)⊥ and (2) since X is dense in Y, ∀ψ ∈ X, Lθ φ, ψ = φ, Lθ ψ = lim φn , Lθ ψ = lim Lθ φn , ψ = 0 . Therefore, Lθ φ = 0. Combining (1) and (2), φ = 0. d2 By the assumption E > 0, let B = − 12 dt 2 + E; we have a constant c > 0 such that

1.3 Lyapunov–Schmidt Reduction and Bifurcation

45

BuY ≥ 2cuX . Noticing that |z(y)| ∼ exp(−α|y|) as |y| → ∞, and that φn 0 in X, in combination with the Sobolev embedding theorem, which implies that φn ∞ is bounded, and φn ⇒ 0 uniformly on any bounded interval, we obtain: Lθ φn Y = Bφn − 3zθ2 φn Y ≥ Bφn Y − 3zθ2 φn Y ≥ 2cφn X − c = c , as n is large. This is a contradiction.

X1 , Y = We have the following Lyapunov–Schmidt reduction: X = Tzθ Z Y1 , and let P : Y → Y1 be the orthogonal projection. The equation Tzθ Z F (h, u) = 0 is equivalent to the system: P F (h, zθ , zθ + ξ) = 0, F (h, zθ + ξ), eθ = 0 . From F (0, zθ ) = 0 and the IFT, for h > 0 small, ∃ a unique ξ = ξ(h, θ) solving the ﬁrst equation, P F (h, zθ + ξ(h, θ)) = 0, and satisfying ξ(h, θ) ≤ CF (h, zθ ), for some constant C > 0. It remains to solve: wh (θ) := F (h, zθ + ξ(h, θ)), eθ = 0 . Once θ = θ(h) is obtained, uh = ξ(h, θ(h)) + zθ(h) solves F (h, u) = 0. We write F (h, uh ) = (F (h, uh ) − F (0, uh )) + F (0, uh ) , and by the Taylor formula, F (0, uh ) = F (0, zθ ) + Lθ ξ(h, θ) − N (h, θ) , where N (h, θ) = 3zθ ξ 2 (h, θ) + ξ 3 (h, θ), and θ = θ(h). Since F (0, zθ ) = Lθ eθ = 0, we have | F (0, uh ), eθ | = | N (h, θ), eθ | ≤ Cξ(h, θ(h))2Y , and

F (h, uh ) − F (0, uh ), eθ | =

(V (x0 + hy) − V (x0 ))(zθ (h) + ξ(h, θ(h))eθ dy = I1 + I2 .

The following estimates hold: I1 = (Vh − V0 )zθ , eθ

= −h zθ , Vh zθ − zθ , (Vh − V0 )eθ

1 = − h zθ , Vh zθ

2

1 = − h Vh (y)|z(y − θ)|2 dy 2

1 = − h V (x0 + h(y + θ))|z(y)|2 dy , 2 I2 = | (Vh − V0 )ξ(h, θ), eθ | ≤ (Vh − V0 )eθ ξ(h, θ) ,

46

1 Linearization

and F (h, zθ )2Y = F (h, zθ ) − F (0, zθ )Y

= |Vh (y + θ) − V0 |2 |z(y)|2 dy

2 2 2 ≤ |Vh (y + θ) − V0 | |z(y)| dy + 4Max|V |

R1 \Bρ 2 −µρ

Bρ

|z(y)|2 dy

≤ Max|y|≤ρ |V (h(y + θ)) − V0 |2 + 4Max|V | e ≤ C1 (h(ρ + |θ|))4 + e−µρ

for every ρ > 0, and for some constant C1 > 0. Let us denote the right-hand side by Mh,θ,ρ . Similarly, we have (Vh − V0 )eθ 2 ≤ Mh,θ,ρ . Since z(y) is even,

1 I1 + 1 V (x0 )z2 hθ = 1 [V (x0 )hθ − V (x0 + h(y + θ))]|z(y)|2 dy h 2 2

1 = [V (x0 )h(y + θ) − V (x0 + h(y + θ))|z(y)|2 dy 2

≤ C2 |h(y + θ)|2 |z(y)|2 dy ≤ C2 (h(|θ| + ρ)|2 + e−µρ ) . 1 2 2 V (x0 )z s,

By rescaling θ = hs , v(s) = h1 wh (θ), v0 (s) = h−ν v(hν s), with ν ∈ (1, 2), we obtain

and vh (s) =

|vh (s) − v0 (s)| ≤ C3 (h−ν ((hν |s| + hρ)2 + e−µρ ) + h−1−ν ((hν |s| + hρ)4 + e−µρ )) . Choosing ρ = h−τ with τ > 0, for |s| ≤ 1, we obtain |vh (s) − v0 (s)| ≤ C3 (h−ν ((hν + h1−τ )2 + h−1−ν ((hν + h1−τ )4 −τ

+ (h−ν + h−1−ν )e−µh

)→0,

uniformly as h → 0. Since v0 (s) changes sign at s = 0, we have a zero of vh in [−1, 1] as h > 0 small. This proves the existence of a zero s0 ∈ [0, 1] of vh and then of wh (θ). Thus we have proved the following theorem: Theorem 1.3.7 Assume that V ∈ C 3 (R1 ) is bounded and that x0 is a nondegenerate critical point of V . If λ < inf V , then ∃h0 > 0 such that ∀h ∈ (0, h0 ) ∃ 0 a nonzero solution φ(x) = z( x−xh0 −s0 ) + ξ(h, s0 (h))( x−x h ) of the equation (1.32), with s0 ∈ [−hν , hν ] ν ∈ (1, 2), and ξ(h, s0 (h)) → 0, as h → 0. 0 This solution φ, which is close to z( x−x h ), becomes more concentrated about x0 as h → 0. It is called a nonspreading wave packet with width α = (2(V (x0 ) − λ))−1/2 .

1.3 Lyapunov–Schmidt Reduction and Bifurcation

47

1.3.4 Gluing Gluing is an important technique in nonlinear analysis. It is a method of joining two heteroclinic trajectories to form a new trajectory such that the end point of the ﬁrst is glued to the starting point of the second, and the new trajectory is closed to the union of the two. For instance, on a Riemannian manifold M , let f ∈ C 1 (M, R1 ). Given (x, y) ∈ K ×K, where K is the critical point set of f , i.e., K = {x ∈ M | f (x) = θ}, let M(x, y) be the space of all trajectories c ∈ H 1 (R1 , M ) satisfying c˙ = −∇f ◦ c,

c(−∞) = x, c(+∞) = y .

The importance of the gluing technique is in the study of the compactness of the totality of the trajectory spaces: {M(x, y) | (x, y) ∈ K × K}. It is proved that the trajectory manifolds M(x, y) are compact up to the ∞ topolexistence of sequences which converge to broken trajectories in the Cloc ogy. And the “gluing” means to map such broken trajectories equipped with an additional suitable parametrization into the appropriate trajectory space: M(x, z) × M(z, y) → M(x, y) . This technique is crucial in Floer homology theory (see Hofer and Zehnder [HZ 2], Floer [Fl 2,3], Taubes [Tau]). In order not to involve over specialized knowledge in that theory, it will suﬃce to introduce the technique by an example of a nonspreading wave packet, which we met in the previous subsection. We have known the existence of a nonspreading wave packet, if V has a single nondegenerate critical point. However, if V has several nondegenerate critical points x1 , x2 , . . . , xn , there are at least n nonspreading wave packets with widthes αj = (2(V (xj ) − λ))−1/2 , j = 1, 2, . . . , n. As h becomes small, these wave packets are separated. Are there multi-peak wave packets? More precisely, are there new solutions of equation (1.32), which are closed to some of these wave packets on their peak intervals simultaneously? For simplicity, we only consider the case n = 2; the result can be extended to any n. Assume that V has onlytwo nondegenerate critical points x± = ±R. Let λ < inf(V ). Deﬁne α± = 2(V (±R) − λ), z± (y) = α± sech (α± y), and ∀(θ+ , θ− ) ∈ R2 , z(θ+ , θ− ) = z+ (θ+ ) + z− (θ− ), where z± (θ± ) = τ±(R+θ± )/h z± . Again let 1 d2 u + (Vh − λ)u − u3 . F (h, u) = − 2 dy 2 We are looking for a solution of the form u = z(θ+ , θ− ) + φ ∈ X for the equation F (h, ·) = 0, in which φ is small as h > 0 is. Deﬁne a two-dimensional manifold: Zh = {(z+ (θ+ ), z− (θ− )) |(θ+ , θ− ) ∈ . R2 }, then Tz(θ+ ,θ− ) Zh = span{τ(R+θ+ )/h e+ , τ−(R+θ− )/h e− }, where e± = z± In contrast with the one-peak solution, now z(θ+ , θ− ) are not zeroes of F (0, u), therefore the zeroes of F (h, u) cannot be obtained directly by the IFT.

48

1 Linearization

Instead, we appeal to the contraction mapping theorem. However, Lyapunov– Schmidt reduction is again useful. Fixing (θ+ .θ− ) ∈ R2 , let u0 = z(θ+ , θ− ), we write down the orthogonal decomposition: X1 , Y = Tu0 Zh Y1 , X = Tu0 Zh where the spaces X and Y were introduced in the previous subsection, and let P : Y → Y1 be the orthogonal projection. Again, P F (h, u0 + ξ) = 0 F (h, u) = 0 ⇔ (I − P )F (h, u0 + ξ) = 0 . In fact, by Taylor’s formula, F (h, u0 + ξ) = F (h, u0 ) + Fu (h, u0 )ξ + N (h, u0 , ξ) , where N (h, u0 , ξ) = 3u0 ξ 2 + ξ 3 . First, ﬁxing (θ+ , θ− ) ∈ R2 , we solve the ﬁrst equation in X1 for h > 0 small. Let L(θ+ , θ− , h) = P Fu (h, u0 ). We have the following lemmas: Lemma 1.3.8 ∃γ > 0, ∃h0 > 0, ∃α0 ∈ (0, 1/2) such that L(θ+ , θ− , h)Y ≥ γuX ∀u ∈ X1 . as h ∈ (0, h0 ), and |α± | < α0 . Lemma 1.3.9 ∀ρ > 0, F (h, u0 )Y ≤ C1 (Σ± Max|y|≤ρh (V (y) − V (±R))2 | ± R ± θ± | + e−2µρ + e−µR/h ), where µ = Min{α+ , α− }. Lemma 1.3.10 N (h, u0 , ξ)Y ≤ C2 ξ2X , N (h, u0 , ξ1 ) − N (h, u0 , ξ2 )Y ≤ C2 Max(ξ1 X , ξ2 X )ξ1 − ξ2 X . Once these lemmas have been proved, according to the contraction mapping theorem, one ﬁnds a ﬁxed point ξ of the operator −L−1 (θ+ , θ− , h)N (h, u0 , ξ) − F (h, u0 ) in a neighborhood of u0 on X1 . Namely, ∃h0 > 0, ∃α0 > 0, ∃C1 > 0 such that ∀h ∈ (0, h0 ), ∀|θ± | < α0 , ∃ a unique ξ = ξ(θ+ , θ− , h) ∈ X1 such that P F (h, u0 + ξ) = 0, and ξ ≤ C1 F (h, u0 )Y . Now the second equation is a two system in two variables (θ+ , θ− ) ∈ [−α0 , α0 ]2 : vh (θ+ , θ− ) := h−1 ( e+ , F (h, z(θ+ , θ− ) + ξ(θ+ , θ− , h) , e− , F (h, z(θ+ , θ− ) + ξ(θ+ , θ− , h) ) = (0, 0) .

1.3 Lyapunov–Schmidt Reduction and Bifurcation

49

Let us deﬁne a two-vector function: v0 (θ+ , θ− ) =

1 (|z+ |2 V (R)θ+ , |z− |2 V (−R)θ− ) . 2

Again we need a Lemma: Lemma 1.3.11 For ν ∈ (1, 2) and 0 < h < Min(h0 , α0 ), we have h−ν vh (hν θ+ , hν θ− ) → v0 (θ+ , θ− ) , uniformly on [−1, 1]2 as h → 0. The diﬀerence between these proofs of the Lemmas 1.3.8 to 1.3.11 and those in the previous subsection lie in the interaction terms (for details refer to [Oh]). From the Brouwer degree theory (cf. Sect. 3.1), deg(h−ν vh (hν θ+ , hν θ− ), [−1, 1]2 , (0, 0)) = deg(v0 (θ+ , θ− ), [−1, 1]2 , (0, 0)) = 0 . This proves the existence of a zero of the above system. Finally we arrive at: Theorem 1.3.12 Suppose that V ∈ C 3 is bounded, and λ < inf (V ). Then for each pair (x1 , x2 ) of nondegenerate critical points of V, ∃h0 > 0, such that for h ∈ (0, h0 ) equation (1.32) has a nonzero solution uh of the form τ(x1 +s+ )/h z+ + τ(x2 +s− )/h z− + ξ(h), where |s± | ≤ h−ν with ν ∈ (1, 2), and ξ → 0 as h → 0. 1.3.5 Transversality Transversality is an important notion in diﬀerential geometry. It is a condition, induced by the IFT, on a map between two manifolds, under which the preimage of a submanifold is again a submanifold. f : X → Y is said to be transversal to the submanifold W ⊂ Y if at every point x ∈ f −1 (W ), Im f (x) + Tf (x) (W ) = Tf (x) (Y ) , and is denoted by f W . Related notions are regular points and regular values deﬁned as follows. Deﬁnition 1.3.13 Suppose that X and Y are C 1 Banach manifolds and f ∈ C 1 (X, Y ). A point x ∈ X is called a regular point of f , if f (x) : Tx (X) → Tf (x) (Y ) is surjective, and is singular (or critical) if it is not regular. The images of the singular points under f are called singular values (or critical values), and their complement, regular values. If y ∈ Y is not in the image of f , i.e., f −1 (y) = ∅, then y is a regular value. The following Sard theorem reveals the smallness of the set of critical values:

50

1 Linearization

z=α sech(h−1α(x−x0))

φ(x)

x0

x

S0

z+

z−

uh x1+S+

x2+S− Fig. 1.8.

Theorem 1.3.14 (Sard) Suppose that X and Y are diﬀerential manifolds with dimensions n and m respectively, and that U ⊂ X is open. If f ∈ C r (X, Y ), where r ≥ 1 and r > max{0, dimX −dimY }, then the set of critical values of f has measure zero in Y . The proof is purely measure theoretic, we shall not give it here. The special case dimX = dimY < ∞ will be given in Sect. 3.1, but for the general case refer to Milnor [Mi 2]. We shall extend the study to maps between inﬁnite-dimensional manifolds. Recall a map f between two Banach spaces X and Y is called Fredholm in U ⊂ X, if f ∈ C 1 (U, Y ) and f (x) is a Fredholm operator ∀x ∈ U . The index of f is deﬁned to be ind f (x) = dim ker f (x) − dim coker f (x) . It is known that if U is connected, then ind f (x) is a constant integer, and then is denoted by ind(f ). Lemma 1.3.15 If f ∈ C 1 (U, Y ) is a Fredholm map, then the set of critical points is closed.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

51

Proof. Since x → dim kerf (x) is u.s.c., and ind(f ) is locally a constant, dim cokerf (x) is also u.s.c. Let S be the critical point set of f , then the set S = {x ∈ U |f (x) is not surjective} = {x ∈ U |dim coker f (x) ≥ 1}

is closed.

Theorem 1.3.16 (Sard–Smale) Suppose that X is a separable Banach space and Y is a Banach space. Let f ∈ C r (U, Y ) be a Fredholm map, where U ⊂ X is open. If r > max (0, ind(f )), then the set of critical values is of ﬁrst category. Proof. Since the ﬁrst category set is a countable union of closed nowhere dense sets, and U is separable, it is suﬃcient to prove that ∀x ∈ U , there is a neighborhood of x, V ⊂ U such that the set of critical values of f |V , S(f, V ) is closed and nowhere dense. By the deﬁnition of Fredholm operators, one has the decomposition X = kerf (x)⊕X1 . Let Q be the projection of Y onto Imf (x); from the IFT, there is a neighborhood U0 × V0 ⊂ kerf (x) × X1 , a neighborhood W ⊂ Imf (x) of f (x), and h ∈ C 1 (U0 × W, V0 ) such that Qf (u + h(u, w)) = w, ∀(u, w) ∈ U0 × W , and ∀u ∈ U0 , h(u, ·) : W → V0 is a diﬀeomorphism. Since U0 is ﬁnite dimensional, it can be chosen compact. We set V = U0 × V0 , 1. First we show that S(f, V ) is closed. Due to Lemma 1.3.15 it is suﬃcient to show that the map f |V is closed. In fact, let xn = un + vn ∈ U0 × V0 be such that yn = f (xn ) → y. From the compactness of U0 , un is subconvergent, and vn = h(un , Qyn ) is convergent, therefore f |V is closed. 2. Next we show that S(f, V ) is a nowhere dense set. Deﬁne H : U0 × W → V by H(u, v) = (u, h(u, w)). Then H is a diﬀeomorphism satisfying Qf ◦ H(u, w) = w. Setting f˜ = f ◦ H, we have S(f, V ) = S(f˜, U0 × W ), and Qf˜ |W = idW , therefore ∀w ∈ W , (x, w) ∈ S(f˜, U0 × W ) ⇐⇒ x ∈ S((I − Q)f˜(·, w), U0 ). From r > ind f (x) = dim kerf (x) − dim cokerf (x) and Sard theorem, S((I − Q)f˜(·, w), U0 ) is a null set. Since S(f, V ) is also closed, it is a nowhere dense set. We conclude that the critical set of f is of the ﬁrst category. Corollary 1.3.17 Let X and Y be Banach spaces, and X be separable. If f ∈ C 1 (X, Y ) is Fredholm with negative index, then f (X) does not contain interior points. Proof. If not, say y0 ∈ f (X) is an interior point, i.e., ∃ a neighborhood V of y0 such that V ⊂ f (X). According to the Sard–Smale lemma, ∃y ∈ V such that f {y}, i.e., ∀x ∈ f −1 (y), f (x) : X → Y is surjective, or codim Im(f (x)) = 0, thus ind(f )(x) = dim kerf (x) ≥ 0. A contradiction. The following transversality theorem will be often used in the sequel.

52

1 Linearization

Theorem 1.3.18 Suppose that X, Z are Banach spaces, where X is separable, and that S is a C r Banach manifold. Assume that F ∈ C r (X × S, Z) satisﬁes (1) F {θ}, and (2) ∀s ∈ S, fs (u) = F (u, s) is a Fredholm map with index satisfying max{0, ind(fs )} < r. Then ∃ a residual set Σ ⊂ S (i.e., the countable intersection of open dense sets) such that ∀s ∈ Σ, fs {θ}. Proof. Deﬁne the injection: i : V −→ X ×S and the projections π : X ×S −→ S, p : X × S → X. Let V = F −1 (θ). In the sequel, both of them are restricted on V . We claim that π ◦ i is a Fredholm map with ind (π ◦ i) = ind (fs ). In fact, ker F (v) = Tv (V ) = Im ((p ◦ i) (v)) ⊕ Im ((π ◦ i) (v)) ∀v = (x, s) ∈ Tv (V ). Since F {θ}, we have a direct sum decomposition: X × Ts (S) = Y ⊕ Tv (V ) such that F (v) : Y → Z is an isomorphism. By the assumption that fs (x) is Fredholm, we have direct sum decompositions: X = Im((p ◦ i) (v)) ⊕ Y1 , Z = Z1 ⊕ Z2 , such that fs (x) : Y1 → Z1 . Thus, Y = Y1 ⊕ Y2 , where Ts (S) = Im(π ◦ i) (v) ⊕ Y2 . From F (v) = fs (x) ⊕ ∂s F (v), we have ∂s F (v) : Y2 → Z2 is an isomorphism. Thus, ker fs (x) = Im(p ◦ i) (v) = ker (π ◦ i) (v), coker fs (x) = Z2 Y2 coker(π ◦ i) (v). Therefore ind (fs ) = ind (π ◦ i). According to the Sard–Smale Theorem, the set of regular values of π◦i Σ is a residual set. We are going to prove that for a regular value s, fs {θ}. From F (x, s) = θ, and F {θ}, we have Im F (x, s) = Z, i.e., ∀a ∈ Z, ∃(α, β) ∈ ∂F X × Ts (S) such that a = ( ∂F ∂x α + ∂s β). We are going to prove Im fs (x) = Z, i.e., ∀ a ∈ Z, ∃y ∈ X such that a = fs (x)y. Thus, if β = 0, we take y = α. Otherwise, since π is a projection, (π ◦ i) (x, s) : X × Ts (S) −→ Ts (S) is a projection. That s is a regular value means that (π ◦ i) (x, s) : T(x,s) (V ) −→ Ts (S) is surjective, i.e., to the given β ∈ Ts (S), ∃w ∈ X such that (w, β) ∈ T(x,s) (V ). Since F −1 (θ) = V , we have F (x, s)(w, β) = θ. Setting y = α − w, we obtain ∂F ∂F α+ β fs (x)y − a = F (x, s)[(α, β) − (w, β)] − a = ∂x ∂s − a − F (x, s)(w, β) = θ . The same conclusion holds if X and Z are Banach manifolds. As an application we study the simplicity of eigenvalues of the Laplacian on bounded domains with Dirichlet data. It is well known that all eigenvalues of second-order ODEs on bounded intervals with Dirichlet data are simple, but it is not true for PDE, e.g., the Laplacian on a ball. However, we shall show that for most domains, this is true. What is the meaning of most domains? Given a bounded open domain Ω0 ⊂ Rn with smooth boundary, we consider the manifold S = Diﬀ 3 (Ω0 ) := {g ∈ C 3 (Ω0 , Rn ) | det (g (x)) =

1.3 Lyapunov–Schmidt Reduction and Bifurcation

53

0 ∀x ∈ Ω0 }. Thus as g ∈ S, the domain Ω = g(Ω0 ) is C 3 . ∀Ω, let X(Ω) = H 2 (Ω) ∩ H01 (Ω) and Y (Ω) = L2 (Ω). ∀g ∈ S, deﬁne g ∗ : X(Ω0 ) → X(Ω) by (g ∗ u)(x) = u(g −1 (x)) ∀x ∈ Ω = g(Ω0 ) . One deﬁnes a map F ∈ C 1 ((X(Ω0 )\{θ}) × S, Y (Ω0 )) by F (u, λ, g) = g ∗−1 (∆ + λI)g ∗ u . We want to show that there exists a residual set Σ of S such that ∀g ∈ Σ, all eigenvalues of the problem: (∆ + λ)u = 0 on H01 (Ω) , are simple, where Ω = g(Ω0 ). Since there are, at most, countable eigenvalues, it is suﬃcient to show that for each eigenvalue λ after suitable perturbation of the domain, i.e., for after a suitable g ∈ S, it is simple. To this end, let us deﬁne fg (u, λ) = F (u, λ, g). We claim that it remains to show fg {θ}, or equivalently, fg (u, λ) : X(Ω) × R1 → Y (Ω) is surjective, i.e., {(∆ + λ)w + µu | (w, µ) ∈ X(Ω) × R1 } = Y (Ω)}, or equivalently, codim Im(∆ + λI) ≤ 1. Since we have assumed that λ is an eigenvalue, i.e., codim (Im(∆ + λI) ≥ 1. Therefore λ is simple. We know that fg (u, λ) is a Fredholm map, from Theorem 1.3.18, it is suﬃcient to verify that F {θ}. To this end, ∀(u0 , λ0 , g0 ) ∈ F −1 (θ), let L = F (u0 , λ0 , g0 ), and Ω = g0 (Ω0 ). With no loss of generality we may assume g0 = Id. It remains to verify that L is surjective. Since L(w, µ, h) = (∆+λ0 )w +µu0 +[h·∇, ∆+λ0 ]u0 = (∆+λ0 )(w −h·∇u0 )+µu0 , L is Fredholm, where [A, B] = AB − BA. Suppose it is not surjective, then codim Im(L) > 0, there exists a nontrivial element φ⊥Im(L). First by taking w = h = θ, we have

φu0 = 0 . Ω

Second, we take h = θ; it follows that

φ(∆ + λ0 I)w = 0 ∀ w ∈ X(Ω) . Ω

¯ α ∈ (0, 1) is a solution of the equation: Then φ ∈ C 2,α (Ω), (∆ + λ0 )φ = 0 , with boundary value zero. Then we have

54

1 Linearization

φ(∆ + λ0 )(h · ∇u0 )

0=

Ω

φ(∆ + λ0 )(h · ∇u0 ) − (h · ∇u0 )(∆ + λ0 I)φ

=

Ω

φ∂n (h · ∇u0 ) − ∂n φ(h · ∇u0 )

=

∂Ω

∂n φ(h · ∇u0 )

=− ∂Ω

¯ Rn ). Thus, for all h ∈ C 3 (Ω, ∂n φ∇u0 = 0 on ∂Ω . According to the uniqueness of the Cauchy problem (see [Hor 1]) for the equa¯ this contions (∆ + λ0 )φ = 0, and (∆ + λ0 )u0 = 0 for φ, u0 ∈ C 2,α ∩ C0 (Ω), tradicts with u0 = θ and φ = θ. We have proved a result due to K. Uhlenbeck [Uh]: Theorem 1.3.19 There exists a residual set Σ ⊂ Diﬀ 3 (Ω0 ) such that ∀g ∈ Σ, all eigenvalues of the equation (∆ + λI)u = 0 on H01 (g(Ω0 )) are simple.

1.4 Hard Implicit Function Theorem The inverse function theorem is established for C 1 mappings between Banach spaces, f : X → Y , where X and Y are Banach spaces, under the assumption f (x0 )−1 ∈ L(Y, X). However, if f (x0 )−1 is not bounded, then the problem will be very complicated. The small divisor problem, arising in the study of the long-time behavior of oscillatory motions in the solar system, is a typical example. The celebrated KAM theory, which states that most (in the sense of measure) quasi-periodic solutions of an integrable smooth Hamiltonian system persist under small perturbations of that Hamiltonian, provided that the Hessian is nondegenerate, is a breakthrough in the progress in this direction. Not only is the result an important conclusion in celestial mechanics, but also the methods developed in this theory have great impact in other nonlinear problems. In fact, Nash–Moser iteration, which was introduced in the study of the isometric embedding problem for Riemann manifolds, see [Na 1,2], and [Mos 1,2], is extensively applied in problems arising in geometry (see for instance Hamilton [Ham]) and in physics (see for instance Hormander [Hor 2]). In recent years, KAM theory has been extended to Hamiltonian systems with inﬁnitely many freedoms (see Poschel [Po], Wayne [Way], Kuksin [Kuk], Bourgain [Bou]etc.)

1.4 Hard Implicit Function Theorem

55

In avoiding too many complicated computations, we take an approach due to Hormander by appealing to the Leray–Schauder degree theory in the proof of the existence of a solution. Readers may read this subsection after Chap. 3. In fact there are other approaches without using topological arguments (see for instance J. Moser [Mos 1,2,3], Hormander [Hor 2] or J. T. Schwartz [Scw]). Actually, the boundedness assumption on f (x0 )−1 guarantees the convergence of the simple iteration method in solving the nonlinear equation: y = f (x) ⇔

(1.34)

x = x − f (x0 )−1 (y − f (x)) .

One produces the iteration process as follows: xn+1 = xn − f (x0 )−1 (y − f (xn )),

n = 1, 2, . . . .

(1.35)

However, if we consider the nonlinear wave equation: utt − F (uxx ) = f u|t=0 = ut |t=0 = 0 , where F is a function with F (0) = 1 and F (0) = 0, then the above simple iteration process fails, because the linearized equation reads as utt − uxx = g u|t=0 = ut |t=0 = 0 . when g ∈ H s , with compact support for some s > 0. We can at most estimate the boundedness of H s+1 norm of u, but not H s+2 , as that in the elliptic case. If we use the simple iteration, then we lose derivatives in each step; after a ﬁnite number of steps, the iteration cannot go on. This phenomenon is called “loss of derivatives”. It occurs in nonelliptic problems. In the analytic category, occurs phenomenon whereby the inverse maps reduce the convergence radii (e.g. the small divisor problem, see below); this is called “loss of convergence radii”. As before, this is also the case that the linearized operator has no bounded inverse. 1.4.1 The Small Divisor Problem For a given f , analytic in a neighborhood of θ, with f (0) = 0, and f (0) = σ, ﬁnd u, analytic in a neighborhood of θ with u(0) = 0, and u (0) = 1, satisfying f (u(z)) = u(σz) . We write it in the IFT form:

(1.36)

56

1 Linearization

Let f (z) = σz + fˆ(z) , u(z) = z + u ˆ(z) . Equation (1.36) is equivalent to Φ(fˆ, u ˆ) = fˆ(z + u ˆ(z)) + σ u ˆ(z) − u ˆ(σz) = θ .

(1.37)

Obviously, Φ(θ, θ) = θ, and Φuˆ (fˆ, u ˆ)w ˆ = fˆ (z + u ˆ(z))w(z) ˆ + σ w(z) ˆ − w(σz) ˆ . Φuˆ (θ, θ) is injective, if σ = e2πiτ , where τ is irrational. In fact, we write vˆ =

∞

vj z j , w ˆ=

j=2

∞

wj z j .

j=2

The equation ˆ = vˆ Φuˆ (θ, θ)w has a unique solution w, ˆ in which wj =

vj j = 2, 3, · · · . σ − σj

Since the denominator tends to zero for a subsequence, one cannot expect a bounded inverse of Φuˆ (θ, θ). A real number τ is of type (b, ν), b > 0, ν > 2, if τ − p ≥ b ∀p, q ∈ Z\{0} . (1.38) q qν Claim. For almost all real numbers τ , there exists (b, ν) depending on τ , such that τ is of type (b, ν). Indeed, for given (b, ν), ﬁxing q, the set of all real numbers τ ∈ [0, 1] such that (1.38) does not hold has a measure less than 2bq −ν+1 . Therefore, the set of all real numbers τ such that (1.38) does not hold has the measure 2b q −ν+1 < ∞. Since b can be arbitrarily small, the conclusion follows. Suppose σ = e2πiτ , where τ is a number of type (b, ν). Then p |σ − σ j | = e2πi(j−1)(τ − j−1 ) − 1 p ≥ sin 2π τ − (j − 1) j−1 p 2 ≥ · 2π(j − 1) τ − π j − 1 b ≥4 , (j − 1)ν−1

1.4 Hard Implicit Function Theorem

therefore

57

j ν−1 1 . |σ − σ j | 4b

In this case, even if vˆ has the convergent radius r, w ˆ can only have a smaller convergent radius. In other words, if we introduce a family of Banach spaces: A(r) = {w(z)| ˆ bounded and analytic in |z| < r, w(0) ˆ =w ˆ (0) = 0} with the norm ˆ , |w| ˆ r = Sup|z| 0 such that ∀(x, y) ∈ Ωrλ , ∀λ ∈ (0, 1] with |Φ(x, y)|λ Cλ

−2(α+τ )

, we have y∗ = u(x) ∈ Y λ satisfying 2

Φ(x, u(x)) = θ . Proof. One chooses sequences λn =

λ (1 + 2−n ), 2

n = 0, 1, 2, . . . ,

and µn+1 =

1 λ (λn + λn+1 ) = (1 + 3 · 2−n−2 ), 2 2

n = 0, 1, 2, . . . .

Then λ0 = λ, λn ↓ λ2 . For simplicity, we write Φ(yn ) = Φ(x, yn ), T (yn ) = T (x, yn ). The Newton iteration sequence reads as yn+1 = yn − T (yn )Φ(yn ) n = 0, 1, 2, . . . (1.48) y0 = y . We want to prove: (1) yn ∈ Brλn (y), ∞ nτ (2) let cn = |Φ(yn )|λn , n=1 2 cn < ∞, (n+3)τ −τ (3) yn+1 − yn µn+1 M cn (2 λ ).

60

1 Linearization

If they are proved, then ∞

∞

−τ 3τ

yn+1 − yn λ M λ 2

n=1

2

2nτ cn < ∞ .

n=1

Therefore ∃y∗ = u(x) such that yn → y∗ in Y λ , and 2

|Φ(x, y∗ )| λ lim |Φ(yn )|λn = lim cn = 0 , 2

n→∞

n→∞

i.e., Φ(x, y∗ ) = θ. First, we prove (3) . In fact, From (1.48) and assumption (2), yn+1 − yn µn+1 T (yn )Φ(yn )µn+1 M (2(n+3)τ λ

−τ

(n+3)τ

= M cn (2

)|Φ(yn )|λn

−τ

λ

).

Next, we turn to proving (2) . By (1.48), we have Φ(yn+1 ) = [Φ(yn+1 ) − Φ(yn ) − Φy (x, yn )(yn+1 − yn )] − [Φy (x, yn )T (yn ) − I]Φ(yn ) .

(1.49)

Combining (3) , with assumptions (1) and (3), for n large, cn+1 = |Φ(yn+1 )|λn+1 M 22α(n+3) λ + M c2n (2n+3 λ where a = λ Set

−1 2(α+τ )

)

−2α

yn+1 − yn 2µn+1

≤ aq n c2n ,

−2(α+τ ) 3

q (M + M 3 ) and q = 4α+τ . αn = aq n cn ;

we obtain αn+1 = aq n+1 cn+1 a2 q 2n+1 c2n = qαn2

∀n .

One chooses k ∈ (1, 2), 0 ∈ (0, 1) satisfying 1

q k−1 0 < 1 , and n+1 = qkn ∀n. Thus n+1 = q 1+k+k

2

+···+kn kn+1 0

1

1

= q − k−1 (q k−1 0 )k

n+1

0, such that yn+1 − y0 µn+1 ac0 M1 . If c0 is so small that yn+1 − y0 µn+1 < r − y − yλ

∀n ,

then yn − yλn yn − yµn < r − y − yλn . Therefore yn − yλn < r .

The proof is complete.

As an application, now we return to the small divisor problem. We have the following theorem. Theorem 1.4.2 (Siegel) Suppose that τ is of type (b, ν), b > 0, ν > 2. Let σ = e2πiτ . If f (z) = σz + fˆ(z) is an analytic function on |z| < 1, with fˆ(0) = fˆ (0) = 0, then there exist γ, r0 ∈ (0, 1) and an analytic function u 3r0 such that on |z| < 4(1+γ) sup |fˆ(z)| γ |z| 0, A(r) = {w(z)|analytic and bounded in |z| < r, with w(0) = w (0) = 0} , with norm |w|r = sup |w(z)| , |z| 0 small. Since all assumptions (1)–(3) of Theorem 1.4.1 are satisﬁed (cf. (1.46), (1.44), and (1.43) respectively), the conclusion follows from Theorem 1.4.1. 1.4.2 Nash–Moser Iteration Let us turn to the “loss of derivative” problem. Besides the simple iteration method, there are other ways to solve equation (1.34). Given c > 0 and a curve x(t) satisfying f (x(t)) = (1 − e−ct )y , x(t) satisﬁes the ODE: x(t) ˙ = (f (x(t)))−1 (y − f (x(t))) . If, for some initial data x(0) = x0 , the solution globally exists, and has a limit x(t) → x∞ , then f (x∞ ) = y . Discretizing the equation, we return to Newton’s approximation scheme: xn+1 − xn = cf (xn )−1 (y − f (xn )) . This is (1.41) for c = 1. There is enough room to generalize the above method. Let f (x0 ) = y0 be a special solution. In order to overcome the problem of “loss of derivatives”, we introduce a family of smoothing operators St , t ∈ [1, ∞), which regularizes the function x, and satisﬁes St → Id, as t → ∞. Find a suitable function g(t), which will be described later so that the curve x(t) tends to a solution x∞ of the equation f (x) = y. We intend to solve the ODE: ⎧ ˙ = f (v(t))−1 g(t) ⎨ x(t) v(t) = St x(t) ⎩ x(0) = x0 .

1.4 Hard Implicit Function Theorem

63

Since d f (x(t)) = f (x(t))x(t) ˙ dt ˙ + g(t) , = (f (x(t)) − f (v(t)))x(t) we should have

f (x∞ ) − f (x0 ) =

∞

e(t)dt + 0

∞

g(t)dt , 0

where e(t) = (f (x(t)) − f (v(t)))x(t). ˙ Design the iteration scheme: ⎧ ⎨ δn := xn+1 − xn = n f (vn )−1 gn , vn = Sθn xn , ⎩ n = θn+1 − θn ,

(1.51)

where θ1 < θ2 < . . . < θn → ∞. Then f (xn+1 ) − f (xn ) = n (en + gn ) , where n en = f (xn + δn ) − f (xn ) − f (xn )δn + (f (xn ) − f (vn ))δn . It follows that f (xn+1 ) − f (x0 ) =

n

j (gj + ej ) .

j=0

Since we are only interested in the limiting result, sometimes we modify it to be: n j gj + Sθn En = Sθn (y − f (x0 )) , j=0

where En =

n−1 j=0

j ej , so that gn can be determined step by step.

gn = −1 n [(Sθn − Sθn−1 )(y − f (x0 ) − En−1 ) − n−1 Sθn en−1 ], n = 1, 2, . . . , (1.52) and g0 = −1 0 Sθ0 (y − f (x0 )) . We start with an abstract framework. {Ea }a≥0 is called a family of Banach spaces with smoothing operators, if Eb → Ea for b ≥ a is an injection, and ∃C = C(a, b) such that ua Cub . Let E∞ = ∩a≥0 Ea be endowed with the weakest topology, such that E∞ → Ea is continuous. Moreover, we assume that ∃ a family of linear operators Sθ : E0 → E∞ , depending on a parameter θ ≥ 1, such that

64

1 Linearization

(1) Sθ ub Cua b a, d Sθ ub Cθb−a−1 ua , (2) dθ where C is a constant depending on a and b. The following inequalities hold: (1) Sθ ub Cθb−a ua , ∀ a < b. (2) (I − Sθ )ub Cθb−a ua ∀ b < a. ∀λ ∈ (0, 1) ∀ a, b. (3) (Interpolation inequality) uλa+(1−λ)b Cuλa u1−λ b Both (1) and (2) follow from the second inequality of the deﬁnition. In fact, if a > b, then ∞ d S u u − Sθ ub = t dt θ b

∞ b−a−1 C t ua dt = Cθb−a ua if b < a . θ

and if a < b, then

θ d Sθ ub = St u + S1 ub 1 dt b

θ C tb−a−1 dtua + Cua Cθb−a ua 1

where C denotes various constants. (3) is a consequence of (1) and (2). In fact, may assume a < b. For c = λa + (1 − λ)b, uc Sθ uc + (I − Sθ )uc C(θc−a ua + θc−b ub ) 1

b b−a , we obtain the desired inequality. By choosing θ = ( C u u a )

Example. H¨ older Spaces with Smoothing Operators Let Ω be a bounded domain in Rn we write H α (Ω) as the H¨older space, deﬁned as follows: H 0 (Ω) = C(Ω). If k ≥ 0 is an integer, k < a k + 1, C α (Ω) if a < k + 1 α H (Ω) = C α−0 (Ω) if a = k + 1 . The seminorm is deﬁned to be |u|a =

|α|=k

|∂ α u|a−k ,

1.4 Hard Implicit Function Theorem

and for 0 < a 1 ,

65

|u(x) − u(y)| , |x − y|α x,y∈Ω

|u|a = sup and the norm is deﬁned as:

ua = uC + |u|a . We now come to deﬁne the smoothing operators Sθ , θ ≥ 1. For the sake of simplicity, we assume Ω = Rn , the bounded domain case can be modiﬁed by standard argument. For a compact set K in Rn , one chooses X ∈ C0∞ (Rn ) to be 1 in a neighborhood of K, a function ψ ∈ C0∞ (Rn ) to be 1 in a neighborhood of 0, and let ϕ be the Fourier transform of ψ, ϕ = Fψ, i.e.,

exp (2πixξ)ψ(ξ)dξ . ϕ(x) = Rn

Let ϕθ (x) = θn ϕ(θx) θ ≥ 1 . If u has support in K, we deﬁne Sθ u = X (ϕθ ∗ u) ∈ C0∞ (Rn ) . Since ϕθ (x) = Fψ(ξ/θ), and ψ( θξ ) → 1 (in the distribution sense), we have ϕθ → δ, so Sθ u → u, as θ → +∞. The operators Sθ , which approximate the identity and map to smooth functions, are call smoothing operators. Theorem 1.4.3 The smoothing operators Sθ have the following properties, for θ > 1 and u ∈ H α : (1) Sθ ub Cua b a, d Sθ ub Cθb−a−1 ua . (2) dθ Proof. (1) following directly from the translation invariance and the convexity of the norms: ϕθ ∗ ua ϕL1 ua . We verify (2). Noticing d Sθ u = X · F −1 (θ−1 ψ1 dθ

ξ · (Fu)(ξ)) θ

where ψ1 (ξ) = −ξ · ∇ψ(ξ) , −1

is the inverse Fourier transform. Again ψ1 ∈ C0∞ , and vanishes in and F the neighborhood of the origin. According to (1), (2) holds for b = a. We shall only need to verify (2) for b = 0 and b = a + k, k ∈ N, because the remaining cases follow from the interpolation inequality.

66

1 Linearization

For b = a + k, d d Sθ u ∂ α S = u θ dθ dθ a+k a |α|=k α −1 k−1 ξ ξ = θ ψ1 · (Fu)(ξ) F θk θ a |α|=k

Cθk−1 ua because ξ α ψ1 (ξ) is again in L1 . For b = 0, we only need to prove for a = k ∈ N. Let us write ψ1 (ξ) =

ξα

ξ α ψα (ξ) where ψα (ξ) =

|α|=k

|α|=k

|ξ α |2

ψ1 (ξ) ,

then ψα ∈ C0∞ and vanishes in the neighborhood of the origin: d −1 −k−1 ξ α F Sθ u θ u)(ξ) ψ (∂ α dθ θ 0 0 |α|=k

[∂ α u(x − y) − ∂ α u(x)](ψα )θ (y)dy θ−k−1 |α|=k

=θ

−k−1

·

|y||(ψα )θ (y)|dyuk+1

= Cθ−k−2 uk+1 .

The proof is complete.

Now we are going to introduce a family of Banach spaces associated with Ea as follows: Let θj = 2j , j = 0, 1, 2, . . . , j = θj+1 − θj , j ≥ 1, 0 = 1: −1 R0 = −1 0 Sθ1 , Rj = j (Sθj+1 − Sθj ), j ≥ 1

According to (2), Rj ub

−1 j

b−a θj+1 d − θjb−a θj+1 St u C ua . θj dt (b − a)j b

We obtain: (4) Rj ub Cab θjb−a−1 ua , and (5)

∞ j=0

j Rj u = u.

1.4 Hard Implicit Function Theorem

67

The series is convergent in Eb if u ∈ Ea , a > b. This is called the Paley–Littlewood decomposition. Conversely, suppose we have a sequence {uj } ⊂ Ea , a ∈ [a1 , a2 ], satisfying uj ai Cθjai −a−1 ,

i = 1, 2 ∀j ;

then, by interpolation inequality, (6) uj b Cθjb−a−1 ∀b ∈ [a1 , a2 ], ∀j, and then the series

j

j uj converges in Eb if b < a.

Deﬁnition 1.4.4 For a ∈ [a1 , a2 ], we deﬁne Ea = {u = Cθjai −a−1 i = 1, 2, ∀j} with norm: ua =

inf u=

j

∞ j=0

j uj | uj ai

sup θj−ai +a+1 uj ai .

j uj i,j

Then Ea is a family of Banach spaces. The following properties hold: (7) ub Cua C1 ua if b < a. To prove the ﬁrst inequality. ∀ > 0 ∀ u ∈ Ea ∃{uj } ⊂ Eai , i = 1, 2 such that u = j j uj , and uj ai Cθjai −a−1 (ua + ) . From (6), uj b Cθjb−a−1 . It follows that ub C(ua + ). The second inequality follows from (4) and (5). (8) The space Ea does not depend on a1 , a2 . It follows from the interpolation inequality directly. for b < a. Indeed, ∀u = Ea one chooses (9) (I − Sθ )ub Cθb−a ua a1 < a2 such that b < a1 < a < a2 . ∀ > 0, ∃{uj } ⊂ Eai , i = 1, 2, such that u = Σj uj and uj ai ≤ Cθjai −a−1 (ua + ). Then j uj − Sθ uj b , (I − Sθ )ub and (I − Sθ )uj b Cθb−ai uj ai Cθb−ai θjai −a−1 (ua + ), i = 1, 2 , therefore (I − Sθ )ub C(ua + ) ⎞ ⎛ j θb−a1 θja1 −a−1 + j θb−a2 θja2 −θ−1 ⎠ . × ⎝ θj >θ

Thus,

θj a. Let α, β, r > 0,and Brα be a ball with radius r centered at the origin in Eα . Assume that f : Brα → Fβ is C 2 with f (θ) = θ, and satisﬁes (1) f (v)−1 exists for v ∈ Brα ∩ E∞ , and ∀g ∈ F∞ , the map (v, g) → f (v)−1 g: (E∞ ∩ Brα ) × F∞ → Ea2 is continuous for some a2 > α > a1 0, and satisﬁes f (v)−1 ga ≤ C(gβ+a−α + g0 vβ+α ) (2) f (u)(v, w)β+δ C

∀a ∈ [a1 , a2 ] ,

max(l−α,0)+max(m,a1 )+n 0. Then ∀y ∈ Fβ with yβ small, ∃u ∈ Eα satisfying f (u) = y . Proof. One uses the iteration scheme (1.51), so one should determine g ﬁrst. As we have seen from above, after decomposition, it satisﬁes a series of recursive equations. ∀g ∈ Fβ we have the decomposition: g=

j gj ,

j

with Deﬁne

gj b Cb θjb−β−1 gβ .

(1.53)

⎧ ⎨ uj+1 = uj + j f (vj )−1 gj , u0 = θ , vj = S θ j u j , ⎩ δj = j f (vj )−1 gj .

(1.54)

We are going to prove f (vj )−1 gj a C1 gβ θja−α−1 vj a C2 gβ θja−α uj − vj a C3 gβ θja−α

a ∈ [a1 , a2 ] ,

(1.55)

a ∈ (α, a2 ] ,

(1.56)

a a2 ,

(1.57)

inductively. Suppose (1.56) and (1.57) are true for j k and (1.55) holds for j < k, we prove (1.55) for j = k. By the assumption (1) and (1.53): f (vk )−1 gk a C(gk β+a−α + gk 0 vk β+a ) # " C θka−α−1 gβ + θk−β−1 gβ · θkβ+a−α gβ Cθka−α−1 gβ ,

1.4 Hard Implicit Function Theorem

69

if gβ is small. Now we prove (1.57) for j = k + 1, from (9), in the case a < α, uk+1 − vk+1 a = uk+1 − Sθk+1 uk+1 a a−α Cθk+1 uk+1 α .

Since uk+1 =

k

j f (vj )−1 gj ,

j=0

and by the deﬁnition of Fα norm, we have uk+1 α Cgβ , ∀α ∈ [a1 , a2 ] . Thus

a−α gβ . uk+1 − vk+1 a Cθk+1

In the case a = a2 , uk+1 − vk+1 a2 Cuk+1 a2 C

k

j f (vj )−1 gj a2

j=0 k

j θja2 −α−1

Cgβ

a2 −α Cθk+1 gβ

j=0

.

The other cases, α a a2 , are veriﬁed by the interpolation property. Finally, we prove (1.56) for j = k + 1: vk+1 a uk+1 a + vk+1 − uk+1 a a−α 2Cgβ θk+1 .

Thus the construction of the sequence uk is possible. And uk , vk are all in Brα if gβ is small. Now f (uj+1 ) − f (uj ) = (f (uj + δj ) − f (uj ) − f (uj )δj ) + (f (uj ) − f (vj ))δj + j gj = j (ej + ej + gj ) where and

ej = −1 j (f (uj+1 ) − f (uj ) − f (uj )(uj+1 − uj )) ,

(1.58)

70

1 Linearization

ej =

1

f (vj + t(uj − vj ))(−1 j δj , uj − vj )dt .

0

We obtain from the assumption (2), for n < α < l, (1 + tuj + (1 − t)vj l )f (vj )−1 gj n uj − vj n (1.59) ej β+δ C Cθj−1− gβ

2

(1.60)

where = 3α − l − 2n. Similarly by Taylor’s formula, ej β+δ θj−1− gβ

2

(1.61)

Let T (g) = j j (ej + ej ). Then T (g)β+δ Cgβ . According to (7), and the assumption on the compactness of embedding, Fb → Fa as b > a, we conclude that T : Fβ → Fβ is compact. According to the recursive formula (1.52), g is uniquely determined by y, therefore I + T is locally injective. Now one can apply the Leray–Schauder invariance of domain theorem (see Chap. 3, Corollary 3.4.12) to conclude that ∀y ∈ Fβ with small yβ , there exists g ∈ Fβ such that (I + T )(g) = y. Substituting this g into the iteration scheme (1.54), un is convergent to some u in Ea , provided by (1.55), and (7). Again, by (1.58), f (u) = (I + T )(g) = y. 2

2 Fixed-Point Theorems

It is well known that the contraction mapping theorem is one of the most important ﬁxed-point theorems in analysis. It is based on the metric of the underlying space. It is simple and is strongly dependent on the chosen metric, but useful. In particular, the unique solution can be computed by iteration. In fact, the implicit function theorem and then the ﬁrst three Sects. of Chap. 1 are based on it. The Brouwer ﬁxed theorem (1911), which says that every continuous selfmapping on a closed ball B n has a ﬁxed-point, is a fundamental ﬁxed-point theorem in topology. It is based on the notion of retraction. Because of its importance, there are a lot of proofs, roughly speaking, divided into three classes according to methods: (1) Combinatorics (Sperner lemma) by which computing methods are developed. (2) Algebraic topology; the topological degree and other algebraic topological invariants are introduced. (3) Diﬀerential topology; the proofs are simple and beautiful (see Dunford and Schwartz [DS], Milnor [Mi 2] etc.). In the study of analysis, we need inﬁnite-dimensional versions of this theorem. A new ingredient – the compactness – is added, while the ball is replaced by its topological equivalent – the convex set. The Schauder ﬁxedpoint theorem and its extensions are all based on convexity and compactness. They are widely used in combining with a priori estimates for solutions in diﬀerential equations. In an ordered space, the Bourbaki–Kneser principle is another basic ﬁxedpoint theorem with applications in analysis, in case compactness is unavailable. The chapter is organized as follows: The order method is studied in Sect. 2.1. Several ﬁxed-point theorems based on the Bourbaki–Kneser principle are derived, by which, the sub- and super-solutions method in ordered Banach space are developed with applications in PDEs. The convexitycompactness method is developed in detail in Sect. 2.3. We start with the KKM map and the Ky Fan inequality. All other ﬁxed-point theorems, including the Schauder ﬁxed-point theorem and its generalizations, the Nash equilibrium, and the Von Neumann–Sion saddle point theorem, are derived

72

2 Fixed-Point Theorems

as consequences. Various applications are studied, in particular, the Ky Fan ﬁxed-point theorem for set valued mappings is applied to free boundary problems. Section 2.4 is devoted to the existence and iteration method for ﬁxed points of nonexpansive maps, which are on the borderline of the contraction mappings. The prototype of the monotone mapping is the subdiﬀerential of a convex function. Due to the special feature of monotonicity, the compactness requirement can be reduced considerably. Since monotone mappings map a Banach space into its dual, one studies the surjectivity of monotone mappings. The main results in this aspect are due to Minty [Min] and Browder [Bd 3,4]. Applications to variational inequalities and quasi-linear elliptic problems are studied as well. We shall show how the monotonicity is applied instead of the compactness in the existence theory. These are the contents of Sects. 2.5 and 2.6. Convex sets and convex functions will be used form time to time; we collect their most important properties in Sect. 2.2.

2.1 Order Method A kind of nonlinear operator deﬁned on ordered spaces is order preserving; this special feature makes the ﬁxed-point problem easy to handle. A set E is said to be ordered if a partial order is deﬁned by the following axioms: ∀x, y, z ∈ E (i) x x, (ii) x y, and y x imply x = y, (iii) x y, and y z imply x z. A chain (or totally ordered set) E is an ordered set, on which ∀x, y ∈ E either x y or y x. The following terminologies are introduced: The smallest (greatest) element x (x resp.): x x (x x), ∀x ∈ E; the minimum (maximum) element x∗ (x∗ resp.): if x x∗ (x∗ x), x ∈ E then x∗ = x (x∗ = x); the lower (upper) bound of a subset F ⊂ E: a ∈ E (a ∈ E) and a x (x a) ∀x ∈ F ; the inﬁmum (supremum) of F inf F (sup F respectively) is the greatest (smallest) element of the subset, which consists of all lower (upper) bounds of F . ∀a ∈ E, we denote S+ (a) = {x ∈ E| a x} , and call it the right section of a. Similarly we deﬁne the left section S− (a). Set [a, b] = S+ (a) ∩ S− (b) for a b, we call it an order interval. Zorn’s lemma, which is equivalent to the Zermelo selection axiom, is our starting point: In a nonempty ordered set (E, ≤), if every chain has an upper bound in E, then the set has a maximum element.

2.1 Order Method

73

Let (E, ) and (F, ) be two ordered sets. A map f : E → F is called (strict) order preserving, if x ( 0 ∃yλ ∈ E satisfying ϕ(yλ ) ϕ(x) ρ(x, yλ ) λ1

(2.2) (2.3)

ϕ(y) > ϕ(yλ ) − ελρ(y, yλ ) ∀y ∈ E\{yλ } .

(2.4)

Proof. Deﬁne a subset of E: E1 = {y ∈ E| ϕ(y) + λερ(x, y) ϕ(x)} , E1 is closed (ϕ is l.s.c.) and nonempty (x ∈ E1 ), so is complete. We claim that ∃yλ ∈ E1 such that ϕ(y) > ϕ(yλ ) − λερ(y, yλ )

∀y ∈ E1 \{yλ } .

Indeed, if not, ∃λ > 0, ∀yλ ∈ E1 , ∃y ∈ E1 \{yλ } such that ϕ(y) ϕ(yλ ) − λερ(y, yλ ) . Deﬁne f (yλ ) = y = yλ , then f : E1 → E1 satisfying λερ(yλ , f (yλ )) ϕ(yλ ) − ϕ(f (yλ )). According to the Caristi theorem, f has a ﬁxed point; this is a contradiction. Since (2.2) and (2.3) are trivially true, it remains to verify (2.4) for y ∈ E\E1 . In fact, if ϕ(y) ϕ(yλ ) − λερ(y, yλ ) and y ∈ E\E1 , then ϕ(y) ϕ(x) − λερ(x, yλ ) − λερ(y, yλ ) ϕ(x) − λερ(x, y) . It follows that y ∈ E1 . This is a contradiction.

2.1 Order Method

75

Theorem 2.1.4 (Amann) Assume that every chain of (E, ) has a supremum, and that f : E → E is order-preserving mapping. If f has a subsolution a f (a) then f has a least ﬁxed point in S+ (a). Proof. (1) We verify the existence of a ﬁxed point. Deﬁne E+ = {x ∈ E| x f (x)} ∩ S+ (a). According to the Bourbaki–Kneser principle, we only want to show: (i) E+ = Ø, (ii) f (E+ ) ⊂ E+ , (iii) every chain of E+ has an upper bound in E+ . Indeed, a ∈ E+ ; (i) follows. Since f is order preserving, f (E+ ) ⊂ E+ . Let X be a chain of E+ . By assumption, it has a supremum b ∈ E. From x f (x) f (b) ∀x ∈ X, we have b f (b), and then b ∈ E+ , i.e., b is an upper bound of X in E+ . (2) Let FixS+ (a) (f ) be the ﬁxed point set of f in S+ (a). Deﬁne G+ = {y ∈ E+ | y z ∀z ∈ FixS+ (a) (f )} . We claim that FixG+ (f ) = Ø, then ∀y0 ∈ FixG+ (f ), y0 is the least ﬁxed point in S+ (a). To this end, from the Bourbaki–Kneser principle, it is suﬃcient to verify: (i) G+ = Ø, (ii) f (G+ ) ⊂ G+ , (iii) every chain of G+ has an upper bound in G+ . In fact, a ∈ G+ ; (i) follows. ∀y ∈ G+ , ∀z ∈ FixS+ (a) (f ), f (y) f (z) = z , which implies f (y) ∈ G+ , and then (ii) follows. Let X be a chain of G+ , by assumption, it has a supremum b ∈ E, so is b ∈ E+ as in (1) and b z ∀z ∈ FixS+ (a) (f ). i.e., b ∈ G+ . This is an upper bound of X in G+ . An ordered set is called chain complete if every chain has an inﬁmum and a supremum. Corollary 2.1.5 Let E be a chain complete ordered set, and let f : E → E be order preserving. Suppose that ∃ a pair of sub- and super- solutions a a. Then f has at least a least and a greatest ﬁxed point in [a, a]. An ordered set E is called a lattice, if for every pair x, y ∈ E, x ∨ y = sup{x, y} and x ∧ y = inf{x, y} are in E. A lattice is called complete if every nonempty subset possesses an inﬁmum and a supremum. Corollary 2.1.6 (Birkhoﬀ–Tarski) Let E be a complete lattice and let f be an order-preserving self-map on E, then there exist a smallest and a greatest ﬁxed point of f . Proof. Let a = inf E and a = sup E, then a and a are sub- and super- solutions of f respectively, satisfying a a. We study chain complete ordered sets. Let X be a Banach space, and let P ⊂ X be a nonempty closed convex positive cone with P ∩ (−P) = {θ}. It reduces a partial order structure on X:

76

2 Fixed-Point Theorems

x y if and only if y − x ∈ P . The order deﬁned above matches the linear structure and the topology on X, but not necessarily the magnitude of the norm. xi yi , i = 1, 2 =⇒ x1 + x2 y1 + y2 , x y, λ 0 =⇒ λx λy , and xn yn , xn → x, yn → y =⇒ x y . A Banach space X with a closed convex positive cone P, induces an ordered Banach space (OBS) (X, P). In particular ∀a ∈ X, S± (a) is closed. Lemma 2.1.7 Every compact subset E of an OBS is chain complete. Proof. For every chain X ⊂ E, we want to show that it has a supremum. In fact ∀x ∈ X, the family of sets: {S+ (x) | x ∈ X} is a family of closed n

subsets with ﬁnite intersection property, i.e., ∀x1 , x2 , . . . , xn ∈ X, ∩ S+ (xi ) = i=1

S+ (x∗ ), where x∗ = sup{x1 , . . . , xn } ∈ X, provided that X is a chain. Since X is compact, ∩ S+ (x) ∩ X = Ø. Let x be an element in the intersection. x∈X

Then (1) x x ∀x ∈ X, i.e., x is an upper bound of X, and (2) ∀ upper ¯ d. Thus x ¯ is a supremum. bound d of X, from x ≤ d and x ¯ ∈ X, it follows x Similarly, one proves that it has an inﬁmum. A positive cone P is called normal if every ordered interval [a, b] in X is bounded in the norm. Lemma 2.1.8 P is normal if and only if ∃ a constant C > 0 such that 0 x y =⇒ x C y . Proof. “=⇒” If not, ∃θ xn yn such that xn n3 yn n = 1, 2, . . . . Let zn = n2x ynn , then zn n. But θ zn n2y ynn . Deﬁne ∞ y = n=1 n12 yynn ; we have θ zn y, i.e., zn ∈ [θ, y]. But zn is unbounded. A contradiction. “⇐=” ∀x ∈ [a, b], we have θ x − a b − a. Since x − a C b − a , it follows that x x − a + a a +C b − a . Corollary 2.1.9 Suppose that (X, P) is a normal OBS, and that D = [a, b] is an order interval. If f is an order-preserving compact self-mapping on D, then it has a smallest and a greatest ﬁxed point.

2.1 Order Method

77

Proof. By normality, D is bounded in the norm. Let E = f (D). E is a compact subset of an OBS, provided by the compactness of f . Since f is a self-mapping on D, a ≤ f (a), f (b) ≤ b, so a < b is a pair of sub- and super-solutions of f . The conclusion follows from Corollary 2.1.5 and Lemma 2.1.7. Noticing that a bounded closed convex set of a reﬂexive Banach space is weakly compact, when we make use of the weak topology, Corollary 2.1.9 can be modiﬁed as follows: Corollary 2.1.10 Suppose that (X, P) is a reﬂexive normal OBS, and that D = [a, b] is an order interval. If f is an order-preserving self-mapping on D, then f has a smallest and a greatest ﬁxed point. Example 1. Let M be a compact space, and let X = C(M ) with norm x = max |x(ξ)| . ξ∈M

Deﬁne P = {x ∈ X| x(ξ) 0}. Then P is normal and (X, P ) is an OBS. Example 2. Let (Ω, B, µ) be a measure space and let Lp (Ω, B, µ), 1 p ∞ be the Lp space over (Ω, B, µ). Deﬁne P = {x ∈ Lp (Ω, B, µ)| x(ξ) 0 a.e. ξ ∈ Ω}. Then (X, P ) is an OBS, and P is normal. Example 3. Let X = C 1 ([0, 1]) with norm x = max |x(t)| + max |x (t)| , t∈[0,1]

t∈[0,1]

and P = {x ∈ X| x(t) 0 ∀t ∈ [0, 1]}. Then (X, P ) is an OBS, but P is not normal. Corollaries 2.1.9 and 2.1.10 are the foundation of the sub- and supersolutions method in the theory of diﬀerential equations. Example 4. Consider the following semi-linear elliptic BVP: −u(x) = ϕ(x, u(x)) in Ω , u|∂Ω = 0

(2.5)

where Ω is a bounded domain with smooth boundary in Rn , and ϕ : Ω × R1 → R1 is continuous. Assume that ∀x ∈ Ω, the function t → ϕ(x, t) is nondecreasing. Deﬁne K = (−)−1 with 0-Dirichlet condition; it is known that K is a positive linear operator from the maximum principle. Let F be the mapping u(x) → ϕ(x, u(x)). Since ϕ is nondecreasing in u, the map F is order preserving. Setting X = C0 (Ω), we consider the following mappings: F

i

K

◦

j

C0 (Ω) → C0 (Ω) → Lp (Ω) → Wp2 ∩ Wp1 (Ω) → C0 (Ω) ,

78

2 Fixed-Point Theorems

where i and j are injections, and p > n2 . The later is compact, provided by the Sobolev embedding theorem. Deﬁne f = j ◦ K ◦ i ◦ F : C0 (Ω) → C0 (Ω); the problem (2.5) is equivalent to ﬁnding the ﬁxed point of f . Since f is compact and order preserving, it has a ﬁxed point if one can ﬁnd a sub-solution u and ◦

super-solution u of (2.5) in Wp2 ∩ Wp1 (Ω). i.e., u u satisfying −u ϕ(x, u(x)) and − u ϕ(x, u(x)) a.e. in Ω . Namely, we have proved: Statement 2.1.11 Suppose that ϕ : Ω × R1 → R1 is continuous and ∀x ∈ Ω, t → ϕ(x, t) is nondecreasing. If there is a pair of sub- and super-solutions ◦

u u in Wp2 ∩ Wp1 (Ω),

n 2

< p < ∞, then equation (2.5) has the smallest and ◦

the greatest solutions in the order interval [u, u] ⊂ Wp2 ∩ Wp1 (Ω). Remark 2.1.12 The nondecreasing condition on ϕ can be weakened as follows: ∃ω 0, such that ∀x ∈ Ω, the function: ϕω (x, t) = ϕ(x, t) + ωt is nondecreasing in t. Indeed, instead of K and F , we introduce the linear positive operator Kω = (− + ωI)−1 and the order-preserving mapping Fω : u(x) → ϕω (x, u(x)). Then we ﬁnd the ﬁxed point of fω = j ◦ Kω ◦ i ◦ Fω . Moreover, the continuity condition on ϕ can be replaced by the Caratheodory condition. In this case, we consider the map f : Lp (Ω) → Lp (Ω), with f = i ◦ j ◦ ◦K ◦ F instead, where F is the Nemytski operator. Remark 2.1.13 The sub- and super-solutions method has an advantage in numerical analysis, because one can get the smallest and the greatest solutions in the order interval [u, u] by iteration. Setting ui+1 = f (ui ), ui+1 = f (ui ) u0 = u, u0 = u .

i = 0, 1, 2, . . . ,

Then one has u0 u1 · · · un un · · · u1 u0 . The sequences {un } and un converge to the least solution u∗ and the greatest solution u∗ respectively. If u∗ = u∗ , then this is the unique solution in [u, u]; and the two sequences provide lower and upper bounds estimates for u∗ = u∗ . Remark 2.1.14 In contrast to Corollaries 2.1.9 and 2.1.10, we consider an order-reversing compact mapping f on a normal OBS (X, P). Let a ∈ X be a

2.1 Order Method

79

sub- (or super-) solution of both f and f 2 . Setting xi+1 = f (xi ), i = 0, 1, 2, . . ., and x0 = a, we have either x0 x2 x4 · · ·

· · · x3 x1 if a f 2 (a) f (a) ,

or x1 x3 x5 · · ·

· · · x2 x0 if f (a) f 2 (a) a, respectively .

Thus x2n # x and x2n+1 $ x if a f (a) , (x2n+1 # x and x2n $ x if f (a) a respectively.) If the ﬁxed-point set of f , Fix(f ) ∩ S+ (a) (or Fix(f ) ∩ S− (a)) is not empty, then it is contained in [x, x]. Example 5. Let (M, g) be a compact Riemannian manifold without boundary, and let 2 1 ∂i (g ij det(g)∂j ) ∆M = det(g) i,j=1 be the Laplace–Beltrami operator with respect to g. Assume the Gaussian curvature k(x) < 0 ∀x ∈ M . Given a function K(x) < 0 on M , we consider the following equation: M u(x) − k(x) + K(x)e2u(x) = 0 x ∈ M .

(2.6)

This equation arises from geometry. The solution u deﬁnes a conformal metric g! = e2u g. Under the new metric g!, K(x) is the Gaussian curvature of M. We construct a pair of sub- and super-solutions: Choose C > 0 suﬃciently large such that −k(x) + K(x)e2C 0 −k(x) + K(x)e−2C

∀x ∈ M .

Then (−C, C) is a pair of sub- and super-solutions. Statement 2.1.15 Suppose that K, k ∈ C(M ) satisfy K(x) < 0, k(x) < 0 ∀x ∈ M , then equation (2.6) has a unique solution. Proof. Since ϕ(x, u) = −k(x) + K(x)e2u is decreasing in u, one cannot apply Statement 2.1.11 directly. Assume K = max(−K(x)) and set ω = 2Ke2C , x∈M

then the function t → ωt + K(x)e2t is nondecreasing. After Remark 2.1.13, Statement 2.1.11 is applied, and we conclude that there is a smallest and a greatest solution of (2.6) in the order interval [−C, C] of C(M ) for any suﬃciently large constant C > 0. However, since K(x) < 0 and e2u is increasing in u, from the maximum principle, the solution is unique. References can be found in Kazdan [Ka], Kazdan and Warner [KW 1][KW 2] and Aubin [Au 1].

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2 Fixed-Point Theorems

Example 6. Consider the following nonlinear heat equation: ⎧ ∂u k 1 ⎪ ⎨ ∂t − u = λu − au in Ω × R+ u=0 on ∂Ω × R1+ ⎪ ⎩ u(·, 0) = ϕ on Ω × {0}

(2.7)

where λ > 0, a > 1, k > 1 and ϕ ∈ Wp2 (Ω) is nonnegative, for some p > n2 . Notice that one can ﬁnd a large constant M > 0 such that λM − aM k < 0 and max ϕ M . Since ∃ω > 0 such that (λ + ω)u − auk is increasing in the interval [0, M ], one can use the sub- and super-solution method to prove the existence of a positive solution of (2.7). In fact, ∀T > 0 deﬁne a mapping A : g → v = Ag, Lp (Ω × [0, T ]) → 2,1 Wp (Ω × [0, T ]) → C(Ω × [0, T ]) as follows: ⎧ ∂v ⎪ ⎨ ∂t − v + ωv = g v=0 ⎪ ⎩ v(·, 0) = ϕ

in Ω × [0, T ] on ∂Ω × [0, T ] on Ω × {0} .

From the maximum principle for heat equations, A is order preserving. Deﬁne F (u) = (λ + ω)u − auk ; again F is order-preserving in the order interval D = [θ, M ] of the OBS C(Ω × [0, T ]). Setting f = A ◦ F ◦ i, where i is the injection C(Ω × [0, T ]) → Lp (Ω × [0, T ]), f is a compact order-preserving map on D. Now θ f (θ), and f (M ) M . From the parabolic maximum principle, we have a ﬁxed point u = f (u) in [θ, M ]. This is a solution of (2.7). The sub- and super-solutions method is extensively used in the study of nonlinear diﬀerential equations. The analytic basis is the maximum principle. As to systems, the method is also applicable if the associated maximum principle is extended to the linearized operator. However, the construction of a pair of sub- and super-solutions is technical, but crucial. It depends on the special feature of the nonlinear term.

2.2 Convex Function and Its Subdiﬀerentials 2.2.1 Convex Functions Let X be a vector space, a function f : X → R ∪ {+∞} is convex, if ∀x, y ∈ X, ∀λ ∈ [0, 1], we have f (λx + (1 − λ)y) λf (x) + (1 − λ)f (y) . It is called concave, if −f is convex. f is called aﬃne if f is both convex and concave. The domain, the epigraph and the level set of f are deﬁned as follows:

2.2 Convex Function and Its Subdiﬀerentials

81

dom(f ) = {x ∈ X| f (x) < +∞} , epi(f ) = {(x, λ) ∈ X × R1 | f (x) λ} , fa = {x ∈ X| f (x) a} ∀a ∈ R1 . f is said to be proper if f ≡ +∞. By deﬁnition, f is convex ⇐⇒ ∀x1 , . . . , xn ∈ X, ∀λ1 , . . . , λn 0, f

n i=1

λi = 1 ,

i=1

λi xi

n

n

λi f (xi ) ⇐⇒ epi(f ) is convex .

i=1

Also f is convex =⇒ ∀a ∈ R1 , fa is convex, but the converse is not true. A function f is called quasi convex (quasi concave) if fa is convex (concave resp.) ∀a ∈ R1 . For a subset C of X, the indicator function is deﬁned to be 0 if x ∈ C χC (x) = +∞ if x ∈ C . Thus C is convex ⇐⇒ χC is a convex function. The following simple propositions hold: (1) If f and g are convex, then ∀α, β 0 αf + βg is convex. (2) If A is a linear mapping: X → Y and if f : Y → R ∪ {+∞} is convex, then f ◦ A : X → R ∪ {+∞} is convex. (3) If {fι | ι ∈ ∧} is a set of convex functions, then sup{fι | ι ∈ ∧} is convex. We study the continuity of convex functions on Banach spaces. Theorem 2.2.1 Suppose that f : X → R ∪ {+∞} is a convex function on a Banach space X. If f is bounded from above in a neighborhood of a point x, then f is continuous at x. Moreover, f is locally Lipschitzian in int(dom(f )). Proof. (1) We may assume x = θ and f (θ) = 0. From the assumption, ∃η > 0, ∃M > 0 such that (2.8) f (y) M ∀y ∈ Bη (θ) . On one hand, ∀z ∈ Bη (θ)\{θ}, we have ηz z ηz z M f z . f (z) = f η z η z η One the other hand, ∀z ∈ X

(2.9)

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2 Fixed-Point Theorems

η z z+ 0 = f (θ) = f η+ z η+ z η M z f (z) + , η+ z η+ z

η z − z

It follows that

M z . η Combining (2.9) with (2.10), we obtain f (z) −

|f (z)|

M z η

(2.10)

∀z ∈ Bη (θ) .

(2) To prove the second assertion, we ﬁrst show that ∀y ∈ int(dom(f )), f is bounded above in a neighborhood of y, i.e., ∃η1 > 0, ∃M1 > 0 such that f (z) M1

∀z ∈ Bη1 (y) .

(2.11)

To this end one chooses w = (1+t)y such that the segment yw ⊂ int(dom(f )). t η and M1 = max {M, f (w)}, we have f (w) M1 , and Setting η1 = 1+t 1 + t t (z − y) η , if z ∈ Bη1 (y). Since t z= 1+t

w 1+t (z − y) + t 1+t

it follows from the convexity that 1+t 1 t f (z − y) + f (w) M1 . f (z) 1+t t 1+t Thus (2.11) holds. (3) We turn to proving that f is locally Lipschitzian in int(dom(f )). In fact, after (2), we only want to show that |f (z) − f (z )|

M z − z η−δ

∀z, z ∈ Bδ (θ), 0 < δ < η .

Dividing the segment zz into n equal length subsegments yi yi+1 , i = 1, . . . , n with y1 = z, yn+1 = z , and n > z−z η−δ , we have yi+1 ∈ Bη−δ (yi ) ⊂ Bη (θ). Applying (1), it follows that |f (yi+1 ) − f (yi )|

M yi+1 − yi η−δ

Then |f (z) − f (z )|

i = 1, 2, . . . , n .

M z − z . η−δ

2.2 Convex Function and Its Subdiﬀerentials

83

We also need to study lower semi-continuity of functions deﬁned on a topological space X. Deﬁnition 2.2.2 A function f : X → R1 ∪ {+∞} is said to be lower semicontinuous (l.s.c.), if ∀λ ∈ R1 , the level set fλ = {x ∈ X| f (x) λ} is closed. It is called sequentially lower semi-continuous (s.l.s.c.), if for any sequence {xn }, with xn → x ∈ X, we have limf (xn ) f (x) . From the deﬁnition, it follows directly that (1) f is l.s.c. ⇐⇒ epi(f ) is closed in X × (R1 ∪ {+∞}). (2) If {fα |α ∈ ∧} is a family of l.s.c. functions. Then f (x) = sup {fα (x)|α ∈ ∧} is l.s.c. (3) If f, g are l.s.c. and λ, µ 0, then λf + µg is l.s.c. Theorem 2.2.3 Let X be a Banach space and let f : X → R1 ∪ {+∞} be quasi convex. Then (1) f is l.s.c. ⇐⇒ f is weakly lower semi-continuous (i.e., l.s.c. in weak topology). (2) If X ∗ is separable, f is weakly l.s.c. ⇐⇒ f is sequentially weakly l.s.c. (3) If X = Y ∗ , where Y is a separable B-space, then f is w∗ l.s.c. ⇐⇒ f is sequentially w∗ l.s.c. Proof. (1) “⇐=” is trivial; we prove “=⇒”. Since ∀λ ∈ R1 , fλ is convex and closed. By the Hahn–Banach theorem, it is also weakly closed, i.e., f is weakly l.s.c. (2) In this case, the weak topolopy restricted to a normed bounded set is metrizable. (3) In this case, the w∗ -topology restricted to a normed bounded set is metrizable. We collect the following theorems on weak compactness, w∗ -compactness, weak sequential compactness and w∗ -sequential compactness for further reference. Theorem 2.2.4 (Banach–Alaoglu) Let X be a Banach space, and let E ⊂ X ∗ . Then E is w∗ -compact if and only if E is normed bounded and w∗ -closed. In particular, if further, X is reﬂexive, then for any E ⊂ X, E is weakly compact if and only if E is normed bounded and weakly closed. See for instance Larsen [La], pp. 254–257. Theorem 2.2.5 Let E be a weakly closed set in a Banach space, then it is weakly compact if and only if it is weakly sequentially compact. See for instance Larsen [La], pp. 303–309.

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2 Fixed-Point Theorems

2.2.2 Subdiﬀerentials Convex functions on Banach spaces are not diﬀerentiable in general. However, the notion of subdiﬀerentials is introduced as a replacement of derivatives of diﬀerentiable functions. Deﬁnition 2.2.6 (Subdiﬀerential) Let f : X → R ∪ {+∞} be a convex function on a vector space X. ∀x0 ∈ dom(f ), x∗ ∈ X ∗ is called a subgradient of f at x0 if x∗ , x − x0 + f (x0 ) f (x) ∀x ∈ X . The set of all subgradients at x0 is called the subdiﬀerential of f at x0 , and is denoted by ∂f (x0 ). Geometrically, x∗ ∈ ∂f (x0 ) if and only if the hyperplane y = x∗ , x − x0 + f (x0 ) lies below the epigraph of f , i.e., it is a support of epi(f ). y

x∗2

f (x)

f (x0) x∗1 x

x0

x

Fig. 2.1.

Obviously, ∂f (x0 ) may contain more than one point. The following propositions hold, if X is a Banach space. (1) ∂f (x0 ) is a w∗ -closed convex set. (2) If x0 ∈ int(dom(f )), then ∂f (x0 ) = Ø. Proof. We apply the Hahn–Banach separation theorem to the convex set epi(f ): ∃(x∗ , λ) ∈ X ∗ × R1 \{(θ, 0)} such that

2.2 Convex Function and Its Subdiﬀerentials

x∗ , x0 + λf (x0 ) x∗ , x + λt

85

∀(x, t) ∈ epi(f ) .

Since (x0 , f (x0 ) + 1) ∈ epi(f ), it follows that λ 0. However, λ = 0. Otherwise, we would have x∗ , x − x0 0 ∀x ∈ dom(f ). From x0 ∈ int(dom(f )), we conclude that x∗ = θ. It contradicts with (x∗ , λ) = (θ, 0). 1 ∗ x , we obtain x∗0 ∈ ∂f (x0 ). Setting x∗0 = −λ (3) ∀λ 0 ∂(λf )(x0 ) = λ∂f (x0 ). (4) If f, g : X → R1 ∪ {+∞} are convex, then ∀x0 ∈ int(dom(f )∩ dom(g)) ∂(f + g)(x0 ) = ∂f (x0 ) + ∂g(x0 ) . Proof. “⊃” is trivial; we are going to prove “⊂”. We may assume x0 = θ, f (θ) = g(θ) = 0, and θ ∈ ∂(f + g)(θ). We want to show that ∃x∗0 ∈ ∂f (θ) such that −x∗0 ∈ ∂g(θ). Since the set C = {(x, t) ∈ dom(g) × R1 | t −g(x)} is convex, and C ∩ int(epi(f )) = (θ, 0), from the fact that f (x) + g(x) 0. According to the Hahn–Banach separation theorem, ∃(x∗ , λ) ∈ X ∗ × R1 such that x∗ , x + λf (x) 0 x∗ , x + λt

∀(x, t) ∈ C .

In the same manner as the proof of (2), we verify λ > 0. Setting x∗0 = we have x∗0 , x f (x), − x∗0 , x g(x) ∀x ∈ X . i.e., x∗0 ∈ ∂f (θ), and −x∗0 ∈ ∂g(θ).

−x∗ λ ,

(5) If f : X → R ∪ {+∞} is convex, and is G-diﬀerentiable at a point x0 ∈ int(dom(f )), then ∂f (x0 ) is a single point x∗0 satisfying x∗0 , h = df (x0 , h) ∀h ∈ X. 1

Proof. We may assume x0 = θ and f (θ) = 0. Deﬁne the functional on X: L(h) = df (θ, h) . It is homogeneous. From the convexity of f , we have L(h1 + h2 ) L(h1 ) + L(h2 ), ∀ h1 , h2 ∈ X . Then L is linear. Again by the convexity: −f (−h) L(h) f (h)

∀h ∈ X .

Combining with Theorem 2.2.1, L is continuous. Therefore ∃x∗0 ∈ X ∗ such that L(h) = x∗0 , h . It follows that x∗0 ∈ ∂f (θ). Now, suppose x∗ ∈ ∂f (θ), i.e., x∗ , h f (h) ∀h ∈ X, and then x∗ , h 1 ∗ t f (th) ∀h ∈ X, ∀t > 0. By taking limit t → +0, it follows that x , h ∗ ∗ ∗ x0 , h ∀h ∈ X. Then x = x0 .

86

2 Fixed-Point Theorems

(6) If f : X → R1 ∪ {+∞} is convex and attains its minimum at x0 , then θ ∈ ∂f (x0 ). Let us present a few examples in computing the subdiﬀerentials of convex functions. Example 1. (Normalized duality map) Let X be a real Banach space, and let f (x) = 12 x 2 . Then ∂f (x) = F (x) := {x∗ ∈ X ∗ | x∗ = x , x∗ , x = x 2 }. It is called the normalized duality map. In the particular case where X is a real Hilbert space, it is known that f (x) = x, and indeed, F (x) = x. However, for Banach spaces, f may not be diﬀerentiable. We verify ∂f (x) = F (x) as follows: “⊂” ∀x∗ ∈ ∂f (x), we have x∗ , y − x

1 ( y 2 − x 2 ) , 2

and then ∀h ∈ X, x∗ , h

1 t ( x + th 2 − x 2 ) x h + h 2 , 2t 2

as t > 0. It follows that x∗ , h x h , and then x∗ x . On the other hand, setting y = λx, we have (λ−1) x∗ , x 12 (λ2 −1) x 2 . Dividing by (λ − 1) as λ ∈ (0, 1), and then letting λ = 1 we obtain x∗ , x ≥ x 2 ; it follows that x x∗ . Thus x = x∗ , and x∗ , x = x2 . “⊃” ∀x∗ ∈ F (x), one has x∗ , y − x = x∗ , y − x 2 x∗ y − x 2 1 ( y 2 − x 2 ) 2 = f (y) − f (x) ∀y ∈ X, i.e., x∗ ∈ ∂f (x). Example 2. Let C be a convex subset of X. The support function of C is deﬁned to be SC (x∗ ) = sup x∗ , x . Geometrically, SC (x∗ ) = x∗ , x0 if and x∈C

only if {x ∈ X| x∗ , x = SC (x∗ )} is a support hyperplane of C at x0 . We consider the subdiﬀerential of the indicator function of C: ⎧ ◦ ⎪ if x0 ∈ C ⎨{θ} ∂χC (x0 ) = {x∗ ∈ X ∗ | SC (x∗ ) = x∗ , x0 } if x0 ∈ ∂C ⎪ ⎩ Ø if x0 ∈ C .

2.3 Convexity and Compactness

87

Example 3. Let β : R1 → R1 be a monotone nondecreasing function. Then

t ϕ(t) = β(s)ds 0

is a continuous convex function, and ∂ϕ(t0 ) = [β(t0 − 0), β(t0 + 0)] ∀t0 ∈ R1 . If further, ∃p > 1, C1 , C2 > 0 such that |β(t)| C1 + C2 |t|p−1 , then the functional

ϕ(u(x))dx J(u) = Ω

is a convex functional on Lp (Ω), where Ω is a measurable set in Rn with bound measure. We conclude that ∂J(u0 ) = [β(u0 (x) − 0), β(u0 (x) + 0)]

:= {v ∈ Lp (Ω)| β(u0 (x) − 0) v(x) β(u0 (x) + 0) ∀ a.e. Ω} where

1 p

+

1 p

= 1. In fact,

v ∈ ∂J(u0 ) ⇐⇒

v(x)(u(x) − u0 (x))

Ω

[ϕ(u(x)) − ϕ(u0 (x))] ∀u ∈ Lp (Ω) Ω

⇐⇒ v(x)(u(x) − u0 (x)) ϕ(u(x)) − ϕ(u0 (x)) a.e. ⇐⇒ v(x) ∈ ∂ϕ(u0 (x)) a.e.

2.3 Convexity and Compactness The Schauder ﬁxed-point theorem is one of the most important ﬁxed-point theorems in nonlinear analysis. In this section we shall study a series of ﬁxedpoint theorems on compact convex set, including the Schauder ﬁxed-point theorem and its extensions. All these theorems are set up by considering the Ky Fan inequality and KKM mapping from the outset. Deﬁnition 2.3.1 Suppose that X is a vector space, E ⊂ X is a subset of X. A set-valued mapping G : E → 2X is called a KKM mapping, if ∀x1 , . . . , xn ∈ n E, conv{x1 , . . . , xn } ⊂ ∪ G(xi ). i=1

Knaster, Kuratowski, and Mazurkiweicz (1929) discovered the following (KKM) theorem: Let [p0 , . . . , pn ] be an n-simplex generated by n + 1 points p0 , p1 , . . . , pn in a vector topological space, and let M0 , . . . , Mn be (n + 1) k

closed sets, satisfying [pi0 , . . . , pik ] ⊂ ∪ Mij ∀ index subset {i0 , . . . , ik } ⊂ j=0

n

{0, 1, . . . , n}. Then ∩ Mi = Ø. i=0

The following version of the KKM theorem is due to Ky Fan.

88

2 Fixed-Point Theorems

G( x1 ) x1 G ( x3 )

x3

x2 G ( x2 ) Fig. 2.2.

Theorem 2.3.2 (FKKM) Suppose that X is a locally convex Hausdorﬀ space (LCS), E ⊂ X, and that G : E → 2$X is a closed-valued KKM map. If ∃x0 ∈ E such that G(x0 ) is compact, then x∈E G(x) = Ø. Proof. Deﬁne F (x) = G(x) ∩ G(x0 ). Then {F (x)| x ∈ E} is a family of closed subsets of G(x0 ). If one can show the ﬁnite intersection property of {F (x)| x ∈ E}, then our conclusion is proved. To this end, we only want to verify the ﬁnite intersection property for n {G(x)| x ∈ E}. If not, ∃{x1 , . . . , xn } ⊂ E, such that ∩ G(xi ) = Ø. Let i=1

us introduce the Euclidean metric d on L = span{x1 , . . . , xn }. Since L is ﬁnite dimensional, the metric derives the same topology on L, from the following facts: Each ﬁnite-dimensional LCS is normable (a consequence of the Kolmogorov normable theorem), and all norms on a ﬁnite-dimensional space are equivalent. Then the function λ(x) =

n

d(x, L ∩ G(xi )) > 0

∀x ∈ L .

i=1

Deﬁne βi (x) =

1 d(x, L ∩ G(xi )) i = 1, . . . , n . λ(x)

∀x ∈ L, we have βi (x) > 0 ⇐⇒ x ∈ G(xi ) i = 1, · · · , n . Again, deﬁne C = conv{x1 , . . . , xn } and a map ϕ : C → C to be ϕ : x →

n

βi (x)xi .

i=1

According to the Brouwer ﬁxed-point theorem, ∃x0 ∈ C such that x0 = ϕ(x0 ). Letting I(x0 ) = {i| βi (x0 ) > 0}, we have

2.3 Convexity and Compactness

89

ϕ(x0 ) ∈ conv{xi | i ∈ I(x0 )} . But x0 ∈ G(xi ) ∀i ∈ I(x0 ) implies x0 ∈

∪

i∈I(x0 )

G(xi ). This contradicts with

the KKM mapping.

Ky Fan’s inequality is the foundation of this section, by which all ﬁxedpoint theorems are derived. Theorem 2.3.3 (Ky Fan’s inequality) Suppose that X is an LCS and that E ⊂ X is a nonempty convex set. Assume φ : E × E → R1 satisfying (1) ∀y ∈ E x → φ(x, y) is l.s.c., (2) ∀x ∈ E y → φ(x, y) is quasi-concave, (3) ∃y0 ∈ E such that {x ∈ E| φ(x, y0 ) sup φ(x, x)} is compact . x∈E

Then ∃x0 ∈ E such that sup φ(x0 , y) sup φ(x, x) .

y∈E

x∈E

Before going to the proof of Theorem 2.3.3, we ﬁrst introduce a geometric version of the statement. Given a set E, let A ⊂ E × E be a subset. Deﬁne the sections: ∀x ∈ E, A1 (x) = {y ∈ E| (x, y) ∈ A} , and ∀y ∈ E, A2 (y) = {x ∈ E| (x, y) ∈ A} . Denote the diagonal {(x, x)| x ∈ E} by . A1 (x)

y

x

A2 (y) Fig. 2.3.

In this manner, we deﬁne two set-valued mappings: E → 2E , A1 : x → A1 (x) and A2 : y → A2 (y) .

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2 Fixed-Point Theorems

Theorem 2.3.4 Suppose that X is an LCS and that E ⊂ X is a nonempty convex set. Assume that Γ ⊂ E × E is a subset satisfying (1 ) (2 ) (3 ) (4 )

∀y ∈ E, Γ2 (y) is open, ∀x ∈ E, Γ1 (x) is convex, ∩ Γ = Ø, ∃y0 ∈ E such that E\Γ2 (y0 ) is compact.

Then ∃x0 ∈ E such that Γ1 (x0 ) = Ø. We are going to show that Theorem 2.3.3 ⇐⇒ Theorem 2.3.4. Proof. “ =⇒” Deﬁne a function on E × E: 1 (x, y) ∈ Γ φ(x, y) = 0 (x, y) ∈ Γ ∀y ∈ E, we see ⎧ ⎪ ⎨E {x ∈ E| φ(x, y) > λ} = Γ2 (y) ⎪ ⎩ Ø

if λ < 0 if 0 λ < 1 if 1 λ ,

then x → φ(x, y) is l.s.c., from (1 ). Similarly, ∀x ∈ E ⎧ ⎪ if λ < 0 ⎨E {y ∈ E| φ(x, y) > λ} = Γ1 (x) if 0 λ < 1 ⎪ ⎩ Ø if 1 λ , then y → φ(x, y) is quasi-concave, from (2 ). Since ∩ Γ = Ø, φ(x, x) = 0 ∀x ∈ E. The set {x ∈ E| φ(x, y0 ) 0} = E\Γ2 (y0 ) is compact, from (4 ). Theorem 2.3.3 is applied to conclude that ∃x0 ∈ E such that φ(x0 , y) 0 ∀y ∈ E, i.e., Γ1 (x0 ) = Ø. “ ⇐=” Deﬁne Γ = {(x, y) ∈ E × E| φ(x, y) > µ} , where µ = supx∈E φ(x, x). Then (1) and (2) imply (1 ) and (2 ) respectively. By deﬁnition Γ ∩ = Ø. Also E\Γ2 (y0 ) = {x ∈ E| φ(x, y0 ) µ} is compact. We then apply Theorem 2.3.4, and conclude that ∃x0 ∈ E such that Γ1 (x0 ) = Ø, i.e., φ(x0 , y) µ ∀y ∈ E.

2.3 Convexity and Compactness

91

Now we turn to giving a proof of Theorem 2.3.4. Proof. Deﬁne a set-valued mapping G : y → E\Γ2 (y) . From (1 ), it is a closed set-valued mapping. (4 ) implies that G(y0 ) is compact. If we can show that G is a KKM map, then the FKKM theorem is applied to conclude that ∩ G(y) = Ø, i.e., y∈E

∃x0 ∈ ∩ G(y) ⇐⇒ (x0 , y) ∈ (E × E)\Γ ∀y ∈ E ⇐⇒ Γ1 (x0 ) = Ø . y∈E

Veriﬁcation of the KKM map. If not, ∃{y1 , . . . , yn } ⊂ E and ∃λ1 ≥ 0, . . . , λn ≥ 0, with Σn1 λi ≤ 1, such that w=

n

n

n

i=1

i=1

λi yi ∈ ∪ G(yi ) ⇐⇒ w ∈ ∩ Γ2 (yi ) ,

i=1

⇐⇒ yi ∈ Γ1 (w) ∀i , by (2 )

=⇒ w ∈ Γ1 (w) , =⇒ ∩ Γ = Ø .

This contradicts with (3 ).

First we mention that this theorem is very useful in game theory. In particular, the Nash equilibrium theorem and Von Neumann–Sion saddle point theorem can be derived from it. In an n-person game, there are n players {1, 2, . . . , n}. The ith person has a strategy set Ei and a payoﬀ function fi , i = 1, 2, . . . , n. The payoﬀ functions depend on the strategies of all players, i.e., fi , i = 1, 2, . . . , n are functions of x = (x1 , . . . , xn ), where xj ∈ Ej , j = 1, . . . , n. Nash’s solution of the game is such a strategy (x∗1 , . . . , x∗n ) ∈ E1 × · · · × En : no player has any incentive to change his strategy as long as his enemies don’t change theirs. Mathematically, let X1 , . . . , Xn be n-LCS, let Ei ⊂ Xi be nonempty compact convex sets, ∀i = 1, . . . , n, and let f1 , . . . , fn be n continuous functions on E = E1 × · · · ×En . A point x∗ = (x∗1 , . . . , x∗n ) ∈ E is called a Nash equilibrium if fi (x∗ ) fi (x∗1 , . . . , yi , . . . , x∗n ) ∀i = 1, 2, . . . , n ∀y = (y1 , . . . , yn ) ∈ E . Corollary 2.3.5 (Nash) If the continuous functions xi → fi (x1 , . . . , xi , . . . , xn )

∀i = 1, . . . , n

are quasi-concave ∀(x1 , . . . , x ˆi , . . . , xn ) ∈ Ej ∀i. Then the Nash equilibrium j=i

exists.

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2 Fixed-Point Theorems

In fact, let us deﬁne a function on E × E: φ(x, y) =

n

(fi (x1 , . . . , yi , . . . , xn ) − fi (x1 , . . . , xn ))

i=1

It is easy to verify that all the assumptions of the Ky Fan inequality are satisﬁed (in particular, y → φ(x, y) is quasi-concave), and φ(x, x) = 0 ∀x ∈ E. Then ∃x∗ ∈ E satisfying φ(x∗ , y) 0 ∀y ∈ E. This is the Nash equilibrium. Corollary 2.3.6 (Von Neumann–Sion) Suppose that X, Y are reﬂexive Banach spaces, E ⊂ X and F ⊂ Y are nonempty closed convex sets. Assume that f : E × F → R1 satisﬁes (1) ∀y ∈ F , x → f (x, y) is l.s.c. and quasi-convex. (2) ∀x ∈ E, y → f (x, y) is u.s.c. and quasi-concave. (3) ∃(x0 , y0 ) ∈ E ×F such that f (x0 , y) → −∞ as y → +∞ and f (x, y0 ) → +∞ as x → +∞. Then there exist (x∗ , y ∗ ) ∈ E × F such that f (x∗ , y) f (x∗ , y ∗ ) f (x, y ∗ )

∀(x, y) ∈ E × F .

(2.12)

Proof. Deﬁne φ(u, v) = f (x, w) − f (z, y) , where u = (x, y), v = (z, w) ∈ E × F . Then we have the following: From Theorem 2.2.3, ∀v ∈ E × F, u → φ(u, v) is w.l.s.c. It is easy to verify directly that∀u ∈ E × F, v → φ(u, v) is quasi-concave, φ(u, v0 ) = f (x, y0 ) − f (x0 , y) → +∞ as x + y → +∞, where v0 = (x0 , y0 ) . The set {u ∈ E × F | φ(u, v0 ) 0} is weakly compact. We apply Ky Fan’s inequality to the space E × F with weak topology. There exists u∗ = (x∗ , y ∗ ) such that f (x∗ , w) − f (z, y ∗ ) 0 ∀(z, w) ∈ E × F , which implies f (x∗ , y) f (x∗ , y ∗ ) f (x, y ∗ ) ∀(x, y) ∈ E × F . The solution (x∗ , y ∗ ) is called the saddle point of f . As a special case where dim Y = 0, we have:

2.3 Convexity and Compactness

93

Corollary 2.3.7 Suppose that X is a reﬂexive Banach space. If f : X → R1 ∪ {+∞} is an l.s.c. convex function, satisfying f (x) → +∞ as x → ∞, then there exists x0 ∈ X such that f (x0 ) = min {f (x)| x ∈ X}. In the case where E and F are compact, the Von Neumann–Sion theorem is also rewritten as follows: min max f (x, y) = max min f (x, y) . x∈E y∈F

y∈F x∈E

(2.13)

Indeed, (2.12) ⇐⇒ (2.13). Proof. “=⇒” We always have β := max min f (x, y) min max f (x, y) =: α . y∈F x∈E

x∈E y∈F

However, (2.12) implies α max f (x∗ , y) f (x∗ , y ∗ ) min f (x, y ∗ ) β . y∈F

x∈E

Therefore α = β, and then (2.13) holds. “⇐=” Since both E and F are compact and the functions x → f (x, y) is l.s.c., y → f (x, y) is u.s.c., the functions x → maxy∈F f (x, y) is l.c.s, and y → minx∈E f (x, y) is u.s.c., we have (x∗ , y ∗ ) ∈ E × F such that min f (x, y ∗ ) = max min f (x, y) x∈E

y∈F x∈E

= min max f (x, y) x∈E y∈F

= max f (x∗ , y) . y∈F

Since

min f (x, y ∗ ) ≤ f (x∗ , y ∗ ) ≤ max f (x∗ , y) , x∈E

these three are equal. It (2.12) follows.

y∈F

In this sense, the Von Neumann–Sion theorem is called the mini-max theorem, which is the fundamental theorem in two-person game theory, and has many applications in other ﬁelds of mathematics (linear programming, convex programming, potential theory, the dual variational theory in mechanics etc.) Next, we turn to ﬁxed-point theorems. For a set-valued mapping Γ : E → 2F , where E, F are Hausdorﬀ topological vector spaces, we say Γ is upper (or lower) semi-continuous (u.s.c. or l.s.c., respectively for short), if for any closed (open resp.) set W ⊂ F , the pre-image Γ−1 (W ) = {x ∈ E| Γ(x) ∩ W = Ø} is closed (open resp.). For single-valued mappings, both u.s.c. and l.s.c. are all reduced to continuity. An alternative description of the upper semi-continuity is that for any open set W ⊂ F , the set {x ∈ E| Γ(x) ⊂ W } is open. A point x0 ∈ E is called a ﬁxed point of Γ if x0 ∈ Γ(x0 ).

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2 Fixed-Point Theorems

Theorem 2.3.8 (Ky Fan–Glicksberg) Suppose that X is an LCS, and that E ⊂ X is a nonempty compact convex set. If Γ : E → 2E is u.s.c., and ∀x ∈ E, Γ(x) is a nonempty closed convex subset of E, then ∃x0 ∈ E such that x0 ∈ Γ(x0 ). Proof. We prove by contradiction. If ∀x ∈ E, x ∈ Γ(x), then from the Hahn– Banach theorem ∃x∗ ∈ X ∗ ∃t ∈ R1 such that x∗ , y < t < x∗ , x

∀y ∈ Γ(x) .

Since Γ is u.s.c. there exists a neighborhood U (x) of x such that x∗ , y < t < x∗ , z

∀y ∈ Γ(z) ∀z ∈ U (x) . n

Since E is compact, we have x1 , . . . , xn ∈ E such that E ⊂ ∪ U (xi ). Let {x∗1 , . . . , x∗n } ⊂ X ∗ be the associated elements; one has x∗i , y < x∗i , x

i=1

∀y ∈ Γ(x) ∀x ∈ U (xi ) .

We construct a partition of unity of E: {βi : E → R1 , i = 1, . . . , n| βi 0,

n

βi = 1 on E and βi (x) > 0 iﬀ x ∈ U (xi )} .

i=1

Deﬁne a continuous map f : E → X ∗ by f (x) =

n

βi (x)x∗i ,

i=1

one has f (x), y =

n

βi (x) x∗i , y

0, substituting y 0 = f (x0 ) − ε, z 0 = f (x), y i = −ε, z i = gi (x) i = 1, . . . , n into (2.17) ∀x ∈ X, and letting ε → 0, we have

ω0 f (x0 )

n

ωi gi (x) + ω0 f (x)

∀x ∈ X .

(2.18)

i=1 n

Claim ω0 > 0. Otherwise, ω0 = 0, thus i=1 ωi gi (x) 0. Since we have assumed gi (x∗ ) < 0, combining with the fact that ωi 0, i = 0, 1, . . . , n, it follows that ωi = 0, i = 1, . . . , n. This is impossible. Dividing (2.18) by ω0 , we obtain f (x0 )

n

ξj gj (x) + f (x)

∀x ∈ X ,

j=1

where ξj =

ωj ω0 .

Then n

ξj gj (x0 ) = 0 ,

j=1

it follows that ξj gj (x0 ) = 0 j = 1, . . . , n. Thus f (x0 ) +

n

yj gj (x0 ) f (x0 ) +

j=1

n

ξj gj (x0 ) f (x) +

j=1

n

ξj gj (x)

j=1

∀(x, y) ∈ X × Rn+ , i.e., (x0 , ξ) is a saddle point of L. Statement 2.3.13 In the convex programming problem, assume (1) f (x) → +∞ as x → ∞, (2) ∃x0 ∈ X such that gi (x0 ) < 0, i = 1, 2, . . . , n. The problem (2.16) has a solution. Proof. Consider the Lagrangian on X × Rn+ , L(x, y) = f (x) +

n

yi gi (x) .

i=1

Obviously, ∀y ∈ Rn+ , x → L(x, y) is l.s.c. and convex, ∀x ∈ X, y → L(x, y) is linear , and by (2) and (1),

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2 Fixed-Point Theorems

L(x0 , y) → −∞ as y → ∞ , L(x, θ) = f (x) → +∞ as x → ∞ . Applying Corollary 2.3.6, there is a saddle point of L, which is the solution of the problem (2.15), according to Statement 2.3.12. 2. Periodic solutions of ODE We study the 2π-periodic solutions of the following ODE: x ¨ + g(x) = f (t)

(2.19)

where g ∈ C(R1 ) and f ∈ L2 ([0, 2π]). Assume that ∃n ∈ N and ε > 0 such that g(x) (n + 1)2 − ε for |x| large . (2.20) n2 + ε x Statement 2.3.14 The problem (2.19) has a 2π-periodic solution under the assumption (2.20). Proof. Let γ = n2 + n + 1/2. Then ∃R > 0 such that 1 |g(x) − γx| n + − |x| as |x| R . 2 Rewrite (2.19) as x ¨ + γx + g(x) − γx − f (t) = 0

(2.21)

and deﬁne the linear operator L : x → −(¨ x + γx) with domain D(L) = {x ∈ H 2 (0, 2π)| x(0) = x(2π), x(0) ˙ = x(2π)}. ˙ Then L has a bounded inverse: L−1 : L2 (0, 2π) → D(L). Since the injection D(L) → L2 (0, 2π) is compact, (2.21) is equivalent to x = L−1 · G(x) , where G : x → g(x) − γx − f is a bounded continuous mapping on L2 (0, 2π). Then F = L−1 ◦ G is a compact map on L2 (0, 2π). 1 , Gx (n + 12 − ε) x + C, where Moreover L−1 n+1/2 √ C = 2π max |g(x) − γx|+ f 2 . Setting R1 Cε , we have F : B R1 (θ) → |x|R

B R1 (θ). The Schauder ﬁxed-point theorem is applied to ensure the existence of x0 ∈ BR1 (θ) satisfying F (x0 ) = x0 , thus x0 ∈ D(L) and satisﬁes (2.21). 3. Obstacle problem Find the equilibrium position u of a membrane with a ﬁxed boundary, acted upon by an external force f , and obstructed by an ﬁxed obstacle ψ.

2.3 Convexity and Compactness

99

The problem is posed as a minimizing problem on a closed convex set. Let Ω ⊂ Rn be a bounded domain, f ∈ L2 (Ω), ψ ∈ H 1 (Ω). Find u ∈ H01 (Ω) such that u is the minimizer of the problem: min {J(v)| v ∈ E}, where

1 |∇v|2 − f · v , J(v) = 2 Ω and

& % E = v ∈ H01 (Ω)| v(x) ψ(x) a.e. .

From J(u) J(u + t(v − u)) ∀v ∈ E and t ∈ (0, 1), and letting t → 0, we obtain the following variational inequality:

∇u · ∇(v − u) f (v − u) ∀v ∈ E . (2.22) Ω

Ω

The existence of a weak solution for (2.22) can be studied by variational method, see Chap. 4. However, we present here a ﬁxed-point approach. First we reduce the problem to a system of inequalities. One has u ψ a.e. ,

(2.23)

Denote the positive cone of H01 (Ω) by C = {w ∈ H01 (Ω)| w(x) 0 a.e.}. Let v∈ H01 (Ω), and v ≤ u, then v ∈ E, and let w = u − v, then (2.22) implies that Ω ∇u∇w Ω f w ∀w ∈ C i.e., −u − f 0 a.e. ,

(2.24)

Substituting v = ψ and 2u − ψ, respectively into (2.22), we obtain

(−u − f ) · (ψ − u) ≥ 0 , Ω

and

(−u − f ) · (u − ψ) ≥ 0 , Ω

respectively, therefore Ω (−u − f )(u − ψ) = 0. If u ∈ Wp2 (Ω) then, (−u − f )(u − ψ) = 0 a.e.

(2.25)

Conversely, from (2.24), (2.25) and v ≤ ψ, we obtain Ω (−∆u−f )(v−u) ≥ 0, i.e., (2.22) Thus, for u ∈ Wp2 ∩ H01 (Ω), it is a solution of the variational inequality (2.22) if and only if it satisﬁes the system of inequalities (2.23), (2.24), (2.25). Second, the system of inequalities is equivalent to a PDE with discontinuous nonlinearity, i.e., for ψ ∈ Wp2 (Ω) the system of inequalities (2.23), (2.24), (2.25) is again equivalent to

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2 Fixed-Point Theorems

min {f (x), −ψ(x)} −u = f (x)

if u(x) ψ(x) if u(x) < ψ(x) .

(2.26)

Veriﬁcation: “=⇒” On the set {x ∈ Ω| u(x) < ψ(x)} the equality follows from (2.25). On the complement set {x ∈ Ω| u(x) = ψ(x)} we have −ψ(x) = −∆u(x) f (x) a.e., and then −∆u(x) = −∆ψ(x) = min {f (x), −ψ(x)}. “⇐=” (2.24) is trivial. If the set U = {x ∈ Ω| u(x) > ψ} = ∅, (2.26) implies that −(u − ψ) 0 on U . This contradicts the maximum principle; then (2.23) holds, and consequently (2.25). Now, (2.26) is a PDE with discontinuous nonlinearity: −u = φ(x, u) where

min {f (x), −ψ(x)} φ(x, u) = f (x)

(2.27) if u ψ(x) if u < ψ(x) .

We shall solve the equation by the Ky Fan ﬁxed-point theorem. First, let us deﬁne a set-valued mapping p

F : C(Ω) → 2L in Lp (Ω) ,

(Ω)

: u → F u = [φ(x, u(x)), φ(x, u(x))], the order interval

where φ(x, t) = lim φ(x, t ), and φ(x, t) = lim φ(x, t ). t →t

t →t

We consider the following maps: F

p

C(Ω) −→ 2L

(Ω)

K

2

1

j

−→ 2Wp ∩H0 (Ω) −→ 2C(Ω) ,

where K = (−)−1 and j is the embedding map. It is not diﬃcult to verify that Γ := j ◦K◦F is u.s.c., and ∀u ∈ C(Ω), Γ(u) is a nonempty closed convex set of C(Ω). Second, we consider the set E = K(B R (θ)) of C(Ω), where R = max { f p , ψ p }. It is a nonempty compact convex set in C(Ω), and Γ : E → 2E . Ky Fan’s ﬁxed-point theorem is applied to ensure the existence of u0 ∈ E such that u0 ∈ Γ(u0 ), i.e., u0 ∈ Wp2 ∩ H01 (Ω) satisfying ⎧ ⎪ ⎨min {f (x), −ψ(x)} −u0 (x) ∈ [min {f (x), −ψ(x)}, f (x)] ⎪ ⎩ f (x)

if u0 (x) > ψ(x) if u0 (x) = ψ(x) if u0 (x) < ψ(x) .

Third, however, on the set U = {x ∈ Ω| u0 (x) = ψ(x)}, −u0 (x) = −ψ(x) a.e. This implies that −ψ(x) f (x) a.e., i.e.,

2.3 Convexity and Compactness

min {f (x), −ψ(x)} −u0 (x) = f (x)

101

if u0 (x) ψ(x) if u0 (x) < ψ(x) .

This is exactly the solution of the PDE with discontinuous nonlinearity (2.26). Statement 2.3.15 If ψ ∈ Wp2 (Ω), f ∈ Lp (Ω), p > ◦

n 2,

then there exists

u ∈ Wp2 ∩ Wp1 (Ω) satisfying (2.26). The obstacle problem is a free boundary problem for PDEs. The boundary where the membrane detaches from the obstacle is unknown, and is to be determined simultaneously with the position of the membrane. A kind of free boundary problem, having the feature that the boundary values and the normal derivatives of the unknown function coincide on both sides of the unknown free boundary, can be studied via the above approach, i.e., pose the free boundary problem as a PDE with discontinuous nonlinearity, in which the unknown boundary is automatically absorbed into the unknown function. Fixed-point theorems for set-valued mappings can be used to study the existence of the solutions. Examples of the obstacle problem are the seepage surface in dams, the free boundary of conﬁned plasma, water cones in oil reservoirs etc. (see K.C. Chang [Ch 1,2]). 4. Stefan problem Consider a container Ω with boundaries Γ and Γ , ﬁlled with ice. Cooling on Γ and heating on Γ (see Fig. 2.4), we study the melting process.

Γ ’’ C(t) Γ’ Ω

Fig. 2.4.

At time t, the domain occupied by water is denoted by C(t), and let θ(x, t) 0 be the temperature at the space–time (x, t) as x ∈ C(t). Both θ and C are unknown, but g(x, t) = θ(x, t)|Γ ×[0,T ] is given ∀T > 0. Let S(t) = ∂C(t) be the unknown boundary. Let us suppose that S(t) is given by an equation σ(x) = t. The temperature distribution is governed by the following equations:

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2 Fixed-Point Theorems

⎧ ∂ ⎪ ⎪ ∂t θ − θ = 0 ⎪ ⎪ ⎪ ⎪ ⎨θ(x, t)|Γ ×[0,T ] = g(x, t), θ(x, t)|S(t) = 0 ⎪ ⎪ ⎪ ∇θ · ∇σ|S(t) = −L, ⎪ ⎪ ⎪ ⎩θ(x, 0) = 0

∀t ∈ [0, T ], ∀x ∈ C(t), ∀t ∈ [0, T ],

(2.28)

∀x ∈ Ω,

where L > 0 is a constant. The boundary conditions on S(t) are derived from the continuity of the temperature and the conservation of the energy, respectively. We set θ(x, t) if x ∈ C(t) ! θ(x, t) = 0 if x ∈ C(t) , and

t

u(x, t) =

! s)ds . θ(x,

0

The latter is called the modiﬁed Baiocchi transformation of θ. Let

t ψ(x, t) = g(x, s)ds . 0

The problem (2.28) is reduced to a parabolic equation with discontinuous nonlinearity: ⎧ ∂u ⎪ ∂t − u = −LH(u) in QT := Ω × (0, T ) , ⎪ ⎪ ⎨u| Γ ×[0,T ] = ψ , (2.29) ⎪ u|Γ ×[0,T ] = 0 , ⎪ ⎪ ⎩ u(x, 0) = 0, on Ω . Again, the advantage of the formulation (2.29) is that the unknown domain C(t) is implicitly involved. The price we paid is the nonlinear term being a discontinuous nonlinear function: the Heaviside function H(u) = 1 as u > 0, and = 0 as u 0. The problem can be solved by the Ky Fan ﬁxed-point theorem as before. Let us introduce the anisotropic Sobolev space Wp1,2 (QT ) = {u ∈ Lp (QT )| ∂t u, ∇x u, ∇2x u ∈ Lp (QT )} , with norm: u Wp1,2 = u p + ∂t u p + ∇2x u p . The boundary values of functions in Wp1,2 (QT ) span a fractional Sobolev 1 1 1− 2p ,2− p

space Wp

(Γ × (0, T )), where Γ = Γ ∪ Γ . In particular, C 1,2 (Γ × 1 1 1− 2p ,2− p

[0, T ]) = {u ∈ C(Γ×[0, T ])| ∂t u, ∇x u ∇2x u ∈ C(Γ×[0, T ])} ⊂ Wp (0, T )).

(Γ×

2.3 Convexity and Compactness 1−

1

103

,2− 1

p Statement 2.3.16 Assume ψ ∈ Wp 2p (Γ × (0, T )), p > n, and ψ 0; the equation (2.29) has a unique solution u ∈ Wp1,2 (QT ).

Proof. Deﬁne a set-valued mapping ⎧ ⎪ ⎨1 u → F (u) = [0, 1] ⎪ ⎩ 0

u(x, t) > 0 , u(x, t) = 0 , u(x, t) < 0 ,

p

then F : C(QT ) → 2L (QT ) is u.s.c. Let Kψ : Lp (QT ) → Wp1,2 (QT ) be the aﬃne operator f → u, where u satisﬁes the linear parabolic equation: ⎧ ∂u − u = f in QT , ⎪ ⎪ ⎪ ∂t ⎨ u|Γ ×(0,T ) = ψ , ⎪ u|Γ ×(0,T ) = 0, ⎪ ⎪ ⎩ u(x, 0) = 0, on Ω . Then the set-valued mapping Γ := Kψ ◦ (L ◦ F ) : C(Ω) → 2C(Ω) is a u.s.c. closed convex set-valued mapping. Deﬁne a compact convex set: 1

E = {Kψ u| u ∈ Lp (QT ), u p Lm(Ω) p } . Obviously Γ : E → 2E . According to Ky Fan’s ﬁxed-point theorem, one ﬁnds u ∈ Wp1,2 (QT ) satisfying:

− u ∈ −LF (u) u|Γ ×(0,T ) = ψ, u|Γ ×(0,T ) = 0, u|Ω×{0} = 0 . ∂u ∂t

On the set u−1 (0) = {(x, t) ∈ QT | u(x, t) = 0}, we have ∂u ∂t − u = 0. This means u automatically satisﬁes (2.29). Now, we prove the uniqueness of the solution u of equation (2.29). In fact, let u1 and u2 be two solutions of (2.29), and u = u1 − u2 , then ∂u ∂t − u = −L(H(u1 ) − H(u2 )) in QT , u|∂QT \(Ω×{T }) = 0 . By the maximum principle for parabolic equations, both sets {(x, t) ∈ QT | ± u(x, t) > 0} are null sets. Thus u1 = u2 a.e. Returning to the original problem (2.28), we deﬁne C(t) = {x ∈ Ω| u(x, t) > ! C(t) ∀t ∈ (0, T ). ! t) = ∂ u(x, t), and θ(·, t) = θ| 0}, θ(x, ∂t The veriﬁcation of (2.28) is omitted.

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2 Fixed-Point Theorems

2.4 Nonexpansive Maps Let X be a Banach space, and E be a closed subset of X. A map T : E → X is called a nonexpansive map, if T (x) − T (y) x − y

∀x, y ∈ E .

In contrast with the fact that a contraction mapping T : E → E, (i.e., ∃α ∈ (0, 1) such that T (x) − T (y) α x − y ) always has a ﬁxed point, the nonexpansive map does not in general. This can be seen by the following example: Example 1. X = c0 , the space consisting of all sequences x = {ξi }∞ 1 converging to zero, with the norm x = max |ξi | . i1

Let T (x) = (1− x , ξ1 , ξ2 , . . .) , and E be the closed unit ball. Then T : E → E is non-expansive: T (x) − T (y) = max | x − y |, max |ξi − ηi | x − y . i1

However, T does not have a ﬁxed point. In fact, if x = T (x), then ξ1 = ξ2 = . . . = 1− x . This is impossible. We are going to study the ﬁxed points for a nonexpansive map on a closed convex subset of a Banach space X with a normal structure, which means that for every closed convex subset E of X containing at least two points, there exists a point x0 ∈ E such that sup x − x0 < diam (E) .

(2.30)

x∈E

It is easy to see that Hilbert spaces are Banach spaces with normal structure: Assume a, b ∈ E, then ∀x ∈ E, by parallelogram identity, 2 2 x − a + b + a − b = 1 ( x − a 2 + x − b 2 ) diam (E) . 2 2 2 Taking x0 = 12 (a + b), (2.30), is satisﬁed. Now, for any closed bounded convex subset E, one deﬁnes a number rE = inf sup x − y . x∈E y∈E

By deﬁnition, rE diam (E). If further, X is a Banach space with a normal structure, then rE < diam (E).

2.4 Nonexpansive Maps

105

Lemma 2.4.1 If X is a reﬂexive Banach space and if E is a bounded closed convex set, then ∃x0 ∈ E such that sup x0 − y = rE .

y∈E

Proof. The function f (x) = sup x − y is convex, l.s.c., and coercive. We y∈E

now apply Corollary 2.3.7; there exists x0 ∈ E such that f (x0 ) = minx∈E f (x). Setting G(y) = {x ∈ E| y − x rE } ∀y ∈ E, we deﬁne ct (E) = ∩ G(y) . y∈E

Provided by Lemma 2.4.1, x0 ∈ ct (E). Therefore ct (E) is a nonempty closed convex subset of E. It is called the center of E. By deﬁnition, we have sup

x − y ≤ rE .

(2.31)

x,y∈ct(E)

Theorem 2.4.2 Suppose that E is a bounded closed convex set of a reﬂexive Banach space with a normal structure, and that T : E → E is nonexpansive, then the ﬁxed-point set of T Fix (T ) is closed and nonempty. Proof. (1) Let X be the set of all nonempty closed convex subsets of E. Deﬁne the partial order on X by inclusion: C1 C2 if C2 ⊂ C1

∀C1 , C2 ∈ X .

Then (X, ) is an ordered set, in which every chain X1 has a supremum: ∩ C. The intersection is not empty, because E is weakly compact and C is C∈X1

weakly closed. (2) Deﬁne f (C) = ct (conv(T (C))) , then f : X → X is order preserving and E ≤ f (E). Applying the Amann theorem, f has a ﬁxed point C0 ∈ X, i.e., C0 ⊂ E is a nonempty closed convex T -invariant set satisfying: C0 = ct(conv(T (C0 ))) . (3) It remains to verify that C0 consists of a single point. If not, from the fact that rC < diam (C) for all bounded closed convex sets C containing more than one point, we obtain: diam(C0 ) = diam(ct(conv(T (C0 ))))

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2 Fixed-Point Theorems

=

sup

x−y

x,y∈ct(conv(T (C0 )))

≤ rconv(T (C0 )) < diam (conv(T (C0 ))) = diam (T (C0 )) diam (C0 ) . This is a contradiction. Thus, C0 is a single-point set in E, say C0 = {x0 }. It follows that x0 = f ({x0 }) = ct(conv (T x0 )) = T x0 . Obviously Fix (T ) is closed.

Remark 2.4.3 A Banach space X is called uniformly convex, if ∀ε > 0 ∃δ > 0 such that ∀x, y ∈ X x , y 1, x − y ε implies that x+y 2 1 − δ. Obviously, a Hilbert space is a uniformly convex Banach space. Moreover, Lp (Ω, B, µ), 1 < p < +∞ are also, but C(Ω) is not. It is known [DS] that a uniformly convex Banach space is a reﬂexive Banach space with normal structure. One has the following: Theorem 2.4.4 (Browder–G¨ ohde) Suppose that X is a uniformly convex Banach space, and that E is a nonempty bounded closed convex set of X. If T : E → E is nonexpansive, then T has a ﬁxed point. A natural question: can we ﬁnd out a ﬁxed point of a nonexpansive map T by iteration as we did for contraction mappings? Starting from any point x0 ∈ E, set xi = T i x0 , ∀i, from xi+1 − xi xi − xi−1 . . . x1 − x0 , we know that { xi+1 − xi }∞ i=0 is a nonincreasing sequence. It is obviously true that even if {xi } has a convergent subsequence xni → x∗ , we still cannot conclude that x∗ is a ﬁxed point of T , e.g., T x = −x, and x0 = θ. To this end, Mann [Man] introduced an iteration method for the following perturbed mappings: ∀α ∈ (0, 1), set Tα x = αx + (1 − α)T x . One easily veriﬁes that if E is convex and T : E → E is nonexpansive then: (1) (2) (3) (4)

Tα : E → E. Tα is nonexpansive. Tα shares the same ﬁxed point set with T . Let X be a uniformly convex Banach space, and E be a bounded closed convex subset of X. Let p be the ﬁxed point of T . If further, I − T is proper (a map f is called proper if the inverse image of any compact set C, f −1 (C) is compact), then ∀α ∈ (0, 1), Tαi x−p → 0 as i → ∞ ∀x ∈ E.

2.4 Nonexpansive Maps

107

The proofs of (1)–(3) are trivial. In order to give a proof of (4). Let zi = Tαi x . Since the sequence zi+1 − zi is nonincreasing, either (1) ∃ > 0 such that zi+1 − zi ≥ , or (2) zi+1 − zi → 0. But case (1) is impossible. In fact, from the deﬁnition of the uniform convexity ∃δ0 = δ(α, ) > 0, such that zi+1 − p = α(zi − p) + (1 − α)(T zi − T p) (1 − δ0 ) max { zi − p , T zi − T p } = β zi − p ... βi x − p , where β = 1 − δ0 . This proves that zi → p, a contradiction. In case (2), by the assumption that I − T is proper, there exists a convergent subsequence zni → p. It follows that p = T p, i.e., p ∈ Fix (T ) = Fix (Tα ). Thus zn − p ≤ zni − p → 0 as n ≥ ni → ∞. We have thus proved: Theorem 2.4.5 Assume that X is a uniformly convex Banach space, and that E is a nonempty bounded closed convex set of X. If T : E → E is nonexpansive and I − T is proper, ∀α ∈ (0, 1), let Tα = αI + (1 − α)T , then ∀x ∈ E, the sequence Tαi x converges to a ﬁxed point of T. Applications (1) The iteration method is an eﬀective computing method in the convex feasibility problem: Given a family of closed convex sets {Ei | i = 1, . . . , n} in a Hilbert space n

H, ﬁnd a point x ∈ ∩ Ei , if the intersection is nonempty. i=1

In fact, given a closed convex set E in H, ∀x ∈ H, there is a projection onto E: PE x ∈ E, which is deﬁned by the minimizer of the problem: c = min x − y . y∈E

The existence of PE x follows from the parallelogram identity: 2 yn + ym , x − yn − ym 2 = 2( x − yn 2 + x − ym 2 ) − 4 2 where {yn } is a minimizing sequence: x − yn → c. Again by the parallelogram identity, the minimizer is unique. We deﬁne the minimizer y ∗ to be the projection PE x. The projection can be characterized by the following variational inequality: (x − PE x, PE y − PE x) 0 ∀x, y ∈ H .

(2.32)

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2 Fixed-Point Theorems

x

E PE x

Fig. 2.5.

In fact, ∀t ∈ (0, 1), tPE y + (1 − t)PE x ∈ E. One has x − tPE y − (1 − t)PE x 2 x − PE x 2 . i.e., −2t(x − PE x, PE y − PE x) + t2 PE y − PE x 2 0 . Letting t → 0, we obtain the desired result (2.32). Statement 2.4.6 PE is a nonexpansive map on H. Proof. By deﬁnition, ∀x, y ∈ H (x − PE x, PE y − PE x) 0 , (y − PE y, PE x − PE y) 0 . Summing the two inequalities: (x − y + (PE y − PE x), PE y − PE x) 0 , it follows that PE y − PE x 2 (y − x, PE y − PE x) x − y PE y − PE x . Therefore PE is nonexpansive.

One may use Mann iteration to compute a feasible solution. Various algorithms have been introduced to improve the convergent rate and the computation. (2) Periodic solution for nonlinear evolution equations: Let X be a real Hilbert space. Given ϕ ∈ X, and f : R1+ × X → X. We consider the following nonlinear evolution equation: Find x ∈ C 1 (R1+ , X) satisfying x(t) ˙ = f (t, x(t)) t ∈ R1+ , x(0) = ϕ .

(2.33) (2.34)

2.5 Monotone Mappings

109

Statement 2.4.7 Assume ω > 0 and that (1) (2) (3) (4)

f is ω-periodic in t, i.e., f (t, x) = f (t + ω, x), (f (t, x) − f (t, y), x − y) 0 ∀t ∈ R1+ , ∀x, y ∈ X, ∃R > 0 such that (f (t, x), x) < 0, ∀t ∈ [0, ω], ∀x ∈ ∂BR (θ), ∀ϕ ∈ B R (θ), the initial-value problem (2.33) (2.34) has a solution.

Then (2.33) has a unique ω-periodic solution, i.e., x(t + ω) = x(t). Proof. From (4), we consider the Poincar´e map on X: T : x(0) → x(ω) , and want to show that T has a unique ﬁxed point. First we show that T : B R (θ) → B R (θ) is a well-deﬁned nonexpansive map: 1. We show that the solution for the initial-value problem is unique, i.e., if x(t), y(t) are solutions with the same initial data ϕ, then x(t) = y(t). Indeed, from d x(t) − y(t) 2 = 2(x (t) − y (t), x(t) − y(t)) dt = 2(f (t, x(t)) − f (t, y(t)), x(t) − y(t)) 0, it follows that x(t) − y(t) x(0) − y(0) .

(2.35)

Thus x(t) = y(t), and then T is well deﬁned. 2. Moreover, (2.35) implies that T is nonexpansive. 3. We turn to showing that T maps B R (θ) into itself. Because of assumption (3) d x 2 = 2(x (t), x(t)) = 2(f (t, x(t)), x(t)) < 0 , dt whenever x(t) ∈ ∂BR (θ). If the conclusion is not true, then ∃t > 0 such that x(t) ∈ / BR (θ) and then ∃t0 ∈ (0, t), such that x(t0 ) ∈ ∂BR (θ) and x(t) ∈ / BR (θ) as t0 < t < t0 + δ for some δ > 0. But this contradicts the diﬀerential inequality. Now, we apply Theorem 2.4.2; there is a ﬁxed point of T in B R (θ). Again applying step 1, the ω-periodic solution is also unique.

2.5 Monotone Mappings Let us consider the subdiﬀerential of a convex function f on a vector space X: ∀x, y ∈ dom (f ), ∀x∗ ∈ ∂f (x), ∀y ∗ ∈ ∂f (y ∗ ), we have

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2 Fixed-Point Theorems

x∗ , y − x + f (x) f (y) , y ∗ , x − y + f (y) f (x) . By addition,

y ∗ − x∗ , y − x 0

(2.36)

This is what we call the monotonicity of ∂f . Deﬁnition 2.5.1 Suppose that X is a real Banach space and that E ⊂ X is ∗ a nonempty subset. A set-valued mapping A : E → 2X is called monotone if ∗ ∗ ∀x, y ∈ E, ∀x ∈ A(x), ∀y ∈ A(y), one has y ∗ − x∗ , y − x 0 . The set {x ∈ E| A(x) = Ø} is called the domain of A, denoted by D(A), and the set ΓA = {(x, x∗ ) ∈ E × X ∗ | x ∈ D(A), x∗ ∈ A(x)} is called the graph of A. A single-valued monotone mapping is called a monotone operator. Example 1. Suppose that X is a real Hilbert space, and that A is a linear positive operator, i.e., ∀x ∈ D(A), (Ax, x) 0. Then A is a monotone operator. Example 2. Suppose that X is a real Hilbert space, and that T is a nonexpansive map, then A = id − T is a monotone operator: (Ay − Ax, y − x) = y − x 2 −(T y − T x, y − x) 0 . Example 3. Let X be a real Banach space, and let f : X → R1 ∪ {+∞} be convex, then ∂f is a monotone mapping. Example 4. (p-Laplacian) Let Ω be a bounded domain of Rn . For 1 < p < ∞, the operator ⎛⎡ ⎞ ⎤ p−2 2 n n 2 ∂ ⎜ ∂u " # ∂u ⎟ ⎦ Au = −div |∇u|p−2 ∇u = − ⎝⎣ ⎠ ∂xi ∂xj ∂xi i=1 j=1 ◦

deﬁnes a map from Wp1 (Ω) to Wp−1 (Ω),

Au, v =

1 p

+

1 p

= 1, as follows:

⎡ ⎤ p−2 2 2 n ∂u ∂u ∂v ⎣ ⎦ dx ∂xj ∂xi ∂xj Ω j=1

n

i=1

◦

∀v ∈ Wp1 (Ω) .

In fact, by the H¨ older inequality:

|∇u|

p−1

Ω

|∇v|

|∇u|

p

Ω

1

p

Ω

|∇v|

p

p1 ,

2.5 Monotone Mappings

111

◦

Au ∈ (Wp1 (Ω))∗ = Wp−1 (Ω). We verify that A is monotone: According to the elementary inequalities: |b − a|p if p 2 , p−2 p−2 (|b| b − |a| a)(b − a) cp p−2 2 (1 + |b| + |a|) |b − a| if 1 < p < 2 , older inequality, we obtain where cp > 0 is a constant, and a, b ∈ Rn , and the H¨ Au − Av, u − v

|∇u − ∇v|p Ω cp 2 2 ( Ω (1 + |∇u| + |∇v|)p )1− p ( Ω |∇u − ∇v|p ) p

if p 2 , if 1 < p < 2 .

The requirement of the continuity for monotone operators is very weak. Deﬁnition 2.5.2 Let X be a real Banach space and let E ⊂ X be a nonempty subset. A map A : E → X ∗ is called hemi-continuous at x0 ∈ E, if ∀y ∈ X, ∗ ∀tn ↓ 0 with x0 + tn y ∈ E, imply that A(x0 + tn y) A(x0 ). It is called demi∗ continuous at x0 ∈ E, if ∀{xn } ⊂ E, xn → x0 implies that A(xn ) A(x0 ), ∗ where ∗ is the w -convergence. Obviously, “continuous” =⇒ “demi-continuous” =⇒ “hemi-continuous” . ◦

The monotone operator in Example 4 is hemi-continuous. In fact, ∀u, v, w ∈

Wp1 (Ω), we consider the function t → A((1 − t)x + ty), w , and verify the continuity. Now ⎡ ⎤ p−2 2 2 n ∂v ∂u ⎣ ⎦ +t (1 − t) A((1 − t)u + tv), w = ∂xj ∂xj Ω j=1

×

n i=1

∂v ∂u +t (1 − t) ∂xi ∂xi

∂w . ∂xi

Since the integrand on the RHS is dominated by an integrable function (|∇u|+ |∇v|)p−1 |∇w|, the Lebesgue dominance theorem is applied. An important property for monotone operators reads as: Lemma 2.5.3 Let E be a convex subset of a real Banach space X. If A : E → X ∗ is hemi-continuous and monotone, then for any sequence {xj } ⊂ E with xj x ∈ E and lim A(xj ), xj − x 0, we have lim A(xj ), xj − y A(x), x − y

∀y ∈ E .

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2 Fixed-Point Theorems

Proof. First we claim that lim A(xj ), xj − x = 0 .

(2.37)

Indeed, since A is monotone, it follows that 0 = lim A(x), xj − x lim A(xj ), xj − x lim A(xj ), xj − x 0 . Again by the monotonicity and (2.37), ∀z ∈ E lim A(xj ), x − z = lim A(xj ), xj − z lim A(z), xj − z = A(z), x − z . (2.38) Now, ∀y ∈ E, ∀tn ↓ 0, substituting z = zn := (1 − tn )x + tn y into (2.38), we obtain (2.39) lim A(xj ), x − y A(zn ), x − y . By using the hemi-continuity, the RHS of (2.39) tends to A(x), x − y . Combining (2.37) (2.39) and the last fact, it follows that lim A(xj ), xj − y A(x), x − y .

(2.40)

Remark 2.5.4 The deduction argument from (2.37) to (2.40) is called Minty’s trick, in which a combination of the monotonicity and the hemicontinuity is applied. This lemma is substantial in the study of monotone operators. The following notion on pseudo-monotonicity is abstracted from it. Deﬁnition 2.5.5 Let X be a reﬂexive Banach space and let E ⊂ X be a nonempty closed convex subset. An operator A : E → X ∗ is called pseudo monotone, if (1) ∀ ﬁnite-dimensional linear subspace L ⊂ X, A|L∩E : L ∩ E → X ∗ is demi-continuous. (2) ∀ sequence {xj } ⊂ E with xj x ∈ E, the condition lim A(xj ), xj −x 0 implies that lim A(xj ), xj − y A(x), x − y , ∀y ∈ E. Thus, a hemi-continuous monotone operator is pseudo monotone. Moreover, a completely continuous mapping A : X → X ∗ (i.e., for any xj x in X, we have Axj → Ax in X ∗ ) is pseudo monotone. In contrast with the ﬁxed-point problem for compact maps and nonexpansive mappings, we shall study the surjection of pseudo monotone operators, because the latter maps a subset of a Banach space into its dual space. The following generalized Ky Fan’s inequality is the basis of this section. Theorem 2.5.6 (Brezis, Nirenberg, Stampacchia) Assume that E is a nonempty convex set of an LCS X. Assume Φ : E × E → R1 satisfying

2.5 Monotone Mappings

113

(1) ∀ ﬁnite-dimensional linear subspace L of X, ∀y ∈ E ∩ L, x → Φ(x, y)|L∩E is l.s.c. (2) ∀x ∈ E, y → Φ(x, y) is quasi concave. (3) ∃ a compact set K ⊂ X, ∃y0 ∈ E such that {x ∈ E| Φ(x, y0 ) 0} ⊂ K. (4) sup Φ(x, x) = 0. x∈E

(5) ∀x, y ∈ E and for any net xα → x, ∀t ∈ [0, 1], Φ(xα , (1 − t)x + ty) 0 implies that Φ(x, y) 0. Then ∃x0 ∈ E such that sup Φ(x0 , y) 0. y∈E

Proof. Again we transform the problem into its geometric version by setting Γ = {(x, y) ∈ E × E| Φ(x, y) > 0}. Then (1)–(5) are transformed into: (1 ) ∀L, letting EL = E ∩ L and ΓL = Γ ∩ (L × L), ∀y ∈ EL ΓL 2 (y) is open in E. (2 ) ∀x ∈ E, Γ1 (x) is convex. (3 ) ∃y0 ∈ E such that E\Γ2 (y0 ) ⊂ K. (4 ) ∩ Γ = Ø, where is the diagonal of E × E. (5 ) Let xy be the segment connecting x and y, then Γ1 (xα ) ∩ xy = Ø implies y ∈ Γ1 (x) ∀y ∈ E for any net xα → x. Now we set X = {L| ﬁnite-dimensional linear subspace, with y0 ∈ L and EL = Ø}. ∀L ∈ X, applying Ky Fan’s inequality, ∃xL ∈ EL such that ΓL 1 (xL ) = Ø. From (3 ), xL ∈ K as L ∈ X. ∀L ∈ X, setting L L N = {x ∈ K| Γ1 (x) = Ø}, from the above discussion, it is nonempty. L We claim that the family {N | L ∈ X} has the ﬁnite intersection property: Indeed, ∀ L1 , . . . , Ln ∈ X, setting L = span {L1 , . . . , Ln } we have n

n

i=1

i=1

Li L Li L ∈ X. Then ΓL 1 (xL ) = ∪ Γ1 (xL ) and N = ∩ N . Since K is compact, L

Z := ∩ N = Ø. L∈X

We shall prove that ∀x ∈ Z, Γ1 (x) = Ø. To this end, ∀y ∈ E, we choose L L L ∈ X such that the segment xy ⊂ L. Since x ∈ N , ∃ a net xL α ∈ N , such L L that Γ1 (xα ) ∩ xy = Ø, and xα → x. From assumption (5 ), y ∈ Γ1 (x). Since y ∈ E is arbitrary, we obtain Γ1 (x) = Ø, Therefore there exists x0 ∈ Z such that sup Φ(x0 , y) 0. y∈E

In many of the applications, the LCS is taken to be a reﬂexive Banach space with its weak topology. Since then for weakly closed set it is weakly compact if and only if it is sequentially weakly compact (cf. Theorem 2.2.5), the net convergence in condition (5) of the above theorem can be replaced by the sequential convergence. The following theorem is the foundation of the theory of variational inequalities, which arise in free boundary problems for partial diﬀerential equations.

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2 Fixed-Point Theorems

Theorem 2.5.7 (Hartman–Stampacchia) Suppose that X is a real reﬂexive Banach space, and that E ⊂ X is a nonempty closed convex set. Assume (1) A : E → X ∗ is pseudo monotone, (2) ϕ : E → R1 is convex and l.s.c., (3) ∃y0 ∈ E ∃R0 > 0 such that Ax, x − y0 + ϕ(x) − ϕ(y0 ) > 0

as x R0 and x ∈ E .

Then ∃x0 ∈ X such that Ax0 , x0 − y + ϕ(x0 ) − ϕ(y) 0,

∀y ∈ E .

Proof. (1) Assign the weak topology on X, and deﬁne Φ(x, y) = Ax, x − y + ϕ(x) − ϕ(y) . It is easy to see that conditions (1)–(4) of Theorem 2.5.6 are satisﬁed. Let us verify (5). ∀x, y ∈ E ∀t ∈ [0, 1], assume xj x and Φ(xj , (1 − t)x + ty) 0. Setting t = 0 and 1 separately, we have lim A(xj ), xj − x lim [ϕ(x) − ϕ(xj )] 0

(2.41)

and lim A(xj ), xj − y lim [ϕ(y) − ϕ(xj )] ϕ(y) − ϕ(x) .

(2.42)

From the pseudo monotonicity, it follows that lim A(xj ), xj − y A(x), x − y .

(2.43)

Combining (2.42) and (2.43), we obtain Φ(x, y) 0. (2) Now we apply Theorem 2.5.6 and conclude the existence of x0 ∈ E satisfying Φ(x0 , y) 0. This is the desired conclusion. Theorem 2.5.8 (F. Browder) Suppose that X is a real reﬂexive Banach space, and that A : X → X ∗ is pseudo monotone and coercive, i.e., x −1 A(x), x → +∞ as x → +∞. Then A is surjective. Proof. We shall prove that ∀z ∈ X ∗ , ∃x0 ∈ X satisfying A(x0 ) = z. Deﬁne T : x → A(x) − z. Then T is pseudo monotone and satisﬁes T x, x > 0 as x > R0 , for some R0 > 0, provided by the coerciveness of A. We apply the Hartman– Stampacchia theorem to conclude the existence of x0 ∈ X satisfying T x0 , x0 − y 0 ∀y ∈ X. Since y is arbitrary in the linear space X, it follows that Ax0 = z.

2.5 Monotone Mappings

115

As a consequence, we return to Example 4, the p− Laplacian −∆p : W01,p (Ω) → W −1,p (Ω), 1 < p < ∞, is a homeomorphism. The surjection follows from Browder’s theorem, and the injection as well as the continuity of the inverse mapping follow from the inequality in the veriﬁcation of the monotonicity. Corollary 2.5.9 (Hartman–Stampacchia) Suppose that A is a hemicontinuous monotone operator deﬁned on the unit ball B centered at θ of a Hilbert space H. If Ax = λx, ∀x ∈ ∂B, ∀λ < 0 then ∃x0 ∈ B such that Ax0 = θ. Proof. Since A is pseudo monotone, we apply Theorem 2.5.7 to conclude the existence of x0 ∈ B satisfying (Ax0 , x0 − y) 0 ∀y ∈ B . ◦

If x0 ∈ B, then we have Ax0 = θ. Otherwise x0 ∈ ∂BR ∀y ∈ BR , we decompose y = (1 − t)x0 + y ⊥ , and Ax0 = λx0 + y0⊥ , where t > 0, y ⊥ ⊥x0 , y0⊥ ⊥x0 , and λ ∈ R1 . Then (2.44) −λt x0 2 +(y0⊥ , y ⊥ ) 0 . Since y ⊥ is arbitrary, ﬁrst letting t → +0, we obtain y0⊥ = θ, i.e., Ax0 = λx0 . By the assumption, λ 0. Again by (2.44) λt x0 2 0 ,

which implies that λ = 0. Therefore Ax0 = θ.

y

θ

x0 y⊥ (1 − t)x0

Fig. 2.6.

Corollary 2.5.10 (Minty) Suppose that H is a real Hilbert space, and that A is a continuous strongly monotone operator, i.e., ∃c > 0 such that (Ax − Ay, x − y) c x − y 2 Then A is a homeomorphism.

∀x, y ∈ H .

(2.45)

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2 Fixed-Point Theorems

Proof. Obviously, A is pseudo monotone and coercive. As a consequence of the Browder theorem, A is surjective. The injectivity of A as well as the continuity of A−1 follows from the inequality (2.45). Applications 1. Elliptic variatiational inequality Let A be the p-Laplacian deﬁned in Example 4 and let Ω ⊂ Rn be a ◦

bounded domain. We consider the closed convex subset of X = Wp1 (Ω) with 1 < p < ∞: ◦

E = {u ∈ Wp1 (Ω)| |∇u(x)| 1 a.e.} . Given f ∈ Wp1 (Ω), where p =

p p−1 ,

ﬁnd u ∈ E such that

Au, v − u f, v − u ∀v ∈ E . The existence of a solution follows from the Hartman–Stampacchia theorem directly. This is a free boundary problem, in fact the solution u, if it is regular, satisﬁes the equation: ⎧ p−2 ⎪ in Ω1 , ⎨−div (|∇u| ∇u) = f |∇u| = 1 in Ω2 , ⎪ ⎩ u and ∇u coincide on the interface of Ω1 and Ω2 , where Ω1 = {x ∈ Ω| |∇u(x)| < 1} , and Ω2 = Ω\Ω1 . The same method can be applied to the obstacle problem and the Stefan problem studied in Sect. 2.3. But in these problems, the working spaces should be taken to be the following Hilbert spaces: H01 (Ω) and H 1 ([0, T ] × Ω) respectively. 2. Weak solutions for quasi-linear elliptic equations Let Ω be a bounded domain of Rn . We study the weak solution of the quasi-linear elliptic equation: n ∂ i=1 ∂xi Ai (x, u(x), ∇u(x)) + B(x, u(x), ∇u(x)) = f (x) in Ω , u|∂Ω = 0 ,

where we make use of the following structural condition on A = {Ai }n1 and B:

(1) B ∈ C(Ω × R1 × Rn , R1 ), A ∈ C(Ω × R1 × Rn , Rn ),

2.5 Monotone Mappings

(2) ∃p ∈ (1, ∞), ∃g ∈ Lp (Ω), p =

p p−1 ,

117

∃C > 0 such that

A(x, u, ξ) g(x) + C(|u| + |ξ|)p−1 , |B(x, u, ξ)| g(x) + C(|u| + |ξ|)p−1 , (3)

[A(x, u, ξ)− A(x, u, ξ )]·(ξ−ξ ) > 0, ∀(x, u) ∈ Ω×R1 , ∀ξ, ξ ∈ Rn , ξ = ξ , and (4) 1 A(x, u, ξ) · ξ → +∞ as |ξ| → ∞, (x, u, ξ) ∈ Ω × R1 × Rn . p−1 1 + |ξ| + |ξ|

Remark 2.5.11 Condition (3) (i.e., the monotone condition) implies the el

lipticity of the diﬀerential operator: u(x) → div A(x, u(x), ∇u(x)). ◦

Let X = Wp1 (Ω); we deﬁne a form on X × X:

a(u, v) = [A(x, u(x), ∇u(x)) · ∇v(x) + B(x, u(x), ∇u(x))v(x)]dx Ω

× ∀(u, v) ∈ X × X . From (1) and (2), |a(u, v)| C( u p−1 1,p + g p ) v 1,p . We deﬁne T : X → X ∗ by T u, v = a(u, v), where , is the duality between X ∗ and X. In order to verify that T is pseudo monotone, we deﬁne A : X × X → X ∗ as follow:

A(x, u, ∇w) · ∇v + B(x, u, ∇u)v ∀u, v, w ∈ X . A(u, w), v = Ω

Ω

We conclude: (a) A(u, u) − A(u, w), u − w 0, ∀u, w ∈ X. (b) ∀u ∈ X, w → A(u, w) is bounded and demi-continuous. (c) If uj u in X and if A(uj , uj ) − A(uj , u), uj − u → 0, then A(uj , w) A(u, w) in X ∗ ∀w ∈ X. (a) and (b) are obviously true; we prove (c). Proof. From uj u in X, we have uj → u(in Lq (Ω)) and uj → u a.e. after a subsequence, and then ∀w ∈ X, → − − → A (x, uj (x), ∇w(x)) → A (x, u(x), ∇w(x))

a.e.

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2 Fixed-Point Theorems

Combining with the growth condition (2), → − → − A (·, uj , ∇w) → A (·, u, ∇w) in Lp (Ω, Rn ) . Thus

− → A (x, uj , ∇w)∇uj →

Ω

and

Ω

− → A (x, u, ∇w)∇u ,

(2.46)

Ω

− → A (x, uj , ∇w)∇v →

A(x, u, ∇w)∇v .

(2.47)

Ω

It remains to verify that gj := B(·, uj , ∇uj ) g := B(·, u, ∇u) (Lp (Ω)). The diﬃculty is that we only have ∇uj ∇u in Lp (Ω, Rn ), and that B is nonlinear with respect to ∇u. How can we pass the limit? If one can show that ∇uj → ∇u a.e., then gj → g a.e., and then we claim that gj g in Lp (Ω). In fact, let Ek = {x ∈ Ω||gj (x)−g(x)| ≤ 1, j ≥ k}; we have E1 ⊂ E2 ⊂ . . . , and mes(Ek ) → mes(Ω). Since ∀χ ∈ Lp (Ω) with support in Ek , we have (g − g)χ → 0, from the Lebesgue dominance theorem, and those χ span a Ω j dense set in Lp (Ω); in combination with the fact that gj p is bounded, the conclusion follows. We turn to verifying that ∇uj → ∇u a.e., and deﬁne the function: → − → − Fj (x) = [ A (x, uj (x), ∇uj (x)) − A (x, uj (x), ∇u(x))] · ∇(uj (x) − u(x)) . By (3), Fj (x) 0. As a consequence of our assumption, Ω Fj (x)dx → 0. From Fatou’s lemma, it follows that Fj (x) → 0 a.e. Therefore there exists a / Z. null set Z such that uj (x) → u(x), and Fj (x) → 0 ∀x ∈ Firstly, we claim that ∃ a measurable function M (x) < +∞, such that |∇uj (x)| M (x) ∀x ∈ Z .

(2.48)

From (2), → − Fj (x) ≥ A (x, uj (x), ∇uj (x))∇uj (x) − C1 (x)(1 + |∇uj (x)| + |∇uj (x)|p−1 ) where C1 (x) depends on u(x) and ∇u(x). If (2.48) were false, then there would be a positive measure set E, on which |∇uj (x)| → +∞. It follows that Fj (x) → +∞, from (4). But, this is a contradiction. Secondly, we claim that ∀x ∈ Z, ∇uj (x) → ∇u(x). In fact, let ξ(x) be a limit point of ∇uj (x). From Fj (x) → 0 it follows that: → − → − [ A (x, u(x), ξ(x)) − A (x, u(x), ∇u(x))] · (ξ(x) − ∇u(x))dx = 0 . Applying the monotone relation (3), we have ξ(x) = ∇u(x). Since all limit points of ∇uj (x) equal ∇u(x), we obtain ∇uj (x) → ∇u(x) a.e. Combining (2.47) with this fact, we obtain A(uj , w) A(u, w) in X ∗ ∀w ∈ X.

2.5 Monotone Mappings

119

(d) If uj u in X and A(uj , v) ψ in X ∗ , then A(uj , v), uj → ψ, u . → − − → In fact, we already have uj → u (Lp (Ω, R1 )) and A (·, uj , ∇v) → A (·, u, ∇v) (Lp (Ω, Rn )). Thus

→ − A (x, uj , ∇v)∇(uj − u) + B(x, uj , ∇uj )(uj − u) A(uj , v), uj = Ω

+ A(uj , v), u → ψ, u . After these preparations, we turn to verifying that the operator T , deﬁned by the form a, is pseudo monotone. Since u → T (u) is obviously demicontinuous, it is suﬃcient to prove that uj u and lim T (uj ), uj − u 0 implies that lim T (uj ), uj − v T (u), u − v ∀v ∈ X. First we show that T (uj ), uj − u → 0, and deﬁne the sequence: bj = A(uj , uj ) − A(uj , u), uj − u . By (a), bj 0. Since A(uj , u) is bounded in X ∗ , there exists a subsequence, for which we don’t change the subscript, such that A(uj , u) ψ in X ∗ . By (d), A(uj , u), uj → ψ, u . Thus, A(uj , u), uj − u → 0. But by assumption, lim A(uj , uj ), uj − u = lim T (uj ), uj − u 0 , therefore bj → 0. It follows T (uj ), uj − u → 0. It remains to show: lim T (uj ), u − v ≥ T (u), u − v , from (a), T (uj ) − A(uj , w), uj − w 0, ∀w ∈ X . Letting w = (1 − θ)u + θv, θ ∈ (0, 1), ∀v ∈ X, we have θ T (uj ), u − v θ A(uj , w), u − v + A(uj , w), uj − u − T (uj ), uj − u . By taking limit, lim T (uj ), u − v A(u, w), u − v

or lim T (uj ), u − v A(u, (1 − θ)u + θv), u − v ∀θ ∈ (0, 1) letting θ → 0, by (b), the RHS = T (u), u − v , this is the desired inequality. Statement 2.5.12 Under the above structural conditions, we assume further: (5) ∃C1 > 0 ∃h ∈ L1 (Ω) such that − → A (x, u, ξ) · ξ > C1 |ξ|p − h(x)

∀(x, u, ξ) ∈ Ω × R1 × Rn . ◦

Then ∀f ∈ Lp (Ω), there exists a weak solution u ∈ Wp1 (Ω) of the quasilinear elliptic equation:

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2 Fixed-Point Theorems

− → A (x, u(x), ∇u(x)) · ∇v(x) + B(x, u(x), ∇u(x))v(x) =

Ω

f (x)v(x) Ω ◦

× ∀v ∈ Wp1 (Ω) . Proof. It is reduced to ﬁnd the ﬁxed point of the pseudo monotone operator T . Since (5) implies the coerciveness: 1 1 T u, u = a(u, u) → +∞ . u 1,p u 1,p The existence of u follows from Browder’s theorem.

It is worth noting that in the above statement, no a priori estimate has been used. In the veriﬁcation of the pseudo monotonicity, a crucial point is the ellipticity condition (or the monotone condition (3)). The statement can be extended to high-order quasi-elliptic equations. This is an approach where no compactness is concerned.

2.6 Maximal Monotone Mapping To a set-valued mapping A, ∀x ∈ D(A), one may take any nonempty subset D(A1 ) of D(A) and a nonempty subset A1 (x) of A(x), and deﬁne a new set-valued mapping A1 : x → A1 (x) ∀x ∈ D(A1 ). If A is monotone, so is A1 . In order to avoid the indetermination of domain and the image of setvalued monotone mappings in the study of their surjectivity, one introduces the notion of maximal monotone mappings. First, we deﬁne the graph ΓA := {(x, y) ∈ X × Y | x ∈ D(A), y ∈ A(x)} of a set-valued mapping A : X → 2Y . Deﬁnition 2.6.1 Assume that A1 and A2 are two monotone set-valued map∗ pings from X to 2X . If ΓA1 ⊂ ΓA2 , then we say that A2 is a monotone extension of A1 . A monotone mapping, which does not have any proper monotone extension, is called a maximal monotone mapping. ∗ In other words, A : X → 2X is maximal monotone iﬀ for (x, x∗ ) ∈ X ×X ∗ , y ∗ − x∗ , y − x ≥ 0 ∀ (y, y ∗ ) ∈ ΓA ⇒ (x, x∗ ) ∈ ΓA . Example 1. Assume that φ : R1 → R1 is a nondecreasing function. The map A : u → [φ(u−0), φ(u+0)] is maximal monotone, but the map B : u → φ(u−0) is not, if φ(u − 0) = φ(u + 0). Example 2. A hemi-continuous monotone operator A is maximal monotone. In fact, for (x, x∗ ) ∈ X × X ∗ , if Ay − x∗ , y − x ≥ 0 ∀ y ∈ X ,

2.6 Maximal Monotone Mapping

121

then ∀z ∈ X, ∀t ∈ [0, 1], we have A((1 − t)x + tz) − x∗ , z − x ≥ 0 . Letting t → 0, it follows that Ax−x∗ , z −x ≥ 0. Since z ∈ X is arbitrary, we obtain Ax = x∗ . Moreover, we shall prove later that the subdiﬀerential of a proper l.s.c. convex function is maximal monotone. Since maximal monotone mappings occur in convex analysis, free boundary boundary problems in mathematical physics, and nonlinear semigroups. we shall extend most of the results of monotone operators to their set-valued counterpart-maximal monotone mappings. The central problem is on the surjectivity. Now we are going to extend the notions of the demi-continuity and the hemi-continuity for single-valued mappings to set-valued mappings. Deﬁnition 2.6.2 Suppose that E is a nonempty subset of a real Banach ∗ space X. A mapping A : E → 2X is called weakly ∗ upper hemi-continuous ∗ (w u.h.c.,) if ∀x, y ∈ E ∀z ∈ X, the function t → A((1 − t)x + ty), z is a u.s.c. set-valued function at t = 0. It is called upper demi-continuous (u.d.c.) 1 if ∀z ∈ X, x → A(x), z is u.s.c. from E to 2R . Obviously, u.d.c. implies w∗ u.h.c., and for single-valued mappings, w∗ u.h.c. is hemi-continuous, and u.d.c. is demi-continuous. ∗ Note that for any set-valued mapping A : D(A) → 2X , the inverse mapping A−1 : X ∗ → 2X , x∗ → {x ∈ D(A)| x∗ ∈ A(x)} is well deﬁned (in case x∗ ∈ rang (A), one deﬁnes A−1 (x∗ ) = Ø). In particular, if X is reﬂexive then A is (maximal) monotone ⇐⇒ A−1 is (maximal) monotone. Lemma 2.6.3 Assume that A : X → 2X ∃ε > 0 such that A is bounded on Bε (x0 ).

∗

is monotone. Then ∀x0 ∈ X,

Proof. We may assume x0 = θ. We prove it by contradiction. Assume that A is not bounded in any neighborhood of θ, i.e., ∃xn → θ ∃x∗n ∈ A(xn ) such that x∗n → ∞. We claim that ∀ε > 0 ∃z ∈ Bε (θ) and a subsequence {n } such that (2.49) x∗n , xn − z → −∞ . Then from the monotonicity, ∀z ∗ ∈ A(z), we have x∗n − z ∗ , xn − z 0 . (2.49) implies that z ∗ , z = +∞. Obviously, this is impossible. Let us return to proving the existence of z satisfying (2.49). If it is not true, then ∃ε > 0 ∀z ∈ B ε (θ) ∃Cz ∈ R1 such that x∗n , xn − z Cz

∀n .

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2 Fixed-Point Theorems

Now, ∀k ∈ N, let Ek = {u ∈ B ε (θ)| x∗n , xn − u −k, ∀n} . ∞

Obviously, Ek are closed, and ∪ Ek = B ε (θ). From the Baire category ark=1

gument, ∃r > 0, ∃y0 ∈ B (θ) and ∃k0 ∈ N such that Br (y0 ) ⊂ Ek0 . Setting C = C−y0 , we have x∗n , xn + y0 C ∀n , and

x∗n , xn − u ≥ −k0 , ∀u ∈ B r (y0 ) .

By addition, x∗n , 2xn + y0 − u C − k0

∀u ∈ B r (y0 ) .

Setting v = 2xn + y0 − u, for suﬃciently large n, such that xn < 4r , we obtain x∗n , v C − k0 ∀v ∈ B r2 (θ) . This means x∗n proved.

2|C−k0 | , r

which contradicts x∗n → ∞; (2.49) is ∗

Proposition 2.6.4 Assume that A : X → 2X is a maximal monotone mapping with D(A) = X on a reﬂexive real Banach space X. Then: (1) ∀x ∈ X A(x) is a nonempty closed convex set. (2) A is u.d.c. (3) If there is a sequence {xj }, and a sequence x∗j ∈ A(xj ) satisfying xj x and lim x∗j , xj − x 0, then ∃x∗ ∈ A(x) such that lim x∗j , xj − y x∗ , x − y

∀y ∈ X .

Proof. (1) If x∗i ∈ A(x) i = 1, 2, then ∀y ∈ X, ∀y ∗ ∈ A(y) we have y ∗ − x∗i , y − x 0,

i = 1, 2 .

Therefore y ∗ − (λx∗1 + (1 − λ)x∗2 ), y − x 0 ∀λ ∈ [0, 1] . By the maximality, λx∗1 + (1 − λ)x∗2 ∈ A(x). Then A(x) is convex. Similarly, A(x) is closed. (2) In order to show that A is u.d.c., i.e., ∀z ∈ X, x → A(x), z is a u.s.c. 1 set-valued mapping: X → 2R . We prove it by contradiction. If ∃z0 ∈ X ∃ε0 > ∗ 0 ∃xn → x0 and ∃xn ∈ A(xn ) such that x∗n , z0 ∈ A(x0 ), z0 +(−ε0 , ε0 ). From Lemma 2.6.3, {x∗n } is bounded, and then it possesses a weakly convergent subsequence: x∗n x∗0 in X ∗ . Since A is maximal, x∗0 ∈ A(x0 ). This is a contradiction. (3) The proof is the same as that for single-valued monotone operators; cf. Lemma 2.5.3.

2.6 Maximal Monotone Mapping

123

Conversely, we have: Proposition 2.6.5 Assume that X is a reﬂexive real Banach space and that ∗ A : X → 2X is a w∗ .u.h.c. monotone mapping. If ∀x ∈ X, A(x) is a nonempty closed convex set, then A is maximal monotone. Proof. We shall verify that for (x, x∗ ) ∈ X × X ∗ , if y ∗ − x∗ , y − x 0 ∀(y, y ∗ ) ∈ ΓA ,

(2.50)

then (x, x∗ ) ∈ ΓA . For otherwise, according to the Ascoli separation theorem, ∃z ∈ X such that x∗ , z > sup { w∗ , z | w∗ ∈ A(x)} . Since A is w∗ .u.h.c. ∃δ > 0 such that x∗ , z > yt∗ , z

∀yt∗ ∈ A(yt ) ∀t ∈ (0, δ) ,

where yt = x + tz. But from the monotonicity, we have t yt∗ − x∗ , z = yt∗ − x∗ , yt − x 0 . This is a contradiction.

The following theorem is a characterization of maximal monotone mappings on Hilbert spaces. Theorem 2.6.6 (Minty) Suppose that H is a Hilbert space, and that A : H → 2H is a set-valued mapping. The following statements are equivalent: (1) A is maximal monotone. (2) A is monotone and I + A is surjective. (3) ∀λ > 0 (I + λA)−1 is nonexpansive. Before going on to prove Theorem 2.6.6, we now introduce the following notion. Deﬁnition 2.6.7 Suppose that E is a nonempty subset of a real Hilbert space H. A set-valued mapping T : E → 2H is called expansive if ∀x, y ∈ E, ∀x∗ ∈ T (x), ∀y ∗ ∈ T (y), x∗ − y ∗ x − y . Obviously, T : H → 2H is surjective and expansive iﬀ T −1 is nonexpansive. In this case, T −1 is single valued and injective. Lemma 2.6.8 A : E → 2H is monotone if and only if ∀λ > 0, Tλ = I + λA is expansive.

124

2 Fixed-Point Theorems

Proof. Since ∀x, y ∈ E, ∀x∗ ∈ A(x), ∀y ∗ ∈ A(y), we have y−x+λ(y ∗ −x∗ ) 2 = y−x 2 +2λ(y ∗ −x∗ , y−x)+λ2 y ∗ −x∗ 2 . (2.51) “=⇒” If A is monotone, then the RHS of (2.51) y − x 2 , i.e., Tλ is expansive. “⇐=” If Tλ is expansive, then the LHS of (2.51) y − x 2 . Letting λ → 0, A is monotone. Lemma 2.6.9 (Deburnner–Flor) Suppose that E is a nonempty closed convex subset of a real Hilbert space H. If A : E → 2H is monotone, then ∀y ∈ H, ∃x ∈ E such that (η + x, ξ − x) (y, ξ − x)

∀(ξ, η) ∈ ΓA .

Proof. ∀(ξ, η) ∈ ΓA , let us deﬁne C(ξ, η) = {x ∈ E| (η + x − y, ξ − x) 0} . $ Ø. Since C(ξ, η) Our conclusion is equivalent to {C(ξ, η)| (ξ, η) ∈ ΓA } = is a bounded closed convex set, it is weakly closed and E is weakly compact. It is suﬃcient to verify that {C(ξ, η)| (ξ, η) ∈ ΓA } has the ﬁnite intersection property. Indeed, ∀ξ1 , . . . , ξn ∈ E ∀ηi ∈ A(ξi ), i = 1, . . . , n, we consider the simplex λi = 1 , n = λ = (λ1 , . . . , λn )| λi 0 ∀i, and the function Φ(λ, µ) =

n

µi (x(λ) + ηi − y, x(λ) − ξi )

∀(λ, µ) ∈ n × n ,

i=1

where x(λ) =

n i=1

λi ξi . It is easy to see that ∀µ ∈ n , λ → Φ(λ, µ) is continuous, ∀λ ∈ n , µ → Φ(λ, µ) is linear ,

and Φ(λ, λ) = =

=

n

[λi (x(λ) − y, x(λ) − ξi ) + λi (ηi , x(λ) − ξi )]

i=1 n

λi λj (ηi , ξj − ξi )

i,j=1 n

1 λi λj (ηi − ηj , ξj − ξi ) ≤ 0 , 2 i,j=1

2.6 Maximal Monotone Mapping

125

because A is monotone. According to Ky Fan’s inequality, ∃λ0 ∈ n such that Φ(λ0 , µ) 0 ∀µ ∈ n , i.e., (x(λ0 ) + ηi − y, x(λ0 ) − ξi ) 0 ∀i . Therefore x(λ0 ) ∈

n $

C(ξi , ηi ). This proves the ﬁnite intersection property for

i=1

{C(ξ, η) | (ξ, η) ∈ ΓA }, and thus the lemma.

Proof of Minty’s theorem. Proof. (1)=⇒(2) We only want to show that ∀y ∈ H, ∃x ∈ H such that y − x ∈ A(x). In fact, according to the Deburnner–Flor lemma, ∃x ∈ H satisfying (η − (y − x), ξ − x) 0, ∀(ξ, η) ∈ ΓA . By maximality, y − x ∈ A(x). (1)⇐=(2) Assume that B is a monotone mapping with ΓA ⊂ ΓB ; we want to show that B = A. Since I + A is surjective, ∀(x, y) ∈ ΓB , ∃x ∈ D(A) such that x + y ∈ (I + A)(x ), so is x + y ∈ x + B(x ), i.e, y = y + x − x ∈ B(x ). Also from (x, y) ∈ ΓB , x + y ∈ x + B(x). Provided by the monotonicity of B, we have x − x 2 = (y − y, x − x ) 0 , i.e., x = x . Therefore B = A and then A is maximal. Since A is maximal monotone if and only if λA is ∀λ > 0, we may assume λ = 1 in proving the equivalence of (2) and (3). Also, by Lemma 2.6.8, (I + A) is expansive ⇐⇒ A is monotone. (2) ⇐⇒ (3) follows from the fact that (I + A) is expansive and surjective ⇐⇒ (I + A)−1 is nonexpansive. In the literature, for a maximal monotone mapping A, ∀λ > 0, setting Jλ = (I + λA)−1 we call the operator Aλ (x) = λ1 (I − Jλ )(x) the Yosida regularization of A. By deﬁnition, Aλ (x) ∈ A(Jλ x), but in general, Aλ (x) = A(Jλ x). Theorem 2.6.10 If f : H → R1 ∪ {+∞} is a proper, l.s.c., convex function, then ∂f is maximal monotone. Proof. The monotonicity has been known since the beginning of the last section. Due to Minty’s theorem, it remains to verify the surjection of I + ∂f , i.e., ∀y0 ∈ H, ∃x0 ∈ H such that y0 − x0 ∈ ∂f (x0 ). To this end, we deﬁne a function: ϕ(x) =

1 x 2 +f (x) − y0 , x . 2

It is proper, l.s.c., and convex, and ϕ(x) → +∞ as x → ∞. There exists R1 > 0 s.t. ϕ(x) > ϕ(θ) as x R1 . Applying the Hartman–Stampacchia theorem, ∃x0 ∈ D(ϕ) = D(f ) such that

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2 Fixed-Point Theorems

ϕ(x0 ) ϕ(x)

∀x ∈ H .

Therefore, θ ∈ ∂ϕ(x0 ), i.e., y0 − x0 ∈ ∂f (x0 ).

In fact, the above conclusion can be extended to general real Banach ∗ spaces. In this extension, the normalized duality map J : X → 2X plays ∗ ∗ ∗ 2 ∗ 2 an important role, J(x) = {x ∈ X | < x , x >= x = x }, (cf. Example 1 in Sect. 2.2). It is easy to verify that J is an odd, positively homogeneous, bounded set-valued mapping, and it is known that J(x) = ∂ϕ(x), where ϕ(x) = 12 x2 , so it is monotone. The Browder theorem (Theorem 2.5.8) is also extended to maximal monotone mappings: Let X be a uniformly convex Banach space, if D(A) ⊂ X, and A : D(A) → X ∗ is maximal monotone and coercive: y,x x → +∞ ∀(x, y) ∈ ΓA , as x → ∞, then A is surjective. For a proper l.s.c. convex function on a real Banach space, f : X → R1 ∪ {+∞}, the conjugate function of f, f ∗ : X ∗ → R1 ∪ {+∞} will be deﬁned in Sect. 4.1.3 to be f ∗ (x∗ ) = sup { x∗ , x − f (x)} . x∈X

It will be proved therein that f ∗ is again a proper l.s.c. convex function, satisfying f (x) + f ∗ (x∗ ) = x∗ , x ⇔ x∗ ∈ ∂f (x) ⇔ x ∈ ∂f ∗ (x∗ ) . Thus we have the following: Corollary 2.6.11 If f : X → R1 is proper, l.s.c., and convex, then ∂f ∗ = (∂f )−1 is maximal monotone. Proof. According to the above discussion, ∂f ∗ = (∂f )−1 . The maximality of ∂f ∗ follows from Theorem 2.6.10.

3 Degree Theory and Applications

The Leray–Schauder degree is an important topological tool introduced by Leray and Schauder in the study of nonlinear partial diﬀerential equations in the early 1930s. The nontriviality of the degree ensures the existence of a ﬁxed point of the compact mapping in the domain. It enjoys the properties of homotopy invariance and additivity, which make the topological tool more convenient in application, and provides more information on ﬁxed points. In Sects. 3.5 and 3.6 we shall see how better results could be obtained by using this tool than by using any of the other methods that we discussed previously. The Leray–Schauder degree is an extension of the Brouwer degree from ﬁnite–dimensional spaces to inﬁnite–dimensional Banach spaces, while the Brouwer degree is a powerful tool in algebraic topology. We introduce the notion of the Brouwer degree in Sect. 3.1, and investigate its fundamental properties in Sect. 3.2. On the one hand, we introduce the deﬁnition of the Brouwer degree from the point of view of diﬀerential topology so that it closely ties with the counting of zeroes of mappings; on the other hand, we build up the relationship between this deﬁnition and that used in algebraic topology. The applications to the Brouwer ﬁxed-point theorem and Borsuk–Ulam theorem, as well as the intersection number etc. are studied in Sect. 3.3. The notion of the Leray–Schauder degree is deﬁned by approximation. All its fundamental properties are transferred from those of the Brouwer degree directly. These are the contents of Sect. 3.4. We emphasize the computation of the degree, because the more precisely we know the degree the sharper we can estimate the number of ﬁxed points. This opens a door to the study of multiple solutions in nonlinear analysis. Rabinowitz’s global bifurcation theorem, based on the computation of the Leray–Schauder degree, is an important part of nonlinear functional analysis and probably the only global result in bifurcation theory. It provides global information on the branch of solutions emanating from a bifurcation point. The applications to nonlinear Sturm–Liouville problems as well as nonlinear elliptic problems are presented. They are studied in Sect. 3.5.

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3 Degree Theory and Applications

In Sect. 3.6, we introduce various applications of the Leray–Schauder degree. Schaefer’s ﬁxed-point theorem is more convenient in application to partial diﬀerential equations. Relying on the degree arguments, multiple solutions for semilinear elliptic equations are studied. In particular, some interesting multiple results are obtained in combination with the super- and sub-solutions method. When we are concerned with positive solutions of diﬀerential equations, one modiﬁes the degree theory to that for compact mappings deﬁned on closed convex sets. Moreover, we present some other applications: The Krein–Rutmann theory for positive linear operators, the existence of a positive solution for the superlinear elliptic problem, and bridging solutions of disjoint domains etc. in order to expose the extensive applicability of the degree theory. In Sect. 3.7, various extensions of the Leray–Schauder degree are discussed, including the degrees for set contraction mappings, condensing mappings, setvalued mappings, Fredholm mappings, etc.

3.1 The Notion of Topological Degree The topological degree was originally introduced by H. Poincar´e in the qualitative study of ODEs. Firstly, he deﬁned the index of a singularity P0 as follows: Let P0 = (x0 , y0 ) be a singularity of the plane vector ﬁeld (F(x,y), G(x,y)), i.e., F (x0 , y0 ) = G(x0 , y0 ) = 0. Let C be a closed curve surrounding P0 in the phase plane. We count the variation of the angles of the tangents of the ﬂow: x˙ = F (x, y) y˙ = G(x, y) generated by (F, G) along C counterclockwise. This amount should be a multiple of 2π, and is denoted by ind(P0 ). One may measure this amount by observing the compass moving along the curve C in a magnetic ﬁeld. Afterwards, he deﬁned the winding number of a closed curve C, on which there are no singularities of the vector ﬁeld. Diﬀerent from the index, at this time, C is not necessarily just enclosing one single singularity. He noticed: (1) The winding number is invariant under deformation of curves C, if there is no singularity on these curves C. (2) The winding number for a given curve C is invariant under deformation of vector ﬁelds, if there is no singularity on C. A direct consequence follows: (3) Let D be the domain enclosed by C. If the winding number of C is nonzero, then there must be a singularity of the vector ﬁeld in D. For a given vector ﬁeld f = (F, G), a closed curve C (or equivalently, the enclosed domain D), we denote the winding number by deg (f, D), More generally, instead of the zeroes of f , sometimes we study the solutions of

3.1 The Notion of Topological Degree

129

f (x, y) = P0 for a given P0 ∈ R2 ; we denote the winding number of f −P0 by deg (f, D, P0 ). In order to extend the concept of winding numbers to higher-dimensional vector ﬁelds, we would rather make some stronger restrictions on f at ﬁrst. It makes the geometric characterization of winding numbers easy to understand. Let us ﬁrst assume that f is an analytic function: f (z) = F (z) + iG(z), where z = x + iy and w0 = p0 + iq 0 are complex numbers. Let Ω be a bounded open domain in C with boundary ∂Ω. Suppose f (z) = w0 ∀z ∈ ∂Ω, then we have

1 d arg(f (z) − w0 ) deg(f, Ω, w0 ) = 2π ∂Ω

1 = d log(f (z) − w0 ) 2πi ∂Ω

f (z) 1 = dz 2πi ∂Ω f (z) − w0 = σj , (3.1) zj ∈f −1 (w0 )∩Ω

where σj is the multiplicity of zj , i.e., f (z) = w0 + cj (z − zj )σj + ◦(|z − zj |σj ),

(3.2)

as |z − zj | → 0, and cj = 0. If f is not analytic but diﬀerentiable, let f = (F, G), z = (x, y) and w0 = (p0 , q0 ). Assume f (z) = w0 , ∀z ∈ ∂Ω and det f (z) = det

∂(F, G) (z) = 0 ∀z ∈ f −1 (w0 ) , ∂(x, y)

then the point set Ω ∩ f −1 (w0 ) is ﬁnite, from the IFT; and

1 d arg(f (z) − w0 ) deg(f, Ω, w0 ) = 2π ∂Ω

G − q0 1 = d arctan 2π ∂Ω F − p0

(F − p0 )dG − (G − q 0 )dF 1 = . 2π ∂Ω (F − p0 )2 + (G − q 0 )2 Applying Green’s formula, it equals

(F − p0 )dG − (G − q 0 )dF = (F − p0 )2 + (G − q 0 )2 ∂Bε (zj ) −1 zj ∈f

(w0 )∩Ω

sgn det f (zj ),

zj ∈f −1 (w0 )∩Ω

(3.3)

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3 Degree Theory and Applications

−1 where ε > 0 is small enough that + Bε (zi )∩Bε (zj ) = Ø ∀zi = zj in f (w0 )∩Ω, provided det f (z) = 0 ∀z ∈ zi ∈f −1 (w0 )∩Ω Bε (zi ). The last formula provides a clue to deﬁning the topological degree for higher-dimensional mappings. For each root of f (z) = w0 , we assign a signed number, which is the signature of the determinant at this point, and then the topological degree is the summation of all these signed numbers. Before going to the deﬁnition, we need a special case of the Sard theorem. Since we have not proved the Sard theorem in Chap. 1, now we shall present a proof for the special case, due to its simplicity. However, we refer the reader to Milnor [Mi 2] for a proof of the general theorem, which can also be obtained by combining our proof for the special case with an inductive argument.

Theorem 3.1.1 (Sard) Let Ω ⊂ Rn be an open set, and let f : Ω → Rn be a C 1 map. Write Z = {x ∈ Ω| det f (x) = 0} . Then f (Z) is a zero measure set. Proof. Consider a closed cube C ⊂ Ω, each side with length a. We divide C (1) (1) (1) into N n conﬁguration cubes Ci , i = 1, . . . , N N , with int(Ci ) ∩ int(Cj ) = a ∅, i = j, each side with length N . Thus on one hand, √ na , f (x) − f (x0 ) KC x − x0 KC N ∀x, x0 ∈ Ci for some i, where KC = max { f (x) | x ∈ C}. On the other hand, ∀ε > 0 ∃ an integer N such that √ na . f (x) − f (x0 ) − f (x0 )(x − x0 ) < ε x − x0 ε N (1)

Let x0 ∈ Z, f (x0 ) is not invertible. Thus f (x0 )Rn cannot span the whole space Rn , but is included in a (n−1)-dimensional linear subspace H. It follows that √ ε na (1) ∀x ∈ Ci , dist(f (x), f (x0 ) + H) N (1)

(1)

in which Ci is the small cube including x0 . √We conclude that f (C ) is √ √i included in a cube, whose sides have length 2ε Nna , 2KCN na , . . . 2KCN na , respectively. Therefore (1) m∗ (f (Ci ∩ Z)) m∗ (f (C ∩ Z)) (1)

Ci

(1)

Ci

∩Z=Ø

m∗ (f (Ci )) (1)

∩Z=Ø n

n−1 2 n 2 KC n a ε, n

3.1 The Notion of Topological Degree

131

where m∗ is the Lebesgue outer measure. Since ε > 0 and the cube C are arbitrary, we obtain m∗ (f (Z)) = 0. Suppose that Y is an n-manifold, a subset W ⊂ Y is called a null set, if for each chart (ϕ, U ) of Y , the set ϕ(U ∩ W ) ⊂ Rn is a null set. Let us recall the deﬁnitions of regular/critical values/points of a map between two Banach manifolds. Sard’s theorem asserts that if X and Y are n-manifolds, then for any f ∈ C 1 (X, Y ), the set of critical values of f is a null set. Let X0 , Y be two oriented smooth n-dimensional manifolds, and X ⊂ X0 be an open subset satisfying the condition that X = X ∪ ∂X is compact. If f ∈ C(X, Y ) ∩ C 1 (X, Y ), and if y0 is a regular value of f then f −1 (y0 ) = {x ∈ X| f (x) = y0 } must be a ﬁnite set, from the IFT. Deﬁne f −1 (y0 ) = {x1 , . . . , xk } . Assuming that y0 ∈ / f (∂X) is a regular value, we deﬁne deg(f, X, y0 ) =

k

sgn det f (xj ) .

(3.4)

j=1

Again we let Z denote the set of critical points of f . We shall extend the deﬁnition to continuous mappings in three steps: I. The special case f ∈ C(X, Y ) ∩ C 2 (X, Y ), y0 ∈ f (Z) ∪ f (∂X). In order to remove the assumptions on the regularity and the condition f ∈ C 1 (X, Y ), we would rather express (3.4) in an integration form. Let Uj be a neighborhood of xj , such that f : Uj → f (Uj ) is a diﬀeomorphism, and that Ui ∩ Uj = Ø, ∀i = j, where i, j = 1, 2, . . . , k. Set k , V = f (Uj ) . j=1

This is a neighborhood of y0 . Let us choose a C ∞ n-form on Y : µ = ψ(y)dy1 ∧ . . . ∧ dyn , (in the following, we write simply dy = dy1 ∧ . . . ∧ dyn ) such that supp ψ ⊂ V ∩ (Y \f (∂X)) ,

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3 Degree Theory and Applications

and

µ=1. Y

Since detf (x) has the same sign in each Uj , we have

µ◦f = X

k

j=1

=

k

j=1

=

k

µ◦f

Uj

ψ(f (x)) det f (x)dx

Uj

sgn det f (xj )

Uj

j=1

=

k

sgn det f (xj )

k

ψ(y)dy f (Uj )

j=1

=

ψ ◦ f (x)| det f (x)|dx

sgn det(f (xj ))

j=1

= deg(x, X, y0 ) .

(3.5)

In the integration form, µ is arbitrarily chosen. We shall prove that the integral X µ ◦ f does not depend on the special choice of µ. We call the chart (g, Ω) a good chart at y0 ∈ Y , if (1) g(y0 ) = θ, (2) g(Ω) = In = (−1, +1)n in Rn . A C ∞ − n-form µ on Y is called admissible if µ = ψ(y)dy satisﬁes (1) supp ψ ⊂ Ω ∩ (Y \f (∂X)) for a good chart (g, Ω) at y0 . (2) Y µ = 1. Lemma 3.1.2 Suppose that µ = ψ(y)dy is a C ∞ − n-form on Y satisfying (1) supp ψ ⊂ Ω, and (g, Ω) is a good chart at y0 , (2) Y µ = 0. Then there exists an (n − 1) form ω such that supp ω ⊂ Ω and dω = µ. Proof. In the coordinates, g

Ω −→ In ∩ ∩ Y Rn one may assume suppψ ⊂ In and ψ(y)dy = 0. What we want to prove is the existence of a function ν ∈ C 1 (Rn , Rn ) such that

3.1 The Notion of Topological Degree

133

supp ν ⊂ In and div ν = ψ . Indeed, let ν = (ν1 , ν2 , . . . , νn ), then ω = ν1 dy2 ∧ . . . ∧ dyn + . . . + (−1)n νn dy1 ∧ . . . ∧ dyn−1 satisﬁes dω = µ . We prove the conclusion by mathematical induction. For n = 1, one takes

y ψ(t)dt , ν(y) = −∞

then ν is as required. Assume the lemma is true for n = k. Let

∞ φ(ˆ y) = ψ(ˆ y , yk+1 )dyk+1 , −∞

where y = (ˆ y , yk+1 ), yˆ = (y1 , . . . , yk ). Then supp φ ⊂ Ik (−1, 1)k ,

and

φ(ˆ y )dˆ y=

ψ(y)dy = 0 .

By the hypothesis of induction, there exists ω ∈ C 1 (Rk , Rk ) satisfying supp ω ⊂ Ik , div ω = φ .

Choose τ ∈ C 1 (R1 ), supp τ ⊂ I1 , then

∞

−∞

τ (t)dt = 1 , −∞

[ψ(ˆ y , yk+1 ) − φ(ˆ y )τ (yk+1 )]dyk+1 = 0 .

Let y , yk+1 ) = νk+1 (ˆ then

∞

yk+1

−∞

[ψ(ˆ y , t) − φ(ˆ y )τ (t)]dt ,

∂ νk+1 (ˆ y , yk+1 ) = ψ(ˆ y , yk+1 ) − φ(ˆ y )τ (yk+1 ) . ∂yk+1

This means that ψ = div ν, where ν = (ω1 (ˆ y )τ (yk+1 ), . . . , ωk (ˆ y )τ (yk+1 ), νk+1 (ˆ y , yk+1 )) , and (ω1 , . . . , ωk ) = ω .

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3 Degree Theory and Applications

In order to prove that the integral X µ ◦ f does not depend on µ, we assume that ν, µ are two admissible n forms with respect to a good chart (g, Ω) at y0 . According to Lemma 3.1.2, there exists an (n − 1) form ω such that dω = ν − µ, and then

ν◦f − µ◦f = dω ◦ f . X

If we can prove

X

X

(dω) ◦ f =

d(ω ◦ f ),

X

(3.6)

X

then by the Stokes theorem, we conclude:

ν◦f = µ◦f . X

X

We are going to prove (3.6). First, we need Lemma 3.1.3 If f ∈ C(X, Y ) ∩ C 2 (X, Y ), then for any (n − 1) form ω, we have d(ω ◦ f ) = dω ◦ f . Proof. In local coordinates, ω=

n

νj (y)dy1 ∧ . . . ∧ dˆ yj ∧ . . . ∧ dyn ,

j=1

where yj ∧ . . . ∧ dyn = dy1 ∧ . . . ∧ dyj−1 ∧ dyj+1 ∧ . . . ∧ dyn . dy1 ∧ . . . ∧ dˆ Thus ω◦f =

n n j=1

νk ◦

f (x)Jfkj (x)

dx1 ∧ . . . ∧ dˆ xj ∧ . . . ∧ dxn ,

k=1

where Jfki is the (k, i) cofactor of the Jacobian determinant det f (x) = Jf (x). Hence d(ω ◦ f ) n

n ∂νk ◦ f (x) ∂Jfki (x) ∂fl ki Jf (x) · + νk ◦ f (x) dx1 ∧ . . . ∧ dxn = ∂yl ∂xi ∂xi i,k=1 l=1 ⎤ ⎡ n n n ∂Jfki (x) ∂ν ◦ f (x) ∂f k l ⎦ =⎣ Jfki (x) + νk ◦ f (x) ∂yl ∂x ∂x i i i=1 i k,l=1

× dx1 ∧ . . . ∧ dxn

k=1

3.1 The Notion of Topological Degree

135

in which f = (f1 , . . . , fn ) . Let g = (−1)k−1 (f1 , . . . , fˆk , . . . , fn ) ,

then Jfki

i−1

= (−1)

det

ˆ ∂g ∂g ∂g ,..., ,..., ∂x1 ∂xi ∂xn

.

Hence n n 2 ˆ ∂ ki g ∂g ∂ ∂g ∂g Jf = (−1)i−1 det ,..., ,..., ,..., ∂x ∂x1 ∂xi ∂xl ∂xi ∂xn i i=1 i=1 l=i n 2 ˆ g ∂ ∂g ∂g ∂g = (−1)i−1+l−2 det , ,..., ,..., ∂xi ∂xl ∂x1 ∂xi ∂xn i=1 l>i - ˆ ∂2g ∂g ∂g ∂g i−1+l−1 (−1) det , ,..., ,..., =0. + ∂xl ∂xi ∂x1 ∂xi ∂xn i>l

From this, together with n ∂fl ki Jf (x) = δkl Jf (x) ∂x i i=1

follows d(ω ◦ f ) =

n ∂νk ◦ f (x) k=1

∂yk

Jf (x)dx1 ∧ . . . ∧ dxn = (dω) ◦ f .

Now we arrive at: / f (Z) ∩ f (∂X) and Theorem 3.1.4 Assume f ∈ C(X, Y ) ∩ C 2 (X, Y ). If y0 ∈ respect to a good chart (g, Ω) at y0 , with µ is an admissible C ∞ n form with supp(µ) ⊂ Ω ∩ (Y \f (∂X)), then X µ ◦ f is a constant independent of µ. II. Removal of the restriction y0 ∈ f (Z) Combining Theorem 3.1.4 and (3.5), we know that

deg(f, X, y0 ) = µ◦f , X

if f ∈ C(X, Y )∩C 2 (X, Y ) and y0 ∈ f (Z)∪f (∂X), where µ is an arbitrary admissible C ∞ n form with respect to a good chart at y0 . Note that the integral on the right-hand side of the above formula does not contain y0 explicitly.

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3 Degree Theory and Applications

Further recall that the set of regular valuesis dense in f (X) according to the Sard theorem, thus if we take the integral X µ ◦ f to be the deﬁnition of deg(f, X, y0 ), the assumption y0 ∈ f (Z) is not necessary. In fact, this integral is deﬁned for all y0 ∈ f (∂X), and is integer valued (the left-hand side of the formula (3.5)) if y0 belongs to the dense (regular value) set. Hence X µ ◦ f is integer valued. Therefore the restriction y0 ∈ f (Z) can be removed when we use X µ ◦ f as the deﬁnition of the deg (f, X, y0 ). III. Extension to continuous mappings We ﬁrst prove: C

Lemma 3.1.5 Let φ : X × [0, 1] −→ Y satisfy φ(·, t) ∈ C 2 (X, Y ) ∀t ∈ [0, 1]. If y0 ∈ φ(∂X × [0, 1]), then deg(φ(·, t), X, y0 ) is a constant independent of t. Proof. Since φ(∂X × [0, 1]) is closed, there is a good neighborhood Ω at y0 such that Ω ∩ φ(∂X × [0, 1]) = Ø. If we choose ψ ∈ C ∞ (Y, R1 ) such that suppψ ⊂ Ω, Y ψdy = 1, then µ = ψ(y)dy is admissible, and

deg(φ(·, t), X, y0 ) = µ ◦ φ(·, t) . (3.7) X

The right-hand side is integer valued and is continuous in t, hence it must be a constant. Now we may extend the deﬁnition of degree to mappings in C(X, Y ) by approximation. By the use of the Whitney embedding theorem, we embed Y into RN as a regular submanifold for large N . Thus there exist a tubular neighborhood W of Y and a C ∞ retract r : W → Y . Since we assumed y0 ∈ f (∂X), one may choose 0 < ε < dRN (r−1 (y0 ), f (∂X)) such that the ε neighborhood of Y is in W , where dRN (·, ·) is the distance in RN . According to the Weierstrass approximation theorem and the partition of unity, there exists g ∈ C 2 (X, RN ) satisfying (3.8) f − g C(X,RN ) < ε . Thus for such g, deg (r ◦ g, X, y0 ) is well deﬁned. Moreover, we shall prove that deg(r ◦ g1 , X, y0 ) = deg(r ◦ g2 , X, y0 ) for g1 , g2 ∈ C 2 (X, RN ) both satisfying (3.8). Indeed, deﬁne h : X × [0, 1] → RN by h(x, t) = (1 − t)g1 (x) + tg2 (x) , and φ : X × [0, 1] → Y by φ = r ◦ h. Then distRN (φ(x, t), y0 ) = 0 ⇔ distRN (h(x, t), r−1 (y0 )) = 0 .

3.2 Fundamental Properties and Calculations of Brouwer Degrees

137

But, distRN (h(∂X, t), r−1 (y0 )) distRN (f (∂X), r−1 (y0 )) − ε > 0 , this proves that y0 ∈ φ(∂X × [0, 1]). Applying Lemma 3.1.5, we obtain deg(r ◦ g1 , X, y0 ) = deg(r ◦ g2 , X, y0 ) . We are now in a position to deﬁne the Brouwer degree for continuous mappings. Deﬁnition 3.1.6 (Brouwer degree) Let X0 , Y be two oriented smooth n− manifolds, and let X ⊂ X0 be an open subset with compact closure X. For / f (∂X), to the triple (f, X, y0 ) we deﬁne f ∈ C(X, Y ), y0 ∈ deg(f, X, y0 ) = deg(r ◦ g, X, y0 ) , where r is a C ∞ retract of a tubular neighborhood W of Y to Y , and g is a map deﬁned in (3.8). This is called the Brouwer degree of f . The Brouwer degree is integer valued. It is not diﬃcult to verify that Brouwer degree does not depend on the special choice of W and r. By deﬁnition, if X = Ø, then deg (f, X, y0 ) = 0. In dealing with mappings between complex manifolds, we identify C n with R2n by the canonical isomorphism z → (x, y), where z = x + iy for z ∈ C n , x, y ∈ Rn . Similarly for the map f → (u, v), where f : Ω → C n , and f = u + iv. As an exercise, readers can verify that if f is analytic then det(f (z)) > 0.

3.2 Fundamental Properties and Calculations of Brouwer Degrees The Brouwer degree has the following fundamental properties. (1) (Homotopy invariance). If φ : X × [0, 1] → Y is continuous and y0 ∈ φ(∂X × [0, 1]), then deg(φ(·, t), X, y0 ) = constant . Proof. Using the above notations, we choose 0 < < dRN (r−1 (y0 ), φ(∂X × [0, 1])) , where dRN (·, ·) is the distance in RN . By the above approximation, there is φˆ ∈ C 2 (X × [0, 1], Y ) such that ˆ 0) . One may choose g0 ∈ C 2 (X, RN ) such that dC(X,RN ) (g, f ) < r ◦ g0 , and choose y1 ∈ g(Z) satisfying dRN (y0 , y1 ) < 2ε , then

ε 2

for g =

dist(y1 , g(X\(X1 ∪ X2 ))) > 0 . On the one hand, by (3.5) we have deg(g, X, y1 ) = sgn det g (xi ) xi ∈g −1 (y1 )

=

+

xi ∈g −1 (y1 )∩X1

sgn det g (xi )

xi ∈g −1 (y1 )∩X2

= deg(g, X1 , y1 ) + deg (g, X2 , y1 ) . On the other hand, by property (2), deg(f, X, y0 ) = deg(g, X, y1 ), deg(f, Xi , y0 ) = deg(g, Xi , y1 ), i = 1, 2 . This is due to the fact that θ = tf (x) + (1 − t)g(x) − ty0 − (1 − t)y1 , as x ∈ ∂X ∪ ∂X1 ∪ ∂X2 ⊂ X\(X1 ∪ X2 ), t ∈ [0, 1] . Corollary 3.2.3 (Excision) If K ⊂ X is compact and y0 ∈ f (K) ∪ f (∂X), then deg(f, X, y0 ) = deg(f, X\K, y0 ) . Proof. Let X1 = X\K, X2 = Ø, then X\(X1 ∪ X2 ) = ∂X ∪ K . Corollary 3.2.4 (Kronecker existence) If y0 ∈ f (∂X) and deg(f, X, y0 ) = 0, then f −1 (y0 ) = Ø. Proof. Assume the contrary: f −1 (y0 ) = Ø, i.e., y0 ∈ f (X). By Corollary 3.2.3, deg(f, X, y0 ) = deg(f, X\X, y0 ) = 0 .

3.2 Fundamental Properties and Calculations of Brouwer Degrees

141

Kronecker existence theorem is often used in the study of the solvability of the equation f (x) = y0 . (5) (Normality). If X ⊂ Y , and y0 ∈ ∂X, then ◦ deg(id, X, y0 ) = 1, y0 ∈ X 0, y0 ∈ /X. More generally, we have: (6) If L is a n × n nondegenerate real matrix, and if X is a bounded open subset and θ ∈ X ⊂ Rn , then deg(L, X, θ) = (−1)β , where β = λj 0 such that deg (f, Bε (x0 ), y0 ) = deg (f (x0 ), Bε (θ), θ) from the homotopy invariance and translation invariance. The conclusion then follows from (6).

3.2 Fundamental Properties and Calculations of Brouwer Degrees

145

(10) If f ∈ C(X, Y ) and y0 ∈ f (∂X) with f −1 (y0 ) = {x1 , . . . , xp }, then deg (f, X, y0 ) =

p

i(f, xj , y0 ) .

i=1

Proof. It follows from the excision property plus the additivity property. At the end of this section we establish the connection between the definition of the Brouwer degree given above with that in algebraic topology. Following Corollary 3.2.2, for f ∈ C 2 (X, Y ), we have µ◦f deg (f, X, y0 ) = X . µ Y Let f ∗ : Ωn (Y ) → Ωn (X), µ → µ ◦ f be the pullback of f , where Ωn (X) is the space of n− forms over X. This means that deg (f, X, y0 ) is the ratio of the integrations of f ∗ µ and µ, ∀µ ∈ Ωn (Y ). In algebraic topology, the Brouwer degree for maps f from the sphere into itself is deﬁned to be the multiplier of the homomorphism f ∗ : H n (S n ) → H n (S n ), in which H n (S n ) stands for the cohomology group of the n− sphere, and the map f maps the generator [ω] into λ[ω] for some integer λ. According to de Rham theory, for a compact, oriented, connected n manifold without boundary X, the cohomology group H k (X) is deﬁned to be ker dk /Im dk−1 , where dk : Ωk (X) → Ωk+1 (X) is the exterior diﬀerentiation, k = 0, 1, . . . , n. We shall prove that Ωn (S n ) and then H n (S n ) is one dimensional. Namely: Lemma 3.2.7 For a compact, oriented, connected n-manifold X without boundary, dim H n (X) = 1. Proof. We want to show that H n (X) is generated by one generator. Let us choose an atlas {(Ui , ψi )| i = 1, . . . , p} such that Ui ∩ Uj = Ø ⇒ Ui ∪ Uj is contained in a good chart . ∈ Ωn (X) with suppµ0 ⊂ U1 , and It is suﬃcient to prove that1 for ∀µ, µ0 n−1 µ = 1, there exist λ ∈ R and ω ∈ Ω (X) such that X 0 µ − λµ0 = dω . Since X is connected, for ∀i, ∃ a curve C starting from U1 ending at Ui . Let p {Ui0 , Ui1 , . . . , Uil } be a chain of neighborhoods in {Ui }1 , such that Ui0 = U1 , Uil = Ui and Uik ∩ Uik−1 = Ø k = 1, 2, . . . , l .

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3 Degree Theory and Applications

We choose αk ∈ Ωn (X) satisfying

supp αk ⊂ Uik , αk = 1, k = 1, 2, . . . , l , X

and α0 = µ0 . Since supp(αk − αk−1 ) ⊂ Uik ∪ Uik−1 is contained in a good neighborhood, and

(αk − αk−1 ) = 0 , X

we have ωki ∈ Ωn−1 (X) satisfying αk − αk−1 = dωki , k = 1, 2, . . . , l , provided by Lemma 3.1.2. Hence αl = µ0 + d

l

ωki .

(3.15)

k=1

Since Uil = Ui , αl depends on the index i, we write it as βi , i = 1, . . . , p. p p Now, let {χi }1 be a partition of unity with respect to {Ui }1 , i.e., supp χi ⊂ Ui χi 0,

p

χi ≡ 1 .

i=1

Set µi = χi µ , then Denote ci =

supp µi ⊂ Ui .

X

µi . There exists ω !i ∈ Ωn−1 (X) such that µi − ci βi = d! ωi ,

from Lemma 3.1.2, i = 1, 2, . . . , p. Combining (3.15) and (3.16), we have µ=

p

χi µ =

i=1

=

p i=1 p i=1

µi ci βi + d

p i=1

ω !i

(3.16)

3.2 Fundamental Properties and Calculations of Brouwer Degrees

=

p

ci µ0 + d

i=1

p

ω !i + ci

i=1

l

147

ωki

k=1

= λµ0 + dω , p p l where λ = µ , and ω = ωi + ci k=1 ωki ). Thus i=1 ci = i=1 (! X i n we have proved that all elements in H (X) are multipliers of [µ0 ], i.e., dim H n (X) = 1. Combining Corollary 3.2.2 and Lemma 3.2.7, we obtain Theorem 3.2.8 Let X and Y be compact, oriented, connected n-manifolds without boundaries. Then for any f ∈ C 2 (X, Y ), the following diagram commutes: f∗ H n (Y ) −−−−→ H n (X) ⏐ ⏐ ⏐ ⏐ Y0 X0 R1

−−−−→ deg f

R1

Finally, we turn to studying the relationship between degrees on balls and those on spheres. Let f : B n → Rn be a continuous map satisfying θ ∈ f (∂B n ), so deg (f, B n , θ) is well deﬁned. Let us deﬁne Φ(x) =

f (x) f (x)

∀x ∈ ∂B n .

Then Φ : S n−1 → S n−1 deﬁnes a Brouwer degree: deg (Φ, S n−1 ). What is the relationship between deg(f, B n , θ) and deg(Φ, S n−1 )? Deﬁne a homotopy θ if x = θ φ(x, t) = x 2 x f ( ) if x = θ . x f ( ) t x x

Since f |∂B n = φ(·, 0)|∂B n , from Corollary 3.2.1 and the homotopy invariance, deg(f, B n , θ) = deg(φ(·, 0), B n , θ) = deg(φ(·, 1), B n , θ) . In the case where f ∈ C 1 , and y0 ∈ S n−1 is a regular value of Φ, for ∀ε > 0 small, ε2 y0 is a regular value of φ(·, 1), and Φ−1 (y0 ) = {x1 , . . . , xk } if and only if φ(·, 1)−1 (ε2 y0 ) = {εx1 , . . . , εxk }. Since sgn JΦ (xj ) = sgn Jφ(·,1) (εxj ), j = 1, 2, . . . , k , we obtain deg(φ(·, 1), B n , θ) = deg(Φ, S n−1 ). The assumptions on f and on y0 can easily be dropped. Namely, we have proved: Corollary 3.2.9 deg(f, B n , θ) = deg(Φ, S n−1 ) = deg (Φ). Thus the notation deg (Φ, S n−1 ) is in coincidence with the notation deg (fˆ, ∂B n , θ) introduced after Corollary 3.2.1.

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3.3 Applications of Brouwer Degree 3.3.1 Brouwer Fixed-Point Theorem The Brouwer degree is a fundamental tool in algebraic topology. It is widely used in topological arguments. We are satisﬁed having a glimpse of these applications. Theorem 3.3.1 (Brouwer ﬁxed-point theorem) A continuous map f n n from B into itself has a ﬁxed point x0 ∈ B , i.e., f (x0 ) = x0 . Proof. Let g = id − f . With no loss of generality, we may assume θ ∈ g(∂B n ). Since ∀x ∈ ∂B n , ∀λ 1, f (x) ∈ λx, we have θ ∈ φ(∂B n ) × [0, 1], where φ(x, t) = (1 − t)g(x) + tx . Thus by the homotopy invariance and the normality, ◦

◦

deg (g, B n , θ) = deg (id, B n , θ) = 1 . Our conclusion follows from the Kronecker existence theorem.

The following topological fact follows from a simple degree argument: Theorem 3.3.2 There is no continuous map f : B n f |∂B n = id|∂B n , i.e., ∂B n is not a retraction of B .

n

→ ∂B n such that

n

Proof. If not, there is a continuous map f : B → ∂B n . f |∂B n = id|∂B n . On one hand, deg (f, B n , θ) = 0 due to the Kronecker existence theorem. On the other hand, by Corollary 3.2.1 and the normality deg (f, B n , θ) = deg (id, B n , θ) = 1 .

This is a contradiction. 3.3.2 The Borsuk-Ulam Theorem and Its Consequences

Let Ω ⊂ Rn be a bounded open set, which is symmetric with respect to the origin θ, i.e., −Ω = Ω. A map f : Ω → Rn is called odd, if f (−x) = −f (x) ∀x ∈ Ω . We are going to study the Brouwer degree of odd mappings. Theorem 3.3.3 (Borsuk) Suppose that Ω ⊂ Rn is a bounded open set containing θ and is symmetric with respect to θ. If f : Ω → Rn is an odd continuous map, then deg (f, Ω, θ) = odd , whenever θ ∈ f (∂Ω).

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149

Proof. One chooses ε > 0 such that Bε = Bε (θ) ⊂ Ω. According to Tietze’s theorem, there exists a continuous map fε : Ω → Rn satisfying x x ∈ Bε fε (x) = f (x) x ∈ ∂Ω . If we use 12 (fε (x) − fε (−x)) to replace fε (x), then one may assume that fε is odd. Since fε |∂Ω = f |∂Ω , according to Corollary 3.2.1, the additivity and the normality, deg (f, Ω, θ) = deg (fε , Ω, θ) = deg (fε , Bε , θ) + deg (fε , Ω\B ε , θ) = 1 + deg (fε , Ω\B ε , θ) . In the following, we want to prove that deg (fε , Ω\B ε , θ) is even. Noticing that for a C 1 odd map g with regular value θ, elements in g −1 (θ) ∩ (Ω\B ε ) occur in pairs. Our strategy is to approximate fε by a C 1 odd map g with regular value θ. Once it is constructed, we have deg(f , Ω\B , θ) = deg(g, Ω\B , θ) ≡ 0 (mod 2) . By the Weierstrass approximation theorem, we may assume that fε ∈ C ∞ . It remains to approximate it such that θ is a regular value. To this end we ¯ (θ))×Mn×n −→ Rn deﬁned by F (x, A) = f (x)+ consider the map F : (Ω\B Ax. If we can show that F {θ}, then according to the transversality theorem (Theorem 1.3.14), gA {θ}, for almost every A ∈ Mn×n , where gA (x) = F (x, A). In fact, F (x, A)(y, B) = f (x)y + Ay + Bx, ∀(y, B) ∈ Rn × Mn×n . Since x = θ, ∀z ∈ Rn , we take B = x 2 z, and y = θ, we have f (x, A)(y, B) = z, i.e., F (x, A) is surjective, or F {θ}. Since A can be chosen arbitrarily small, the proof is complete.

Theorem 3.3.4 (Borsuk–Ulam) Suppose that Ω ⊂ Rn is a symmetric bounded open set including θ, and that g : ∂Ω → Rm , m < n, is odd and continuous. Then there exists x0 ∈ ∂Ω such that g(x0 ) = 0. Proof. We prove by contradiction. Suppose θ ∈ g(∂Ω). We deﬁne a continuous extension g! : Ω → Rm by the Tietze theorem. With no loss of generality, one may assume g! is odd. Now we apply Theorem 3.3.3, deg (! g , Ω, θ) = 0. Choosing y0 ∈ Rn \Rm , with small norm, it follows that deg (! g , Ω, y0 ) = 0 Thus, g!−1 (y0 ) = Ø, but this is impossible.

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Corollary 3.3.5 Suppose that Ω ⊂ Rn is a symmetric bounded open set including θ, and that g : ∂Ω → Rm , with m < n, is continuous. Then there exists x0 ∈ ∂Ω such that g(x0 ) = g(−x0 ) Proof. Set g1 (x) = g(x) − g(−x), then apply Theorem 3.3.4 directly.

Remark 3.3.6 We consider a continuous vector ﬁeld on S n , i.e., a continuous map g : S n → Rn . Corollary 3.3.5 means that there exists a pair of antipodal points ±x0 , which have the same vector g(x0 ) = g(−x0 ). Theorem 3.3.7 (Ljusternik–Schnirelmann–Borsuk) Suppose that Ω ⊂ Rn is a symmetric bounded open set including θ, and that {A1 , . . . , Ap } is a closed covering of ∂Ω, satisfying Ai ∩ (−Ai ) = Ø ∀i. Then p n + 1. Proof. We may assume that ∩pi=1 Ai = Ø. For otherwise, if ∃x0 ∈ ∩pi=1 Ai , then by covering ∂Ω = ∪pi=1 (−Ai ), there must be a j such that x0 ∈ Aj ∩ (−Aj ). But this is impossible. Now we set di (x) = dist (x, Ai ), i = 1, 2, . . . , p , and f : ∂Ω → Rp−1 ⊂ Rn−1 as follows: f (x) = (d1 (x), d2 (x), . . . , dp−1 (x)) . Supposing p n, we apply Corollary 3.3.5, ∃x0 ∈ ∂Ω satisfying f (x0 ) = f (−x0 ). Since {A1 , A2 , . . . , Ap } is a covering, there exists j p such that x0 ∈ Aj . There are two possibilities: 1. j p − 1. It follows that dj (−x0 ) = dj (x0 ) = 0, thus −x0 ∈ Aj or x0 ∈ Aj ∩ (−Aj ); this is a contradiction. +p−1 / j=1 Aj , thus j = p. In this case, dj (−x0 ) = dj (x0 ) > 0, ∀j = 2. x0 ∈ +p−1 / j=1 (−Aj ). Again, by covering, x0 ∈ Ap ∩ (−Ap ), 1, 2, . . . , p − 1, so x0 ∈ a contradiction. The following theorem is an extension of the open mapping theorem to continuous mappings in ﬁnite-dimensional spaces. Theorem 3.3.8 (Invariance of domains) If Ω ⊂ Rn is a nonempty open set, and if f : Ω → Rn is continuous and locally injective, then f is an open map. Proof. We want to show that ∀x0 ∈ Ω, ∀ε > 0 with Bε (x0 ) ⊂ Ω, ∃δ > 0 such that f (Bε (x0 )) ⊃ Bδ (f (x0 )). One may assume x0 = θ = f (θ), and that f : B (θ) → f (B (θ)) is 1–1. Since θ ∈ f (∂B (θ)), ∃δ > 0 such that B δ (θ) ⊂ Rn \f (∂B (θ)). Our problem is reduced to proving that ∀y0 ∈ Bδ (θ), ∃x ∈ B (θ) such that f (x) = y0 . Due to the Kronecker existence theorem, it is suﬃcient to prove that deg (f, B (θ), y0 ) = 0 .

3.3 Applications of Brouwer Degree

151

Since δ > 0 can be chosen arbitrarily small, according to property (2) of the Brouwer degree, deg (f, B (θ), y0 ) = deg (f, B (θ), θ) .

Let H(x, t) = f

x 1+t

−f

−tx 1+t

∀(x, t) ∈ B (θ) × [0, 1], .

It is continuous on B (θ) × [0, 1], and satisﬁes H(x, 0) = f (x), H(x, 1) = f ( 12 x) − f (− 12 x). We claim that θ ∈ H(∂Bε × [0, 1]). For otherwise, there are (x0 , t) ∈ ∂B (θ) × [0, 1] satisfying x0 −tx0 f =f . 1+t 1+t Since f is 1 − 1 on B (θ), we would have x0 −tx0 = , 1+t 1+t i.e., x0 = θ. But, this is impossible. Therefore by the homotopy invariance and Borsuk–Ulam theorem deg (f, Bε (θ), θ) = deg (H(·, 1), Bε , θ) = 0 . and then deg (f, Bε (θ), y0 ) = 0. This proves our conclusion.

Comparing with the global implicit function theorem, the following theorem is an extension to continuous mappings in ﬁnite-dimensional spaces: Corollary 3.3.9 If f : Rn → Rn is continuous and locally injective, and if f (x) → +∞ as x → +∞, then f is surjective. Proof. From Theorem 3.3.8, f (Rn ) is open. We shall prove that f (Rn ) is also closed. Suppose yn ∈ f (Rn ) with yn → y0 . We shall prove that ∃x0 ∈ Rn satisfying f (x0 ) = y0 . Indeed we have xn ∈ Rn satisfying f (xn ) = yn ∀n. According to the assumption, {xn } is bounded, and then there exists a convergent subsequence xn → x0 . Thus y0 = f (x0 ). f (Rn ) is both open and closed. Therefore f (Rn ) = Rn . 3.3.3 Degrees for S 1 Equivariant Mappings We now turn to a computation of the degree of an S 1 group action equivariant map. Let us consider an S 1 representation on C n : Tφ = ei diag {λ1 φ,...,λn φ} , where φ ∈ [0, 2π], and λ1 , . . . , λn ∈ Z.

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3 Degree Theory and Applications

Theorem 3.3.10 Suppose that Ω ⊂ C n is a T (S 1 ) invariant open domain containing θ, and that f ∈ C(∂Ω, C n \{θ}) satisﬁes f (Tφ z) = eikφ f (z), ∀z ∈ ∂Ω, ∀φ ∈ [0, 2π] . Then deg (f, Ω, θ) =

kn . λ1 . . . λn

Proof. 1. We may assume that λ1 = · · · = λn = 1. In fact, if it is proved for the special case, then for general λ1 , . . . , λn ∈ Z+ , we deﬁne the transform ˜ = g −1 (Ω), then the map f ◦ g : Ω ˜ → C n satisﬁes g(z) = (z1λ1 , . . . , znλn ), let Ω Tφ ◦ g(z) = g ◦ T˜φ z, where T˜φ = eiφ In . Therefore f ◦ g(T˜φ z) = f (Tφ ◦ g(z)) = eikφ f ◦ g(z) . Provided by the product formula, we have ˜ θ) = deg (f, Ω, θ) · deg (g, Ω, ˜ θ) . deg (f ◦ g, Ω, We assume the conclusion holds for λ1 = · · · = λn = 1, therefore ˜ θ) = k n . deg (f ◦ g, Ω, But, ˜ θ) = λ1 · · · λn , deg (g, Ω, we obtain our conclusion. If λ1 , . . . , λn are not all positive, then we introduce a transform g : (z1 , . . . , zj , . . . , zn ) → (z1 , . . . , z¯j , . . . , zn ) for all negative λj s. Again by the product formula, we have deg (f ◦ g, g −1 (Ω), θ) = (−1)sgn λ1 ···λn deg (f, Ω, θ) . Thus it is suﬃcient to prove the theorem for λ1 = · · · = λn = 1. 2. We may assume that f is smooth. This can be done by a molliﬁer; the argument is standard, so we omit it. The only thing that has to be veriﬁed is the equivariance of the molliﬁed map. 1 : η(t) = 1, as t ≤ , and 0, as t > 2, 3. Taking a smooth function on R+ 1 where < 2 dist (∂Ω, θ), and deﬁne fˆ = η(|z|)(z1k , . . . , znk ) + (1 − η(|z|))f (z) . Then fˆ has the same degree as f on Ω. In summary, we may assume that f is smooth and S 1 -equivariant, with λ1 = . . . = λn = 1, and has the form (z1k , . . . , znk ) in B (θ). ¯ε (θ). Deﬁne a map F : 4. Now, we consider the degree of f on Ω\B ¯ε (θ)) × Mn×n → C n by (Ω\B

3.3 Applications of Brouwer Degree

153

F (z, A) = f (z) + Az k , where z k = (z1k , . . . , znk ). We show that F {θ}. In fact, n n k ¯ B) = Fz · w + Fz¯ · w ¯+ zi Bij . F (z, A)(w, w, 1

1

¯ (θ), ∀ξ ∈ C n , the system Since z ∈ /B n

Bij zjk = ξi

1

has a solution Bij . By taking w = w ¯ = 0 the map F (z, A) is surjective. Applying the transversality theorem, fA := F (·, A) {θ} for almost all A ∈ Mn×n . Since A can be chosen small, the critical set S(fA , Ω\B (θ)) of fA on Ω\B (θ) consists of isolated points, but ∀z ∗ ∈ S(fA , Ω\B (θ)), Tφ z ∗ ∈ S(fA , Ω\B (θ)). This is impossible if it is nonempty. This proves that ¯ε (θ), θ) = 0. Therefore deg(fA , Ω\B deg (fA , Ω, θ) = deg (fA , B (θ), θ) . But on Bε (θ), fA (z) = z k + Az k , for suﬃciently small A, we obtain: deg (fA , B (θ), θ) = deg (f, B (θ), θ) = k n . Combining all together, we have proved the conclusion. 3.3.4 Intersection The Brouwer degree is useful in the intersection theory. We brieﬂy introduce it here. Let Ω1 , Ω2 ⊂ Rn be two compact manifolds with ∂Ω1 = ∅ and ∂Ω2 = ∅, where dim Ω1 = k + 1, dim Ω2 = n − k − 1. Let φ1 : ∂Ω1 → Rn , φ2 : Ω2 → Rn be two continuous mappings with φ1 (∂Ω1 ) ∩ φ2 (Ω2 ) = ∅. We say φ1 and φ2 link, if for any continuous extension of φ1 , φ!1 : Ω1 → Rn , we have φ!1 (Ω1 ) ∩ φ2 (Ω2 ) = ∅ . In other words, 11 (x) = φ2 (y) . ∃(x, y) ∈ Ω1 × Ω2 , such that φ Deﬁne a mapping F : Ω1 × Ω2 → Rn by F (x, y) = φ!1 (x) − φ2 (y); it is / F (∂Ω1 × Ω2 ) = equivalent to saying that F has a zero in Ω1 × Ω2 . Since θ ∈ F (∂(Ω1 ×Ω2 )), deg (F, Ω1 × Ω2 , θ) is well deﬁned and depends on (φ1 , φ2 ) only,

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3 Degree Theory and Applications

on account of Corollary 3.2.1. It is suﬃcient to verify deg (F, Ω1 × Ω2 , θ) = 0. Thus the Brouwer degree is a tool in the study of linking. Example 1. Let r1 > 1, r2 > 0. Let Ω1 , Ω2 ⊂ Rn be deﬁned as follows: Ω1 = Brk2 (θ) × [0, r1 ] & % := (x1 , x2 , . . . , xk+1 , 0, . . . , 0) x21 + . . . x2k = r22 , 0 ≤ xk+1 ≤ r1 , % & Ω2 = S n−k−1 := (0, 0, . . . , xk+1 , . . . , xn ) x2k+1 + . . . + x2n = 1 , and let φ1 = id|∂Ω1 , φ2 = idΩ2 . Then φ1 and φ2 link. 2 In fact, let F (x1 , x2 , . . . , xn ) = (x1 , . . . , xk , xk+1 − (1 − x2k+2 − . . . − x2n ), xk+2 , . . . , xn ), it has a unique zero in Ω1 × Ω2 : x1 = · · · = xk = xk+2 = · · · = xn = 0, xk+1 = 1. Therefore, deg(F, Ω1 × Ω2 , θ) = 1 . x3

Ω2 r1 θ

x1

Ω1 r2 x2 Fig. 3.1.

The Borsuk–Ulam theorem is applied to the study of the intersection of symmetric sets. Assume that X is a Banach space. To a closed symmetric set A ⊂ X\{θ}, we deﬁne the genus γ(A) of A to be inf{k ∈ N | ∃ an odd φ ∈ C(A, Rk \{θ})}. Example 2. Assume that Ω ⊂ Rn is a symmetric bounded open set containing θ, and that there is an odd homeomorphism h : A → ∂Ω. Then γ(A) = n. In fact, by deﬁnition, γ(A) ≤ n. Suppose γ(A) < n, then there exist k < n and an odd φ such that φ ∈ C(A, Rk \{θ}). Let g = φ ◦ h−1 , then g : ∂Ω → Rk \{θ} is continuous and odd. According to the Borsuk–Ulam theorem, there exists x0 ∈ A such that g(x0 ) = θ. This is a contradiction.

3.4 Leray–Schauder Degrees

155

3.4 Leray–Schauder Degrees In analysis we study continuous mappings in inﬁnite-dimensional spaces, so we should be concerned with extending the Brouwer degree from ﬁnitedimensional spaces to inﬁnite-dimensional Banach spaces. However, the generalization cannot suit arbitrary continuous mappings. This can be seen from the fact that if this sort of degree theory, which possesses the fundamental properties homotopy invariance, additivity and normality established in Sect. 3.2, had been built up, then the Brouwer ﬁxed-point theorem would be extended directly to Banach spaces as follows. Let B be the unit open ball at the origin in an inﬁnite-dimensional real Banach space X, φ ∈ C(B, B). Let ft = id − tφ, t ∈ [0, 1] and θ ∈ ft (∂B). If deg (ft , B, θ) had been extended such that all the above properties of the Brouwer degree hold, then there would be a zero of f1 = id − φ, i.e., a ﬁxed point x ∈ B of φ: φ(x) = x . The “proof” is the same as in the previous section; we repeat it as follows: From the hypothesis θ ∈ ft (∂B), according to the homotopy invariance and the normality, deg (f1 , B, θ) = deg (id, B, θ) = 1 . Hence by the Kronecker existence, f1−1 (θ) ∩ B = Ø, i.e., ∃x ∈ B such that φ(x) = x. However, for an inﬁnite-dimensional Banach space X, the above conclusion cannot be true for any continuous mapping φ. Example. Let X=l = 2

x = (x1 , x2 , . . . , xn , . . .)| x = 2

∞

x2n

0, there is a continuous operator Kε taking values in a ﬁnite-dimensional linear subspace Emε such that K(x) − Kε (x) ε

∀x ∈ Ω .

Proof. Let Bε (yj ) j = 1, . . . , mε , be ﬁnitely many ε− balls covering K(Ω). Let ψj (x) = (ε− x − yj )+ λ, λ 0 , λ+ = 0, λ < 0 .

where

Let ϕi (x) =

ψi (x) , mε j=1 ψj (x)

and Kε (x) =

mε

i = 1, 2, . . . , mε ,

ϕi (K(x))yi ,

i=1

then Kε (x) ∈ span {y1 , . . . , ymε } and K(x) − Kε (x)

m

ϕi (K(x)) K(x) − yi

i=1 mε

ε

ϕi (K(x)) = ε .

i=1

The idea in extending the degree to mapping f = id − K on bounded open set Ω ⊂ X, where K is a compact map, is by approximation: ∀ε > 0 arbitrarily small, we already have the Brouwer degree for fε = id − Kε , then we want to deﬁne the degree for f by Brouwer degrees for these fε . In doing so, we should verify: (1) If y0 ∈ X satisﬁes y0 ∈ f (∂Ω), then for small ε > 0, y0 ∈ fε (∂Ω). (2) For any two such fε1 and fε2 , the Brouwer degrees for fε1 , fε2 are equal. In proving (1), we need: Lemma 3.4.3 Let K : Ω → X be compact, f = id − K and S ⊂ Ω be closed. Then f (S) is closed. Proof. Let {xn } ⊂ S with f (xn ) → z ∗ . We want to show that z ∗ ∈ f (S). Indeed, there are a subsequence {ni } and y ∗ ∈ X such that Kxni → y ∗ , and then xni → z ∗ − y ∗ , which we write as x∗ . Then x∗ ∈ S. By the continuity of K, Kx∗ = y ∗ , and hence f (x∗ ) = z ∗ , i.e., z ∗ ∈ f (S).

3.4 Leray–Schauder Degrees

157

Thus, y0 ∈ f (∂Ω) implies dist (y0 , f (∂Ω)) > 0. Choosing 0 < ε < dist (y0 , f (∂Ω)), we have dist (y0 , fε (∂Ω)) > 0 . Now we turn to (2). Consider the mapping fε : Ω ∩ Rmε → Rmε where mε = dim span{ Kε (Ω)}. The Brouwer degree deg (f , Ω ∩ Rm , y0 ) is well deﬁned. Since mε depends on ε, we should compare all the degrees for these fε . Namely, we have: Lemma 3.4.4 Let Ω ⊂ Rn be a bounded open set, Rm ⊂ Rn , and i : Rm → Rn be the canonical immersion: ˆ = (x1 , . . . , xm , 0, . . . , 0) . x = (x1 , . . . , xm ) → x Let K : Ω → Rm be continuous, f = id − K and p ∈ Rm satisfying pˆ ∈ f (∂Ω). Then deg (f, Ω, pˆ) = deg (f |Rm ∩Ω , Rm ∩ Ω, p) .

ˆ ∈ C 1 (Ω, Rm ) satisfying Proof. Let ε = dist (ˆ p, f (∂Ω)) > 0. Choose K ε ˆ ˆ K − K C(Ω) < 2 . Let g = id − K. Then idm − Jg (y) = det 0

ˆi ∂K ∂xj

ˆ

Ki − ∂∂x k idn−m

i, j = 1, . . . , m, ¯. , ∀y ∈ Ω k = m + 1, . . . , n,

By the Sard theorem the critical value set of g|Rm ∩Ω : Rm ∩ Ω → Rm is an m-dimensional set of measure zero. It has a regular value q ∈ Rm such that p − q < ε/2, and qˆ ∈ g(∂Ω). Thus deg (f, Ω, pˆ) = deg (g, Ω, qˆ) (homotopy invariance) sgn Jg (yi ) = yi ∈g −1 (ˆ q)

=

yi ∈g|−1 m R

∩Ω

sgn Jg |Rm ∩Ω (yi ) (g −1 (ˆ q ) ⊂ Rm ∩ Ω) (ˆ q)

= deg (g|Rm ∩Ω , Rm ∩ Ω, q) = deg (f |Rm ∩Ω , Rm ∩ Ω, p) (homotopy invariance)

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3 Degree Theory and Applications

Now we are ready to deﬁne the topological degree for f = id − K when K is compact. Deﬁnition 3.4.5 (Leray–Schauder degree) Let X be a real Banach space, and Ω ⊂ X be bounded and open. Let K : Ω → X be compact, f = id − K and p ∈ X\f (∂Ω). Deﬁne deg (f, Ω, p) = deg (fε , Ω ∩ Eε , p) , where ε ∈ (0, dist(p, f (∂Ω))), fε = id − Kε , and Kε is a continuous operator assuming values in a ﬁnite-dimensional space Eε with p ∈ Eε and satisﬁes Kx − Kε x ε

∀x ∈ Ω .

Let us explain the legitimacy of the deﬁnition. 1. By Lemma 3.4.3, dist(p, f (∂Ω)) > 0, according to Theorem 3.4.2, there exist such Kε and Eε . 2. We verify that deg (fε , Ω ∩ Eε , p) is well deﬁned for all such fε and Eε and takes the same value. Since dist(p, fε (∂Ω ∩ Eε )) dist(p, f (∂Ω)) − ε > 0 , deg (fε , Ω ∩ Eε , p) is well deﬁned. Let (fε0 , Eε0 ) and (fε1 , Eε1 ) be two arbitrary pairs of mappings and ﬁnite-dimensional linear subspaces satisfying the hypotheses of the deﬁnition, and let ˆε , ˆ = span{Eε , Eε }, fˆε = id − K E 0 1 i i ˆ ε (x) = (Kε (x), 0) and 0 is the zero element in E ˆ & Eε , i = 0, 1. where K i i i By Lemma 3.4.4, we have ˆ ∩ Ω, p), i = 0, 1 . deg (fεi , Eεi ∩ Ω, p) = deg (fˆεi , E ˆ × [0, 1] → E ˆ by Let us deﬁne φ : (Ω ∩ E) φ(x, t) = tfˆε0 (x) + (1 − t)fˆε1 (x),

∀t ∈ [0, 1] .

Applying the homotopy invariance of the Brouwer degree we obtain ˆ ∩ Ω, p) = deg (fˆε , E ˆ ∩ Ω, p) . deg (fˆε0 , E 1 This proves deg (fε0 , Eε0 ∩ Ω, p) = deg (fε1 , Eε1 ∩ Ω, p) .

3.4 Leray–Schauder Degrees

159

Remark 3.4.6 A set Ω ⊂ X is called ﬁnitely bounded, if for all linear ﬁnitedimensional subspaces E ⊂ X, E ∩ Ω is bounded. In Deﬁnition 3.4.5, one may assume that Ω is ﬁnitely bounded. More generally, for a given {e1 , . . . , ek } ⊂ E, if for all linear ﬁnitedimensional subspaces E ⊂ X with {e1 , . . . , ek } ⊂ E, E ∩ Ω is bounded, then Ω is called {e1 , . . . , ek }- ﬁnitely bounded. Again, the Leray–Schauder degree is well deﬁned on open sets, which are {e1 , . . . , ek }-ﬁnitely bounded. Similarly to the Brouwer degree, the Leray–Schauder degree enjoys the following fundamental properties: (1) (Homotopy invariance) Let K : Ω × [0, 1] → X be compact and p ∈ (id − K)(∂Ω × [0, 1]), then deg (id − K(·, t), Ω, p) = constant . (2) (Translation invariance) deg (id − K, Ω, p) = deg (id − K − p, Ω, θ) . (3) (Additivity) Let Ω1 , Ω2 ⊂ Ω, Ω1 ∩ Ω2 = Ø and p ∈ (id − K)(Ω\(Ω1 ∪ Ω2 )), then deg (id − K, Ω, p) = deg (id − K, Ω1 , p) + deg (id − K, Ω2 , p) . (4) (Normality)

1, deg (id, Ω, p) = 0,

p∈Ω, p ∈ Ω .

Similarly, all the other properties are consequences of these properties. Since the proofs are standard (reducing to ﬁnite-dimensional spaces by ε-approximations, and applying the corresponding properties of the Brouwer degree), we omit them. In particular, we have: (5) (Kronecker existence) Let Ω ⊂ X be a bounded open set and let K : Ω → X be compact. If y0 ∈ (id − K)(∂Ω) and deg (id − K, Ω, y0 ) = 0, then there exists x0 ∈ Ω satisfying x0 = Kx0 + y0 . (6) Let K be a compact linear operator, 1 ∈ σ(K) (the spectrum of K) and θ ∈ Ω, then deg (id − K, Ω, θ) = (−1)β , where β=

λj >1,λj ∈σ(K)

k βj , βj = dim ∪∞ k=1 ker (λj I − K) .

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3 Degree Theory and Applications

Proof. According to the Riesz–Schauder theory, σ(K) has only a point spectrum except 0. Let E1 be the ﬁnite-dimensional subspace spanned by all the generalized eigenvectors corresponding to eigenvalues > 1, then we have the direct sum decomposition X = E1 ⊕ E2 . Both E1 and E2 are invariant subspaces of K. Deﬁne KEi = Ki , then, for ∀t ∈ [0, 1], for ∀x ∈ E2 \{θ}, tK2 x = x. The homotopy invariance and the excision ensure that ∃ε > 0 such that deg (id − K, Ω, θ) = deg (id − K, Bε , θ) = deg (id − (K1 ⊕ tK2 ), Bε , θ) = deg (id − K1 , Bε , θ) = (−1)β . Now, let us return to the discussion in the beginning of this section. We have the special form of the Schauder ﬁxed-point theorem: if K : B → B is compact, then K has a ﬁxed point in B. However, the unit ball B may be replaced by any bounded closed convex set. Because it is known from Dugundji’s theorem (see Sect. 3.6) that for any closed convex subset C of X, there is a retract r : X → C, i.e., r is continuous, and satisﬁes r ◦ i = id|C , and i ◦ r ∼ id|X , where i is the injection C → X. The following Schauder ﬁxed-point theorem, which we have met in Sect. 2.2 is a consequence of the Kronecker existence property of the Leray–Schauder degree. Theorem 3.4.7 (Schauder ﬁxed point) Let C be a bounded closed convex set in X. If K : C → C is compact. Then K has a ﬁxed point. Proof. We choose R > 0 large enough such that C ⊂ BR (θ). Let r : B R (θ) → C be a retract, and i : C → B R (θ) be the injection. We have a compact mapping f = i ◦ K ◦ r : B R (θ) → B R (θ): B R⏐(θ) ⏐ r0 C

B R3(θ) ⏐ i⏐ K

−→

C

Following the steps in the proof of the Brouwer ﬁxed-point theorem, there is a ﬁxed point x0 ∈ B R of f , i.e., x0 = i ◦ K ◦ r(x0 ). Thus x0 ∈ C and then r(x0 ) = x0 . It follows that x0 = Kx0 . Another ﬁxed-point theorem, which is also frequently used in the theory of diﬀerential equations in studying the existence of solutions by a priori estimates, reads as: Theorem 3.4.8 (Schaefer) Suppose that K : X × [0, 1] → X is a compact mapping, satisfying K(x, 0) = θ. If the set S = {x ∈ X| ∃t ∈ [0, 1] such that x = K(x, t)} is bounded. Then K = K(·, 1) has a ﬁxed point.

3.4 Leray–Schauder Degrees

161

◦

Proof. Taking r > 0 large such that S ⊂ B r , we deﬁne f (x, t) = x − K(x, t)

∀(x, t) ∈ B r × [0, 1]

then θ ∈ f (∂Br × [0, 1]). According to the homotopy invariance deg (id − K, Br , θ) = deg (id, Br , θ) = 1 . Corollary 3.4.9 Suppose that K : X → X is compact and that the set S = {x ∈ X| ∃t ∈ [0, 1] such that x = tK(x)} is bounded. Then K has a ﬁxed point. Proof. Set K(x, t) = tK(x) As an application of Schaefer’s ﬁxed-point theorem, we study the following semi-linear elliptic BVP, which we have met in Sect. 1.2. Let Ω ⊂ Rn be a bounded open domain with smooth boundary. Theorem 3.4.10 Suppose that f ∈ C γ (Ω×R1 ×Rn , R1 ), for some γ ∈ (0, 1), satisﬁes (1) ∃ an increasing function c : R1+ → R1+ such that |f (x, η, ξ)| c(|η|)(1 + |ξ|2 )

∀(x, η, ξ) ∈ Ω × R1 × Rn .

(2) ∃ a constant M > 0 such that f (x, η, θ)

0

as η > M as η < −M.

Assume φ ∈ C 2,γ (∂Ω). Then the equation −u = f (x, u(x), ∇u(x)) u|∂Ω = φ

(3.17)

has a solution u ∈ C 2,γ (Ω). Proof. Fixing γ ∈ (0, γ), we deﬁne a map T from C 2,γ (Ω) = X into itself by u → v, where v is the solution of the following BVP: −v = f (x, u(x), ∇u(x)) v|∂Ω = φ .

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3 Degree Theory and Applications

By the Schauder estimate, we obtain v C 2,γ (Ω) C( u C 2,γ (Ω) , φ C 2,γ (∂Ω) ) . Thus, T is compact. Let us introduce a parameter t, and deﬁne v = T (u, t) to be the solution of the equation: −v = tf (x, u(x), ∇u(x)) v|∂Ω = φ . We intend to apply Schaefer’s ﬁxed-point theorem; it is suﬃcient to verify the boundedness of the set S = {u ∈ X| ∃t ∈ [0, 1] such that u = T (u, t)} , i.e., ∃ a constant C > 0 such that solutions ut of the equations −ut = tf (x, ut (x), ∇ut (x)) ut |∂Ω = φ

(3.18)

satisfy the following a priori estimate of the solution of (3.17): ut C 2,γ C

(3.19)

However, this has been done in Sect. 1.2.

Comparing the continuity method with the ﬁxed-point method based on Schaefer’s theorem, the latter does not require the invertible of the linearized equation. Moreover, less regularity on the nonlinear term is assumed. Of course, there is only the existence of a solution but not uniqueness. The Borsuk theorem is also extended. Theorem 3.4.11 Let Ω ⊂ X be a symmetric bounded open set including θ. If K : Ω → X is an odd compact map with θ ∈ f (∂Ω), where f = id − K, then deg (f, Ω, θ) is odd. Proof. We approximate f by ﬁnite-dimensional odd maps. Indeed, in Theorem 3.4.2, let K be approximated by Kε , with Im Kε ⊂ span{y1 , . . . , ymε }, and let ˆ ε (x) = 1 [Kε (x) − Kε (−x)] . K 2 Then ˆ ε (x) K(x) − K

1 1 K(x) − Kε (x) + K(−x) − Kε (−x) < ε , 2 2

and ˆ ε ⊂ span{y1 , . . . , ym } . Im K ε The conclusion follows from Theorem 3.3.3 directly.

3.4 Leray–Schauder Degrees

163

Corollary 3.4.12 Let Ω ⊂ X be a symmetric bounded open set including θ. If K : Ω → X is compact, f = id − K, and if f (x) = tf (−x)

∀(x, t) ∈ ∂Ω × [0, 1] ,

then deg (f, Ω, θ) is odd. Proof. Deﬁne

1 t K(x) − K(−x) 1+t 1+t 1 t = f (x) − f (−x). 1+t 1+t

φ(x, t) = id −

Thus θ ∈ (∂Ω × [0, 1]), φ(·, 0) = f and φ(·, 1) = 12 (f (x) − f (−x)) is odd. We obtain deg (f, Ω, θ) = deg (φ(·, 1), Ω, θ) = odd number . By the same proof of the invariance of domains theorem for the Brouwer degree, we have its inﬁnite-dimensional version: Corollary 3.4.13 (Invariance of domains) Assume that Ω ⊂ X is a nonempty open set, and that K : Ω → X is compact. If f = id − K is locally injective, then f is an open map. The notion of the index of an isolated solution is also extended. For f = id − K, where K is a compact map, we deﬁne the index of f at an isolated ﬁxed point x0 as follows: i(f, x0 , θ) = deg (f, Bε (x0 ), θ) for suﬃciently small ε > 0. (7) In particular, if K is diﬀerentiable at x0 , where f (x0 ) = θ, and if f (x0 ) is invertible, then we have i(f, x0 , θ) = (−1)β , where β =

{λj >1 | λj ∈σ(K (x0 ))}

k βj and βj = dim ∪∞ k=1 ker (λj I − K (x0 )) .

Proof. By homotopy invariance, ∃ε > 0 such that i(f, x0 , θ) = deg (f, Bε (x0 ), θ) = deg (id − K (x0 ), Bε (x0 ), θ) . It is suﬃcient to verify that K (x0 ) is a compact linear operator, because if it is so, then the conclusion follows directly from (6). The veriﬁcation is as follows:

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3 Degree Theory and Applications

If T = K (x0 ) were not compact, then there would be {xj } ⊂ B1 (θ) and ε > 0 such that T xi − T xj ε as i = j . Choose δ > 0 small such that ∀k = 1, 2, . . . , K(x0 + δxk ) − K(x0 ) − δT xk

εδ , 4

thus εδ K(x0 + δxi ) − K(x0 + δxj ) − δT (xi − xj ) 2 δ T (xi − xj ) − K(x0 + δxi ) − K(x0 + δxj ) δε− K(x0 + δxi ) − K(x0 + δxj ) ,

that is K(x0 + δxi ) − K(x0 + δxj )

1 εδ, i = j . 2

It contradicts with the compactness of K.

The Leray-Schauder degree theory is also applied to the study of the intersections of inﬁnite-dimensional manifolds, in which the continuous mappings should be replaced by compact vector ﬁelds.

3.5 The Global Bifurcation We have studied in Sect. 1.3, the local bifurcation phenomenon, which describes a branch of solutions splitting into several branches. Having the topological tools at hand, we are able to provide more precise local information and to investigate the global behavior of bifurcating branches. Consider F : X × R1 → X in the following form: F (x, λ) = Lx − λx + N (x, λ) ,

(3.20)

where X is a real Banach space, L ∈ L(X, X), λ ∈ R1 , and N (x, λ) = o ( x ) as x → θ uniformly in any ﬁnite interval of λ. As we know, bifurcation points (θ, λ0 ) only occur at λ0 ∈ σ(L), i.e., λ0 is a spectrum of L. In particular, if L is compact, and λ0 = 0, then λ0 is an eigenvalue of L. Conversely, we have: Theorem 3.5.1 (Knasnoselski) Suppose that L is a linear compact operator on X,+and that λ0 = 0 is an eigenvalue of L with odd multiplicity, ∞ i.e., β = dim k=1 ker(L − λ0 I)k is odd. If ∀λ, N ( · , λ) is compact, and N is continuous in x and λ, and satisﬁes N (x, λ) = o ( x )

3.5 The Global Bifurcation

165

uniformly in any ﬁnite interval of λ, then (θ, λ0 ) is a bifurcation point of the equation F (x, λ) = θ. Proof. Set S = {(x, λ) ∈ X × R1 | F (x, λ) = θ} , and S+ = S \ ({θ} × R1 ) . If (θ, λ0 ) is not a bifurcation point, then there is a closed interval [λ− , λ+ ] not including 0 and satisfying: 1. σ(L) ∩ [λ− , λ+ ] = {λ0 }, 2. ∃r > 0 such that (Br (θ) × [λ− , λ+ ]) ∩ S+ = ∅. In the sequel we write Br = Br (θ) brieﬂy. Deﬁne a deformation: Φ(x, t) = x −

1 (Lx + N (x, λ(t))) λ(t)

t ∈ [0, 1] ,

where λ(t) = tλ− + (1 − t)λ+ . Hence θ ∈ / Φ(∂Br × [0, 1]). It follows that 1 1 (L + N (· , λ+ )), Br , θ = deg id − (L + N (· , λ− )), Br , θ . deg id − λ+ λ− For suﬃciently small r > 0, again by the homotopy invariance, we obtain 1 1 L, Br , θ = deg id − L, Br , θ , deg id − λ+ λ− i.e., βj

(−1)λj >λ+

βj

= (−1)λj >λ−

,

where λj ∈ σ(L), and βj is the multiplicity of λj . This is (−1)β = 1 . By assumption β is odd. This is impossible. The oddness of the algebraic multiplicity of λ0 is a suﬃcient condition for bifurcation points, but not necessary (see Chap. 5, Theorem 5.1.37.) We turn to studying some global results. Lemma 3.5.2 Let K be a compact metric space, K1 , K2 be disjoint closed subsets. Then the following alternatives hold: either 1. ∃ a component of K intersecting K1 and K2 , or 4 2 such that 41, K 2. ∃ compact subsets K 4 i , i = 1, 2, Ki ⊂ K 41 ∪ K 42, K=K and

41 ∩ K 42 = ∅ . K

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3 Degree Theory and Applications

Proof. If (1) is not true, then ∃ ε0 > 0 such that each ε0 − chain cannot intersect with K1 and K2 simultaneously. For otherwise, ∀ ε > 0 ∃ aεi ∈ Ki 1

and ε-chain Cε connecting aεi , i = 1, 2. Since Ki , i = 1, 2, are compact, {ain } has a limiting point ai ∈ Ki , i = 1, 2. Set Ca1 = {x ∈ K| ∀ ε > 0 a1 can be connected with x by an ε-chain} . Since K is compact, Ca1 is a closed connected set. Therefore, a2 ∈ Ca1 . This means that (1) holds. We arrive at a contradiction. 4 1 = {y ∈ K| ∃ x ∈ K1 and ∃ε0 chain connecting x and y}. To the ε0 , let K 41. 4 1 ∩ K2 = ∅ and K1 ⊂ K Obviously, K 4 1 is both open and closed. We shall prove that K 4 1 , Bε (x) ∩ K 4 1 = ∅, which implies x ∈ K 41, On the one hand, ∀ x ∈ K 0 4 1 , Bε (x) ⊂ K 4 1 , so it is also 4 1 is closed. On the other hand, ∀x ∈ K therefore K 0 4 4 4 4 open. We set K2 = K\K1 . Then K1 and K2 meet all conditions we need. Theorem 3.5.3 (Leray-Schauder) Let X be a real Banach space, T : X × R1 → X be a compact map satisfying T (x, 0) = θ, and f (x, λ) = x − T (x, λ). Let S = {(x, λ) ∈ X × R1 | f (x, λ) = θ} ,

$

ζε S

X

O ζ+

U

λ λ∗

(θ, 0)

Fig. 3.2.

3.5 The Global Bifurcation

167

and let ζ be the component of S passing through (θ, 0). If ζ ± = ζ ∩ (X × R1± ) , then both ζ + and ζ − are unbounded. Proof. Since T is compact, if ζ + (or ζ − ) is bounded, then it is compact. We consider a ε− neighborhood ζε of ζ + . Let K = ζε ∩ S. This is a compact metric space. 4 i , i = 1, 2, such Set K1 = ζ + , K2 = ∂ζε ∩ S. According to Lemma 3.5.2, ∃K that 4i i = 1, 2, Ki ⊂ K 41 ∪ K 42, K=K and Choosing

41 ∩ K 42 = ∅ . K

4 2 ), dist(K 4 1 , ∂ζε )} , 41, K 0 < δ < min{dist(K

4 1 , we have and deﬁning a δ/2− neighborhood O of K ∂O ∩ S = ∅, ζ+ ⊂ O . Setting

2 U = O ∪ {(x, λ) ∈ X × R1− x + λ2 < δ 2 } ,

and λ∗ larger than the projection of O onto R1+ , we consider the map F : (X × R1 ) × [0, 1] → X × R1 as follows: (x, λ, t) → (f (x, λ), λ − tλ∗ ) .

Then ⇐⇒

F (x, λ, t) = (θ, 0)

f (x, λ) = θ . λ = tλ∗

Now, ∀(x, λ) ∈ ∂O, by construction, f (x, λ) = θ; ∀(x, 0) ∈ U, if f (x, 0) = θ, then x = θ; ∀(x, λ) ∈ X × R1− with x + λ2 = δ 2 , we have λ = tλ∗ . 2

Since ∂U ⊂ ∂O ∪

δ 2 (x, 0) ≤ x ≤ δ ∪ {(x, λ) ∈ X × R1− x + λ2 = δ 2 } , 2

we have (θ, 0) ∈ / F (∂U × [0, 1]). Thus deg( F ( · , · , 0), U, (θ, 0) ) = deg( F ( · , · , 1), U, (θ, 0) ) .

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3 Degree Theory and Applications

On the one hand, since λ = λ∗ on U , from Knonecker existence theorem, we have deg( F ( · , · , 1), U, (θ, 0) ) = 0 . On the other hand, ∀t, x − T (x, tλ) = θ, λ = 0,

⇐⇒

x = θ, λ = 0.

This implies (x − T (x, tλ), λ) = (θ, 0) on ∂U . Then, by the homotopy invariance and the excision property, deg( F ( · , · , 0), U, (θ, 0) ) = deg( idX×R1 , B δ (θ, 0), (θ, 0) ) = 1 . 2

This is the contradiction.

Although the Leray–Schauder theorem is not directly related to bifurcation problems (if T (θ, λ) = θ, then the conclusion is trivial: ζ = {θ} × R1 ), but in applications, it can be used in the spirit of the following global bifurcation theorem due to P. Rabinowitz and improved by Ize [Iz]. Theorem 3.5.4 (Rabinowitz) Let X be a real Banach space and F (x, λ) = x − λLx − N (x, λ), where L ∈ L(X, X) and N : X × R1 → X are compact. Let S be the solution set of F (x, λ) = θ, S+ = S \ ({θ} × R1 ), and let ζ be the component of S+ , containing (θ, λ1 ). Assume that N (x, λ) = o( x) uniformly on any ﬁnite interval in λ and that λ−1 1 ∈ σ(L) is an eigenvalue of odd multiplicity. Then the following alternatives hold: Either 1. ζ is unbounded; or 2. there are only ﬁnite number of points {(θ, λi ) i = 1, , . . . , l} lying on ζ, where λ−1 ∈ σ(L), i = 1, 2, . . . , l. Furthermore, if βi is the algebraic muli l tiplicity of λ−1 i , then i=1 βi is even. Proof. If ζ is bounded, then ζ is compact, because both L and N are compact. Since L is a compact linear operator, there are at most ﬁnitely many points {(θ, λj ) λj ∈ σ(L−1 ), j = 1, . . . , l } lying on ζ. Let 0 < ε < dist(ζ, {λ | λ−1 ∈ σ(L), λ = λ1 , λ2 , . . . , λl }), and let ζε be the ε− neighborhood of ζ. Set K = ζε ∩ S+ . Then K is a compact metric space, and ζ ∩ ∂ζε = ∅. Set K1 = ζ, 4 i , i = 1, 2, such K2 = S+ ∩ ∂ζε . According to Lemma 3.5.2, ∃ compact sets K 4 i , i = 1, 2, K = K 41 ∪ K 4 2 and K 41 ∩ K 4 2 = ∅. Set 0 < δ < that Ki ⊂ K 4 2 ), dist(K 4 1 , ∂ζε )}, and let O be the 1 δ neighborhood of K 41. 41, K min{dist(K 2 Then we have ζ ⊂ O ⊂ O ⊂ ζε , . S+ ∩ ∂O = ∅ If λ−1 ∈ σ(L), and λ = λ1 , λ2 , . . . , λl , then (θ, λ) ∈ / O. We consider the following map:

3.5 The Global Bifurcation

169

X

λ1

λ2

λ3

λl+1

λl

λ

ζ

O

Fig. 3.3.

Φ : O × R1 → X × R1 , Φ(x, λ; t) = (F (x, λ), x2 − t2 ) . We claim that (θ, 0) ∈ / Φ(∂O × {t}) as t = 0. Indeed, if Φ(x, λ; t) = (θ, 0), i.e., F (x, λ) = θ ; x2 = t2 = 0 then (x, λ) ∈ S+ , but S+ ∩ ∂O = ∅. By the homotopy invariance, for 0 < r < R, we have deg(Φ( · , R), O, (θ, 0)) = deg(Φ( · , r), O, (θ, 0)) . We choose R large enough such that O ⊂ BR (θ, 0), the ball in X × R1 . From the Knonecker existence theorem, deg(Φ( · , R), O, (θ, 0)) = 0 . Then we choose ε > 0 such that the ε-balls Bε ((θ, λj )) ⊂ O, j = 1, 2, . . . , l. For small r > 0, one wishes to prove:

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3 Degree Theory and Applications

deg(Φ( · , r), O, (θ, 0)) =

l

deg(Φ( · , r), B√r2 +ε2 ((θ, λj )), (θ, 0)) .

(3.21)

i=1

Let K > 0 be an upper bound: (id − λL)−1 K , l +

for λ in the projection of O onto R1 subtracting the set

(λj − ε, λj + ε).

j=1

We choose r > 0 such that N (x, λ)

0. Let us extend this conclusion to nonlinear problems (3.22). Set Sj = {v ∈ C01 ( [0, π] ) v has exactly (j − 1) simple roots in (0, π), and v (0), v (π) = 0} j = 1, 2, . . .. Thus Sj is an open set in C01 ( [0, π] ), and Si ∩ Sj = ∅

as

i = j ,

i, j = 1, 2, . . .. We shall prove that ζi ∩ ζj = ∅ as i = j. It is suﬃcient to prove that ζj ⊂ (Sj × R1 ) ∪ {(θ, λj )},

j = 1, 2, . . .

3.5 The Global Bifurcation

173

Lemma 3.5.5 There exists a neighborhood Oj of (θ, λj ) such that if (u, λ) ∈ Oj \({θ} × R1 ) is a solution of (3.22) then u ∈ Sj , ∀j. Proof. We prove by contradiction. If ∃ solutions (un , αn ) of (3.23), with un = / Sj . Let vn = uunn , then θ, satisfying (un , αn ) → (θ, λj ), but un ∈ vn = αn Lvn + un −1 N (un , αn ) . Since vn = 1, αn → λj , un −1 N (un , αn ) = o(1), and L is compact, we have vn → v, v = 1, satisfying v = λj Lv . According to the Sturm-Liouville theorem, v ∈ Sj . This contradicts the open ness of Sj . Now, we are going to prove: Theorem 3.5.6 Suppose that h satisﬁes (3.24). Then the branch of nontrivial solutions ζj passing through (θ, λj ) is included in (Sj × R1 ) ∪ {(θ, λj )}. In particular, ζj is unbounded. Proof. From Lemma 3.5.5, ζj ∩ Oj ⊂ (Sj × R1 ) ∪ {(θ, λj )}. We shall prove that ζj is entirely contained in the above set. If not, ∃(u∗ , λ∗ ) ∈ ζj ∩ ∂(Sj × R1 )\(θ, λj ). Since ∂(Sj × R1 ) = ∂Sj × R1 , and u∗ ∈ ∂Sj implies that u∗ has at least a double zero, i.e., ∃ξ ∈ [0, π] such that u∗ (ξ) = u∗ (ξ) = 0, it follows from ∈ σ(L), i = j, the uniqueness of ODE, u∗ = θ. Thus (u∗ , λ∗ ) = (θ, λi ), λ−1 i according to the necessary condition of bifurcation. Again, by Lemma 3.5.5, i = j. This is a contradiction. Our conclusion follows from Theorem 3.5.4 directly. Remark 3.5.7 One may consider other boundary conditions than the Dirichlet condition. Let us return to the example in Sect. 1.3: x ∈ (0, π) , −u = λ sin u (3.25) u(0) = u(π) = 0 . Applying Theorem 3.5.6, we conclude that ∀ n = 1, 2, . . . , there is an unbounded branch of solutions ζn passing through the bifurcation point (θ, n2 ). Again, one may prove that if (u, λ) ∈ ζn , then λ n2 . Indeed, if λ < n2 , then sin u < n2 . u Comparing (3.25) with the equation λ

(3.26)

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3 Degree Theory and Applications

−v = n2 v x ∈ (0, π) v(0) = v(π) = 0,

(3.27)

we would have v ∈ Sn+1 , on account of Sturm’s theorem. But v ∈ Sn . This is a contradiction. Moreover, ∀(u, λ) ∈ ζn ,

ξ u (x) = λ sin u(t)dt , x

where ξ ∈ (0, π) is a root of u (x) = 0. Thus u C λπ , and uC 1 λπ(1 + π) .

(3.28)

Combining (3.26) with (3.28), we have ζn ⊂ {(u, λ) ∈ C01 ([0, π]) × R1 λ n2 , uC 1 λπ(1 + π)} . Hence, the projection of ζn onto R1 is the interval (n2 , ∞). For any λ0 ∈ (n , ∞), there exist at least n branches ζ1 , ζ2 , . . . , ζn that meet λ = λ0 . On (j ) each ζj there exists at least a pair of nontrivial solutions ±uλ0 , j = 1, 2, . . . , n. Namely: 2

C01([0, π])

1

4

9

Fig. 3.4.

λ

3.6 Applications

175

Theorem 3.5.8 ∀λ ∈ (n2 , (n + 1)2 ] equation (3.26) has at least n distinct pairs of nontrivial solutions. Remark 3.5.9 It is natural to ask if we can extend the above method to study elliptic nonlinear eigenvalue problems as that for ODE. The main obstruction is that there is no counterpart of the Sturm–Liouville theory for PDEs. There are only a few partial results in this direction. Remark 3.5.10 (Bifurcation at inﬁnity) We study the bifurcation at inﬁnity by transforming the problem into the bifurcation at zero. We consider the equation: x = λLu + M (x, λ) ,

(3.29)

where L is a linear compact operator on X, and M : X × R1 → X satisﬁes M (x, λ) = o(x) uniformly in any ﬁnite interval as x → ∞. (λ0 , ∞) is called a bifurcation point at inﬁnity, if there exists a sequence of solutions (λn , xn ) of (3.29), such that xn → ∞, and λn → λ0 . Now we deﬁne y = x−2 x, then y = x−1 . Equation (3.29) is transformed into: y = λLy + N (y, λ) , (3.30) where N (y, λ) = y2 M (y−2 y, λ). Since N (y, λ) = o(y) as y → θ , (3.30) is exactly the equation we studied previously.

3.6 Applications 3.6.1 Degree Theory on Closed Convex Sets In applications of the Leray–Schauder degree theory, boundary conditions should be carefully investigated, i.e., on the boundary there is no ﬁxed point of the compact vector ﬁeld. However, for mappings with images in a closed convex set, the notion of the boundary can be reduced. Let us present an extension version of the Tietze theorem, Dugundji’s theorem, concerning the extension of continuous mappings on closed convex sets, which we have met in Sect. 3.4. Theorem 3.6.1 (Dugundji) Suppose that A is a closed subset of a metric space X, and that C is a convex subset of a Banach space Y . Then for any continuous map f : A → C, there exists a continuous extension f! : X → C. Proof. Since X \ A is a metric subspace, it is paracompact. ∀x ∈ X\A, let 0 < r(x) < 12 dist(x, A), and let Br(x) (x) be the ball with radius r(x) and

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3 Degree Theory and Applications

center x . All these balls form a covering of X\A. The covering has a locally ﬁnite reﬁnement {Uα | α ∈ ∧}. On the set X\A, we deﬁne dist(x, X\Uα ) (β∈Λ) dist(x, X\Uβ )

λα (x) =

∀α ∈ Λ .

They satisfy:

supp λα ⊂ Uα , 0 λα (x) 1, and

λα (x) = 1 .

α∈Λ

∀α ∈ Λ we choose aα ∈ A, such that dist(aα , Uα ) < 2dist(A, Uα ) ,

and let f!(x) =

λα (x)f (aα ) f (x)

x ∈ X\A x∈A.

Since {Uα | α ∈ Λ} is a locally ﬁnite covering, the sum is ﬁnite. We claim that f! is continuous. It is suﬃcient to prove that f! is continuous on the boundary ∂A, i.e., for ∀x0 ∈ ∂A. ∀ε > 0, ∃δ > 0 such that f!(x) − f (x0 ) < ε , whenever dist(x, x0 ) < δ. In fact, ∃δ > 0, such that if dist(x, x0 ) < 6δ and x ∈ A, then f (x) − f (x0 ) < ε . Now, ∀x ∈ X, if dist(x, aα ) < 3δ, from dist(x0 , aα ) ≤ dist(x0 , x) + dist(x, aα ) , it follows that dist(x0 , aα ) < 4δ, then f (aα ) − f (x0 ) < .

(3.31)

Otherwise, dist(x, aα ) ≥ 3δ, then dist(x, aα ) ≥ 3dist(x, x0 ) ≥ 3dist(x, A) . In this case we claim that x ∈ Uα . If not, i.e., x ∈ Uα , then we would have dist(x, aα ) diam(Uα ) + dist(Uα , aα ) 2r(x) + 2dist(A, Uα ) dist(A, x) + 2dist(A, x) = 3dist(A, x)

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177

This is a contradiction, therefore, x ∈ Uα and then λα (x) = 0 .

(3.32)

Combining equations (3.31) and (3.32), we have λα (x) f (aα ) − f (x0 ) < ε . f!(x) − f (x0 ) dist(x,aα ) 0 is a large number such that U ⊂ BR (θ). Firstly, we point out that the deﬁnition makes sense, i.e, θ ∈ (id − K ◦ r)(∂(BR (θ) ∩ r−1 (U ))). In fact, if it is not true, then ∃p ∈ ∂(BR (θ) ∩ r−1 (U )) such that p = K ◦ r(p). Thus, p ∈ C and then p = Kp, which implies p ∈ C ∩ ∂(BR (θ) ∩ r−1 (U )), i.e., p ∈ ∂U . This is a contradiction. Secondly, one veriﬁes that the degree so deﬁned does not depend on the retraction r, and the radius R. In fact, let R1 < R2 such that U ⊂ int(BR1 (θ)). All ﬁxed points of id − K in U must be in BR1 (θ). By excision, deg (id − K ◦ r, r−1 (U ) ∩ BR2 (θ), θ) = deg (id − K ◦ r, r−1 (U ) ∩ BR1 (θ), θ) . Again let r1 , r2 be two retractions: X → C, by excision, deg (id − K ◦ ri , ri−1 (U ) ∩ BR (θ), θ) = deg (id − K ◦ ri , r1−1 (U ) ∩ r2−1 (U ) ∩ BR (θ), θ) . for i = 1, 2. We deﬁne F (x, t) = id − (tK ◦ r1 + (1 − t)K ◦ r2 ) ∀t ∈ [0, 1] . From the homotopy invariance, degC (id − K, U, θ) is independent of ri , i = 1, 2. In particular, if X is a real OBS, i.e., X is a real Banach space with a closed positive cone P (see Sect. 2.1), then P is a closed convex subset, and then degP (id − K, U, θ) makes sense. Now, we present a few degree computations for special boundary conditions in an OBS.

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3 Degree Theory and Applications

Lemma 3.6.2 Suppose that (X, P ) is an OBS, and that U ⊂ P is bounded and open. Assume that K : U → P is compact, satisfying ∃y ∈ P \{θ}, such that x − Kx = ty ∀t ≥ 0, ∀x ∈ ∂U ; then degP (id − K, U, θ) = 0 . Proof. From the homotopy invariance, deg (id − K − ty, U, θ) = constant, ∀t ≥ 0 . Since K is compact, ∃C > 0 such that x − Kx ≤ C ∀x ∈ U . One chooses C t > y , then x − Kx = ty ∀x ∈ U . It follows that degP (id − K, U, θ) = 0. Theorem 3.6.3 Suppose that (X, P ) is an OBS, U ⊂ P is bounded open and contains θ. Assume that ∃ρ > 0 such that Bρ (θ) ∩ P ⊂ U and that K : U → P is compact and satisﬁes: (1) ∀x ∈ P, with x = ρ, ∀λ ∈ [0, 1), x = λK(x), (2) ∃y ∈ P \{θ}, such that x − K(x) = ty, ∀x ∈ ∂U, ∀t ≥ 0. Then K possesses a ﬁxed point on Uρ , where Uρ = U \Bρ (θ). Proof. We may assume that K has no ﬁxed point on ∂Bρ (θ) then degP (id − K, U, θ) = degP (id − K, Uρ , θ) + degP (id − K, Bρ (θ) ∩ P, θ) . According to Lemma 3.6.2 and assumption (2), LHS = 0, and from assumption (1) and the homotopy invariance, the second term on the RHS = 1. Therefore, degP (id − K, Uρ , θ) = −1. Theorem 3.6.4 Suppose that (X, P ) is an OBS, U ⊂ P is bounded and open, and contains θ. Assume that ∃ρ > 0 such that U ⊂ Bρ (θ) ∩ P , and that Uρ := (Bρ (θ) ∩ P )\U possesses an interior point. If K : B ρ (θ) ∩ P → P is compact and satisﬁes: (1) ∀x ∈ P, x = ρ, ∀λ ∈ [0, 1), x = λK(x). (2) ∃y ∈ P \{θ} such that x − K(x) = ty, ∀t ≥ 0, ∀x ∈ ∂U. Then K has a ﬁxed point in U ρ . Proof. The proof is similar to the previous theorem: deg (id − K, Uρ , θ) = 1. We omit it. Remark 3.6.5 Let (X, P ) be an OBS, and let 0 < ρ1 < ρ2 , and Pρi = P ∩Bρi (θ). We write ∂Pρi = P ∩∂Bρi (θ), i = 1, 2. A map f : Pρ2 \Pρ1 → P is called a cone compression, if x f (x), ∀x ∈ ∂Pρ2 , and f (x) x, ∀x ∈ ∂Pρ1 . It is called a cone expansion if f (x) x, ∀x ∈ ∂Pρ2 , and x f (x) ∀x ∈ ∂Pρ1 . Noticing that

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179

f (x) x implies ∀y ∈ P \{θ}, ∀t ≥ 0, x − f (x) = ty , and x f (x) implies ∀λ ∈ [0, 1], x = λf (x) , Theorems 3.6.3 and 3.6.4 extend the earlier results of Krasnoselski on cone compression and cone expansion mappings. Let (X, P ) be an OBS satisfying X = P − P . A map T : P → P is called positive. We shall study the bifurcation problem for positive mappings on OBS. Let R+ = [0, +∞) and f : P × R+ → P be a continuous mapping satisfying f (θ, λ) = θ ∀λ ∈ R+ . We are looking for λ0 ∈ R+ , such that ∀ > 0, ∃(x, λ) ∈ (B+ \{θ}) × (λ0 − , λ0 + ) satisfying x = f (x, λ) , where B+ = B (θ) ∩ P . At this time, (θ, λ0 ) is a bifurcation point of positive solutions. Lemma 3.6.6 Let (X, P ) be deﬁned above. Let f : P × R+ → P be of the form: f (x, λ) = λT x + g(x, λ) , where T ∈ L(X, X) is a positive and compact operator, and g : P × R+ → P is compact, satisfying g(x, λ) = o(x) as x → 0 and g(x, 0) = θ. If (θ, λ0 ) is an eigenvalue of is a bifurcation point with λ0 ≥ 0, then λ0 > 0, and λ−1 0 T with positive eigenvector. Proof. By deﬁnition, ∃(xn , λn ) ∈ (P \{θ}) × R+ , such that (xn , λn ) → (θ, λ0 ), with xn = f (xn , λn ). Setting yn = xxnn , we have yn − λ0 T yn = (λn − λ0 )T yn + g(xn , λn )xn −1 . Deﬁne S+ = ∂B1 (θ) ∩ P , then yn ∈ S+ , and then, after a subsequence we have T yn → z0 . Let y0 = λ0 z0 , then y0 ∈ S+ . Obviously, y0 = λ0 T y0 , and λ0 > 0. One has the following variant of the global bifurcation theorem. Theorem 3.6.7 (Dancer) Let (X, B) and f, T, g be as in Lemma 3.6.6. Assume that T has ﬁnitely many positive eigenvalues with positive eigenvectors. Let Σ = {(x, λ) ∈ P ×R+ | x = f (x, λ)} and Σ+ = Σ ∩ ((P \{θ}) × (R+ \{0})). Then Σ+ contains an unbounded connected branch emanating from the reciprocal of one of these eigenvalues. be the smallest positive eigenvalue of T , and let ζ be the Proof. Let λ−1 0 connected component of Σ+ ∪ ({θ} × [0, λ0 ]), containing {θ} × [0, λ0 ]. If ζ is bounded, then ∃µ > λ0 such that ζ ⊂ Qµ := Bµ+ × [0, µ], and ζ ∩ ∂Qµ = Ø, where Bµ+ = Bµ (θ) ∩ P .

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3 Degree Theory and Applications

Since µ−1 is not an eigenvalue with positive eigenvector of T , ∃α > 0 such that x − µT x ≥ αµx, ∀x ∈ P . One chooses ρ ∈ (0, min{ αµ 2 , µ}), such that g(x, µ) ≤

αµ + 2 x as x ∈ Bρ . ((Bµ+ \B+ × {0}) ∪

Choose ∈ (0, min{µ, ρ}). Let ζ1 = ζ ∪ ({θ} × [0, µ]), D = ((∂Bµ (θ) ∩ P ) × [0, µ]) ∪ ((Bµ+ \B+ ) × {µ}). Then D ∩ ζ1 = Ø. Similar to the proof in the global bifurcation theorem, ∃U ⊂ P × [0, µ] open, such that Σ ∩ ∂U = Ø, ζ1 ⊂ U , and U ∩ D = Ø. Thus, 1 = degP (id, B+ , θ) = degP (id − f (·, 0), U0 , θ) = degP (id − f (·, µ), Uµ , θ) = degP (id − f (·, µ), B+ , θ) , where Uλ := {x ∈ P |(x, λ) ∈ U }. with norm 1. Let φ ∈ S + be a positive eigenvector associated with λ−1 0 We claim that the equation x − µT x = βφ ∀β > 0 has no positive solution. For otherwise, if ∃β0 > 0, x0 ∈ P, satisfying x0 − µT x0 = β0 φ . Then

x0 ≥ (µτ0 λ−1 0 + β0 )φ > (τ0 + β0 )φ ,

where τ0 = sup {τ ∈ R+ |x0 ≥ τ φ}. This contradicts the deﬁnition of τ0 . Setting β ∈ (0, ρ 2 ), we have x − µT x + (1 − λ)βφ + λg(x, µ) ≥ αµx −

ρ αµ − x > 0, as x = ε . 2 2

Then, by the homotopy invariance, degP (id − f (·, µ), B+ , θ) = degP (id − µT − βφ, B+ , θ) = 0 . The contradiction shows that ζ is unbounded.

3.6.2 Positive Solutions and the Scaling Method The above degree computation is applied to the study of the positive solution of certain superlinear elliptic equations. Let Ω ⊂ Rn be an open bounded domain with smooth boundary. Assume that aij , bj ∈ C(Ω), i, j = 1, . . . , n, φ ∈ C 1 (∂Ω) is nonnegative, and f ∈ C(Ω × R1 ). Let

3.6 Applications

Lu(x) =

n n ∂ ∂u ∂u bj (x) . ai,j (x) + ∂x ∂x ∂x i j j i,j=1 j=1

We study positive solutions of the equation: Lu + f (x, u) = 0, in Ω , u = φ,

on ∂Ω ,

181

(3.33)

(3.34)

in which f is superlinear in u. In order to apply degree theory, we should estimate a priori bounds for positive solutions u. We introduce here a useful method – the scaling method – based on the following global results of Liouville-type theorems, cf. Gidas and Spruck [GS], Chen and Li [ChL 1] and Y. Li [Li 1]. n+2 ) for n ≥ 3, and p > 1 for n = 1, 2. If u ∈ Theorem. Assume p ∈ (1, n−2 2 n C (R ) is a nonnegative solution of the equation:

∆u + up = 0 in Rn ,

(3.35)

then u ≡ 0. n , Proof. (We only prove the theorem under a stronger condition: 1 < p < n−2 for general case see the above-mentioned references.) Let ϕ1 > 0 be the ﬁrst eigenfunction of the Laplacian on B1 (θ) ⊂ Rn , i.e., −∆ϕ1 = λ1 ϕ1 , in B1 (θ) , (3.36) ϕ1 = 0 on ∂B1 (θ) . x ), we have For ∀R > 0, let ϕR (x) = ϕ1 ( R λ1 in BR (θ) , −∆ϕR = R 2 ϕR ,

ϕR = 0

on ∂BR (θ) .

Integrating by parts,

up ϕpR = − ∆uϕpR BR BR

∇u∇ϕpR = BR

∂ϕR p−1 ϕ =− u∆ϕpR + p u ∂n R BR ∂BR

u∆ϕpR . =−

(3.37)

(3.38) (3.39) (3.40) (3.41)

BR

(3.42)

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3 Degree Theory and Applications

Since

BR

u∆ϕpR

2 uϕp−2 R |∇ϕR |

= p(p − 1) BR

it follows that

λ1 p (uϕR ) ≤ 2 R BR

BR

uϕpR ,

(3.43)

p

λ1 p ≤ 2 R

pλ1 − 2 R

BR

uϕpR

(3.44)

p

p1

(uϕR ) BR

BR

ϕpR

p−1 p .

(3.45)

ϕp1 .

(3.46)

Therefore,

(uϕR )p ≤ BR

We obtain limR→0

λ1 p R2

BR

p p−1

BR

p

2p

ϕpR = (λ1 p) p−1 Rn− p−1

B1

(uϕR )p = 0. Thus u ≡ 0.

Similarly, one has the Liouville theorem for half space: n+2 ) for n ≥ 3, and p > 1 for n = 1, 2. If u ∈ Theorem. Assume p ∈ (1, n−2 n C 2 (Rn ) ∩ C(R+ ) is a nonnegative solution of the equation:

n

∆u + up = 0, in R+ , u=0 on xn = 0 ,

(3.47)

then u ≡ 0. n+2 ) for n ≥ 3, and p > 1 for n = 1, 2. Theorem 3.6.8 Assume p ∈ (1, n−2 1 ¯ Suppose that f ∈ C(Ω × R ) satisﬁes

lim

t→+∞

f (x, t) = h(x) uniformly in x ∈ Ω, tp

(3.48)

where h is a positive function. If u ≥ 0 is a solution of equation (3.34), then there exists a constant C depending on p, Ω, φ and f only such that u ≤ C. Proof. One proves the theorem by contradiction. If the conclusion is not true, then there exist a sequence of solutions uk of equation (3.34) and points Pk ∈ Ω satisfying Mk := sup uk (x) = uk (Pk ) → +∞ as k → ∞ . x∈Ω

We may assume Pk → P ∈ Ω as k → ∞. There are two possibilities: either (1) P ∈ Ω◦ , or (2) P ∈ ∂Ω. 2

In case (1): Let d = 12 dist(P, ∂Ω), λkp−1 Mk = 1, y =

x−Pk λk ,

and set

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183

2

vk (y) = λkp−1 uk (x) . It is deﬁned on B

d λk

(θ) and satisﬁes vk (y) = vk (θ) = 1 .

sup y∈B

d λk

(θ)

and n ∂ ∂vk (y) k aij (y) ∂yj ∂yi i,j=1 + λk

n j=1

bkj (y)

2p ∂vk (y) + λkp−1 f ∂yj

−2 λk y + Pk , λkp−1 vk (y) = 0 ,

where akij (y) = aij (λk y + Pk ) → aij (P ), λk bkj (y) = λk bj (λk y + Pk ) → 0, and the last term is asymptotic to h(λk y + Pk )vkp . ∀R > 0, ∃k such that BR (θ) ⊂ B d (θ). Since vk is uniformly bounded λk

on BR (θ), from the Lq estimates, 1 < q < ∞, we have vk W 2,q (BR (θ)) ≤ C, a uniform constant. According to the Sobolev embedding theorem and the diagonal principle, one has a subsequence vkj → v in C 1,β ∩ W 2,q (BR (θ)) for some β ∈ (0, 1), ∀R > 0, and v(θ) = 1. Thus as a weak solution v satisﬁes n ∂ 2 v(y) p n i,j=1 aij (P ) ∂yi ∂yj + h(P )v (y) = 0, in R . v(θ) = 1. After scaling and rotation, it is reduced to equation (3.35). Again by bootstrap iteration, v ∈ C 2 (Rn ), it follows from the Liouville-type theorem that v ≡ 0. This contradicts v(θ) = 1. In case (2), P ∈ ∂Ω. Without loss of generality, we may assume ∃δ > 0 such that Bδ (P ) ∩ ∂Ω is contained in the hyperplane xn = 0. One deﬁnes vk , λk as before. Let dk = d(Pk , ∂Ω), then dk = Pk · en , where en is the unit normal vector of ∂Ω near P . Therefore vk is well deﬁned in B δ (θ) ∩ {yn > − λdkk }. λk

Again we have sup vk (y) = vk (θ) = 1. We claim that there is a constant C1 > 0 such that λdkk ≥ C1 . In fact, according to elliptic regularity up to the boundary, |∇vk | is uniformly bounded, so is vk (θ) − vk 0, . . . , 0, − dk ≤ C −1 dk , 1 λk λk i.e., 2

1 − λkp−1 sup φ(x) ≤ C1−1 x∈∂Ω

Since λk → 0, our claim is proved.

dk . λk

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3 Degree Theory and Applications dkj λkj

If there is a subsequence

→ ∞, then the discussion is reduced to case

(1). Therefore we may assume that there is another constant C2 such that C1 ≤ λdkk ≤ C2 , after a subsequence, λdkk → s. By taking the limit, we obtain:

n i,j=1

2

v aij (P ) ∂y∂i ∂y + h(P )v p (y) = 0 in yn > −s , j v(y) = 0 on yn = −s .

After scaling and rotation, it is reduced to equation (3.47). By the same reasoning, we have v ≡ 0, which contradicts v(θ) = 1. Combining these two cases, we have proved the theorem. Lemma 3.6.9 Under the assumptions of Theorem 3.6.8, if further f (x, t) ≥ 0, and f (x, t) > λ1 , uniformly in x ∈ Ω , (3.49) lim inf t→+∞ t where λ1 is the ﬁrst eigenvalue of the linear elliptic operator −L (see equation (3.33)) with eigenfunction ϕ1 . Then there exists a constant C > 0 such that for all solutions u of the equation: Lu + f (x, u) + tϕ1 = 0, in Ω, ∀ t ≥ 0 , (3.50) u=0 on ∂Ω ,

we have

f (x, u(x))ϕ1 (x)dx ≤ C , Ω

and t≤C. Proof. We may assume ϕ1 L2 = 1. By the condition (3.49), there are constants M > 0, k ∈ (λ1 , lim inf t→+∞ f (x,t) t ) such that f (x, t) ≥ kt − M, ∀ t > 0 . For any solution u of equation (3.50), by integration, we have

ϕ1 (x)Lu(x)dx = ϕ1 (x)f (x, u(x))dx + t , − Ω

Ω

thus

ϕ1 (x)u(x)dx ≥ k

λ1 Ω

ϕ1 (x)u(x)dx − M

Ω

i.e.,

ϕ1 (x) + t , Ω

ϕ1 (x)u(x)dx ≤ M

t + (k − λ1 ) Ω

ϕ1 (x)dx . Ω

Since f (x, t) ≥ 0, from the maximum principle, u ≥ 0. The conclusions follow.

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185

Theorem 3.6.10 Suppose that f ∈ C(Ω × R1 ) is positive, which satisﬁes the conditions (3.48) (3.49) and lim sup t→0

f (x, t) < λ1 uniformly in t

x∈Ω.

(3.51)

Then equation (3.34) has a positive solution in which φ = 0. Proof. Deﬁne F : u(x) → f (x, u(x)) from C(Ω) into itself, and K = (−L)−1 being the linear compact operator on C(Ω). Then T = K ◦ F is a compact mapping from C(Ω) into its positive cone. Applying Lemma 3.6.9 and Theorem 3.6.8 we take R > 0 large enough such that equation (3.50) has no positive solution u with C− norm u = R, and it follows from Lemma 3.6.2 that degP (id − T, BR (θ) ∩ P, θ) = 0. According to condition (3.51) and the homotopy invariance, we have ρ > 0 small such that degP (id − T, Bρ (θ) ∩ P, θ) = 1. Therefore there is a nontrivial positive solution. 3.6.3 Krein–Rutman Theory for Positive Linear Operators Let X be a real Banach space, and P ⊂ X be a positive closed cone with ◦

nonempty interior P . A linear continuous operator L ∈ L(X, X) is called ◦

positive if L(P ) ⊂ P ; it is called strictly positive if L(P \{θ}) ⊂ P . Example. Let M be a compact topological space with a Radon measure µ. Let X = C(M ), P = {u ∈ C(M )|u ≥ 0}, then (X, P ) is an OBS. Given a nonnegative continuous function K : M × M → R1 , we deﬁne:

(Lu)(x) = K(x, y)u(y)dµ . M

Then L is a positive operator. Example. We consider the second-order elliptic operator: 2 Lu = −Σni,j=1 aij ∂ij u + Σni=1 bi ∂i u + cu ,

(3.52)

where aij , bi , c ∈ C(Ω), i, j = 1, 2, . . . , n, Ω ⊂ Rn is an open domain with smooth boundary (cf. Protter and Weinberger [PW]). Assume the ellipticity condition, i.e., ∃α > 0 such that Σni,j=1 aij (x)ξi ξj ≥ α|ξ|2 , ∀ξ = (ξ1 , . . . , ξn ) ∈ Rn , ∀x ∈ Ω , and c≥0. It is known that ∀f ∈ Lp , 1 < p < ∞, the equation:

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3 Degree Theory and Applications

Lu = f , u|∂Ω = 0 , has a unique solution u ∈ W 2,p (Ω); the inverse operator K = L−1 : f → u is positive and bounded on Lp (Ω)(and also on C0 (Ω) the subspace of C(Ω) with boundary value zero), but not strictly positive. However, we have: Lemma 3.6.11 Let X = C01 (Ω), and P = {u ∈ X|u ≥ 0}. Then K = L−1 is a strictly positive compact operator. Proof. We only want to prove the strict positivity, i.e., ∀f ∈ C01 (Ω) if f ≥ 0, ◦

but f = θ, then u = Kf ∈ P . In fact, by the strong maximum principle, u(x) > 0 ∀x ∈ Ω. Again by the Hopf maximum principle, ∂n u|∂Ω < 0, where ∂n is the outer normal derivative. Since ∂Ω is compact, ∃δ > 0 such that supx∈∂Ω ∂n u(x) < −δ < 0. Now, u ∈ C 1 (Ω), there is a neighborhood N of ∂Ω on which ∂ν u ≤ −δ/2, where ν is the direction connecting x ∈ N to the closest point on ∂Ω. Setting α = inf {u(x)|x ∈ Ω\N } and β = Min{α, δ/2}, then the open ball, centered ◦

at u with radius β, is contained in P .

The ﬁrst eigenvalue λ1 and the ﬁrst eigenfunction φ play important role in the study of second-order elliptic equations. It is known that λ1 > 0 is algebraically simple, and that the normalized φ > 0 is unique. In fact, the conclusion is a special case of the following Krein–Rutman theorem: Theorem 3.6.12 (Krein–Rutman) Suppose that (X, P ) is an OBS with non◦

empty interior P , and that T is a compact strictly positive operator. Then T ◦

possesses a unique positive eigenvector φ ∈ P with φ = 1, associated with a 1 algebraically simple eigenvalue λ11 = limn→∞ T n n > 0, satisfying: λ1 ≤ |λ|

∀λ−1 ∈ σ(T ) .

(3.53)

Proof. (1). (The existence of a positive eigenvector.) ∀x ∈ P \{θ}, ∃M > 0, P, ∀n ∈ N large, such that M T x ≥ x. For otherwise, it must be T x − n1 x ∈ ◦

and then T x ∈ P ; this contradicts the strict positiveness of T . According to Dugundji’s theorem, there exists a retraction r : X → P . We deﬁne T : X × R1 → P as follows: T (y, λ) = λr ◦ T (y + x) . Thus, ∀λ, T (·, λ) is compact, and satisﬁes T (y, 0) = θ, ∀y ∈ X. We notice that for λ ≥ 0, y = T (y, λ) is equivalent to y ∈ P , and y = λT (y + x). From the Leray–Schauder Theorem, there is an unbounded connected branch of solutions ζ + ⊂ P × R1+ passing through (θ, 0).

3.6 Applications

187

We claim that if (y, λ) ∈ ζ + , then λ ∈ [0, M ]. In fact, since M T x ≥ x, y = λ x, we obtain λT (y + x) ≥ M y ≥ λT y ≥

λ M

n x, ∀n ∈ N .

If λ > M , then x = θ. This is a contradiction. Therefore, ∃(yn , λn ) ∈ ζ+ such that λn → λ∗ ∈ [0, M ], yn > n. Setting zn = yynn , we have zn = λ n T z n +

λn Tx . yn

Since T is compact, we have zn → z ∗ = θ satisfying z ∗ = λ∗ T z ∗ . This implies ◦

that z ∗ ∈ P , λ∗ > 0, z ∗ = 1. Let φ = z ∗ , and λ1 = λ∗ ; these are what we need. ◦ (2). (The uniqueness of the positive eigenvector.) ∀x ∈ P, ∀y ∈ P , let us deﬁne δy (x) = sup {λ ≥ 0|y + λx ∈ P }. It is easily seen that δy (x) > 0 is a continuous function of x ∈ X\P . By deﬁnition, ⎧ ⎨λ ∈ [0, δy (x)] implies y + λx ∈ P , ⎩λ > δ (x) implies y + λx ∈ P . y ◦

Now, suppose that φi ∈ P satisfy φi = 1, φi = λi T φi , i = 1, 2. Set γ1 = δφ1 (−φ2 ), γ2 = δφ2 (−φ1 ) . Then

⎧ ⎨T (φ1 − γ1 φ2 ) =

1 λ1 (φ1

− γ1 λλ12 φ2 ) ,

⎩T (φ − γ φ ) = 2 2 1

1 λ2 (φ2

− γ2 λλ21 φ1 ) . ◦

If φ2 − γ2 φ1 = θ, then T (φ2 − γ2 φ1 ) ∈ P . This implies λ2 < λ1 . But T (φ1 −γ1 φ2 ) ∈ P implies λ1 ≤ λ2 . This is a contradiction. Therefore, φ1 = φ2 . Again, by the normality, λ1 = λ2 . (3). (The inequality (3.52)) Assume that ψ ∈ X\(P ∪ (−P )) satisfying λT ψ = ψ, and ψ = 1. In the case where λ is real: From φ ± δφ (±ψ)ψ = θ, it follows that ◦ 1 λ1 φ ± δφ (±ψ)ψ = T (φ ± δφ (±ψ)ψ) ∈ P . λ1 λ Therefore λ1 < |λ|. In the case where λ is not real: We write λ = |λ|eiθ , then ∃x1 , x2 ∈ X, such that x1 + x2 = λ(T x1 + iT x2 ) .

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1 Let H = span{x1 , x2 }, then T |H = |λ| Rθ , where Rθ is the following matrix: cos θ − sin θ sin θ cos θ

We claim that P ∩ H = {θ}. For otherwise, since both P and H are invariant with respect to T , T |P ∩H is positive, according to the conclusion of the ﬁrst paragraph, it should have a positive eigenvalue. But this contradicts the representation of Rθ . Now, ∀z ∈ H\{θ}, ◦ λ1 1 δφ (z)Rθ z = T (φ + δφ (z)z) ∈ P . φ+ λ1 |λ| λ1 Thus |λ| δφ (z) < δφ (Rθ z). We choose z0 ∈ H\{θ} such that δφ (z0 ) = sup {δφ (z)|z ∈ C}, where C = {cos θx1 + sin θx2 |θ ∈ [0, 2π]}, it follows that λ1 < |λ|. (4). (The algebraic simplicity of λ1 ). From (1), we have shown that ker(id− λ1 T ) = span{φ}. According to the Riesz–Schauder theory, it is suﬃcient to show that ker(id−λ1 T )2 = ker(id−λ1 T ), i.e., if x ∈ X satisﬁes x−λ1 T x = −φ, then x = θ. To this end, it is suﬃcient to show that x ∈ P ∩ (−P ). In fact, if x ∈ P , let γ = δφ (−x), then

T (φ − γx) =

1 γ φ − (x + φ) . λ1 λ1

It follows that

◦ γ 1 (φ − γx) = T (φ − γx) + φ ∈ P . λ1 λ1 It contradicts the deﬁnition of γ, therefore x ∈ P . Similarly, x ∈ −P . Therefore x ∈ P ∩ (−P ) = θ.

Corollary 3.6.13 Let L be the linear second-order elliptic operator (3.52) with the Dirichlet boundary condition, deﬁned on a bounded open domain Ω in Rn , the boundary ∂Ω being smooth. Then the eigenvalue problem: Lu = λu in Ω u = 0 on ∂Ω . has a positive eigenfuction φ with positive, algebraically and geometrically simple eigenvalue λ1 , satisfying λ1 ≤ |λ| ∀λ ∈ σ(L) . Corollary 3.6.14 ([Tu]). Let (X, P ) be an OBS, and let f, T, g, Σ+ be as in Theorem 3.6.7, in which T is a strictly positive compact operator. Then (θ, λ1 ) is a bifurcation point and Σ+ contains an unbounded connected component in P emanating from (θ, λ1 ). Also, we have the following result on the bifurcation at inﬁnity:

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189

Corollary 3.6.15 Let (X, P ) be an OBS. Let T ∈ L(X, X) be strictly positive and compact, and g : P ×R+ → P be compact, satisfying g(x, 0) = θ, g(x, λ) = o(x), as x → ∞ uniformly on each ﬁnite interval. Deﬁne f (x, λ) = λT x + g(x, λ) ∀(x, λ) ∈ P × R+ . Let Σ = {(x, λ) ∈ P × R+ |f (x, λ) = x} and Σ+ = Σ ∩ (P \{θ} × R+ \{0}). Then there are only two possibilities for the connected component ζ of Σ+ , passing through the point (∞, λ1 ), where λ1 is the eigenvalue of T with positive eigenvector: 1 (1) The projection of ζ onto R+ is unbounded. 1 (2) ζ ∩ ({θ} × R+ ) = Ø.

3.6.4 Multiple Solutions In this subsection, we present a few examples showing that degree theory is applicable to the study of multiple solutions for certain nonlinear elliptic equations. Example 1. Let L be the second-order elliptic operator deﬁned in (3.52), and let g ∈ C(R1 ) satisfy g(0) = 0, |g(t)| ≤ C, and g(t) − λt = o(t), as |t| → 0, where C is a constant, and λ ∈ R1 is between the ﬁrst and the second eigenvalues of L. We consider the equation: Lu = g(u) in Ω , (3.54) u = 0. on ∂Ω . Obviously, u = 0 is a trivial solution. We shall show that there exists a nontrivial solution. This can be seen by the following degree computation. Let X = C01 (Ω), since λ is between λ1 and λ2 , by homotopy invariance of the index, i(id − L−1 g, θ) = i(id − λL−1 , θ) = −1 . Since all solutions are bounded in X, again by the homotopy invariance of the degree, deg (id − L−1 g, BR (θ), θ) = deg (id, BR (θ), θ) = 1 . If there were no nontrivial solution, the above identities would contradict the additivity of the degree. If one slightly changes the above assumptions then there exists a positive solution. Example 2. Let L be as in (3.52), and let g ∈ C 1 (R1 ) satisfy g(0) = 0, 0 ≤ g ≤ C, and g(t) − λt = o(t), as t → 0 with λ > λ1 . Then there exists a positive solution of (3.54). This can be shown by the sub- and super-solutions method. Obviously, u = L−1 C is an super-solution. Let u = φ1 , where φ1 is the positive eigenfunction

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of L, and > 0 small. By the condition of g near zero, it is a sub-solution of L. From the Hopf maximum principle, we have u ≤ u. Thus we obtain a positive solution. We are inspired by Example 2 to compute the degree of an order-preserving compact map on an order interval. ◦

Theorem 3.6.16 Let (X, P ) be an OBS with P = ∅, and let [u, u] ⊂ X be a {e1 , . . . , ek } ﬁnitely bounded order interval for some {e1 , . . . , ek } ⊂ X. If f : [u, u] → X is order preserving, compact and satisfying: ◦

◦

f (u) ∈ u + P , f (u) ∈ u − P ; Then U := [u, u]◦ = ∅ and deg (id − f, U, θ) = 1 .

Proof. Since f is order preserving, ∀u ∈ [u, u], f (u) ≤ f (u) ≤ f (u). Moreover, we have f ([u, u]) ⊂ U . In fact, ∃ > 0, such that f (u) + B (θ) ⊂ u + P, f (u) + B (θ) ⊂ u − P . Therefore f (u) + B (θ) ∈ (u + P ) ∩ (u − P ) , i.e., f (u) ∈ U . This implies that U is a nonempty open convex set. Choosing arbitrarily x ∈ U, ∀t ∈ [0, 1], deﬁne φ(u, t) = t(u − f (u)) + (1 − t)(u − x) . Since f (∂U ) ⊂ U, tf (u) + (1 − t)x ∈ U ∀(u, t) ∈ ∂U × [0, 1]. Thus θ ∈ φ(∂U × [0, 1]). From the homotopy invariance of the degree, deg (id − f, U, θ) = deg (φ(·, 1), U, θ) = deg (φ(·, 0), U, θ) = deg (id, U, x) = 1 . The above theorem is due to Amann [Am 3]. ◦

Corollary 3.6.17 (Amann) Let (X, P ) be an OBS with P = ∅, and let ui , ui , i = 1, 2, satisfy u1 ≤ u1 ≤ u¯2 , u1 ≤ u2 ≤ u2 ; and [u1 , u1 ] ∩ [u2 , u2 ] = Ø. If [u1 , u¯2 ] is {e1 , . . . , ek }- ﬁnitely bounded for some {e1 , . . . , ek } ⊂ X, and if f : [u1 , u2 ] → X is an order-preserving compact map satisfying: ◦

◦

f (ui ) ∈ ui + P , f (ui ) ∈ ui − P , i = 1, 2 . Then there are at least three distinct ﬁxed points of f on the order interval [u1 , u2 ].

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191

Proof. Set Ui = [ui , ui ])◦ , i = 1, 2, and U = [u1 , u2 ]◦ . From the additivity, deg (id − f, U, θ) = deg (id − f, U1 , θ) + deg (id − f, U2 , θ) + deg (id − f, U \(U1 ∪ U2 ), θ) . Applying the above theorem, we have deg (id − f, U \(U1 ∪ U2 ), θ) = −1 . According to the Kronecker existence property, ∃ ﬁxed points ui , i = 1, 2, 3, such that ui ∈ Ui , i = 1, 2. and u3 ∈ U \(U1 ∪ U2 ). In the applications to PDEs, let Ω ⊂ Rn be a bounded domain with smooth boundary. We take X = C01 (Ω), and e ∈ X is a positive function in Ω and satisﬁes ∂n e(x) < 0, ∀ x ∈ ∂Ω, where ∂n is the outer normal derivative. Then it is easy to verify that any order interval [u, v] is e− ﬁnitely bounded. Example 3. Let L be the operator deﬁned in (3.52), and let g ∈ C 1 (R, R1+ ) be nondecreasing and satisfying g(0) = 0, g (0) = 1, t < g(t) ∀t ∈ (0, t0 ] for some small t0 > 0, and g(t) ≤ C as t > 0, where C > 0 is a constant. Then ∃λ∗ ∈ (0, λ1 ) such that the equation: Lu = λg(u), in Ω (3.55) u = 0, on ∂Ω , possesses at least one positive solution ∀λ ≥ λ∗ and at least two distinct positive solutions ∀λ ∈ (λ∗ , λ1 ). Indeed, we extend g to R1 oddly; the mapping f (u) = L−1 g(u) is order preserving. ∀λ > 0, it has a pair of strict sub- and super-solutions {−λCe, +λCe}, where e = L−1 1. Thus, vλ := limn→∞ (λf )n (λC) exists and is the maximal solution of (3.55). Obviously, vλ is upper semi-continuous and monotone non1 |vλ > 0}. We claim that λ∗ < λ1 , decreasing in λ. Deﬁne λ∗ = inf {λ ∈ R+ where λ1 is the ﬁrst eigenvalue of L. In fact, near the bifurcation point (θ, λ1 ) we ﬁnd uλ > 0, a solution of (3.55) with small uλ C < t0 . One has

Luλ φ1 dx = λ g(uλ )φ1 dx , Ω

Ω

where φ1 > 0 is the ﬁrst normalized eigenfunction of L∗ . Then

λ1 uλ φ1 dx > λ uλ φ1 dx . Ω

It follows that λ < λ1 , Since vλ ≥ uλ > 0, then λ∗ < λ1 . For ∀λ ∈ (λ∗ , λ1 ) ∃δ > 0, such that λg(t) < λ1 t, as |t| < δ. Let φ1 be the normalized ﬁrst eigenfunction; as ∈ (0, δ), ±φ1 is a pair of sub∗ and super-solutions of (3.55). However, as λ = λ 2+λ , vλ is a sub-solution of (3.55). Set > 0 small enough such that φ1 < vλ . Then we ﬁnd two pairs of sub- and super solutions: [−φ1 , +φ1 ], [vλ , λCe]. According to the Amann theorem, there must be two distinct nontrivial solutions of (3.55): u1λ ∈ [−φ1 , λCe]\([−φ1 , φ1 ] ∪ [vλ , λCe]), and u2λ ∈ [vλ , λCe]. Since > 0 can be arbitrarily small, both of u1λ , u2λ are positive. This is our conclusion.

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3.6.5 A Free Boundary Problem We turn to an application of the bifurcation at inﬁnity to a free boundary problem for the ﬂux equation in the conﬁned plasma. Let Ω ⊂ R2 be the section of the container, a bounded domain with smooth boundary rotated along and oﬀ the z axis in R3 , which contains plasma conﬁned by an external magnetic ﬁeld. Let u be the ﬂux function of the magnetic ﬁeld. The current I is a given positive constant. The following equation is derived from the Maxwell equation: ⎧ ∂2 ∂ 1 ∂ ⎪ Lu := −[ ∂r ( r ∂r ) + 1r ∂z ]u = λu+ in Ω, ⎪ ⎪ ⎨ (3.56) u = C, on ∂Ω, ⎪ ⎪ ⎪ ⎩ ∂u − ∂Ω 1r ∂n = I, where C is a constant to be determined, and u+ = max {u, 0}. The domain Ωp := {x ∈ Ω | u(x) > 0}, which is the space occupied by the plasma, is unknown. It will be solved simultaneously with the ﬂux function u. Physically, we are interested in the case where Ωp ⊂ Ωp ⊂ Ω, and Ωp = ∅. From Green’s formula,

1 ∂u =I. u+ = Lu = − λ Ω Ω ∂Ω r ∂n We see that the problem is solvable only if λ > 0. Again, from Green’s formula,

1 ∂φ1 , u + φ1 = λ 1 uφ1 + C λ Ω Ω ∂Ω r ∂n where φ1 is the ﬁrst eigenfunction of L with respect to Dirichlet boundary condition. Combining with the maximum principle it is seen that C > 0 ⇒ λ < λ1 , and C = 0 ⇒ λ = λ1 . In all these cases, the solution u(x) > 0 ∀ x ∈ Ω, from the maximum principle. i.e., Ωp = Ω. They are not of interest in physics. We reduce the problem (3.56) to the following: Given a constant C1 < 0, ﬁnd (v, λ) satisfying ⎧ ⎨ Lv = λv+ in Ω , v = C1 on ∂Ω , (3.57) ⎩ λ>0. Indeed, if (v, λ) is a solution of (3.57), let v = αu, C1 = αC, where α > 0 is an adjustment constant satisfying

1 ∂u = −I . ∂Ω r ∂n Then (u, λ) is the solution of equation (3.56). We have the following conclusion:

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193

Statement 3.6.18 ∀I > 0, ∃u ∈ C 2,γ (Ω) γ ∈ (0, 1) with C < 0, which solves equation (3.56) if and only if λ > λ1 . Proof. After the above discussion, we just take C1 = −1. Let w = v + 1, then equation (3.57) is reduced to ⎧ ⎨ Lw = λ(w − 1)+ in Ω , w = 0 on ∂Ω , (3.58) ⎩ λ>0. Let us take X = C(Ω), F : w → (w − 1)+ , K = L−1 , our problem is reduced to ﬁnding the solution of w = λK ◦ F (w). It is easily seen that there is no 1 . According to Corollary 3.6.15, the bifurcation point located at {θ} × R+ projection of the connected component ζ of the positive solution set passing 1 is unbounded. It through the bifurcation point at inﬁnity (∞, λ1 ) onto R+ must include the interval (λ1 , +∞). Therefore, ∀λ ∈ (λ1 , +∞), there is a solution wλ on ζ. On the other hand, if there is a nontrivial solution wλ of equation (3.58), then wλ ≥ 0 and F (wλ ) < wλ . Thus

1 wλ φ1 dx = wλ Lφ1 dx λ1 Ω Ω

1 Lwλ φ1 dx = λ1 Ω

λ = F (wλ )φ1 dx λ1 Ω

λ < wλ φ1 dx . λ1 Ω This proves λ > λ1 . The proof of the regularity is standard.

3.6.6 Bridging We present an example to express the fact that solutions of a diﬀerential equation on disjoint domains can be glued up as a new solution in a larger domain. A typical example we consider here is a dumbbell D ⊂ R2 , i.e., two disks B± = B1/2 ((±1, 0)) connected by a segment E = {(x, y) ∈ R2 | y = 0, x ∈ [−1/2, 1/2]}. Let B = B+ ∪B− , and D = B ∪E. Assuming that f ∈ C 1 is of power growth with f (0) = 0, and that u± ∈ H01 (B± ) are solutions of the equation: − u = f (u) . (3.59) We consider a sequence of domains Dn squeezing to D, and want to ﬁnd a solution un ∈ H01 (Dn ) of (3.59) such that un is closed to u± in certain sense.

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Ω Dn -1

0

1

Fig. 3.5.

Let Ω be a bounded domain containing Dn , and D ⊂ Dn . Assuming that U ⊆ Ω is a bounded domain, the composition operator u → f (u) is C 1 from H 1 (U ) to Lq (U ), ∀q ∈ (1, ∞). Let r : H 1 (Ω) → H 1 (U ) be the restriction u → u|U , and e : H01 (U ) → H01 (Ω) be the extension (eu)(x) = u(x) if x ∈ U and 0 elsewhere. Let K = (−∆)−1 : Lq (U ) → H01 (U ), 1 ≤ q < ∞, then K is linear and compact. We deﬁne T = e · K · f · r : H01 (Ω) → H01 (Ω). The following statements are easy to verify: (1) ∀u ∈ H01 (U ), we have (r · e)u = u. (2) ∀v ∈ H01 (Ω), if rv ∈ H01 (U ), then (e · r)v = v. (3) For u0 ∈ H01 (U ), u0 is a solution of (3.59) if and only if e · u0 is a ﬁxed point of T . (4) If u0 is nondegenerate (i.e., ker−∆ − f (u0 )I) = {θ}), then I − T (u0 ) is invertible on H01 (Ω). Let us choose X = H01 (Ω). Let Tn , TB and en , eB be the operators and the extensions on Dn ∀n ∈ N and B = B+ ∪ B− , respectively. Statement 3.6.19 For any nondegenerate ﬁxed point eB u0 of TB , u0 ∈ H01 (B), there exists a ﬁxed point en un of Tn for n large, such that en un − e0 u0 X → 0 as n → ∞. Proof. By the nondegeneracy assumption, eB u0 is isolated and then i(I − T, eB u0 ) = ±1. In order to show the existence of a ﬁxed point en un of Tn , un ∈ H01 (Dn ), it is suﬃcient to verify that for all small δ > 0, I − Tn is homotopic to I − TB on ∂V , where V = Bδ (eB u0 ) is a ball in H01 (Ω). If it is so, then deg(I − Tn , V, θ) = deg(I − TB , V, θ) = 0 . Our conclusion follows. Now, we verify it by contradiction. Suppose that ∃tn ∈ [0, 1], ∃un ∈ H01 (Dn ), en un ∈ ∂V , such that en un = tn TB en un + (1 − tn )Tn en un . Since

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195

en un is bounded, and Tn , TB are compact, there exists a subsequence unj such that enj unj converges to some v and then v ∈ ∂V . We verify that v is a solution of (3.59), and that v(x) = 0, a.e., on Ω\B. Since V can be chosen arbitrarily close to eB u0 , it contradicts the nondegeneracy of u0 . (1) ∀φ ∈ H01 (B), from

∇un ∇φ = f (un )φ , B

it follows that

B

∇v∇φ = B

f (v)φ ; B

i.e., v is a solution (3.59) on B. (2) From en un = 0 a.e., on Ω\Dn , we have v = 0 a.e., on Ω\D. Since |E| = 0, v = 0, a.e., on Ω\B. Therefore, v ∈ H01 (Ω) is in fact a solution of (3.59). In other words, u+ and u− are nondegenerate solutions of (3.59) on H01 (B+ ) and H01 (B− ), respectively. We deﬁne u0 (x) = u± (x) as x ∈ B± , then eB u0 is a degenerate ﬁxed point of TB . By Statement 3.6.19 there are a sequence of domains Dn squeezing to D and a sequence of solutions un ∈ H01 (Dn ) of the equation (3.59) such that un is close to u0 .

3.7 Extensions We brieﬂy introduce a few directions in generalizing the Leray-Schauder degree. 3.7.1 Set-Valued Mappings We have used the ﬁxed-point theorem for set-valued mappings in Chap. 2. It is natural to ask if the degree theory for set-valued mappings can be set up. Let X be a Banach space, let Γ(X) be the set of all nonempty closed convex subsets of X. Deﬁnition 3.7.1 (Compact convex set-valued mapping) Let Ω ⊂ X be a bounded subset, and let φ : Ω → Γ(X). It is called a compact convex setvalued mapping, if (1) it is upper semi-continuous, and (2) φ(Ω) is compact. Obviously, any compact single-valued mapping is a compact convex setvalued mapping. The main idea in extending the Leray–Schauder degree to compact convex set-valued mappings is to approximate the set-valued mapping by singlevalued mappings. The following notion is useful:

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Deﬁnition 3.7.2 (Single-valued approximation) Let (M, d) be a metric space, and let Y be a Banach space. Let T : M → 2Y and T˜ : M → Y. We say that T˜ is a − single-valued approximation of T for > 0, if 1. T˜(M ) ⊂ convT ¯ (M ), 2. ∀x ∈ M, ∃y ∈ M and z ∈ T (y) satisfying d(x, y) < , and T˜(x) − z < . The following picture shows the − approximation of a set-valued mapping.

T

T˜

Fig. 3.6.

Theorem 3.7.3 Let (M, d) be a metric space, X be a Banach space, and let φ : M → Γ(X) be upper semi-continuous. Then ∀ > 0 there exists an − approximation φ of φ. Proof. By the upper semi-continuity, ∀x ∈ M, ∃δx > 0, such that φ(Bδx (x)) ⊂ φ(x) + B (θ). One may choose δx < 2 . Then A = {Bδx (x) | x ∈ M } is an open covering of M . It has an open locally ﬁnite star-reﬁnement B = {Vβ | β ∈ Λ} (see Dugundji [Du 2]). The so-called star-reﬁnement means that + ∀β ∈ Λ, ∃xβ ∈ M such that St(Vβ , B) ⊂ Bδxβ (xβ ), where St(Vβ , B) = {Vα ∈ $ B | Vα Vβ = ∅}. Deﬁne a partition of unity with respect to B : {λβ | β ∈ Λ}, and φ (x) = Σβ∈Λ λβ (x)zβ , where zβ ∈ φ(Vβ ). We are going to verify that φ is an − approximation of φ. Indeed, ∀x0 ∈ M , if x0 ∈ Vβ , then there must be y ∈ M and δy < 2 such that Vβ ⊂ Bδy (y). Thus, d(x0 , y) < 2δy < . However, zβ ∈ φ(Vβ ) ⊂ φ(Bδy (y)) ⊂ φ(y) + B (θ), we have φ (x0 ) = Σni=1 λβi (x0 )zβi ∈ φ(y) + B (θ), provided by the convexity of φ(y), and the fact {Vβi }ni=1 = {Vβ | x0 ∈ Vβ , β ∈ Λ}. We may ¯ ). choose z ∈ φ(y) such that φ (x0 ) − z < . Obviously, φ (M ) ⊂ convφ(M What is the homotopy equivalence for compact convex set-valued mappings? Deﬁnition 3.7.4 Let Ω be a bounded open set in a Banach space X, and let ¯ → Γ(X) be two compact convex set-valued mappings. We say that φ1 , φ2 : Ω φ1 is homotopically equivalent to φ2 , denoted by φ1 φ2 , if there exists a ¯ → Γ(X), such family of compact convex set-valued mappings Φ : [0, 1] × Ω that θ ∈ / F ([0, 1] × ∂Ω), and Φ(i, ·) = φi (·), where F (t, x) = x − Φ(t, x).

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197

¯ → Γ(X) is a compact convex set-valued Lemma 3.7.5 Suppose that φ : Ω mapping and f = id − φ. If θ ∈ / f (∂Ω), then ∃0 > 0 such that ∀ ∈ (0, 0 ), for any − single-valued approximation φ˜ of φ, we have θ ∈ / F ([0, 1] × ∂Ω), ˜ where F (t, x) = x − (1 − t)φ(x) − tφ(x). Proof. We prove the lemma by contradiction. If there exist single-valued continuous mappings φ˜n , and tn ∈ [0, 1], xn ∈ ∂Ω, yn ∈ Ω, zn ∈ φ(yn ) such that: xn ∈ (1 − tn )φ˜n (xn ) + tn φ(xn ) , 1 xn − yn < , n 1 ˜ . φn (xn ) − zn < 2n Since φ(Ω) is compact, and φ˜n (Ω) ⊂ convφ(Ω), after a subsequence, we have xn → x∗ ∈ ∂Ω, tn → t∗ , and then yn → x∗ . Since φ is u.s.c., for n large, we ˜n (xn ) ∈ 1 (θ). It follows, φ have φ(xn ) ⊂ φ(x∗ ) + B n1 (θ), φ(yn ) ⊂ φ(x∗ ) + B 2n ∗ φ(x ) + B n1 (θ), then (1 − tn )φ˜n (xn ) + tn φ(xn ) ⊂ φ(x∗ ) + B n1 (θ) , and then xn ∈ φ(x∗ ) + B n1 (θ) as n large. Thus x∗ ∈ φ(x∗ ). This is a contradiction. Lemma 3.7.6 If φ1 , φ2 are two homotopically equivalent compact convex setvalued mappings, then there exists > 0 such that for any − approximation φ˜1 , φ˜2 of φ1 and φ2 , respectively, we have φ˜1 φ˜2 . Proof. According to Lemma 3.7.5, we have φ˜0 φ0 φ1 φ˜1 in the sense of set-valued mappings. We shall prove that φ˜0 φ˜1 is also true in single-valued sense. ¯→ In fact there exists a compact convex set-valued mapping h : [0, 1] × Ω Γ(X) satisfying h(i, x) = φ˜i (x), i = 0, 1 and := dist(H([0, 1] × ∂Ω), θ) > 0, where H = id − h. By Lemma 3.7.3, and Lemma 3.7.5, there exists a 2 − ˜ : [0, 1] × Ω ˜ ¯ → X of h such that h(µ, ·) h(µ, approximation h ·). More precisely, let ˜ p(λ, µ; x) = (1 − λ)h(µ, x) + λh(µ, x) , we have θ ∈ / P ([0, 1] × [0, 1] × ∂Ω), where P (λ, µ; x) = x − p(λ, µ; x). Deﬁne a family of single-valued mappings: ⎧ 1 ⎪ ⎨p(1 − 3t, 0; x) if 0 ≤ t ≤ 3 g(t, x) = p(0, 3t − 1; x) if 13 ≤ t ≤ 23 ⎪ ⎩ p(3t − 2, 1; x) if 23 ≤ t ≤ 1 . It is easy to verify that this is the required homotopy.

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Now, we are ready to deﬁne the degree of id − φ, where φ is a compact convex set-valued mapping. Deﬁnition 3.7.7 Let Ω be an open bounded set in a Banach space X, let ¯ → Γ(X) be a compact convex-valued mapping, and f = id − φ. If φ : Ω θ∈ / f (∂Ω), deﬁne deg (f, Ω, θ) = lim deg (f˜ , Ω, θ) , →0

where f˜ = id − φ˜ and φ˜ is an − approximation of φ. According to Lemmas 3.7.5 and 3.7.6, the degree is well deﬁned. Readers can easily verify that the degree enjoys the same basic properties as the Leray–Schauder degree: Homotopy invariance, additivity, translation invariance and normality. So do the Kronecker existence and the excision. The results of Examples 1–3 in Sect. 3.6 can be extended to case where g has jump discontinuity by the use of the degree of set-valued mappings, see K. C. Chang [Ch 1], [Ch 2]. 3.7.2 Strict Set Contraction Mappings and Condensing Mappings The motivation of introducing the strict set contraction mapping is to extend the notion of compactness and then to extend the L-S degree. Let X be a Banach space, and let A ⊂ X be a bounded subset. ∀ > 0, we consider a − net N of A. Let α(A) = inf{ > 0 | ∃a ﬁnite − net N of A} . By deﬁnition, for α(A) = 0 if and only if A¯ is compact. It is easy to verify the following simple properties of α: If A ⊂ B, then α(A) ≤ α(B). α(A ∪ B) ≤ α(A) + α(B). ¯ = α(A). α(A) α(A + B) ≤ α(A) + α(B), α(λA) = |λ|α(A), ∀λ ∈ C. For a sequence of bounded closed nonempty subsets · · · ⊂ An ⊂ · · · ⊂ A2 ⊂ A1 satisfying α(An ) → 0, we have A = ∩∞ n=1 An = ∅, and α(A) = 0. 6. (Mazur) α(conv(A)) ¯ = α(A).

1. 2. 3. 4. 5.

Deﬁnition 3.7.8 (k-set contraction map) Let X be a Banach space, Ω ⊂ X be a bounded subset. A continuous map φ : Ω → X is called a k− set contraction mapping, k ≥ 0, if α(φ(A)) ≤ kα(A) ∀ bounded A ⊂ Ω . Remark 3.7.9 We assign ∞ · 0 = 0, then continuous mappings can be considered as ∞ – set contraction mappings. Example 1. If φ : Ω → X is compact, the φ is a 0− set contraction. Example 2. If φ : Ω → X is a Lipschitz mapping with Lipschitz constant L, then φ is an L− set contraction.

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199

Deﬁnition 3.7.10 (Strict set contraction mapping and condensing mapping) A k− set contraction mapping φ is called strict if k ∈ [0, 1). It is called a condensing mapping if α(φ(A)) < α(A) ∀ bounded A . According to (4): If φ, ψ : Ω → X are k1 − and k2 − set contraction mappings respectively, then φ + ψ is a k1 + k2 − set contraction. In particular, if φ is a contraction mapping (i.e., a Lipschitz mapping with Lipschitz constant L < 1), and if ψ is compact, then φ + ψ is a strict set contraction mapping. By the deﬁnition, if φ : Ω → X is a k1 − set contraction, and ψ : φ(Ω) → X is a k2 − set contraction. (In case, k1 = ∞, we assume further φ maps bounded set to bounded set); then ψ ◦ φ is a k1 k2 − set contraction. First we are going to deﬁne the degree for strict set contraction mappings. We intend to reduce it to compact mappings. Let f = id − φ, where φ is a k− contraction mapping, where 0 ≤ k < 1. ¯ then we deﬁne deg (f, Ω, θ) = 0. If θ ∈ / f (Ω), ¯ Otherwise the ﬁxed-point set of φ is not empty. Deﬁne A1 = convφ(Ω), ¯ ∩ Ak ), k = 1, 2, . . .. Since the ﬁxed point set of φ is not Ω and Ak+1 = convφ( $∞ empty, A := n=1 An = ∅. However, we have ¯ α(An+1 ) = α(convφ(An ∩ Ω)) ≤ kα(An ) ≤ ... ≤ k n α(A1 ) → 0 . ¯ ∩ A → A. Therefore α(A) = 0, A is a compact convex set, and then φ : Ω According to the Dugundji extension theorem, there exists a continuous ˜ Ω∩A ¯ → A such that φ| = φ|Ω∩A . Then φ˜ is compact and shares the same φ˜ : Ω ¯ ¯ ﬁxed point set with φ. ˜ We claim that θ ∈ / (id − φ)(∂Ω) ⇒ θ ∈ / (id − φ)(∂Ω). In fact, if x ∈ ∂Ω ˜ ˜ satisﬁes x = φ(x), then x ∈ ∂Ω ∩ A, so is φ(x) = φ(x) and then θ ∈ (id − φ)(∂Ω). With the above notations, we deﬁne the degree for strict set contraction mappings. Deﬁnition 3.7.11 Let φ be a strict set contraction mapping; we deﬁne ˜ Ω, θ) . deg (id − φ, Ω, θ) = deg (id − φ, To verify that the degree is well deﬁned, we shall prove that the def˜ i.e., if φ˜1 , φ˜2 are two inition does not depend on the special choice of φ, ˜ ˜ ¯ = φ|Ω∩A , i = 1, 2, then we deﬁne such extensions: φi : Ω → A, φi |Ω∩A ¯ ¯ / (id − φ)(∂Ω), it follows that F (t, x) = x − [(1 − t)φ˜1 (x) + tφ˜2 (x)]. From θ ∈

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3 Degree Theory and Applications

θ∈ / (id−F )([0, 1]×∂Ω), i.e., φ˜1 φ˜2 . According to the homotopy invariance of the Leray–Schauder degree, we have deg (id − φ˜1 , Ω, θ) = deg (id − φ˜2 , Ω, θ). It is easy to verify that the degree enjoys the homotopy invariance, additivity, translation invariance and normality. They are left to readers as exercises. Accordingly, this enables us to apply the degree theory to a map, which can be decomposed into a summation of a contraction mapping and a compact map. Finally, we extend the degree to condensing mappings. If φ is a condensing ¯ is bounded, say it is included in the ball centered at θ with mapping, φ(Ω) radius R > 0. For any > 0, setting λ ∈ (1 − R , 1), and φλ = λφ, we have φ(x) − φλ (x) ≤ , and α(φλ (A)) ≤ λα(φ(A)) ≤ λα(A) ∀ bounded A , i.e., φλ is a strict set contraction mapping which is closed to φ. We deﬁne deg (id − φ, Ω, θ) = deg (id − φλ , Ω, θ) . Again, it is easy to verify that the degree is also well deﬁned and enjoys all basic properties of the Leray–Schauder degree. Again, this are left to readers. 3.7.3 Fredholm Mappings We know that the Leray–Schauder degree can be applied to quasilinear elliptic ¯ γ ∈ (0, 1) satisfying equations: Find u ∈ C 2,γ ∩ C0 (Ω), n

aij (x, u, ∇u)

i,j=1

∂2u + f (x, u, ∇u) = 0, in Ω . ∂xi ∂xj

(3.60)

¯ → C 1 (Ω) ¯ such that It is executed as follows. Deﬁne a mapping K : C01 (Ω) 0 1 ¯ 2,γ ¯ ∀u ∈ C0 (Ω), v = Ku ∈ C ∩ C0 (Ω) satisﬁes the following linear equation: n

aij (x, u, ∇u)

i,j

∂2v + f (x, u, ∇u) = 0 in Ω . ∂xi ∂xj

Thus the ﬁxed points of K are the solutions of equation (3.60). However, if we consider a general elliptic equation: A(u) := A(x, u, ∇u, ∇2 u) = g(x) in Ω ,

(3.61)

where g is a given function and the quadratic form is positive deﬁnite: n

∂A ξi ξj ≥ α|ξ|2 , α > 0 , ∂u x x i j i,j=1 then it seems that the Leray–Schauder degree argument is not applicable, because we do not know how to recast it as a ﬁxed-point problem for compact

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201

mappings. However, the linearization of A(u) is a linear second-order elliptic operator, so is a Fredholm operator with index zero. People have made great eﬀorts in deﬁning an integer-valued degree theory for C 1 Fredholm mappings of index 0 between Banach manifolds. Among them we should mention Caccioppoli [Cac 2], Smale [Sm 3](for Z2 valued), Elworthy and Tromba [ElT 1],[ElT 2], Borisovich, Zvyagin, and Sapronov [BZS] (for C 2 mappings and Z valued), and Fitzpatrick, Pejsachowicz, and Rabier [FP],[FPR] and [PR 1] [PR 2] etc. We are satisﬁed to introduce the idea of the deﬁnition; details are to be found in the above-mentioned references. The main diﬃculty in extending the Leray–Schauder degree theory to Fredholm mappings lies in the fact discovered by Kuiper [Kui] that the general linear group of inﬁnite-dimensional separable Hilbert space is connected and even contractible. A new ingredient has to be introduced to deﬁne the orientation of Fredholm mappings. Following [FPR], one deﬁnes the parity of curves of Fredholm operators. Let us denote by K(X) the space of all compact linear operators, and by Φ0 (X, Y ), (GL(X, Y )) the space of all Fredholm operators of index 0 (and isomorphisms resp.) between Banach spaces X and Y . For a curve of linear compact operators K ∈ C([0, 1], K(X)), if I−K(i), i = 0, 1 are invertible, then we deﬁne the parity by σ(I − K) = iLS (I − K(0), θ)iLS (I − K(1), θ) , where iLS (I − K(i), θ), i = 0, 1, are the Leray–Schauder indices. The notion of parity is extended to curves of Fredholm mappings with index 0: ∀A ∈ C([0, 1], Φ0 (X, Y )), if A(i) ∈ GL(X, Y ), then there exists N ∈ C([0, 1], GL(Y, X)) such that N (t)A(t) = I − K(t), ∀t ∈ [0, 1], where K ∈ C([0, 1], K(X)). If A(i) ∈ GL(X, Y ), i = 0, 1, then we deﬁne σ(A) = σ(I − K) . One can show that σ(A) does not depend on the special choice of N . In particular, if A ∈ C([0, 1], GL(X, Y )), then we take N (t) = A−1 (t), and then σ(A) = 1. If X = Y is a ﬁnite-dimensional Banach space, by deﬁnition, σ(A) = ±1 if and only if A(0) and A(1) lie in the same/diﬀerent connected component(s) in GL(X). But this is not true for inﬁnite-dimensional space due to the above-mentioned Kuiper’s theorem. A geometric interpretation is given in [FP]: Let Sj = {L ∈ Φ(X, Y +)∞| dimkerL = j}, j = 1, 2, . . ., and S = Φ0 (X, Y )\GL(X, Y ), then S = j=1 Sj , and σ(A) is the number of points of transversal intersection of a generic path A with S1 . The parity of the curve A ∈ C([0, 1], Φ0 ) enjoys the following homotopy invariance and the invariance under reparametrizations: (Homotopy invariance) Let H ∈ C([0, 1]2 , Φ0 (X, Y )) and suppose that H(t, i) ∈ GL(X, Y ), i = 0, 1, ∀t ∈ [0, 1]. Then σ(H(t, ·)) is a constant.

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(Invariance under reparametrizations) Let A ∈ C([0, 1], Φ0 (X, Y )) with A(i) ∈ GL(X, Y ), i = 0, 1, and let γ ∈ C([0, 1], [0, 1]) satisfy γ(i) = i, i = 0, 1. Then σ(A ◦ γ) = σ(A). Let Ω ⊂ U ⊂ X be open subsets, and U be connected and simply con¯ and the boundary ∂Ω are understood nected. In the following the closure Ω relative to U . Deﬁnition 3.7.12 A Fredholm mapping F ∈ C 1 (U, Y ) of index 0 is called Ω-admissible if F |Ω¯ is proper. A Fredholm homotopy H ∈ C 1 ([0, 1] × U, Y ) of index 1 is called Ω-admissible if H|[0,1]×Ω¯ is proper. Let p ∈ U be a regular point of a C 1 Ω-admissible Fredholm mapping F ; we take it as a base point. Let y ∈ Y \F (∂Ω) be a regular value of F |Ω , then F −1 (y) ∩ Ω is a ﬁnite set (may be empty): {x1 , x2 , . . . , xk }. If it is not empty, let γi ∈ C([0, 1], U ), with γi (0) = p, γi (1) = xi , be curves connecting p and xi , i = 1, 2, . . . , k. Then the parities σi = σ(F ◦ γi ), i = 1, 2, . . . , k, are all well deﬁned, and are independent of γi . Thus for a regular value y, we deﬁne the degree of F with the base point p by degp (F, Ω, y) =

k

σi .

i=1

If F −1 (y) ∩ Ω = ∅, then we deﬁne degp (F, Ω, y) = 0. Quinn and Sard [QS] avoided the requirement of the separability of the spaces X and Y , and obtained an improved version of Sard–Smale Theorem, by which one shows that for any y ∈ Y \F (∂Ω) there exists > 0 such that B (y) ⊂ Y \F (∂Ω) contains a regular value z of F . In combining with an approximation theorem for C 1 Fredholm mappings of any index in [PR 2], we can deﬁne Deﬁnition 3.7.13 Assume that F ∈ C 1 (U, Y ) is Ω-admissible, p ∈ U is a base point of F , and y ∈ Y \F (∂Ω). We deﬁne the degree degp (F, Ω, y) = degp (F, Ω, z), where z ∈ B (y) ⊂ Y \F (∂Ω) is a regular value of F |Ω . Again, the degree is independent of the choice of z. For diﬀerent base points p0 , p1 , one has degp0 (F, Ω, y) = σ(F ◦ γ) degp1 (F, Ω, y) , where γ ∈ C([0, 1], U ), γ(i) = i, i = 0, 1. The following fundamental properties hold: (Homotopy invariance) Let H be an Ω-admissible homotopy. Suppose that y ∈ Y \([0, 1] × ∂Ω) and that p is a base point of H(t, ·) ∀ t ∈ [0, 1]. Then degp (H(1, ·), Ω, y) = degp (H(0, ·), Ω, y) . (Additivity) Suppose the Ω = Ω1 ∪ Ω2 , where Ωi , i = 1, 2 are disjoint open subsets of U , and that F ∈ C 1 (U, Y ) is Ω-admissible, p ∈ U is a base point

3.7 Extensions

203

of F . Then F is Ωi -admissible for i = 1, 2. Moreover, if y ∈ Y \F (∂Ω), then y ∈ Y \F (∂Ωi ), i = 1, 2, and degp (F, Ω, y) = degp (F, Ω1 , y) + degp (F, Ω2 , y) . (Normality) Let Ω be an open subset of X, ∀p ∈ X, and y ∈ / ∂Ω. Then / Ω. degp (id, Ω, y) = 1 if y ∈ Ω, = 0 if y ∈ Obviously, the excision property and the Kronecker existence theorem hold as well. The generalized degree has been used to extend Rabinowitz global bifurcation theorem [PR1], and is applied to the study of bifurcation problems for semi-linear elliptic equations on Rn (see [JLS]).

4 Minimization Methods

The calculus of variations studies the optimal shape, time, velocity, energy, volume or gain etc. under certain conditions. Laws in astronomy, mechanics, physics, all natural sciences and engineering technologies, as well as in economic behavior obey variational principles. The main object of the calculus of variations is to ﬁnd out the solutions governed by these principles. Tracing back to Fermat, who postulated that light follows a path of least possible time, this is a subject in ﬁnding the minimizers of a given functional. Starting from the brothers Johann and Jakob Bernoulli and L. Euler, the calculus of variations has a long history, and renews itself according to the developments of mathematics and other sciences. The problem is formulated as follows: Assume that f : Rn × RN × RnN → 1 R is a continuous function, and that E is a set of N-vector functions. Let J be a functional deﬁned on E:

J(u) = f (x, u(x), ∇u(x))dx . Find u0 ∈ E, such that J(u0 ) = Min{J(u) | u ∈ E} . The central problems in the calculus of variations are the existence and the regularity of the minimizers. These are the 19th and the 20th problems among the 23 problems posed by Hilbert in his famous lecture delivered at International Congress of Mathematicians in 1900. This chapter is devoted to an introduction of the minimization method. We pay attention only to the existence of minimizers, but not to the regularity, although the latter is a very important and rich part of the theory of the calculus of variations. The direct method, studied in Sect. 4.2, is the core of the minimization method, in which w∗ -compactness and w∗ lower semicontinuity (w∗ l.s.c.) play crucial roles. A necessary and suﬃcient condition on the integrand f for the w∗ l.s.c. of the functional J on the Sobolev space W 1,p , p ∈ (1, ∞] is studied in Sect. 4.3.

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4 Minimization Methods

In the case when w∗ l.s.c. fails, either the minimizing sequence does not converge or it does not converge to a minimizer. The Young measure and the relaxation functional are introduced in Sect. 4.4. In the spaces W 1,1 and L1 the closed balls are no longer w∗ compact. Instead, we consider the BV space and the Hardy space, respectively. They are studied in Sect. 4.5. Two interesting applications are given in Sect. 4.6. One is on the phase transitions and the other is the segmentation in the image processing. The concentration phenomenon, which happens in many problems from geometry to physics, concerns the lack of compactness. We give a brief introduction to the method of managing this phenomenon in Sect. 4.7. The minimax method dealing with saddle points is brieﬂy introduced in Sect. 4.8. With the aid of the Ekeland variational principle and the Palais– Smale condition, it is studied in the spirit of the minimization method. Section 4.1 is an introduction, where various variational principles and their reductions are introduced.

4.1 Variational Principles Let X be a real Banach space, and U ⊂ X be an open set. A point x0 ∈ U is called a local maximum (or minimum) point of f : U → R1 , if f (x) f (x0 ) (or f (x) f (x0 ))

∀x ∈ Bε (x0 ) ⊂ U ,

for some ε > 0. If further, f is G-diﬀerentiable at x0 , then df (x0 , h) =

d f (x0 + th)t=0 = θ ∀h ∈ X , dt

or simply df (x0 ) = θ .

(4.1)

Moreover, if f has second-order G-derivatives at x0 , then d2 f (x0 )(h, h) 0

(or 0)

∀h ∈ X .

In particular, if X is a Hilbert space, and f ∈ C 2 then d2 f (x0 ) is a selfadjoint operator. We conclude that d2 f (x0 ) is nonnegative (nonpositive) if x0 is a local minimum (or maximum) point. Conversely, x0 is a local minimum (or maximum) point if d2 f (x0 ) is positive (or negative) deﬁnite. 4.1.1 Constraint Problems Let X, Y be real Banach spaces, U ⊂ X be an open set. Suppose that f : U → R1 , g : U → Y are C 1 mappings. Let

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207

M = {x ∈ U g(x) = θ} . Find the necessary condition for min f (x) .

x∈M

(4.2)

Theorem 4.1.1 (Ljusternik) Suppose that x0 ∈ M solves (4.2), and that Im g (x0 ) is closed. Then ∃(λ, y ∗ ) ∈ R1 × Y ∗ such that (λ, y ∗ ) = (0, θ), and λf (x0 ) + g (x0 )∗ y ∗ = θ .

(4.3)

Furthermore, if Im g (x0 ) = Y , then λ = 0. Proof. In the case where Y1 = Im g (x0 ) Y , the conclusion (4.3) is trivial; one may choose λ = 0 and y ∗ ∈ Y1⊥ := {z ∗ ∈ Y ∗ | z ∗ , z = 0 ∀z ∈ Y1 }. We assume Y1 = Y . The tangent space Tx0 (M ) of M at x0 is as follows: Tx0 (M ) ˙ = h} . = {h ∈ X ∃ε > 0, ∃v ∈ C 1 ((−ε, ε), X), x0 + v(t) ∈ M, v(0) = θ, v(0) We want to prove that Tx0 (M ) = ker g (x0 ). In order to avoid technical complication, we make an additional assumption: Either g (x0 ) is a Fredholm operator, or X is a Hilbert space. The assumption is superﬂuous, because a modiﬁed IFT has been studied in [De] (pp. 334) to improve the proof. Indeed, from g(x0 + v(t)) = θ , it follows that g (x0 )h =

d g(x0 + v(t)) t=0 = 0 ∀ h ∈ Tx0 (M ) , dt

i.e., Tx0 (M ) ⊂ ker g (x0 ). On the other hand, if h ∈ ker g (x0 ), one solves the equation: g(x0 + th + w(t)) = θ , for w ∈ C 1 ((−ε, ε), X1 ), w(0) = θ, where X1 is the complement of ker g (x0 ) and ε > 0 is small. Since g(x0 ) = θ, g (x0 ) : X1 → Y is an isomorphism, one may apply the IFT to obtain such a solution w. Setting v(t) = th + w(t), we have v(0) = θ and v(0) ˙ = h + w(0). ˙ From ˙ =θ, g (x0 )(h + w(0)) it follows that w(0) ˙ ∈ ker g (x0 ), but w(0) ˙ ∈ X1 which implies that w(0) ˙ = θ, i.e., v(0) ˙ = h. Now, ∀ h ∈ Tx0 (M ), ∃v(t) satisfying v(0) = θ, v˙ = h and x0 + v(t) ∈ M , so that f (x0 + v(t)) f (x0 ) ,

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4 Minimization Methods

which implies that

f (x0 ), h = 0 ,

i.e. f (x0 ) ∈ ker g (x0 )⊥ . By the closed range theorem, ∃y ∗ ∈ Y ∗ such that −f (x0 ) = g (x0 )∗ y ∗ . This completes the proof. Corollary 4.1.2 Suppose that g1 , . . . , gm : U → R1 are C 1 functions, and that x0 solves (4.2) with M = {x ∈ U gi (x) = 0, i = 1, 2, . . . , m} . If {gi (x0 )}m 1 is linearly independent, then ∃ λ1 , . . . , λm such that f (x0 ) +

m

λi gi (x0 ) = 0 .

i=1

Moreover, we may also consider inequality constraints: Given f, g1 , . . . , gm , h1 , . . . , hl ∈ C 1 (X, R1 ) ﬁnd min {f (x)|gi (x) = 0, i = 1, . . . , m; hj (x) ≤ 0, j = 1, . . . , l} . In the same manner, we ﬁnd the necessary condition of an extremum point x0 :

∃ λ1 , . . . , λm ∈ R1 , ∃ µ0 , µ1 , . . . , µl ≥ 0, but not all zero ,

such that µ0 f (x0 )+

m i=1

λi gi (x0 )+

l

µj hj (x0 ) = θ, and µj hj (x0 ) = 0, ∀ j = 1, . . . , l .

j=1

This is called the Kuhn–Tucker condition in the mathematical programming. In particular, if all the functions hj , j = 1, . . . , l are convex, which implies that the set C = {x ∈ X|hj (x) ≤ 0, j = 1, . . . , l} is convex, then a necessary condition for an extremum x0 ∈ C is the following variational inequality: 9 8 m λi gi (x0 ), y − x0 ≥ 0, ∀y ∈ C . f (x0 ) + 1

Remark 4.1.3 λ1 , . . . , λm ; µ0 , µ1 , . . . , µl , are called the Lagrangian multipliers.

4.1 Variational Principles

209

4.1.2 Euler–Lagrange Equation In case f is a functional of the following form:

f (u) = ϕ(x, u(x), ∇u(x))dx

Ω ⊂ Rn ,

(4.4)

Ω

where ϕ : Ω × RN × RnN → R1 is a C 2 function, let us turn to the 1st and 2nd variations of the functional (4.4). ∀u ∈ C 1 (Ω) f (u), v

n N N ∂ϕ(x, u(x), ∇u(x)) ∂vi ∂ϕ(x, u(x), ∇u(x)) + vi , = ∂ξαi ∂xα i=1 ∂ui Ω i=1 α=1 and

N n ∂ 2 ϕ(x, u(x), ∇u(x)) ∂vi ∂vk f (u), v ⊗ v = ∂xα ∂xβ ∂ξαi ∂ξβk Ω i,k=1 α,β=1

+2

n N i,k=1

N ∂ 2 ϕ ∂vi ∂2ϕ · v + vi vk k i ∂ξα ∂uk ∂xα ∂ui ∂uk α=1

,

i,k=1

∀v ∈ C 1 (Ω, RN ). Thus the Euler–Lagrange equation under certain boundary conditions reads as: n ∂ϕ(x, u(x), ∇u(x)) ∂ ∂ϕ(x, u(x), ∇u(x)) = 0, i = 1, 2, . . . , N . + − i ∂x ∂ξ ∂ui α α α=1 It is a second-order diﬀerential system. If the functional f is only deﬁned on a closed convex subset C of the function space C 1 (Ω), and u is a minimizer in C, then we only have a variational inequality:

n N ∂ϕ(x, u(x), ∇u(x)) ∂(vi − ui ) ∂ϕ(x, u(x), ∇u(x)) + (vi − ui ) ∂ξαi ∂xα ∂ui Ω i=1 α=1 ≥0, for all v ∈ C. In case N = 1, the Euler–Lagrange equation reads as −divϕξ (x, u(x), ∇u(x)) + ϕu (x, u(x), ∇u(x)) = 0 . This is a diﬀerential equation. We give here a few examples.

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4 Minimization Methods

1. (Geodesics) Let U be an open set of Rn , and let gij : U → R1 , ∀i, j = 1, 2, . . . , n be symmetric and positive deﬁnite. For u = (u1 , . . . , un ) ∈ C 1 ([0, 1], U ), one deﬁnes ϕ(u, ξ) =

n

gij (u)ξi ξj , ∀ ξ = (ξ1 , . . . , ξn ) ∈ Rn .

i,j=1

Then the Euler–Lagrange equation is the following diﬀerential system: j k d2 u i i du du = 0, + Γ j,k dt2 dt dt

where Γij,k =

∀i = 1, 2, . . . , n, t ∈ [0, 1] ,

1 [∂j glk + ∂k glj − ∂l gjk ]g li , 2

and (g li ) is the inverse matrix of (gil ), i.e., gij g ik = δjk . 2. (Poisson equation) Let ϕ(x, u, ξ) = c(x)u + 12 |ξ|2 . The Euler–Lagrange equation is the Poisson equation: u = c(x)

in Ω .

3. (Hamiltonian systems) For H ∈ C 1 (R1 × Rn × Rn , R1 ), the following ODE system x˙ = −Hp (t, x, p), p˙ = Hx (t, x, p), with (t, x, p) ∈ R1 × Rn × Rn , is called a Hamiltonian system, and H is called a Hamiltonian function. Sometimes we use the notations: z = (x, p) ,

and J=

0 I

−I 0

,

where I is the n × n unit matrix. The above system then has a simple form: z˙ = J grad H(t, z) . J grad sometimes is also called a symplectic gradient. The 2π-periodic solution can be seen as a critical point of the following functional:

2π 1 f (z) = z, J z

˙ R2n + H(t, z) 2 0 on the space C 1 ([0, 2π], R2n ). In other words, the Euler–Lagrange equation of this functional is the Hamiltonian system.

4.1 Variational Principles

211

1

4. (Minimal surface) Let u ∈ C 1 (Ω, R1 ), and ϕ(ξ) = [1 + |ξ|2 ] 2 . Then the Euler–Lagrange equation reads as ∇u =0. div 1 [1 + |∇u|2 ] 2 5. (Obstacle problem revisit) Let Ω ⊂ Rn be a bounded domain, g ∈ L2 (Ω), ψ ∈ H 1 (Ω). Find a minimizer of the problem: Min{f (u) | u ∈ E}, where

1 2 |∇u| − g · u , f (u) = 2 Ω and E = {u ∈ H01 (Ω) | u(x) ≤ ψ(x) a.e. x ∈ Ω } . The variational inequality reads as

∇u · ∇(v − u) ≥ g(v − u), ∀ v ∈ E . Ω

Ω

Now we derive the necessary condition on ϕ for a minimizer of f under Dirichlet boundary conditions. If u is a minimizer, then by Taylor expansion, f (u), v ⊗ v 0,

∀v ∈ W01,∞ (Ω, RN ) .

(4.5)

Let ρ(t) = 1 + t for t ∈ [−1, 0], and ρ(t) = 1 − t for t ∈ : [0, 1]. ∀x0 = n (x0,1 . . . , x0,n ) ∈ Ω, ∃ε > 0 such that the cube centered at x0 : α=1 [x0,α − , x0,α + ] ⊂ Ω. ∀λ = (λ1 , . . . , λn ) ∈ Sn−1 and ∀ξ = (ξ1 , . . . , ξN ) ∈ RN , we deﬁne α λ (xα − x0,α ) vε (x) = Πnα=1 ρ ·ξ , ε where ρ is understood to be 0 outside (−1, 1). Substituting vε into (4.5), and letting ε → 0, we obtain n N ∂ 2 ϕ(x0 , u(x0 ), ∇u(x0 )) α β λ λ ξi ξk ≥ 0 ∀(λ, ξ) ∈ (Rn \{θ}) × RN . i ∂ξ k ∂ξ α β i,k=1 α,β=1

This is called the Legendre–Hardamard condition, which can be rewritten in a compact form: (λ ⊗ ξ)T ∂ 2 ϕ(x, u(x), ∇u(x))(λ ⊗ ξ) 0 ,

(4.6)

where λ ⊗ ξ denotes the rank-one n × N matrix. In the next section, this condition on ϕ is called rank-one convexity. Thus, rank-one convexity is the necessary condition for ϕ at a minimizer. In particular, if N = 1, condition (4.6) is reduced to n ∂ 2 ϕ(x, u(x), ∇u(x)) j l λ λ 0, ∂ξj ∂ξl

j,l=1

∀λ ∈ Rn \{θ}, ∀x ∈ Ω .

(4.7)

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4 Minimization Methods

However, if the strict inequality in (4.7) holds for all λ ∈ Rn \{θ}, i.e., ϕ is strictly convex with respect to ξ. This is the ellipticity condition. To the vectorial case, the condition: N n

∂2ϕ (u, u(x), ∇u(x))ξαi ξβk > 0, ∀ ξ = {ξαi } ∈ Mn×N , with |ξ| = 0, i ∂ξ k ∂ξ α β i,k=1 α,β=1 is called the strong ellipticity. 4.1.3 Dual Variational Principle The following notion was introduced by Fenchel, and is very important in convex analysis. Deﬁnition 4.1.4 Suppose that X is a real Banach space and that f : X → R1 ∪ {+∞} is proper. The conjugate function of f, f ∗ : X ∗ → R1 ∪ {+∞} is deﬁned by f ∗ (x∗ ) = sup { x∗ , x − f (x)} . x∈X

The notion was initiated by Young’s inequality: if f (x) = p1 |x|p , 1 < p <

∞, then f ∗ (x) = p1 |x|p , where p1 + p1 = 1. In particular, if f (x) = 12 Ax, x Rn , where A is positive deﬁnite, then ∗ f (p) = 12 p, A−1 p Rn . The following propositions hold: 1. f ∗ is convex and l.s.c. Moreover, it is proper, if f is proper l.s.c. and convex. Proof. Only the properness of f ∗ needs to be proved, i.e., one should ﬁnd x∗0 ∈ X ∗ such that f ∗ (x∗0 ) < +∞. We consider the closed convex set epi(f ) = {(x, t) ∈ X × R1 | f (x) t}. Since f is proper, ∃x0 ∈ X such that f (x0 ) < +∞. One chooses t0 < f (x0 ), then (x0 , t0 ) ∈ epi(f ). By the Ascoli separation theorem, ∃(x∗0 , λ) ∈ X ∗ × R1 , ∃α ∈ R1 , satisfying x∗0 , x + λt > α > x∗0 , x0 + λt0 , In particular,

∀(x, t) ∈ epi(f ) .

x∗0 , x0 + λf (x0 ) > x∗0 , x0 + λt0 .

It follows that λ > 0, and then < ; α 1 − x∗0 , x − f (x) < − λ λ

∀x ∈ D(f ) ,

(4.8)

4.1 Variational Principles

i.e., f

∗

x∗ − 0 λ

0, then x∗0 , x + λf (x) > α It follows that

∀ x ∈ dom(f ) .

∗ x α f∗ − 0 − . λ λ

However, by the deﬁnition of f ∗∗ , ; ∗ < ∗ x0 x ∗∗ ∗ f (x0 ) − , x0 − f − 0 . λ λ It follows that

x∗0 , x0 + λf ∗∗ (x0 ) α .

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4 Minimization Methods

This contradicts (4.8), in which t0 = f ∗∗ (x0 ). In the case where f (x0 ) < +∞, from (4.8), λ > 0. It remains to verify that f ∗∗ (x0 ) = +∞ if f (x0 ) = +∞ and λ = 0. Again from (4.8), ∃ > 0 such that x∗0 , x − x0 ≥ ∀x ∈ dom(f ) . Since f ∗ is proper, there is an x∗1 ∈ X ∗ such that f ∗ (x∗1 ) < +∞, and x∗1 , x − f (x) − f ∗ (x∗1 ) ≤ 0 ∀x ∈ dom(f ) . Putting them together, ∀n ∈ N, we have x∗1 − nx∗0 , x + n x∗0 , x0 + n − f (x) − f ∗ (x∗1 ) ≤ 0, ∀x ∈ dom(f ) , it follows that f ∗ (x∗1 − nx∗0 ) + n x∗0 , x0 + n − f ∗ (x∗1 ) ≤ 0 , or n + x∗1 , x0 − f ∗ (x∗1 ) ≤ x∗1 − nx∗0 , x0 − f ∗ (x∗1 − nx∗0 ) ≤ f ∗∗ (x0 ) . Letting n → ∞, we obtain f ∗∗ (x0 ) = +∞. Therefore, f = f ∗∗ .

Corollary 4.1.6 For a proper, l.s.c., convex function f , x∗ ∈ ∂f (x) ⇐⇒ x ∈ ∂f ∗ (x∗ ). Proof. It is a direct consequence of proposition 4 and Theorem 4.1.5.

In the case where both ∂f and ∂f ∗ are single valued, Corollary 4.1.6 means that they are mutually inverse. In this sense, the conjugate function f ∗ of f is called the Legendre transform of f . Recall the variational formulation of the Hamiltonian system, which describes the motion of particle systems. The functional

2π 1 z, J z

˙ R2n + H(t, z) f (z) = 2 0 is very indeﬁnite. One cannot pose the minimization problem. However, in some cases, the Legendre transform may help. (1) Assume that H(t, x, p) is proper, l.s.c., and convex in the variables p, we deﬁne the Legendre transform of H with respect to p, i.e., L(t, x, q) = sup{ p, q Rn − H(t, x, p)} . p

It is called the Lagrangian. In other words, ∀(x, t) as a function of q, L is the conjugate function of the function H(t, x, p). From proposition 4 and the Hamiltonian system, one has

4.1 Variational Principles

215

−x, ˙ p Rn = L(t, x, −x) ˙ + H(t, x, p) . If both H and L are diﬀerentiable, then, Lx (t, x, −x) ˙ = −Hx (t, x, p) . But from Corollary 4.1.6, x˙ = −Hp (t, x, p) ⇔ p = Lq (t, x, −x) ˙ . This shows that (x, x) ˙ satisﬁes the system: d Lq (t, x, −x) ˙ + Lx (t, x, −x) ˙ =0. dt

(4.9)

However, (4.9) is the Euler–Lagrange equation of the functional:

2π

L(t, x, −x)dt ˙ .

I(x) = 0

The system (4.9) is called the associate Lagrange system. For example, in a system of particles with generalized coordinates q = (q1 , . . . , qn ), the kinetic energy of the system is a positive deﬁnite quadratic aij qi qj , where aij = aij (t, q), and the potential energy is a form: T = 12 continuous function bounded from below: U = U (q). The total energy H = T + U is called the Hamiltonian, and L = T − U is called the Lagrangian. In this case, the functional I is bounded from below. (2) Assume that H(t, x, p) is strictly convex in z = (x, p), then the Legendre transform of H with respect to z reads as G(t, w) = sup{ z, w R2n − H(t, z)} . z

Assume Hz (t, θ) = θ. We study nontrivial 2π-periodic solutions of the Hamiltonian system. Let us consider a constraint variational problem:

2π G(t, −J w)dt ˙ , I(w) = 0

with

g(w) = 0

2π

w, J w

˙ R2n dt = c = 0 ,

where c is a parameter. The Euler–Lagrange equation reads as Gw (t, −J w) ˙ = λw . Claim: λ = 0. If not, provided by w = Hz (t, z) ⇔ z = Gw (t, w) ,

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4 Minimization Methods

we have −J w˙ = Hz (t, θ) = θ. This contradicts c = 0. Let z = λw. Again by duality, we have z˙ = λJHz (t, z) . One can adjust c such that λ = 1. For example, if H(t, z) is bounded by two quadratic functions: 0 < m|z|2 ≤ H(t, z) ≤ M |z|2 , then G(t, w) is also. Thus the functional I is bounded from below.

4.2 Direct Method 4.2.1 Fundamental Principle Given a topological space X and a function f : X → R1 ∪ {+∞}, which is bounded from below, we seek the minimizers of f on X, if they exist. It is natural to require the l.s.c. of f (i.e., ∀t ∈ R1 , the level set ft := {x ∈ X| f (x) t} is closed) and certain compactness on X. However, if X is not compact, the coercive condition on f should be assumed as a replacement: f is proper and ∀t ∈ R1 ∃ a compact subset Kt ⊂ X such that the level set ft ⊂ Kt . Indeed, we have: Theorem 4.2.1 If f : X → R1 ∪ {+∞} is l.s.c. and coercive, then it attains a minimizer on X. In particular, f is bounded from below. Proof. Since f is coercive, it is proper. Then −∞ ≤ m := inf f < +∞. By the assumption, ∀t > m, the set ft is compact. Therefore by the ﬁnite intersection property, ∩ ft = Ø. A point x0 in the intersection achieves f (x0 ) = m. Since t>m

f does not assume the value −∞, m > −∞, and then f is bounded from below. However, in analysis, people prefer to use a minimizing sequence (i.e., a sequence {xj } ⊂ X such that f (xj ) → m := inf f ) to approach the minimizer. In this case, it is required that f is sequentially lower semi-continuous (s.l.s.c., for short) and we assume that Kt is sequentially compact, ∀t ∈ R1 . These two notions are deﬁned as follows: f is s.l.s.c. ⇐⇒ limf (zj ) f (z) if zj → z . and K is sequentially compact ⇐⇒ any sequence {zj } ⊂ K contains a subsequence zj → z ∈ K . Namely,

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217

Theorem 4.2.2 If f : X → R1 ∪ {+∞} is s.l.s.c., and that ∀t ∈ R1 ∃ a sequentially compact set Kt ⊂ X such that ft ⊂ Kt , then f attains a minimizer in X. Recall the Eberlein–Schmulian theorem, for weakly closed subsets of a Banach space, weakly compact = sequentially weakly compact. According to the Banach–Alaoglu theorem, every w∗ -closed norm bounded set in the dual space of a Banach space is w∗ compact. Thus, for every weakly closed subset of a reﬂexive Banach space, weakly closed plus norm bounded = weakly compact = sequentially weakly compact. In this case, the coerciveness of f is equivalent to that ∀t ∈ R1 the level set ft of f is bounded in norm, or equivalently, that f is proper and f (x) → +∞ as x → ∞. However, how do we verify the weakly* lower semi-continuity (w∗ l.s.c., in short) or sequentially weakly* lower semi-continuity (s.w∗ l.s.c., in short) of f? We notice that in a Banach space, closed convex set = weakly closed convex set (Hahn–Banach theorem) = sequentially weakly closed convex set (Mazur theorem). Thus, if f is convex, then l.s.c. = w.l.s.c. = s.w.l.s.c. Combining the above discussions, both Theorem 4.2.1 and Theorem 4.2.2 imply Corollary 2.3.7 as a special case. Because of the importance of this statement, we rewrite it as follow: Theorem 4.2.3 Let X be a reﬂexive Banach space and let E ⊂ X be a weakly (or weakly sequentially) closed nonempty subset. If f : E → R1 ∪ {+∞} is a l.s.c., convex and coercive function, then f has a minimizer on E. In fact, Theorem 4.2.3 is a general principle in the proof of the existence of a minimizer. We shall present several classical examples to show how it works. 4.2.2 Examples Example 1. (Dirichlet problem for Poisson equation) Given f ∈ L2 (Ω), where Ω ⊂ Rn is a bounded domain, ﬁnd a minimizer of the following functional on H01 (Ω):

1 2 |∇u| − f · u dx . J(u) = 2 Ω

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4 Minimization Methods

Obviously, J is convex and l.s.c.. We verify the coerciveness: From the Poincar´e inequality, we have a constant C > 0 such that

|u|2 dx C |∇u|2 . Ω

Ω

1 Thus, we may assume uH01 = ( Ω |∇u|2 ) 2 . Then ∃C1 > 0, such that

1 J(u) |∇u|2 − f 2 · u 2 2 Ω

1 1 2 |∇u| − |∇u|2 − C1 f 22 2 Ω 4 Ω 1 = u 2H 1 −C1 f 22 → +∞ 0 4 as u H01 → +∞. It is well known that the Euler–Lagrange equation of J is the Poisson equation with Dirichlet boundary conditions: −u = f in Ω , u=0 on ∂Ω . Example 2. (Harmonic map) Let Ω ⊂ Rm be a bounded open domain with smooth boundary ∂Ω. A map u = (u1 , . . . , un+1 ) : Ω → S n ⊂ Rn+1 , where S n is the unit sphere, is called harmonic if −uk = uk |∇u|2 where |∇u|2 =

n+1

in Ω, k = 1, 2, . . . , n + 1 ,

|∇uk |2 .

k=1 1

Given ϕ = (ϕ , . . . , ϕn+1 ) : ∂Ω → S n , ﬁnd a harmonic map u with prescribed boundary condition u|∂Ω = ϕ. Let us consider a subset of the Banach space X = H 1 (Ω, Rn+1 ): M = {u ∈ X| u|∂Ω = ϕ, u(x) ∈ S n a.e., x ∈ Ω} and deﬁne the functional 1 E(u) = 2

|∇u|2 dx . Ω

Firstly, M is a weakly sequentially closed set. Indeed, if uj u in H 1 (Ω, ), then modulo a subsequence {uj } we have uj → u in L2 (Ω, Rn+1 ) and R then uj (x) → u(x) a.e. From uj ∈ M ∀j, we have u ∈ M . Obviously, E is l.s.c., convex and coercive. n+1

4.2 Direct Method

219

We are then able to apply Theorem 4.2.3 to obtain the existence of a minimizer u∗ ∈ M . What is the Euler–Lagrange equation for E? From

∇u · ∇v = 0 , dE(u, v) = Ω

∀v ∈

H01 (Ω, Rn+1 )

satisfying v(x) ∈ Tu∗ (x) S n

we obtain

a.e. in Ω ,

(u∗ )T (x) = 0 a.e. in Ω ,

where (u∗ )T (x) is the tangential projection of u∗ (x) at u(x). Noticing that the normal projection of u(x) at u(x) reads as (u)N (x) = u(x) · u(x) , and then by diﬀerentiation twice of the constraint |u(x)|2 = 1, we obtain u(x) · u(x) = −|∇u(x)|2 , from which follows

−u∗ (x) = u∗ (x)|∇u∗ (x)|2 .

However, we have only proved that u∗ ∈ H 1 (Ω, Rn+1 ). The rest of the problem is about the regularity of u∗ . For m = 2, the harmonic map u∗ with minimal energy is smooth if ϕ is, according to a result due to Morrey [Mo 2], but for m > 2, generally speaking, there is no such regularity (see Schoen and Uhlenbeck [ScU 1, ScU 2], Lin [Lin]. The deﬁnition and an existence result for harmonic maps between two Riemannian manifolds can be found in Eells and Sampson [ES]. The regularity problem for harmonic maps has attracted many authors. As a special case for elliptic systems, this regularity problem is related to the Hilbert 20th problem. It is proved in Helein [Hel 1] that for m = 2 any harmonic map is regular, but for m = 3 Riviere [Ri] showed that there exists a non-minimal energy harmonic map discontinuous everywhere. Example 3. Nonlinear eigenvalue problem Let Ω ⊂ Rn be a bounded open domain with smooth boundary, and let φ ∈ C(Ω × R1 , R1 ) satisfy q < 2∗ =

(1) |φ(x, t)| C(1 + |t|q−1 ), (2) tφ(x, t) > 0 for t = 0;

2n n−2

if n 3,

then ∃c0 > 0 such that ∀c ∈ (0, c0 ] the equation −u(x) = λφ(x, u(x)) x ∈ Ω , u|∂Ω = 0 , has a solution (λc , uc ) satisfying

uc (x) φ(x, t)dtdx = c . Ω

0

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4 Minimization Methods

t Proof. Set X = H01 (Ω), g(u) = Ω Φ(x, u(x))dx, where Φ(x, t) = 0 φ(x, s)dx = O(|t|q ), Mc = g −1 (c), and f (u) = 12 Ω |∇u|2 . Since Φ(x, t) 0 but not identical to 0, we have g(u) 0 but not identical to 0. We have c0 > 0 such that g −1 (c0 ) = Ø; let u0 ∈ g −1 (c0 ), t → g(tu0 ) is continuous on [0, 1], and the function ranges over [0, c0 ]. Therefore g −1 (c) = Ø ∀c ∈ [0, c0 ]. Since the embedding H01 (Ω) → Lq (Ω) is compact, g is completely continuous, i.e., uj u∗ =⇒ g(uj ) → g(u∗ ). Thus Mc is a sequential weakly closed subset. Obviously, f is l.s.c., convex and coercive, so we have proved the existence of a minimum point uc . It remains to verify that g (u∗ ) = φ(x, u∗ (x)) = θ. In fact, θ ∈ Mc for c = 0, then u∗ = θ, therefore, φ(x, u∗ (x)) = θ. As a corollary, we consider the nontrivial solution of the equation: −u = |u|q−2 u in Ω, 2 < q < 2∗ u|∂Ω = 0 . Let (λ0 , u0 ) ∈ R1 × H01 (Ω) be a solution of the nonlinear eigenvalue problem: −u = λ|u|q−2 u u|∂Ω = 0 ,

with

|u0 |q > 0 . Ω

Let

|∇u0 |2 /

λ0 = Ω

Setting

|u0 |q > 0 . Ω

1

u∗ = λ0q−2 u0 , we obtain

−u∗ = |u∗ |q−2 u∗ u∗ |∂Ω = 0 .

in Ω

Remark 4.2.4 If −∆ is replaced by −∆ + I, then the same conclusion holds. Example 4. (Prescribing constant mean curvature problem) Given a constant H > 0 and a curve γ : S 1 → R3 . Suppose diam (γ) = R C2

with HR < 1. Find a disc-type surface u : D2 −→ R3 satisfying u = 2Hux ∧ uy in D u|∂D = γ , where

(4.10)

4.2 Direct Method

221

u = (u1 , u2 , u3 ) and

2 u x ux ∧ uy = 3 ux

u2y u3x , u3y u1x

u3y u1x , u1y u2x

u1y . u2y

This is a geometric problem, in which H is a prescribed constant mean curvature, and γ is a prescribed boundary value. Write the problem in its variational form; deﬁne

4H 2 J(u) = |∇u| + u · (ux ∧ uy ) on H 1 ∩ L∞ (D, R3 ) . 3 D D Since

J (u), v = 2 −u + 2H(ux ∧ uy ), v ,

∀v ∈ C0∞ (D, R3 ). The Euler–Lagrange equation is exactly (4.10). In appearance, inf J(u) = −∞, there is no minimum. However, if we add a constraint condition: u L∞ (D,R3 ) ≤ R , where R < R , and HR < 1, then by |ux ∧ uy | |ux ||uy |

we have J(u)

D

1 (|ux |2 + |uy |2 ) , 2

2 |∇u|2 − |∇u|2 3

=

1 3

|∇u|2 . D

Moreover, on the convex set CR = {u ∈ H 1 (D, R3 )| u|∂D = γ, u L∞ ≤ R } , which is sequentially weakly closed in H 1 (D, R3 ), J is nonnegative and coercive. We verify the w.s.l.s.c. of J on CR under H 1 norm. Suppose uj u∗ (H 1 (D, R3 )), with uj ∈ CR ; we have a subsequence, for which we do not change the subscripts: ⎧ ∗ 2 3 ⎪ ⎨uj → u , strongly in L (D, R ) , uj,x u∗x , weakly in L2 (D, R3 ) , ⎪ ⎩ uj,y u∗y , weakly in L2 (D, R3 ) . and

uj (x) → u∗ (x)

a.e. in D .

Rewrite uj · (ujx ∧ ujy ) − u∗ · (u∗x ∧ u∗y ) = vj · (u∗x ∧ u∗y ) + uj · (vjx ∧ u∗y + u∗x ∧ vjy ) + uj · (vjx ∧ vjy ) ,

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4 Minimization Methods

where vj = uj − u∗ . Noticing |uj ∧ u∗y | R |u∗y | ,

we obtain

|uj ∧ u∗y − u∗ ∧ u∗y |2 → 0 ,

D

provided by Lebesgue’s dominance theorem. Therefore

∗ uj · (vjx ∧ uy ) = − vjx · (uj ∧ u∗y ) → 0 . D

D

Similarly

uj · (u∗x ∧ vjy ) → 0 .

D

1 |∇u∗ |2 , 2 D D and |vj | 2R , again by Lebesgue’s dominance theorem, we have

vj · (u∗x ∧ u∗y ) → 0 .

Since

|u∗x ∧ vy∗ |

D

Hence, ∗

4H uj · (vjx ∧ vjy ) + ◦(1) |∇vj |2 + 2∇vj ∇u∗ + 3 D

1 |∇vj |2 + ◦(1) ; 3 D

J(uj ) − J(u ) =

it follows that

limj→∞ J(uj ) J(u∗ ) .

How do we drop the constraint? By deﬁnition, if u∗ ∈ CR achieves the minimum point of J on CR , then u∗ satisﬁes the variational inequality:

∇u∗ ∇v + 2Hv · (u∗x ∧ u∗y ) 0, ∀v = w − u∗ , (4.11) D

where w ∈ CR . In particular, for any η ∈ C0∞ (D, R1 ), η ≥ 0, one chooses ε > 0 such that (1 − εη)u∗ ∈ CR . Substituting v = −εηu∗ into (4.11), we have 1 − |u∗ |2 + |∇u∗ |2 + 2Hu∗ · (u∗x ∧ u∗y ) 0 in D . 2 It follows that −|u∗ |2 0, in D , |u∗ | = |γ| on ∂D . Thus, |u∗ |2 is subharmonic. Applying the maximum principle, we have u∗ L∞ < R < R , i.e., u∗ is strictly in the interior of CR under the L∞ (D, R3 ) norm. Therefore we may choose arbitrarily v ∈ C0∞ (D, R3 ) with v L∞ < R − R as variations. Thus u∗ is a weak solution of equation (4.10).

4.2 Direct Method

223

4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement We turn to studying a problem arising in diﬀerential geometry. Let (M, g0 ) be a smooth compact two-dimensional Riemannian manifold with metric g0 ; let k(x) be its Gaussian curvature. Given a function K(x) on M , does there exist a metric g, which is pointwisely conformal to g0 , such that K is the Gaussian curvature with respect to g? Setting a function u on M satisfying g = e2u g0 , the problem is reduced to solving the following PDE: u = k − Ke2u on M , where is the Laplace–Beltrami operator with respect to g0 . According to the Gauss–Bonnet formula:

KdVg = 2πχ(M ) ,

(4.12)

(4.13)

M

where χ(M ) is the Euler characteristic of M , and Vg is the volume form with respect to g. The curvature K is restricted by the topology χ(M ); e.g., if χ(M ) > 0 (or < 0) then max K > 0 (or min K < 0 resp.), and if χ(M ) = 0, then either K ≡ 0 or K changes sign. A result in the case where χ(M ) < 0 has been studied in Sect. 2.1, Example 5. Namely: Statement. Assume k(x) < 0 and K(x) < 0 ∀x ∈ M .

(4.14)

Then (4.12) possesses a solution. However, (4.14) is not a necessary condition; the problem remains open if one merely assumes min K < 0. In contrast, the case χ(M ) = 0 was completely solved by a variational argument. A necessary condition for the solvability of (4.12) with χ(M ) = 0 reads as either K ≡ 0 , (4.15) or K changes sign and M Ke2v dVg0 < 0 , where v is the solution of v = k, v =

1 Vol(M )

v=0.

(4.16)

M

In fact, if u is a solution of (4.12), and let w = u − v, then −w = Ke2(w+v) .

(4.17)

224

4 Minimization Methods

(4.15) follows from

M

Ke2v dVg0 = −

we−2w dVg0

M

= −2 M

|∇w|2 e−2w dVg0 < 0 .

Conversely, the condition (4.15) is also suﬃcient. Without loss of generality, we assume K ≡ 0 (otherwise, this is a linear problem). Let us consider a variational problem associated with (4.17):

1 |∇w|2 dVg0 J(w) = 2 M deﬁned on M = {w ∈ H 1 (M )| g1 (w) = g2 (w) = 0} ,

where g1 (w) =

M

wdVg0 ,

and g2 (w) =

M

Ke2(w+v) dVg0 .

If M is a nonempty weakly closed Banach manifold, and if w is a local minimum point, then there exist Lagrange multipliers λ, µ ∈ R1 such that

∇w · ∇ϕ + λϕ + 2µKe2(w+v) ϕ = 0 , M

∀ϕ ∈ H (M ), i.e., 1

−w + λ + 2µKe2(w+v) = 0 .

From g2 (w) = 0, it follows that λ = 0; and according to (4.15) and integration by parts, we have µ < 0. Let us choose r = 12 log(−2µ), then u = v +w +r solves (4.12). We introduce the notion of the Schwarz symmetric rearrangement of a function and discuss its main properties, which will be often used in analysis. For a nonnegative measurable function u deﬁned on a n-dimensional measurable set Ω with m(Ω) < ∞, the Schwarz symmetric rearrangement of u is deﬁned to be the following: u∗ (y) = sup {t ≥ 0 | |y|n ≤ Cn−1 m{x ∈ Ω | u(x) ≥ t}} , where Cn is the volume of the unit ball in Rn . It possesses the following properties: 1. Let u∗ (y) = g(|y|) ∀ y ∈ Ω∗ := BR (θ), where Rn = nonincreasing and g(R) = 0.

1 Cn m(Ω).

Then g is

4.2 Direct Method

225

2. Let ut ={x ∈ Ω | u(x) ≥ t} ∀ t ≥ 0. Then m(ut ) = m(u∗t ) ∀ t ≥ 0. 3. Ω up = Ω∗ u∗p ∀ p ∈ [1, ∞). 4. (Faber–Krahn inequality, [Fa], [Krh]) If u ∈ H 1 (Ω), where Ω ⊂ Rn is a bounded domain, then

|∇u∗ |2 ≤ |∇u|2 . Ω∗

Ω

Proof. Let H n−1 (E) denote the (n − 1)-dimensional Hausdorﬀ measure. From 2, the isoperimetric inequality, and the rotational invariance of u∗ , we have H

n−1

(u

−1

2

2

(t)) =

≤

1 u−1 (t)

H

n−1

(u

∗−1

2

2

(t)) =

1

u−1 (t)

|∇u|

∗ 2

=

u∗−1 (t)

and

u∗−1

u−1

1 |∇u|

|∇u |

u∗−1 (t)

,

1 |∇u∗ |

,

H n−1 (u∗−1 (t)) ≤ H n−1 (u−1 (t)) .

By the co-area formula and 2, it follows that

dm(ut ) dm(u∗t ) 1 1 − = = =− . ∗| |∇u| dt dt |∇u u−1 (t) u∗−1 (t)

Thus

u∗−1 (t)

∗

|∇u | ≤

Again by the co-area formula

∞

|∇u∗ |2 = |∇u∗ |dt ≤ Ω∗

0

u∗−1 (t)

|∇u| .

u−1 (t)

∞

0

u−1 (t)

|∇u|dt =

|∇u|2 . Ω

We now turn to the variational problem. Lemma 4.2.5 (Trudinger) Assume that Ω is a bounded planar domain with smooth boundary. Then ∀u ∈ H01 (Ω) with u 2 := Ω |∇u|2 ≤ 1, ∀β < 2 4π, ∃γ = γ(β) such that Ω eβu ≤ γ. Proof. Since |u| ≤ u , we may assume u ≥ 0. Let u∗ (y) be the Schwarz symmetric rearrangement of u. Thus u∗ (y) is radially symmetric, nonincreasing and satisﬁes |{y ∈ R2 | u∗ (y) ≥ t}| = |{x ∈ Ω| u(x) ≥ t}| ∀t ∈ R1 .

226

4 Minimization Methods 1

Let u∗ (y) = g(|y|), and R = (π −1 |Ω|) 2 . We have g(R) = 0. By changing variables: 6 r 72 and f (s) = g(r) , e−s = R we have f (0) = 0, f (s) = − R2 e− 2 g (r) and

s |f (σ)|dσ |f (s)| ≤ s

0

√ ≤ s

s

|f (σ)| dσ 2

12

0

12 = R s 2 ≤ |g (r)| rdr 2 0 =

s = |∇u∗ |2 4π BR = 12

s 2 ≤ |∇u| , 4π Ω provided by the Faber–Krahn inequality. Thus

exp[βu2 ] = exp[β(u∗ )2 ] Ω

BR

R

exp[βg(r)2 ]rdr

= 2π 0

= |Ω|

exp[βf (s)2 − s]ds

0

≤

∞

1−

β 4π

−1 |Ω| .

Corollary 4.2.6 There are constants β > 0, and γ > 0 such that

exp[βu2 ]dVg0 ≤ γ , M

for all u ∈ H 1 (M ) with u :=

1 Vol(M )

M

u = 0 and u 2 =

M

|∇u|2 dVg0 = 1. n

Proof. By the use of the partition of unity, we have u = j=1 χj u, where χj ≡ 1. the support of the function χj ≥ 0 ∀j is contained in a chart, and Since u = 0, from Poincar´e’s inequality, there is a constant c > 0 such that 1 |u|2 := ( M |u|2 ) 2 ≤ c u , and then χj u ≤ ∇χj ∞ |u|2 + |χj |∞ u ≤ c1 u for some constant c1 . We obtain from Lemma 4.2.5:

4.2 Direct Method

exp[βu2 ] ≤ M

227

n 1 exp[n2 β(χj u)2 ] ≤ γ n M j=1

for suﬃciently small β. Corollary 4.2.7 There exist constants β > 0, and γ > 0 such that

e2u dVg0 ≤ γ exp[β −1 u 2 +2u] ∀u ∈ H 1 (M ) .

(4.18)

M

Proof. Let a = u, v = a−1 (u − u), then v = 0 and v = 1. From 2av ≤ 2 βv 2 + aβ and Corollary 4.2.6, we obtain

M

e2(u−u) dVg0 ≤ γ exp(β −1 u 2 ) .

The proof is complete.

Lemma 4.2.8 If un u0 in H 1 (M ), then after a subsequence, eun → eu0 in L2 (M ). Proof. We have

|eun − eu0 |22 = ≤

e2u0 |e(un −u0 ) − 1|2 e2u0 |un − u0 |2 (1 + e2(un −u0 ) )

≤ 2|e2u0 |4 (1 + |e2(un −u0 ) |4 )|un − u0 |24 . On account of Corollary 4.2.7 the ﬁrst two factors are bounded. Then by the compactness of Sobolev embedding, one has a subsequence such that |un − u0 |4 → 0. Corollary 4.2.9 The g2 is C 1 and is weakly continuous on H 1 (M ) functional 2(w+v) ϕ ∀ϕ ∈ H 1 (M ). with g2 (w), ϕ = 2 Ke It remains to verify the following: (1) M is a nonempty weakly closed Banach manifold, (2) J|M is weakly lower semi-continuous, (3) J|M is coercive. Indeed, (2), (3) are trivial and the nonemptyness of M is easy to verify. The weak closedness of M follows from Corollary 4.2.9. Since the functions 1 and Ke2(w+v) are linear independent ∀w ∈ M, M is a submanifold of H 1 (M ). Namely, we have:

228

4 Minimization Methods

Statement. (M. S. Berger) Assume χ(M ) = 0. (4.12) is solvable if and only if K satisﬁes (4.15). For χ(M ) > 0, topologically M = S 2 or RP 2 . As we have seen in (4.13), a necessary condition of the solvability of the problem (4.12) is max K > 0. If we follow the above procedure, let v be the solution of −v = k − k, v = 0 . We turn to the equation as before: −w = Ke2(w+v) − k , and introduce a functional 1 J(w) = 2

(4.19)

M

(|∇w|2 + 2kw)dVg0

deﬁned on M = {w ∈ H 1 (M )| g2 (w) = k} . If w is a local minimum point of J, then there exists µ ∈ R1 such that −w = 2µKe2(w+v) − k . From w ∈ M, it follows that µ = 1/2, i.e., (4.19) is the Euler–Lagrange equation for J. From the Gauss–Bonnet formula, k¯ > 0, therefore if max K > 0, M = ∅. Similarly, one can show that M is a weakly closed Banach manifold, and that J is weakly lower semi-continuous. If J were coercive, then (4.12) would be solvable. But this is not true in general. In fact, set w ! = w − w, we have

! 2w = log kVol(M ) − log Ke2(v+w) dVg0 , M

and

1 ! |∇w|2 + k Vol(M ) log kVol(M ) − k Vol(M ) log Ke2(v+w) 2 M M

1 4π ≥ |∇w|2 + const. 1− 2 β M

J(w) ≥

due to (4.18) and the Gauss–Bonnet formula

k Vol(M ) = kdVg0 = 4π . M

If β > 4π, then the coerciveness follows. But Moser [Mos 4] proved that the best constant β = 4π for M = S 2 with the canonical metric g0 . This is the point where our argument breaks down. However, for some special K, e.g., if K is even, we shall improve the estimate in (4.18). In this case, the sphere S 2 is reduced to the real projective space RP 2 geometrically.

4.2 Direct Method

229

Lemma 4.2.10 (Aubin) For a function w ∈ H 1 (S 2 ), if there exist fj ∈ C 1 (S 2 ), j = 1, 2 . . . , k and α > 0, satisfying

fj e2w dVg0 = 0, j = 1, 2, . . . , k , S2

and

k

|fj (x)| ≥ α ∀x ∈ S 2 ,

j=1

then ∀ > 0 ∃C > 0 such that

1 2w 2 e ≤ C exp |∇w| + 2w ∀w ∈ H 1 (S 2 ) . 8π − S 2 S2 Proof. Set

α 2 , Ω± j = x ∈ S | ± fj (x) ≥ k

1 2 and gj± , h± j ∈ C (S ) satisfying − + ± ± suppgj+ ∩ supph− j = suppgj ∩ supphj = ∅, 0 ≤ gj , h ≤ 1 , ± ± h± j (x) = 1 ∀x ∈ Ωj , and gj (x) = 1 if ± fj (x) ≥ 0 ∀j .

Then and then

− S 2 = ∪kj=1 (Ω+ j ∪ Ωj ) ,

S2

e2w ≤

k j=1

Ω+ j

+

Ω− j

Without loss of generality, one many assume that e2w , j = 1, 2, . . . . Ω±

e2w . Ω+ 1

e2w is the largest among

j

− First, ∀w ∈ H 1 (S 2 ), if h+ 1 w ≤ g1 w , then ∀1 > 0, ∃C1 > 0, such that + − 2 2 2 2 h+ 1 w ≤ h1 w + g1 w

≤ w 2 +C1 ( w |w|2 + |w|22 ) ≤ (1 + 1 ) w 2 +C1 |w|22 . Following Lemma 4.2.5 and Corollary 4.2.7,

+ 1 2 2 e2h1 w ≤ C1 exp h+ w +2C |w| 2 2 1 4π − 1 S2 1 + 1 ≤ C1 exp w2 + C3 (1 )|w|22 . 2(4π − 1 )

230

4 Minimization Methods

Otherwise, we have 2 2 g1− w 2 ≤ g1− w 2 + h+ 1w ≤ (1 + 1 ) w 2 +C1 |w|22 ,

and

2g1− w

e S2

1 − 2 2 ≤ C1 exp g1 w +2C2 |w|2 4π − 1 1 + 1 2 2 w + C3 (1 )|w|2 . ≤ C1 exp 2(4π − 1 )

Now let u = w − w. ∀0 > 0, ∃C4 = C4 (0 ) such that |u|1 ≤ 0 u 2 +C4 .

(4.20)

∀2 > 0 small, we choose a = a(2 ) > 0 such that |{x ∈ S 2 | u(x) ≥ a}| = 2 . Therefore, we obtain a≤

0 1 1 |u|1 ≤ u 2 + C4 , 2 2 2

(4.21)

from (4.20), and 1

1

|(u − a)+ |22 ≤ 22 |(u − a)+ |24 ≤ C5 22 u 2 .

(4.22)

Setting 0 = 12 22 , and choosing 1 , 2 > 0 so small that 1 1 + 1 1 + 2 + C3 (1 )C5 22 < , 2(4π − 1 ) 8π −

+ Substituting w = (u − a)+ in the estimates of the integrals: S 2 e2h1 w − + + − and S 2 e2g1 w , since S 2 e2h1 u ≤ e2a S 2 e2(h1 u−a)+ and S 2 e2g1 u ≤ e2a S 2 − 1 e2(g1 u−a)+ , the right-hand sides become B exp[ 8π− u2 ]. Next, for the speciﬁed w in the assumption, we have

+ e2u ≤ 2k e2u ≤ 2k e2h1 u . S2

Ω+ 1

S2

In particular, according to f1 e2u = 0, we also have

k k e2u ≤ (f1 )+ e2u = (f1 )− e2u + α α 2 2 Ω1 S S

4.3 Quasi-Convexity

≤ Max{|fi | | i = 1, . . . , k}

k α

231

−

e2g1 u . S2

In both cases,

S2

i.e.,

e2u ≤ C exp

S2

e2w ≤ C exp

1 u 2 8π −

.

(4.23)

1 w 2 +2w . 8π −

Now, we return to the case (M, g0 ) = (S 2 , g0 ) where g0 is the canonical metric, but assuming K(x) = K(−x), ∀x ∈ S 2 . Let us deﬁne Me = {w ∈ H 1 (S 2 )| g2 (w) = k and w(x) = w(−x)} . Then, again, Me is a nonempty weakly closed Banach manifold. The functional

1 (|∇w|2 + 2kw)dVg0 J(w) = 2 M is again weakly lower semi-continuous. Since ∀w ∈ Me

xi e2w(x) dVg0 = 0 i = 1, 2, 3 , S2

where xi are the Euclidean coordinates, when we embed S 2 into R3 canonically, Lemma 4.2.10 is applicable, and then the coerciveness of J on Me follows. Thus, we have: Statement (J. Moser) For M = RP 2 with the canonical metric g0 , (4.12) is solvable if and only if max K > 0. Remark 4.2.11 Under certain symmetric conditions on K other than the evenness, Hong [Hon 2] obtained the existence of solutions for equation (4.12). Remark 4.2.12 The best constant β = 4π is due to Moser; there are several diﬀerent proofs and extensions, see Adams [Ad], Carleson and Chang [CC] and Ding and Tian [DiT].

4.3 Quasi-Convexity In the calculus of variations, we pay attention to the following functional:

232

4 Minimization Methods

f (x, u(x), ∇u(x))dx ,

J(u) =

(4.24)

Ω

where f : Ω × RN × RnN → R1 is a Caratheodory funciton, and Ω ⊂ Rn is an open domain. As we have seen, the w.l.s.c. (or w∗ l.s.c.) condition plays an important role in the proof of the existence of a minimizer. 4.3.1 Weak Continuity and Quasi-Convexity From the abstract theory, we know that an l.s.c. function is w.l.s.c. and s.w.l.s.c. if it is convex. But to the above integral functional, what is the relationship between the s.w.l.s.c. (or s.w∗ .l.s.c.) with the convexity of the function f with respect to its variables (x, u, ξ)? To this end we would like to understand better the weak convergence in the spaces W 1,p (Ω, Rn ), 1 p ∞. Perhaps the simplest example of a weakly convergent but not strongly convergent sequence in our mind is {sin(mt)}∞ 1 in L2 ([0, 2π]). The phenomenon is caused by the large oscillations of the sequence of functions. It can be extended as follows: n

Lemma 4.3.1 Let D =

Π (aj , bj ) be a rectangle in Rn , and let ϕ ∈

j=1

Lp (D), 1 p ∞, which is extended periodically to Rn . Let ϕm (x) = ϕ(mx), ∀m ∈ N, and

1 ϕ= ϕ(x)dx . m(D) D ∗

Then ϕm ϕ in Lp (D) 1 < p < ∞ and ϕm ϕ in L∞ (D). Proof. First, we may assume ϕ = 0. Otherwise, we consider ϕ ! = ϕ−ϕ instead. Second,

1 ϕm pp = |ϕ(mx)|p dx = n |ϕ(y)|p dy = ϕ pp , ∀1 p ∞ . m D m·D Third, we deﬁne a signed set function Φ(E) = E ϕ for any measurable set E. It is σ-additive, and satisﬁes Φ(x + D) = 0, ∀x ∈ Rn . n

Now, ∀ rectangles Q = Π (ci , di ), by cancelling the nonoverlapping transi=1

lations of D in kQ, we have the estimate:

ϕk χQ = ϕk | = 1 |Φ(kQ) n |ϕ| . k kn D Q D Thus, for simple functions of the form ξ = αi χQi , Qi ∩ Qj = Ø i = j, we have

ϕk · ξ → 0 as k → ∞ . D

4.3 Quasi-Convexity

233

As 1 < p ∞, the simple functions of the above form consist of a dense p . ∀f ∈ Lp (D), ∃ξ a simple function as above such subset of Lp (D), p = p−1 that f − ξ p < ε/(2 ϕ p ). Therefore

ϕk · f ϕk p f − ξ p + ϕk · ξ < ε D

D

for k large enough.

Remark 4.3.2 Lemma 4.3.1 also holds for p = 1. In fact, the only problem in the above proof for p = 1 is that the simple functions ξ cannot approximate any f ∈ L∞ in L∞ norm. But, one can choose ξ such that f − ξ 1 < ε, with |αi | 2 f ∞ ∀i. Now, ∀λ > 0, let Ek,λ = {x ∈ D| |ϕk (x)| λ}, ∀k. Since ϕ1 = ϕ ∈ L1 (D), ∀ε > 0, ∃λ = λ(ε) such that E1,λ |ϕ| < ε. From the deﬁnition of ϕk , so is Ek,λ |ϕk | < ε. Again,

ϕk · f |ϕk (f − ξ)| + ϕk · ξ = I + II D

D

D

where II → 0 as k → ∞, and

I= |ϕk (f − ξ)|

D |ϕk ||f − ξ| + = Ek,λ

|ϕk (f − ξ)|

D\Ek,λ

3 f ∞ ·

|ϕk | + λ f − ξ 1 . Ek,λ

Again, we have proved that ϕk ϕ in L1 (D). Corollary 4.3.3 Let Ω = (0, 1), 0 < λ < 1, α, β ∈ R1 , and α if x ∈ (0, λ) ϕ(x) = β if x ∈ (λ, 1) , then ϕk (∗ )λα + (1 − λ)β in Lp , p ∈ [1, ∞), (p = ∞, resp.). Now, we turn to studying the s.w.l.s.c. (or s.w∗ .l.s.c.) of J in W 1,p (Ω), 1 ≤ p ≤ ∞. For simplicity, we assume D = [0, 1]n , and f = f (ξ). ∀k ∈ N , let Dlk be a sub-cube of D with length 2−k on each side and centered at ckl = 2−k (y1l + 1 1 l l l k 2 , . . . , yn + 2 ), where (y1 , . . . , yn ) runs over the lattice points (0, 1, . . . , 2 − kn + 2 1)n , l = 1, . . . , 2kn . Then D = l=1 Dlk , ∀k ∈ N . ∀v ∈ C0∞ (D, RN ). Let us deﬁne 1 (4.25) wk (x) = k v(2k (x − ckl )), ∀x ∈ Dlk , ∀l = 1, . . . , 2kn . 2 Then

234

4 Minimization Methods

∇wk (x) = ∇v(2k (x − ckl )) ∀x ∈ Dlk , ∀l = 1, . . . , 2kn and

wk → 0 L∞ (D) , ∇wk (∗ )0 in Lp (D) p ∈ [1, ∞)(p = ∞ resp.) .

provided by Lemma 4.3.1. Now we arrive at:

Lemma 4.3.4 Let Ω ⊂ Rn be a domain, and let J(u) = Ω f (∇u). If J is s.w.l.s.c. on W 1,p (Ω, RN ) 1 p < ∞ (s.w∗ l.s.c. on W 1,∞ (Ω, RN )), then for ∀A ∈ M n×N , ∀ cube D ⊂ D ⊂ Ω, we have

1 f (A) f (A + ∇v) ∀v ∈ W01,∞ (D, RN ) , m(D) D where M n×N denotes the n × N matrix space. Proof. We choose a sequence in the Banach space X = W 1,p (Ω, RN ) as follows: uk (x) = Ax + wk (x) k = 1, 2, . . . , where the sequence {wk } is deﬁned in (4.25) and equals to zero outside D. Then it weakly converges to u(x) = Ax (w∗ -converges as p = ∞,). It follows that m(Ω)f (A) = J(u) lim inf J(uk ) . k→∞

But

f (A + ∇wk (x))dx +

J(uk ) =

f (A)dx Ω\D

D

2 kn

=

Dlk

l=1

f (A + ∇v(2k (x − ckl )))dx + f (A)m(Ω\D)

f (A + ∇v(x))dx + f (A)m(Ω\D) .

= D

This is our conclusion. According to Lemma 4.3.4, we introduce the following:

Deﬁnition 4.3.5 A Borel measurable and locally integrable function f : RnN → R1 is called quasi-convex (in the Morrey sense), if for ∀A ∈ M n×N , ∀v ∈ W01,∞ (D, RN ), and for any cube D ⊂ R , the inequality

m(D)f (A) f (A + ∇v(x))dx (4.26) D

holds. In the sequel of this chapter, we shall specify the terminology “quasiconvexity” to be quasi-convexity in the Morrey sense.

4.3 Quasi-Convexity

235

Remark 4.3.6 The cube D in the above deﬁnition can be replaced by any bounded domains, from a simple scaling and covering argument. Let us study the relationship between the convexity and the quasiconvexity. On the one hand, according to Jessen’s inequality, it is easy to see that if f is convex, then

1 1 f (A + ∇v(x)) dx f (A + ∇v(x) dx) = f (A) ; m(D) D m(D) D i.e., convexity =⇒ quasi-convexity. On the other hand, ∀B, C ∈ M n×N , with rank (B −C) = 1, ∀λ ∈ (0, 1), let A = λB + (1 − λ)C. After suitable translation and rotation, we may assume B = (1 − λ)a ⊗ e1 and C = −λa ⊗ e1 for some a ∈ RN and e1 ∈ Rn . Let us introduce a 1-periodic sawtooth function: (1 − λ)t, t ∈ [0, λ] , ϕ(t) = −λ(t − 1), t ∈ [λ, 1] . For ∀x ∈ D = [0, 1]n , let uk (x) = ak−1 ϕ(kx1 ) , then

(1 − λ) {kx1 } ∈ (0, λ) ∇uk (x) = a ⊗ e1 −λ {kx1 } ∈ (λ, 1) ,

where {y} denotes the fractional part of y ∈ R1 ; and let vk (x) = a min {k −1 ϕ(kx1 ), dist (x, ∂D)} , where dist (x, ∂D) = inf { sup xi − yi | y = (y1 , . . . , yn ) ∈ ∂D}. Then we 1in

have vk |∂D = 0 and there is a constant K > 0 such that |vk (x) − vk (y)| ≤ x − y. Therefore vk ∈ W01,p (D, RN ). Furthermore, we have m{x ∈ D| ∇uk (x) = ∇vk (x)} → 0 . as k → ∞. Let us divide D into two disjoint parts D1 and D2 , where D1 = {x ∈ D|∇uk (x) = (1 − λ)a ⊗ e1 } and D2 = {x ∈ D|∇uk = −λa ⊗ e1 }. Thus D = D1 ∪ D2 . If f is quasi-convex, then

f (λB + (1 − λ)C + ∇vk ) . m(D)f (λB + (1 − λ)C) ≤ D

On the account of the local integrability of f and the absolute continuity of the integral, we have

236

4 Minimization Methods

f (λB + (1 − λ)C + ∇vk ) = lim

lim

f (λB + (1 − λ)C + ∇uk )

f (B) + f (C) = lim

D

D

D1

D2

= λm(D)f (B) + (1 − λ)m(D)f (C) . Finally, we obtain: f (λB + (1 − λ)C) λf (B) + (1 − λ)f (C) , i.e., f is convex along segments connecting two matrices with rank-1 diﬀerence. Deﬁnition 4.3.7 A function f : Mn×N → R1 ∪ {+∞} is said to be rank one convex, if f (λB + (1 − λ)C) λf (B) + (1 − λ)f (C) ∀λ ∈ [0, 1], ∀B, C ∈ Mn×N with rank {B − C} 1. As a direct consequence, we see that either n = 1 or N = 1, rank one convex = convex. In summary, we have proved: convex =⇒ quasi-convex =⇒ rank one convex. In particular, either n = 1 or N = 1, quasi-convex = convex. A natural question: are there any counterexamples of these reverse implications? 1. Quasi-convex in Morrey sense =⇒ convex. For A ∈ Mn×n , det A is quasi-convex in Morrey sense, but not convex. It is easily seen that det A is not convex, even for n = 2. Let us show that it is quasi-convex in Morrey sense. More precisely, we have

1 det (A + ∇v) = det (A) ∀D ⊂ Ω, ∀v ∈ C0∞ (D, Rn ) . m(D) D For simplifying the computation, we only verify the cases n 3. Higherdimension cases can be shown by induction. For n = 2, let v1 a11 a12 , v= ∈ C0∞ (D, R2 ) . A= a21 a22 v2 From det (∇v) = ∂x1 (v1 ∂x2 v2 ) − ∂x2 (v1 ∂x1 v2 ) , we have

det (∇v) = 0 , D

and then

4.3 Quasi-Convexity

237

1 det (A + ∇v) m(D) D

1 = [det (A) + a11 ∂x2 v2 + a22 ∂x1 v1 − a12 ∂x1 v2 − a21 ∂x2 v1 + det(∇v)] m(D) D = det (A) . For n = 3, let A = (aij )1i,j3 , adj2 (A) = (Aij )1i,j3 where Aij is the (ij) minor of A, det (A + ∇v) = (A + ∇v)1 , (adj2 (A + ∇v))1

= A1 , (adj2 (A + ∇v))1 + (∇v)1 , (adj2 (A + ∇v))1

3 ∂v1 (adj2 (A + ∇v))1i , A1i (adj2 (A + ∇v))1i + = ∂xi i=1 where B 1 = (b11 , . . . , b1n ) is the ﬁrst row vector of the matrix B = (bij )1i,jn , and , is the scalar product in Rn . By the conclusion for case n = 2, we have 1 m(D) i=1 3

A1i (adj2 (A + ∇v))1i D

1 1 = A (adj2 (A))1i m(D) i=1 i 3

= det (A) . Since 3 ∂ (adj2 (A + ∇v))1i ∂x i i=1 ∂v2 ∂v2 ∂v2 ∂ ∂x3 ∂ ∂x2 ∂x3 = + ∂x1 ∂v3 ∂v3 ∂x2 ∂v3 ∂x2

∂x3

∂x3

∂v2 ∂x1 ∂v3 ∂x1

∂v2 ∂ ∂x1 + ∂x3 ∂v3 ∂x1

∂v2 ∂x2 ∂v3 ∂x2

=0, after integration by parts, it follows that

3 ∂v1 (adj2 (A + ∇v))1i = 0 . ∂x i D i=1 This proves the conclusion. 2. Rank one convex =⇒ quasi-convex. There is a counterexample due to Sverak [Sv 1]. 4.3.2 Morrey Theorem The importance of the notion of quasi-convexity is due to the following:

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4 Minimization Methods

Theorem 4.3.8 Suppose that f : M n×N → R1 is continuous and quasiconvex. If the following growth condition holds: |f (A)| α(1 + |A|), (Cp )

as p = 1 ,

− α(1 + |A|q ) f (A) α(1 + |A|p ), as 1 q < p < ∞ , |f (A)| η(|A|), as p = +∞ ,

where η is continuous and increasing, and α > 0, ∀A ∈ M n×N , then for every bounded open domain Ω ⊂ Rn ,

J(u) = f (∇u)dx Ω

is s.w.l.s.c. in W 1,p (Ω, RN ) (s.w∗ .l.s.c. in W 1,∞ (Ω, RN )). Before going to the proof, we need the piecewise aﬃne function approximation in Sobolev spaces. Deﬁnition 4.3.9 (Triangulation) Let Ω ⊂ Rn be a bounded domain with piecewise aﬃne boundary. A triangulation τ˜ of Ω is a collection of ﬁnitely many n-simplices {Ki |i = 1, 2, . . . , I}, such that ∀i =+j, Ki ∩ Kj is either empty or equal to a p-simplex, 0 ≤ p ≤ n − 1, and Ω = 0≤i≤I Ki . We call hτ˜ = max{diam(Ki )|i = 1, . . . , n} the mesh size of τ˜. To an n-simplex, K = {p0 , p1 , . . . , pn }, (in which, pi ∈ Rn , i = 0, 1, . . . , n, and {pi − p0 |i = 1, 2, . . . , n} are linearly independent). We deﬁne n + 1 aﬃne functions {λ0 , λ1 , . . . , λn } such that λi (pj ) = δij , i, j = 0, 1, . . . , n . ∀v ∈ C(K), we deﬁne a piecewise aﬃne function by the following interpolation formula: n v˜K (x) = v(pi )λi (x) . (4.27) i=0

One shows that ∀v ∈ C (K), ∀x ∈ K, 2

∇v − ∇˜ vK ∞ ≤

n2 (n + 1) h2K vC 2 (K) , 2 ρK

(4.28)

where hK = diam K, and ρK = sup{2R|BR (x) ⊂ K, x ∈ K}. In fact, let pi = (p1i , . . . , pni ), i = 0, 1, . . . , n. We consider the functions vj (x) = xj , j = 1, 2, . . . , n, where x = (x1 , . . . , xn ), and v0 (x) ≡ 1 respectively. We obtain: xj =

n 0

pji λi (x), j = 1, 2, . . . , n, and

n 0

λi (x) = 1 .

4.3 Quasi-Convexity

239

Diﬀerentiating these formulas, it follows that n i=0

pji

∂λi = δkj , j, k = 1, 2, . . . , n, ∂xk

n ∂λi = 0, k = 1, 2, . . . , n . ∂xk i=0

Thus by Taylor expansion, ∂λi (x) ∂˜ vK (x) = v(pi ) ∂xk ∂xk i=0 n 1 2 2 ∂λi (x) = v(x) + ∇v(x)(pi − x) + ∇ v(ξx )(pi − x) 2 ∂xk i=0 n

∂v(x) ∂λi (x) + Ri (x) , k ∂x ∂xk i=0 n

=

where ξx ∈ K, and Ri (x) = 12 ∇2 v(ξx )(pi − x)2 . Since |Ri (x)| ≤

n2 vC 2 h2K ∀i , 2

and |∇λi | ≤ ρ1K . We obtain the desired estimate (4.28). A triangulation τ˜ is called regular, if ∃ a constant C > 0, such that C, ∀K ∈ τ˜.

hK ρK

≤

Lemma 4.3.10 Suppose that Ω ⊂ Rn is a bounded domain with piecewise aﬃne boundary. Then for ∀u ∈ W 1,p (Ω, RN ), ∀ > 0, ∃ a piecewise aﬃne function v on Ω, satisfying: 1. u − v1,p < , 2. |∇v(x)| ≤ |∇u(x)|, and |v(x)| ≤ |u(x)| a.e. Proof. With the aid of the extension operator and the molliﬁer, we have a function u ˜ ∈ C 2 (Ω, RN ), such that 1. ˜ u − u1,p < 3 , 2. |˜ u(x)| ≤ |u(x)|, |∇˜ u(x)| ≤ |∇u(x)| a.e., 3. ˜ uC 2 (Ω) ≤ Cu1,p , where C > 0 is a constant depending on Ω. To any regular triangulation τ˜ of Ω, we deﬁne a linear operator π from W 1,p (Ω, RN ) into itself as follows: (πu)(x) = u ˜K (x) ∀x ∈ K, ∀K ∈ τ˜ , where u ˜K is deﬁned as in equation (4.27).

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4 Minimization Methods

From |(πu)(x)| ≤ |u(x)|, and |∇(πu)(x)| ≤ |∇u(x)| a.e., it follows that π ≤ 1. On account of (4.28) ˜ u − πu ˜1,p is small as hτ˜ > 0 is small. Therefore, ˜1,p + ˜ u − πu ˜1,p + π˜ u − πu1,p < . u − πu1,p ≤ u − u

Set v = πu. This is what we need.

Now, let us return to the proof of Theorem 4.3.8 for the case p = ∞. Other cases are similar, so are omitted. Proof. Suppose u ∈ W 1,∞ (Ω, RN ) and uj ∗ u in W 1,∞ , we shall prove that

lim inf f (∇uj ) ≥ f (∇u) . Ω

Ω

We may assume |∇uj |, |∇u| ≤ C, M = sup{f (ξ)||ξ| ≤ 3C + 1}. ∀ > 0∃δ > as |ξ −ξ | ≤ δ and |ξ|, |ξ | ≤ 3C. One chooses 0 such that |f (ξ)−f (ξ )| ≤ 6m(Ω) . There Ω1 ⊂ Ω1 ⊂ Ω, in which ∂Ω1 is piecewise aﬃne with m(Ω\Ω1 ) < 8M k k 1,∞ (Ω, RN ) exists a regular triangulation of Ω1 : τk = {D1 , . . . , DIk }, ∃vk ∈ W such that vk is a piecewise aﬃne function on Ω1 with vk = u on ∂Ω, and vk − u1,∞ < . Let vk = Aki ∀x ∈ Dik . This can be realized by considering a domain Ω2 : Ω1 ⊂ Ω2 ⊂ Ω2 ⊂ Ω, with a piecewise aﬃne boundary ∂Ω2 , which is parallel to ∂Ω1 . We deﬁne vk in Ω1 according to Lemma 4.3.10, vk (x) = u(x), ∀x ∈ Ω\Ω2 , and linearly interpolate u|∂Ω2 and vk |∂Ω1 in Ω2 \Ω1 . According to Lemma 4.3.10, we may choose τk with hτk → 0 such that ∇vk → ∇u in measure. Let ujk = vk + uj − u, then ujk ∗ vk in W 1,∞ as j → ∞; and |∇ujk | ≤ 3C. One has k0 ∈ N , as k > k0 ,

(4.29) |f (∇vk ) − f (∇u)| < 6 Ω

and

|f (∇ujk ) − f (∇uj )| < Ω

. 6

(4.30)

Fixing k > k0 , let pki be the center of Dik , ∀r ∈ (0, 1); we consider a similar +I subsimplex Cik = r(Dik − {pki }) + pki , i = 1, 2, . . . , Ik , and Ω1 = 1k Cik . Deﬁne a smooth function ρ on Ω1 , such that ρ(x) = 1 ∀x ∈ Ω1 , ρ(x) = 0 ∀x ∈ +Ik k 1 ∂Di , and |∇ρ| ≤ C(τ˜k , r), a constant depending on τ˜k and r. Deﬁne vjk = vk + ρ(ujk − vk ) = vk + ρ(uj − u); we have

f (∇ujk ) = f (∇vjk ) + f (∇ujk ) Ω1

Ω1

Ω1 \Ω1

f (∇vjk ) +

= Ω1

= A+B+C .

Ω1 \Ω1

[f (∇ujk ) − f (∇vjk )]

4.3 Quasi-Convexity

241

On account of the quasi-convexity of f ; we have

f (∇vjk ) =

A= Ω1

Ik

i=1

≥

Ik

Dik

f (Aki + ∇[ρ(ujk − u)])

f (Aki )m(Dik )

i=1 f (∇vk ) .

= Ω1

It is easily seen that

B= Ω1 \Ω1

f (∇ujk ) ≥ −M (1 − rn )m(Ω1 ) .

As to C, from uj − u∞ → 0 as j → ∞, we obtain ∇vjk ∞ ≤ ∇vk ∞ + ∇uj ∞ + ∇u∞ + ∇ρ∞ uj − u∞ ≤ 3C + 1 , as j > j0 (τ˜k , r). Thus

Ω1 \Ω1

f (∇vjk ) ≤ M (1 − rn )m(Ω1 ) .

Putting them together, we have

f (∇ujk ) ≥ f (∇vk ) − 2M m(Ω1 )(1 − rn ), Ω1

(4.31)

Ω1

as j > j0 (k, r). Combining (4.29), (4.30) and (4.31), and suitably choosing r ∈ (0, 1), we have

f (∇uj ) ≥ f (∇ujk ) + f (∇ujk ) − 6 Ω Ω\Ω1 Ω1

f (∇vk ) + [f (∇ujk ) − f (∇vk )] − − 2M (1 − rn )m(Ω) ≥ 6 Ω Ω\Ω1

≥ f (∇u) − , Ω

as j > j0 (k, r). Since > 0 is arbitrarily small, we have proved the s.w∗ .l.s.c. of J.

Remark 4.3.11 Theorem 4.3.8 is initially due to Morrey under some additional conditions. It was reﬁned by Meyers, and greatly improved by Acerbi and Fusco [AF] and Marcellini [Mar]. In the ﬁnal version, f can be a Caratheodory function: Ω × RN × Mn×N → R1 , and (Cp ) is replaced by the following:

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4 Minimization Methods

|f (x, u, A)| α(1 + |u| + |A|)

as p = 1 ,

−α(1 + |u|r + |A|q ) f (x, u, A) α(1 + |u|r + |A|p ) as 1 < p n , np where 1 q < p, and r ∈ [1, n−p ),

−α(1 + |A|q ) f (x, u, A) α(1 + |A|p ) as n < p < ∞ , where 1 ≤ q < p, and f (x, u, A) η(x, |u|, |A|) as p = ∞ , where η is an increasing function in each of its argument. Theorem 4.3.12 Suppose that f : Mn×N → R1 is continuous, quasi-convex and satisfying c|A|p f (A) C(1 + |A|p )

for C > c > 0 ,

for 1 < p < ∞. Then for ∀v ∈ W 1,p (Ω, RN ), the functional

f (∇u) J(u) = Ω

achieves its minimum on E = Wv1,p := {u ∈ W 1,p (Ω, RN )| u|∂Ω = v|∂Ω }. Proof. In fact, J is s.w.l.s.c. and coercive, and E is weakly closed.

4.3.3 Nonlinear Elasticity The body of a given elastic material occupies a bounded domain Ω in R3 ; it is called the reference conﬁguration. External force gives the material a de∂ui formation u : Ω → R3 . The gradient of u, ∇u = ( ∂x )1≤i,α≤3 is called the α deformation gradient. To the hyper-elastic materials, a stored energy density → R1 is introduced with certain symmetric properfunction W : Ω × M3×3 + ties (e.g., it is left invariant under rotations and right invariant under certain denotes all 3 × 3 matrices with positive deterisotropic groups), where M3×3 + minants. The elastic energy reads as

W (x, ∇u)dx . I(u) = Ω

The basic assumption of the variational approach to this problem is that the observed deformations correspond to minimizers of the elastic energy. After the reductions from these invariances, the stored energy density is, in fact, dependent on the determinant, the minors and the eigenvalues of the deformation gradient ∇u only. A function f : Mn×N → R1 ∪ {+∞} is called poly-convex if there exists g : Rτ (n,N ) → R1 convex such that

4.3 Quasi-Convexity

243

f (A) = g(T (A)) where τ (n, N ) =

n∧N s=1

and T : M

n×N

→R

τ (n,N )

n!N ! , (s!)2 (n − s)!(N − s)!

is of the form:

T (A) = (A, adj2 A, . . . , adjn∧N A) , and adjs A denotes the matrix of all s × s minors of A. Examples. Let A = (aij ) ∈ Mn×n , then A 2 =

n

|aij |2 ,

i,j=1

det (A) , and adj2 A p p ≥ 1 , are poly-convex functions. What are the relationships between poly-convexity and other convexities? We claim that convex =⇒ poly-convex =⇒ quasi-convex. In fact, the ﬁrst implication is trivial. To verify the second implication, let us recall

1 det (B + ∇v) = det (B), ∀B ∈ Mn×n , ∀D ⊂ Ω, ∀v ∈ C0∞ (D, Rn ) . m(D) D It implies that ∀A ∈ Mn×n ,

1 adjs (A + ∇v) = adjs (A) m(D) D and then

1 m(D)

2sn,

T (A + ∇v) = T (A) . D

Thus if f is poly-convex, and f (A) = g(T (A)), where g is convex, then by Jessen’s inequality

1 1 f (A + ∇v) = g(T (A + ∇v)) m(D) D m(D) D

1 g T (A + ∇v) m(D) D

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4 Minimization Methods

= g(T (A)) = f (A) , i.e., f is quasi-convex. In summary, we have the following implications: convex =⇒ poly-convex =⇒ quasi-convex =⇒ rank-one convex . One may ask: Are these implications strict? The following example is due to Dacorogna and Marcellini [DM]: A ∈ M2×2 , f (A) = A 4 − γ A 2 det A, then 4√ 2, 3 f is poly-convex ⇐⇒ |γ| 2 , f is quasi-convex ⇐⇒ |γ| 2 + ε, for some ε > 0 , 4 f is rank-one convex ⇐⇒ |γ| √ . 3

f is convex ⇐⇒ |γ|

It is not known whether 2 + ε = √43 . In addition to the counterexample due to Sverak [Sv 1], all the reverse implications are false.

4.4 Relaxation and Young Measure Not all variational integrands are quasi-convex. In this case a minimizing sequence may not converge to a minimizer, there may even be no minimizer! For example, let Ω = (0, 1), we consider the functional

J(u) =

1

[u2 (x) + (u (x)2 − 1)2 ]dx

(4.32)

0

on the Sobolev space W01,4 (Ω). The integrand is not quasi-convex. It is easily seen that inf J = 0. Indeed, on the one hand, we deﬁne a sequence of sawtooth functions: if x ∈ [ nk , 2k+1 x − nk 2n ] , un (x) = k+1 2k+1 k+1 if x ∈ [ n , n ] . −x + n From |un (x)| = 1 a.e., and |un | J(un )

1 2n ,

we conclude that

1 →0. 4n2

On the other hand, J 0. Then {un } is a minimizing sequence with inf J = 0.

4.4 Relaxation and Young Measure

245

However, un θ in W01,4 (Ω), because

1

un

·ϕ=−

0

1

un · ϕ → 0, ∀ϕ ∈ C0∞ (Ω), as n → 0 .

0

But J(θ) = 1. Moreover, the functional does not have a minimizer in W01.4 (Ω). Because, if u0 ∈ W01,4 (Ω) achieves J(u0 ) = 0, then u0 (x) = 0, a.e., and u0 (x) = 0, a.e., which imply that J(u0 ) = 1. This is impossible. Nevertheless, to this kind of problem, the minimizing sequences must provide some useful information in understanding the variational problems. There are several ways to studying these minimizing sequences: (1) Relaxation method: Introduce a relaxed functional with a quasi-convex integrand. It shares a common minimizing sequence with J. (2) Young measure: Extend the working Sobolev space to a measure space so that the minimizing sequences converge in some sense to a measure. (3) Change the working space to something other than the Sobolev space W 1,p (Ω, RN ). 4.4.1 Relaxations In the ﬁrst aspect, let us recall the relationship between s.w∗ .l.s.c. and the quasi-convexity. We are inspired to study various convex envelopes for nonconvex functions. Deﬁnition 4.4.1 Let f : Mn×N → R1 ∪ {+∞}, we call Cf = sup {g f | g convex} , P f = sup {g f | g poly-convex} , Qf = sup {g f | g quasi-convex} , Rf = sup {g f | g rank-one convex} , the convex, poly-convex, quasi-convex and rank-one convex envelope of f , respectively. Obviously, (1) Cf, P f, Qf and Rf are convex, poly-convex, quasi-convex and rank-one convex functions, respectively. (2) Cf P f Qf Rf f . In particular, if n = 1 or N = 1, then Cf = P f = Qf = Rf . By deﬁnition, f Cf f ∗∗ , the biconjugate function of f . But one can show that if f is proper, then Cf = f ∗∗ . Since Qf is quasi-convex, in addition to some growth condition, the associated functional

246

4 Minimization Methods

! J(u) =

(Qf )(x, u(x), ∇u(x))

(4.33)

Ω

is s.w∗ .l.s.c. in a certain Sovolev space W 1,p (Ω, RN ), 1 < p ≤ ∞. The following theorem is very basic in this aspect. Theorem 4.4.2 Suppose that f : Mn×N → R1 is measurable and locally integrable, and satisﬁes C2 |A|p f (A) C1 (1 + |A|p )

(4.34)

for some C1 > C2 > 0 and p ∈ (1, ∞). For ∀v ∈ W 1,p (Ω, RN ), deﬁne Wv1,p = {u ∈ W 1,p (Ω, RN )| u|∂Ω = v|∂Ω }, then ! u ∈ Wv1,p } . inf {J(u)| u ∈ Wv1,p } = inf {J(u)| Moreover, a function u0 ∈ Wv1,p is a minimizer of J! if and only if it is a cluster point (with respect to the weak topology of W 1,p ) of a minimizing sequence for J. Before going to the proof of Theorem 4.4.2, we give a characterization of the quasi-convex envelope. Lemma 4.4.3 If f : Mn×N → R1 is locally bounded and Borel measurable, then

1 1,∞ N f (A + ∇ϕ(x))dx| ϕ ∈ W0 (D, R ) , (4.35) (Qf )(A) = inf m(D) D where D ⊂ Rn is a bounded domain with m(∂D) = 0. Proof. We denote the RHS of (4.35) by (qf )(A, D). By deﬁnition, f ≤ g implies that qf ≤ qg; and if f is quasi-convex, then qf = f . We want to show: (1) (qf )(A, D) is independent of D, then we denote it by (qf )(A). (2) (qf )(A) is quasi-convex. We assume these two conclusions at this moment, then by the deﬁnition of the quasi-convex envelope, (qf )(A) (q(Qf ))(A) = (Qf )(A) ∀A ∈ M n×N . From (2), qf is quasi-convex and qf f . Since Qf is the largest quasi-convex function f , it follows that qf Qf . Therefore qf = Qf .

4.4 Relaxation and Young Measure

247

1. Veriﬁcation of (1). One observes that for any two bounded domains D1 and D2 , if after translation and scaling, D1 is transformed into D2 (denoted by D1 ∼ D2 ), then (qf )(A, D1 ) = (qf )(A, D2 ) . This proves that (qf )(A, D) are equal for all D ∼ D1 . Moreover, if D1 ⊂ D2 , then (qf )(A, D2 ) ≤ (qf )(A, D1 ). Now for any two bounded domains D1 and D2 in Rn , we may assume that D1 is the unit cube. And for any ε > 0, we ﬁnd {Ej | ∼ D1 , j = 1, 2, . . . , m}, m

with Ei ∩ Ej = Ø i = j and m(D2 \ ∪ Ej ) < ε. On each Ej one may ﬁnd a j=1

piecewise aﬃne function ϕεj ∈ W01,∞ (Ej , Rn ), such that

f (A + ∇ϕεj (x))dx (ε + (qf )(A, D1 ))m(Ej ) , Ej

and let ϕε (x) =

⎧ ⎪ ⎨ϕεj (x) x ∈ Ej ⎪ ⎩0

Thus

m

ε f (A + ∇ϕ (x))dx D2

j=1

m

x ∈ D2 \ ∪ Ej . j=1

f (A +

∇ϕεj (x))dx

+

Ej

(ε + (qf )(A, D1 ))

m

f (A)dx.

D2 \ ∪ Ej j=1

m

m m(Ej ) + f (A)m D2 \ ∪ Ej .

j=1

j=1

Since ε > 0 is arbitrary, it follows that (qf )(A, D2 ) (qf )(A, D1 ) . The positions of D1 and D2 are symmetric, thus (qf )(A, D1 ) = (qf )(A, D2 ) , i.e., (qf )(A, D) is independent of D, and will be written as (qf )(A). 2. Veriﬁcation of (2). First, we verify:

1 (qf )(A + ∇ψ(x))dx (qf )(A) , m(D) D for all piecewise aﬃne ψ ∈ W01,∞ (D, RN ), where D is the unit cube. In fact, m

let D = ∪ Di , where Di , i = 1, . . . , m are simplexes with Dj◦ ∩ Di◦ = Ø ∀i = j i=1

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4 Minimization Methods

and ∇ψ|Di = Bi ∀i. By the deﬁnition of qf , for ∀ε > 0, ∃ϕi ∈ W01,∞ (Di , RN ) such that

1 f (A+∇ψ+∇ϕi )−ε, ∀x ∈ Di . (qf )(A+∇ψ(x)) = (qf )(A+Bi ) m(Di ) Di One extends ϕi to be zero outside Di , and deﬁne ϕ = ψ + ϕ ∈ W01,∞ (D, RN ), and

(qf )(A + ∇ψ(x))dx = D

m

i=1 Di m

i=1

m i=1

ϕi . Then

(qf )(A + Bi ) f (A + ∇ϕ) − εm(D)

Di

f (A + ∇ϕ) − εm(D)

= D

m(D)[(qf )(A) − ε] . Since ε > 0 is arbitrary, it follows that

1 (qf )(A + ∇ψ)dx| ψ ∈ W01,∞ (D, RN ) inf m(D) D piecewise linear (qf )(A) .

(4.36)

By the same procedure as in Sect. 4.3, we conclude that (qf ) is rank-one convex. Therefore qf is continuous, and in fact, is locally Lipschitzian (Theorem 2.2.1, Sect. 2.2). Since piecewise aﬃne functions are dense in W01,∞ (D, RN ), (4.36) is extended to all ψ ∈ W01,∞ (D, RN ). Thus qf is quasi-convex. Proof of Theorem 4.4.2. ! inf {J}. We only want to show that any Proof. 1. Since Qf f , inf {J} ! minimizing sequence of J is indeed a minimizing sequence of J. Thus it is 1,∞ (Ω, RN ) such suﬃcient to prove that ∀u ∈ Wv1,∞ (Ω, RN ) ∃{ul }∞ 1 ⊂ Wv that ! as l → ∞ . J(ul ) → J(u) Due to the assumption (4.34) on f and the density of piecewise aﬃne functions in W01,∞ (Ω, RN ), we may assume that u is piecewise aﬃne in a smaller domain Ω0 ⊂ Ω0 ⊂ Ω. By a standard argument, it is suﬃcient to replace Ω by Ω0 . m 2. Let Ω0 = ∪ Di , where Di , i = 1, . . . , m are simplexes with Di◦ ∩ Dj◦ = i=1

(j)

i ⊂ Di Ø i = j and let ∇u|Di = Bi ∀i. Then we choose small cubes {Ci }pj=1

(j)◦

with Ci

(k)◦

∩ Ci

= Ø as j = k, and

4.4 Relaxation and Young Measure

pi

(j) Di \ ∪ C i j=1

(1 + |∇u(x)|)p dx

249

ε . mC1 (j)

We are going to construct a sequence ul on each cube C = Ci . By Lemma 4.4.3, ∃ϕl ∈ W01,∞ (C, RN ) such that for ∀x ∈ C

1 1 f (∇u(x) + ∇ϕl (x))dx (Qf )(∇u(x)) + . (4.37) (Qf )(∇u(x)) m(C) C l Since C is a cube and ϕl ∈ W01,∞ (C, RN ), we may extend ϕl periodically to RN . Let 1 ul (x) = u(x) + ϕl (lx), ∀x ∈ C . l Then ul |∂C = u|∂C , and

f (∇ul (x))dx = f (Bi + ∇ϕl (lx))dx C C

1 f (Bi + ∇ϕl (y))dy = n l lC

= f (Bi + ∇ϕl (y))dy . (4.38) C

Combining (4.37) and (4.38), we obtain

(j) f (∇ul (x))dx → (Qf )(Bi )m(Ci ) , (j)

Ci

as l → ∞. 3. “Putting them together” we deﬁne ⎧ m pi (j) ⎪ ⎨u x ∈ Ω0 \ ∪ ∪ Ci , i=1 j=1 ul (x) = m pi (j) ⎪ ⎩ul (x) x ∈ ∪ ∪ Ci . i=1 j=1

It follows that

f (∇ul (x))dx =

Ω0

m pi

(j)

f (∇u(x))dx

Ω0 \ ∪ ∪ Ci i=1j=1

+

pi m

(j)

m(Ci )(Qf )(Bi ) + o(1)

i=1 j=1

(Qf )(∇u(x))dx + o(1) .

= Ω0

After a diagonal process, we obtain

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4 Minimization Methods

! J(ul ) → J(u) as l → ∞ . ! then by the above argument, we obtain a 4. If u0 is a minimizer of J, minimizing sequence {ul } of J, with ul |∂Ω = u0 |∂Ω . Combining the ﬁrst inequality of (4.34) with the second inequality of (4.37), we obtain

p |∇ul | f (∇ul ) C2 Ω

Ω

! 0 ) + m(Ω) J(u J(u0 ) + m(Ω) . Therefore there exists a weakly convergent subsequence {uli } in Wv1,p . Recalling the construction of {ul }, we have ul (x) → u0 (x), a.e. Therefore, uli u0 in Wv1,p . ! l ) J(ul ) → Conversely, if {ul } is a minimizing sequence of J, then J(u ! ! inf J = inf J, i.e., {ul } is also a minimizing sequence of J. Again by the ﬁrst inequality of (4.34), there exists a subsequence {uli } that weakly converges to ! u0 in W01,p . Since J! is s.w.l.s.c., u0 is a minimizer of J. Now, let us return to the example we met in the beginning of this section. The function f (u, ξ) = u2 + (ξ 2 − 1)2 is obviously not convex in ξ. Since in the case where N = 1, convex = quasiconvex, the convex envelope of f (u, ξ) is the following: u2 + (ξ 2 − 1)2 if |ξ| 1 Qf (u, ξ) = 0 if |ξ| < 1 , and then ! J(u) =

1

Qf (u(x), u (x))dx .

0

In fact, the minimizing sequence introduced previously weakly converges ! to 0 in W 1,4 ([0, 1]), and u = 0 is exactly the minimizer of J. Since the initial functional J is in the lack of weakly lower semi-continuity, we then minimize another functional J! instead, which is s.w.l.s.c. and shares a common minimizing sequence with the original one. This passage is called relaxation. The solution of J! is called a generalized solution of J. Noticing that the representation of the quasi-convex envelope in Lemma 4.4.3 can be seen as an average over ﬁne scale oscillations, we give a physical interpretation of this phenomenon: J represents the microscopic energy, while J! is the macroscopic. There are models in elastic crystals, in which this phenomenon occurs. The quasi convex envelope of a non-convex function is not easy to compute. People have developed various methods in approaching the computations. Readers who are interested in this problem are referred to the book of

4.4 Relaxation and Young Measure

251

Dacorogna [Dac]. In particular, to the interesting example on the two well problem: K = {A, −A}, the quasi convex envelope of the square distance function Qdist(P, K)2 has been expressed explicitly by K. Zhang [Zh 2]. 4.4.2 Young Measure In a variational problem, there may be many minimizing sequences. If minimizers exist, then the minimizers describe the common feature of these sequences. Otherwise, the sequences do not converge to a minimizer as an element of the given function space. In his study of generalized solutions of optimal control problems, L. C. Young [Yo] introduced a family of measures in certain measure space as a replacement of the minimizer. The idea is as follow: Let Ω ⊂ Rn be a measurable set, and let zj : Ω → RN , j = 1, 2, . . . , , be a sequence of measurable functions. To each zj , we relate a family of probability measures νxj = δzj (x) on RN , ∀x ∈ Ω. Instead of the convergence of zj , we consider the convergence of the family of probability measures: {νxj }; i.e., whether there exists a family of measures νx on RN , ∀x ∈ Ω, such that for ∀f ∈ C0 (RN ), the space of continuous functions with compact support,

j f (y)dνx (y) → f (y)dνx (y) , RN

RN

as j → ∞. The family of measures {νx |x ∈ Ω} is a Young measure. However, the function zj is well deﬁned in a.e. sense, what is the meaning of the measure: δzj (x) ? What is the exact meaning of the family of measures {νx |x ∈ Ω} ? In order to clarify the meaning, we appeal to the slicing measures. Let µ be a ﬁnite nonnegative Radon measure on Rn+N . Denote by σ the canonical projection of µ onto Rn : σ(E) = µ(E × RN ) ∀ Borel set E ⊂ Rn . Lemma 4.4.4 For σ − a.e. point x ∈ Rn , ∃ a Radon probability measure νx on RN such that for each bounded continuous function f , 1. x → RN f (x, y)dνx (y) is σ − measurable, 2. Rn ×RN f (x, y)dµ(x, y) = Rn ( RN f (x, y)dνx (y))dσ(x). Proof. Recall that the dual space of C0 (RN ), the space of continuous functions with compact support, is the signed measure space M(RN ). The family of measures νx on RN , ∀x ∈ Rn , should be determined with the aid of functions on C0 (RN ). Since C0 (RN ) is separable, there is a countable dense subset {gk }∞ 1 . Now, ∀ Borel sets E ⊂ Rn , we deﬁne

γk (E) = gk (y)dµ(x, y), ∀k . E×RN

Obviously, γk is absolutely continuous with respect to σ. There exists a σnull set N such that the derivatives exist for ∀x ∈ / N:

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4 Minimization Methods

Dσ γk (x) = lim

r→0

γk (Br (x)) , ∀k , σ(Br (x))

and then x → Dσ γk (x) are bounded and σ measurable. Moreover, for all Borel sets E ⊂ Rn ,

gk (y)dµ(x, y) = γk (E) = Dσ γk (x)dσ(x), ∀k . E×RN

(4.39)

(4.40)

E

N N Since {gk }∞ 1 is dense in C0 (R ), ∀g ∈ C0 (R ), we have a subsequence gkj → g N / N , Γx (g) = limj→∞ Dσ γkj (x) exists, and uniformly on R . Thus for ∀x ∈

|Γx (g)| ≤ Cµ gC0 (RN ) ,

(4.41)

where Cµ = µ(Rn × RN ). Since |γk (Br (x)) − γl (Br (x))| ≤ gk − gl C0 (RN ) σ(Br (x)) , the limit Γx (g) does not depend on the choice of the subsequence converging to g. Also, the map: g → Γx (g) is linear and positive. Provided by the Riesz representation theorem, ∃νx ∈ M(RN ), which is nonnegative, such that for ∀g ∈ C0 (RN ):

Γx (g) =

RN

g(y)dνx (y), ∀x ∈ /N .

By taking limits, we see νx (RN ) = Γx (1) = 1. Therefore, it is a probability measure. As a function of x on RN \N , x → Γx (g) is bounded and σ-measurable, because it is the pointwise limit of a sequence of σ-measurable functions Dσ γkj (x). Therefore, the conclusion (1) is obtained by approximation. Following (4.40), by taking limits we obtain

g(y)dµ(x, y) = Γx (g)dσ(x) = g(y)dνx (y) dσ(x) . (4.42) E×RN

E

E

From (4.42) ∀h ∈ C0 (Rn ), we have

g(y)h(x)dµ(x, y) = Rn ×RN

Since C0 (Rn ) mation.

Rn

>

RN

RN

g(y)dνx (y) dσ(x) .

(4.43)

C0 (RN ) is dense in C0 (Rn+N ), (2) is proved by approxi

Remark 4.4.5 By standard measure theoretic argument, one can show that the statements of Lemma 4.4.4 hold, if the bounded continuous function f is replaced by f ∈ L1 (Rn × RN , µ). Moreover, we have f (x, ·) ∈ L1 (RN , νx ) for a.e., x ∈ Rn , and the function x → RN f (x, y)dνx (y) is in L1 (Rn , σ).

4.4 Relaxation and Young Measure

253

Amap ν : Rn → M(RN ) is called weakly ∗ σ-measurable, if the functions x → RN f (y)dνx (y) are σ-measurable for all f ∈ C0 (RN ) on Rn . In the context of the duality between M(RN ) and C0 (RN ), the map ν : n R → M(RN ), obtained in Lemma 4.4.4 is w∗ σ-measurable. After the preparations, we are able to identify an L∞ (Ω, RN ) function with a w∗ -measurable map ν : Ω → M(RN ). Suppose that Ω ⊂ Rn is a Lebesgue measurable set with a ﬁnite Lebesgue measure. For ∀z ∈ L∞ (Ω, RN ), one deﬁnes a measure µ on Rn × RN , provided by the duality:

f (x, y)dµ(x, y) = f (x, z(x))dx, ∀f ∈ C0 (Rn × RN ) , (4.44) Rn ×RN

Ω

and then a family of measures on RN : δz(x) for a.e. x ∈ Ω. Since the righthand side of (4.44) is linear and positive in f , and

f (x, z(x))dx ≤ m(Ω)f C , 0 Ω

the existence of µ follows from the Riesz representation theorem. Obviously, the canonical projection of µ onto Rn is the Lebesgue measure restricted on the measurable set Ω, and the w∗ -measurable map ν : Ω → M(RN ) satisﬁes νx = δz(x) for a.e. x ∈ Ω. Now we turn to the fundamental theorem for Young measures. Theorem 4.4.6 Let Ω ⊂ Rn be a bounded open domain and let {zk } ⊂ L∞ (Ω, RN ) be a bounded sequence. Then there exist a subsequence {zkj } and a w∗ -measurable map ν : Ω → M(RN ) such that 1. νx is a Radon probability measure for a.e. x ∈ Ω. 2. For ∀g ∈ C0 (RN ), one has

g(zkj ) ∗ g(x) := νx , g = g(y)dνx (y), in L∞ (Ω) . RN

3. Let K ⊂ RN be closed, and let dist(zkj , K) → 0 in measure as j → ∞. Then supp νx ⊂ K for a.e. x ∈ Ω. Proof. According to (4.44), ∀zk , ∃µk on Rn × RN , for which the canonical projection onto Rn is the Lebesgue measure restricted to Ω. Since µk (Rn × RN ) ≤ Ln (Ω), there exist a measure µ on Rn × RN and a subsequence such that µkj ∗ µ, according to the Banach–Alaoglu theorem. We claim that the canonical projection σ of µ into Rn is the Lebesgue measure Ln restricted on Ω. On one hand, if U ⊂ Ω is open, then by the w∗ -convergence: σ(U ) = µ(U × RN ) ≤ lim inf µkj (U × RN ) = Ln (U ) .

254

4 Minimization Methods

According to the Radon–Nikodym theorem, there exists a ∈ L1 (Ω, Ln ) such that σ = aLn . On the other hand, for any compact subset K ⊂ Ω, we show that σ(K) ≥ Ln (K). Since {zk } is bounded in L∞ (Ω, RN ), ∃R > 0, such that supp(µkj ), and then supp(µ) are contained in Ω × BR (θ).∀ > 0, ∃f ∈ C0 (Rn × RN ), with f = 1 on K × BR (θ) and f ≥ 0, such that

f (x, y)dµ(x, y) < µ(K × BR (θ)) + . Rn ×RN

Thus, σ(K) = µ(K × RN ) = µ(K × BR (θ))

> f (x, y)dµ(x, y) − Rn ×RN

= lim f (x, y)dµkj (x, y) − j→∞

Rn ×RN

> Ln (K) − . Since > 0 is arbitrary, we have proved that σ(K) ≥ Ln (K) .

(4.45)

Therefore a = 1, a.e., and σ is the restriction of the Lebesgue measure on Ω. The ﬁrst conclusion follows directly from Lemma 4.4.4. We turn to proving the second conclusion. ∀h ∈ C(Ω), and ∀g ∈ C0 (RN ), ∃ ˜ ∈ C0 (Rn ), from (4.44), we have an extension of h, h

˜ h(x)g(y)dµ kj (x, y) Rn+N

h(x)g(zkj (x))dx . = Ω

By taking limits, and from (4.43), we have

˜ h(x)g(zkj (x))dx → h(x)g(y)dµ(x, y) Ω Rn ×RN

= h(x) g(y)dνx (y) dx . RN

Ω

Since C(Ω) is dense in L1 (Ω) and g(zkj ) is bounded in L∞ (Ω), the above equality implies that

g(y)dν(y) =< ν, g > in L∞ (Ω, RN ) , g(zkj ) ∗ RN

4.4 Relaxation and Young Measure

255

i.e., (2) holds. We prove the third conclusion. ∀g ∈ C0 (RN \K), ∀ > 0, ∃C > 0, such that |g(y)| ≤ + C dist(y, K), ∀y ∈ RN , which implies that |g(zk (x))| ≤ + C dist(zk (x), K), a.e., x ∈ Ω . From dist(zkj (x), K) → 0 in measure, we have | νx , g | ≤ , a.e. x ∈ Ω Since > 0 is arbitrary, supp(νx ) ⊂ K, a.e. x ∈ Ω.

Deﬁnition 4.4.7 The w∗ -measurable map ν : Ω → M(RN ), deﬁned in Theorem 4.4.6, is called a Young measure generated by the sequence {zkj } ⊂ L∞ (Ω, RN ). Remark 4.4.8 The conclusion (2) has a probability interpretation: In the limit, g(zkj ) takes the value g(y) with probability νx (y) at x. Therefore the Young measure can be used to describe the local phase proportions in an inﬁnitesimally ﬁne mixture. Corollary 4.4.9 Assume that ν is the Young measure associated with the sequence {zk } and that zk → z in measure. Then νx = δz(x) a.e. Proof. ∀g ∈ C0 (RN ), g(zk ) → g(z) in measure. But by Theorem 4.4.6, g(zk ) ∗ g = ν, g . Therefore, νx , g = g(z(x)) = δz(x) , g , a.e . Remark 4.4.10 In Theorem 4.4.6, the L∞ -bounded sequence {zk } ⊂ L∞ (Ω, RN ) can be replaced by the L1 bounded sequence {zk } ⊂ L1 (Ω, RN ). In this case, νx is Ln *Ω a.e. a probability measure with

|y|dνx (y)dx ≤ lim inf zk L1 . Ω

k→∞

RN

Again we claim that σ is the restriction of the Lebesgue measure on Ω. In fact, only (4.45) should be modiﬁed: ∀R > 0 σ(K) ≥ µ(K × BR (θ)) ≥ lim sup µk (K × BR (θ)) k→∞

≥ lim sup Ln ({x ∈ K | |zk (x)| ≤ R}) k→∞

256

4 Minimization Methods

≥ Ln (K) −

1 sup zk L1 . R k

As R → ∞, it follows that σ(K) ≥ Ln (K). Finally, we use Remark 4.4.5, and choose a sequence of positive functions fk (x, y) ↑ |y| pointwisely; the monotone convergence theorem implies the conclusion. Corollary 4.4.11 Assume that {zk } ⊂ L∞ (Ω, RN ), satisﬁes zk → z, a.e., and that {wk } ⊂ L∞ (Ω, RM ) generates the Young >measure ν. Then {zk , wk } : Ω → RN +M generates the Young measure δz(x) νx , x ∈ Ω. Proof. ∀ϕ ∈ C0 (RN ), ∀ψ ∈ C0 (RM ), ∀η ∈ L1 (Ω), by deﬁnition and the Lebesgue’s dominance theorem, we have ϕ(zk ) → ϕ(z), a.e. , ηϕ(zk ) → ηϕ(z) in L1 (Ω) , ψ(wk ) ∗ ψ = ν, ψ in L∞ (Ω) . This implies that

? η(ϕ ψ)(zk , wk )dx = ηϕ(zk )ψ(wk )dx → η(x)ϕ(z(x)) νx , ψ dx . Ω

Ω

Ω

Then,

? ? ? ν, ϕ ψ in L∞ (Ω) . (ϕ ψ)(zk , wk ) ∗ δz > Since C0 (RN ) C0 (RM ) is dense in C0 (RN +M ), it follows that ? f (zk , wk ) ∗ δz ν, f in L∞ (Ω) , for all f ∈ C0 (RN +M ).

Example 4.4.12 Let ϕ be the function deﬁned in Corollary 4.3.3, and ϕk (x) = ϕ(kx), ∀k ∈ N . It is known that ϕk

∗

1

ϕ(x)dx = λα + (1 − λ)β .

0

Similarly, ∀f ∈ C(R1 ), we have f ◦ ϕk

∗

1

f (ϕ(x))dx = λf (α) + (1 − λ)f (β) .

0

Therefore the sequence {ϕk } is associated with a Young measure: ν = λδα + (1 − λ)δβ ; i.e., νx = λδ(x − α) + (1 − λ)δ(x − β), ∀x ∈ R1 .

4.4 Relaxation and Young Measure

257

Example 4.4.13 Let {un } be the sequence of sawtooth functions: x − nk if x ∈ [ nk , 2k+1 n ] un (x) = k+1 2k+1 k+1 −x + n if x ∈ [ n , n ] , which is a minimizing sequence of the functional J (see equation 4.32). Let 1 if x ∈ [ nk , 2k+1 n ], zn (x) = un (x) = −1 if x ∈ [ 2k+1 n ]. Let ν be the Young measure associated with to the sequence zk . Conclusion: ν = 12 (δ−1 + δ+1 ). In fact, g(zn (x))

∗

νx , g ,

for all g ∈ C0 (R ). Let us take g(y) = min{(y 2 − 1)2 , 1}. Since J(un ) → 0, νx , g = 0 a.e. This implies that supp(νx ) ⊂ {−1, 1}, i.e., νx = λ(x)δ−1 + (1 − λ(x))δ+1 . Also, zn ∗ 0 in L∞ (R1 ) and the relation g1 (zn (x)) ∗ νx , g1 in ∞ L (R1 ), holds for all g1 ∈ C0 (R1 ) with g1 (y) = y, as |y| < 2. Thus 1

λ(x)δ−1 + (1 − λ(x))δ+1 , g1 = 0 . This implies that 1 − 2λ(x) = 0, or λ(x) = 12 . Example 4.4.14 Let B, C ∈ M n×N satisfy rank(B − C) = 1. Assume ∃λ ∈ (0, 1), such that λB + (1 − λ)C => 0. It is known that without loss of generality, > one may assume B = (1 − λ)a e1 , and C = −λa e1 for some a ∈ RN and e1 ∈ Rn . Let (1 − λ)t if t ∈ [0, λ] , ϕ(t) = −λ(t − 1) if t ∈ [λ, 1] . ∀x = (x1 , . . . , xn ) ∈ D := [0, 1]n , let uk (x) = ak−1 ϕ(kx1 ), k = 1, 2, . . . , and zk = ∇uk . Then {zk } is associated with the Young measure ν = λδB + (1 − λ)δC , where δB and δC are the probability measures concentrated at the matrices B and C, respectively. In fact,

B if {kx1 } ∈ (0, λ) , zk (x) = C if {kx1 } ∈ (λ, 1) ,

so is dist(zk , {B, C}) = 0. By Theorem 4.4.6, the probability measure ν satisﬁes supp(νx ) ⊂ {B, C} a.e. Therefore, ∃µ(x) ∈ [0, 1] such that νx =

258

4 Minimization Methods

µ(x)δB + (1 − µ(x))δC , and ∀g ∈ C0 (M n×N ), g(zkj ) ∗ ν, g for some subsequence > {zkj }. In particular, we take g(y) = y in a ball centered at θ, containing a e1 . Again, from zk ∗ 0, we obtain µ(x)B + (1 − µ(x))C = 0, i.e., µ(x) = λ. Therefore, ν = λδB + (1 − λ)δC . A w∗ -measurable map ν : Ω → M(M n×N ) is called a gradient Young measure, if ∃{uj } ⊂ W 1,∞ (Ω, RN ) such that uj

∗

u in W 1,∞ (Ω, RN ), and δ∇uj

∗

ν.

A central problem for the gradient Young measure is the following: Given a set K ⊂ M n×N , how do we characterize all W 1,∞ gradient Young measures ν such that supp(νx ) ⊂ K for a.e. x ∈ Ω? Indeed, letting ν be the Young measure generated by {uj }, the range of {∇uj } must be contained in a big ball BR (θ) of M n×N . Then by taking K = B R (θ), and according to (3) of Theorem 4.4.6, it follows that: (1). There exists a compact set K ⊂ M n×N such that supp(νx ) ⊂ K, a.e. If we choose g(y) = y on BR (θ) and g ∈ C0 (M n×N ), then g(∇uj ) = ∇uj ∗ ∇u in L∞ (Ω, M n×N ). From the second conclusion of Theorem 4.4.6, we have g(∇uj ) ∗ ν, g in L∞ (Ω, M n×N ). Since g is arbitrary outside the ball BR (θ), we may simply write ν, g as ν, id . Thus, (2) νx , id = ∇u(x) a.e. Finally, assume that g : M n×N → R1 is continuous and quasi-convex, then by the Morrey theorem, for all open sets U ⊂ Ω, we have

g(∇uk (x))dx ≥ g(∇u(x))dx . lim inf k→∞

U

U

Note that g(∇uk (x)) ∗ νx , g , and g(∇u(x)) = g( νx , id ) ; we obtain

νx , g dx ≥ U

g( νx , id )dx . U

Since U ⊂ Ω is arbitrary, we arrive at (3) νx , g ≥ g( νx , id ) a.e. In fact, all these propositions together characterize the W 1,∞ gradient Young measure, i.e., the converse is also true. Namely, one has the following: Theorem 4.4.15 A w∗ -measurable map ν : Ω → M(M n×N ) is a gradient Young measure if and only if νx ≥ 0 a.e., ∃ a compact set K ⊂ M n×N and ∃u ∈ W 1,∞ (Ω, RN ) such that 1. supp(νx ) ⊂ K, a.e. 2. νx , id = ∇u(x), a.e. 3. νx , g ≥ g( νx , id ) a.e., ∀ continuous quasi-convex g: M n×N → R1 . The suﬃcient part of the proof can be found in [KP 1].

4.4 Relaxation and Young Measure

259

Remark 4.4.16 In the modeling of microstructure, the reason people use W 1,∞ sequences approaching a compact set K is based on a truncation lemma due to Kewei Zhang [Zh 1]: Let K be a compact set in M n×N , and let K∞ = 1,1 (Ω, RN ) sup{|A| | A ∈ K}. Let Ω ⊂ Rn be a bounded domain. If {uj } ⊂ Wloc satisﬁes dist (∇uj , K) → 0 in L1 (Ω), and uj → u in L1loc (Ω), then there ex1,1 ists a sequence {vj } ⊂ Wloc (Ω, RN ) such that |∇vj | ≤ cn,N K∞ , |{x ∈ Ω | uj (x) = vj (x)}| → 0, and vj = u near ∂Ω. As an application, we extend that Morrey theorem to the case where the integrand is a function depending on (x, u, ∇u). Corollary 4.4.17 Suppose that f : Ω × RN × M n×N → R1 is a nonnegative Caratheodory function, and that uj u in W 1,∞ (Ω, RN ) generates a gradient Young measure ν. Then

f (x, uj (x), ∇uj (x))dx ≥ f (x, u(x), ξ)dνx (ξ) dx lim inf j→∞

Ω

Ω

Mn×N

If further, ∀(x, u) ∈ Ω × RN , ξ → f (x, u, ξ) is quasi-convex, then

lim inf f (x, uj (x), ∇uj (x))dx ≥ f (x, u(x), ∇u(x))dx , j→∞

Ω

Ω

i.e., the functional J(u) = RN ).

Ω

f (x, u(x), ∇u(x))dx is s.w∗ .l.s.c. on W 1,∞ (Ω,

Proof. According to Corollary 4.4.13, the sequence {zj = (uj , ∇uj )} generates > the Young measure δu ν. Since the range of {zj } are contained in a big ball BR (θ) ⊂ RN × M n×N , we may assume that f = 0 outside BR (θ). Since f is a Caratheodory function, by the Scorza–Dragoni–Vainberg theorem (a version of the Luzin theorem with parameters, see also Vainberg [Va], there exists an increasing sequence of compact sets Ck ⊂ Ω such that m(Ω\Ck ) → 0, and f |Ck ×RN ×Mn×N is continuous. Let gk (x, ·) = χCk (x)f (x, ·) : Ω → C0 (RN × Mn×N ). We obtain

f (x, uj (x), ∇uj (x))dx ≥ gk (x, uj (x), ∇uj (x))dx Ω

Ω ? νx , gk (x, ·) dx, as j → ∞ . → δu(x) Ω

But,

RHS =

δu(x) Ck

?

= Ck

M n×N

νx , f (x, ·, ·) dx f (x, u(x), ξ)dνx (ξ) dx

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4 Minimization Methods

→ Ω

f (x, u(x), ξ)dνx (ξ) dx, as k → ∞ .

M n×N

This proves the ﬁrst conclusion. If further, f is quasi-convex in ξ, then from the conclusions (2) and (3) of Theorem 4.4.15, we have

f (x, u(x), ξ)dνx (ξ) ≥ f (x, u(x), ∇u(x)), a.e. x ∈ Ω . Mn×N

The s.w∗ .l.s.c. of J is proved.

Remark 4.4.18 A gradient Young measure ν, if it is constant up to a null set, is called homogeneous. This is a useful notion in studying oscillations of sequences because it isolates the oscillation near a point in a domain. Examples 4.4.13 and 4.4.14 are homogeneous.

4.5 Other Function Spaces We have seen that the w∗ -compactness plays an important role in the calculus of variations. That is why we always choose Sobolev spaces W m,p , 1 < p ≤ ∞ as working spaces. The case p = 1 is diﬀerent, even if to the Lebesgue space L1 , the Banach–Alaoglu theorem cannot be applied. In fact, the positive deltatype sequence does not weakly * converge to any L1 function. In the extremal case, we study some other function spaces as replacements. In this section, the BV space, the Hardy space and some other related spaces are introduced in the applications to variational problems. 4.5.1 BV Space We study the minimal surface problem. Let Ω ⊂ Rn be a bounded open domain; a hypersurface u ∈ C 1 (Ω, R1 ) with prescribed boundary condition has the area:

(1 + |∇u|2 ) . A(u) = Ω

This leads to the following variational problem: Min{A(u)} in the above set of hypersurfaces. One might expect that the Sobolev space W 1,1 would be a working space for the direct method. Unfortunately, the closed ball in this space is not weakly closed, and so a minimizing sequence may fail to converge. We are introduced to the space of bounded variations, which is widely used in the so-called free discontinuous boundary problems, in which the boundary of the domain is to be determined in the extremum problems. Recall that a continuous linear functional deﬁned on the space of continuous functions on [0, 1] can be represented by a BV function u as follows:

4.5 Other Function Spaces

1

φ(t)du(t), ∀φ ∈ C([0, 1]) .

L(φ) =

261

(4.46)

0

If φ ∈ C01 ([0, 1]), then

L(φ) = −

1

u(t)φ (t)dt .

(4.47)

0

We shall extend this relation to higher dimensional space. Let Ω ⊂ Rn be an open domain, and let C0 (Ω) be the space of continuous functions of compact support with respect to Ω. First, we note that L is a linear continuous functional on C0 (Ω) if and only if there exists a Radon measure µ and a µ-measurable function ν on Ω with |ν(x)| = 1, a.e., such that

φ(x)ν(x)dµ(x), φ ∈ C0 (Ω) . L(φ) = Ω

Second, for each compact subset K ⊂ Ω, let LK = L|C(K) , then LK := sup{L(f ) | f C(K) ≤ 1} = µ(K) . Motivated by (4.46), and (4.47), one deﬁnes the counterpart of a BV function u in higher dimensional spaces, as being a function u ∈ L1 (Ω) such that the functional

u divφ L(φ) := Ω

C01 (Ω, Rn ),

and can be extended continuously to is deﬁned for all φ ∈ C0 (Ω, Rn ). Then by Riesz representation theorem, there exist Radon measures ρi , and ρi measurable functions τi (x) on Ω, with |τi (x)| ≤ 1, ρi − a.e., i i = 1, . . . , n, and let µ = Σni=1 ρi , νi = τi dρ dµ , i = 1, . . . , n. such that

n u divφ = − Σ1 φi τi dρi = − (φ · ν)dµ , (4.48) Ω

Ω

Ω

where ν(x) = (ν1 (x), . . . , νn (x)) with ν(x)∞ ≤ 1 µ− a.e. We deﬁne · := · Rn . Deﬁnition 4.5.1 Let Ω ⊂ Rn be open. A function u ∈ L1 (Ω) is said to be of bounded variations, if the total variation of u on Ω is:

u divφ | φ ∈ C01 (Ω, Rn ), φ(x) ≤ 1, for a.e. x ∈ Ω < ∞ . Du(Ω) := sup Ω

The space BV (Ω) consists of all bounded variation functions on Ω with norm: uBV = uL1 + Du(Ω) .

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4 Minimization Methods

Since .Rn and .∞ are equivalent, for u ∈ BV (Ω), the total variation Du can be regarded as a measure µ: √ 1 √ µ(Ω) ≤ Du(Ω) ≤ nµ(Ω) . n

(4.49)

Moreover, the distributional derivative Du makes sense, it is related to the Radon measure µ and vector measurable function ν(x). Example 4.5.2 If u ∈ W 1,1 (Ω), then u ∈ BV (Ω), and uBV = uW 1,1 . In fact, we shall verify that Du(Ω) = Ω |∇u|dx. On one hand,

u divφ dx| = | ∇u · φ dx| ≤ |∇u|dx | Ω

Ω

Ω

with φ(x) ≤ 1 ∀ x ∈ Ω, it follows that Du(Ω) ≤ for all φ ∈ |∇u|dx. Ω On the other hand, ∀u ∈ W 1,1 (Ω), ∀ > 0, ∃φ ∈ C01 (Ω, Rn ) with φ (x) ≤ 1, ∀x ∈ Ω satisfying:

|∇u| dx ≤ ∇u · φ dx + = u · divφ dx + ≤ Du(Ω) + . C01 (Ω, Rn )

Ω

Ω

Since > 0 is arbitrary, we obtain

Ω

Ω

|∇u|dx ≤ Du(Ω).

Example 4.5.3 Let S ⊂ Rn be a C ∞ compact (n − 1)-dimensional hypersurface with the induced metric, and let Hn−1 be the (n−1)-dimensional Hausdorﬀ measure in Rn . The area of S is Hn−1 (S). Let Ω be the body bounded by S, and let χΩ be the characteristic function of Ω. Then χΩ ∈ BV (Rn ), and DχΩ (Rn ) = Hn−1 (S) . Indeed, by the Gauss formula, ∀φ ∈ C01 (Rn , Rn )

χΩ divφdx = divφdx = n(x) · φ(x)dHn−1 , Rn

Ω

S

where n(x) is the unit exterior normal. Thus DχΩ (Rn ) ≤ Hn−1 (S) . On the other hand, one extends n to be a C ∞ vector ﬁeld V over Rn with V (x) ≤ 1 ∀ x ∈ Rn . This can be done by a partition of unity. Then, ∀ρ ∈ C0∞ (Rn , R1 ) with |ρ(x)| ≤ 1, ∀x ∈ Rn , let φ = ρV , we have

4.5 Other Function Spaces

263

ρdHn−1 .

χΩ divφ dx = Rn

S

Thus, DχΩ (Rn )

= sup χΩ divφdx | φ ∈ C0∞ (Rn , Rn ), with φ(x) ≤ 1, ∀x ∈ Rn n R ≥ sup ρdHn−1 | ρ ∈ C0∞ (Rn , R1 ), |ρ(x)| ≤ 1, ∀x ∈ Rn S

=H

n−1

(S) .

These two examples show that W 1,1 (Ω) is strictly contained in BV (Ω), since for n = 1, the characteristic function χ[0,1] ∈ BV (R1 ) but not in W 1,1 . This leads us to extend the deﬁnition of the co-dimensional one area to the boundary of more general domains. Deﬁnition 4.5.4 Let E be a Borel set in an open domain Ω ⊂ Rn . We call ∂E(Ω) = DχE (Ω) the perimeter of E in Ω. As a function space with the norm uBV = uL1 + Du(Ω), BV (Ω) is a Banach space. Only the completeness remains to be veriﬁed. Lemma 4.5.5 (Lower semi-continuity) If {uj } ⊂ BV (Ω), and uj → u in L1 ; then for every open U ⊂ Ω Du(U ) ≤ lim inf Duj (U ) . j→∞

(4.50)

If further, sup{Duj (Ω) | j ∈ N } < ∞, then u ∈ BV (Ω). Proof. ∀φ ∈ C01 (U, Rn ) with φ(x) ≤ 1, one has

u divφ dx = lim uj divφ dx ≤ lim inf Duj (U ) . U

j→∞

U

j→∞

(4.50) is proved. Theorem 4.5.6 BV (Ω) is complete.

Proof. For a Cauchy sequence {uj } in the BV norm, it is obvious that uj → u in L1 . By the previous lemma, Du(Ω) < ∞, and then u ∈ BV (Ω). It remains to show that D(uj − u)(Ω) → 0. Again, from the lower semi-continuity lemma, ∀ > 0, ∃j0 ∈ N such that D(uj − u)(Ω) ≤ lim inf D(uj − uk )(Ω) < , as j ≥ j0 . k→∞

264

4 Minimization Methods

Now we consider the possibility of C ∞ approximation of the BV functions. Since the W 1,1 norm equals the BV norm for C 1 functions, and C ∞ is dense in W 1,1 , we can only have: Theorem 4.5.7 Let Ω be an open domain of Rn . Then for ∀u ∈ BV (Ω), ∃uj ∈ BV (Ω) ∩ C ∞ (Ω) such that 1. uj → u in L1 (Ω), 2. Duj → Du in the sense of Radon measure. In particular, Duj (Ω) → Du(Ω). We omit the proof, but refer to Giusti [Gi], p. 14. Theorem 4.5.8 (Compactness) Let Ω be a bounded open domain of Rn . Any sequence {uj } ⊂ BV (Ω) with uj BV ≤ M < ∞ possesses a convergent subsequence in the L1 norm, and the limit u ∈ BV (Ω), with uBV ≤ M . Proof. We take a sequence {vj } ⊂ BV (Ω) ∩ C ∞ (Ω) such that uj − vj L1 < 1 1 1,1 = j , and Dvj (Ω) ≤ M + j . From Example 4.5.2, we have vj W Dvj BV . According to the Rellich–Kondrachev compactness theorem, there is a subsequence vjk L1 -converges to u. From the lower semi-continuity lemma, Du(Ω) ≤ M , and u ∈ BV (Ω). Obviously, ujk → u in L1 . Besides the above preparations, in order to study the minimal surface problem stated at the beginning of the subsection we have to deﬁne the trace of BV functions. The following theorem can be found in Giusti [Gi] and Evans and Gariepy [EG]: Theorem 4.5.9 (Trace) Let Ω ⊂ Rn be a bounded domain with Lipschitzian boundary ∂Ω. Then the trace operator T : BV (Ω) → L1 (∂Ω, Hn−1 ) is bounded, and we have

u div φ = − (φ · ν)dµ + T u(φ · n)dHn−1 , ∀φ ∈ C01 (Ω, Rn ), Ω

Ω

∂Ω

where µ is the Radon measure, ν is the µ-measurable vector function with ν(x) ≤ 1, a.e. with respect to µ, and n is the unit normal vector ﬁeld over ∂Ω, which is well deﬁned almost everywhere. We omit the proof and turn to the existence of the minimal surface problem. Theorem 4.5.10 (Nonparametric minimal surface) Let Ω ⊂ Rn be a bounded Lipschitzian domain, and let u0 ∈ BV (Ω). Then there exists a minimizer of the problem: Min{A(u) | u ∈ BV (Ω), T u = T u0 }, where A(u) =

Ω

1 + |Du|2

4.5 Other Function Spaces

265

(φ0 + udivφ)dx | (φ0 , φ) ∈ C01 (Ω, R1+n ) with |φ0 (x)| + φ(x) ≤ 1 ∀x ∈ Ω .

:= sup

Ω

Proof. One considers the set: XM := {u ∈ BV (Ω) | T u = T u0 , and Du(Ω) ≤ M }, for any M > 0. For suﬃciently large M > 0, XM = ∅ since we have Du(Ω) ≤ A(u) ≤ Du(Ω) + mes(Ω) . Combining with Theorems 4.5.7 and 4.5.8, XM is compact in L1 -topology. By the same proof as for Lemma 4.5.5, A is l.s.c. on XM in L1 -topology. The proof follows from the general principle of the direct method, i.e., Theorem 4.2.1. The problem stated above is called the nonparametric minimal surface problem, there is another formulation called parametric minimal surface problem: Again, let Ω be a bounded open domain in Rn , and let L be a set of ﬁnite perimeters. Deﬁne F = {F is Borel measurable | F \Ω = L\Ω} . Theorem 4.5.11 (De Giorgi) The problem Min{DχF (Rn ) | F ∈ F} has a solution. Proof. Since Ω is bounded, ∃R > 0 such that Ω ⊂ BR (θ). If F \Ω = L\Ω, then F \BR (θ) = L\BR (θ). It follows that DχF (Rn ) = DχF (BR (θ)) + DχL (Rn \BR (θ)) . Thus the problem is reduced to Min{DχF (BR (θ)) | F ∈ F} . The functional J(u) := Du(BR (θ)) is bounded from below on the set X := {u = χF | Du(BR (θ)) ≤ ∂χL (BR (θ))}. We shall verify that X is a compact subset under L1 -topology. According to Theorem 4.5.8, it remains to verify that any limit point u of X is a characteristic function of a measurable set of ﬁnite perimeter. In fact, it is the a.e. limit of a sequence of X. Since J is l.s.c. on X under L1 -topology, there exists a minimizer u. Although we have proved the existence of minimal surfaces, the solutions we obtained are very weak. Thus, in order to verify that the weak solutions are geometric minimal surfaces, the diﬃculties lie in the regularity, cf E. Giusti [EG] and De Giorgi [DG 1].

266

4 Minimization Methods

4.5.2 Hardy Space and BMO Space It is known that a bounded sequence in L1 space may not have a w∗ -convergent subsequence. Therefore L1 is not a good space for the variational method on which we work. Comparing with the Lp space p ∈ (1, ∞), the latter has many important properties in analysis: the uniform convexity, the reﬂexivity, the smoothness of the norm, the boundedness of singular integral operators in Calderon–Zygmund theory, as well as the Lp estimates in the elliptic theory, which L1 space does not have. The Hardy space H1 is in some sense a replacement of L1 . It is deﬁned as follows: Deﬁnition 4.5.12 (Hardy space) H1 (RN ) = {f ∈ L1 (RN ) | Rj f ∈ L1 (RN ), j = 1, 2, . . . , N } . 1

where Rj = ∂j (−∆)− 2 is the Riesz transform, j = 1, 2, . . . , N . The norm is deﬁned to be N f H1 = f L1 + Rj f L1 . 1

There are many equivalent characterizations of the Hardy space. The fol lowing is one of them. ∀h ∈ C0∞ (RN ), with h ≥ 0, and h = 1. ∀t > 0 let ht (x) = t1N h( xt ). For every distribution f , one deﬁnes f ∗ (x) = sup |(ht ∗ f )(x)| . t>0

If f ∗ ∈ L1 , then from the Lebesgue dominance theorem, we have f ∈ L1 and f L1 ≤ f ∗ L1 . By deﬁnition f ∗ depends on h, but it is proved that f ∈ H1 if and only if f ∗ ∈ L1 for any such h, and f = f ∗ L1 is an equivalent norm. H1 is a Banach space. In order to study the H1 space, we need some knowledge on harmonic analysis, in particular, the notion of Hardy-Littlewood maximal functions: ∀f ∈ L1loc , the function

1 |f (y)|dy M (f )(x) = sup r>0 |Br (x)| Br (x) is called the maximal function of f . It possesses the following properties: 1. M (cf ) = |c|M (f ), ∀c ∈ R1 , and M (f1 + f2 ) ≤ M (f1 ) + M (f2 ), ∀f1 , f2 ∈ L1loc . 2. |f (x)| ≤ M (f )(x) a.e. 3. If M (f ) ∈ L1 (Rn ), then f = 0 a.e.

4.5 Other Function Spaces

267

Proof. If |f | > 0 on a positive measure set E, then we may assume that ∃R > 0, ∃ > 0 such that E ⊂ BR (θ), |f (x)| ≥ , ∀x ∈ E. Thus, M (f )(x) ≥ CN |E|(|x| + R)−N , where CN is a constant depending on N only. It contradicts M (f ) ∈ L1 (RN ). 4. The maximal function operator is of weak (1, 1) type, i.e., there exists a constant C > 0 such that |Eλ | ≤

C f L1 ∀λ > 0, f ∈ L1 (RN ) , λ

where Eλ = {x ∈ RN | M (f )(x) > λ}. Proof. For ∀x ∈ Eλ , ∃rx > 0 such that

|f |dx ≥ λ|Brx (x)| . Brx (x)

Thus, {Brx (x) | x ∈ Eλ } is a covering of Eλ . According to the Vitali covering theorem, there exist a family of countable disjoint balls {Brxj (xj )} and a constant cN such that

|Eλ | ≤ cN |Brxj (xj )| ≤ cN λ−1 |f |dx ≤ cN λ−1 f L1 . j

j

Brx (xj ) j

5. For p ∈ (1, ∞), if f ∈ Lp (RN ), then M (f ) ∈ Lp (RN ) and there exists a constant Cp > 0 such that M (f )Lp ≤ Cp f Lp . Proof. Since M is of (∞, ∞) type, in combination with property 4, the conclusion follows from the interpolation theory. 6. (Kolmogorov’s inequality) For any δ ∈ (0, 1), there is a constant C = Cδ > 0 such that

1 C δ |E|1−δ f δL1 , ∀ measurable E ⊂ RN . M (f )δ dx ≤ 1−δ E Proof. In fact from property 4,

∞

M (f )δ dx = δ |{x | M (f )(x) > λ} ∩ E|λδ−1 dλ E 0 t ∞ + |{x | M (f )(x) > λ} ∩ E|λδ−1 dλ =δ 0

t

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4 Minimization Methods

≤ |E|t + Cδ δ

∞

λδ−2 dλf L1

t

≤ |E|tδ +

Cδ δ−1 t f L1 /; , 1−δ

where C is the weak (1, 1) norm of M (f ). Setting t = desired inequality.

C f L1 |E|

, we obtain the

We investigate the behavior of H1 functions. 1. There exists a constant C1 > 0 such that f ∗ ≤ C1 M (f ). If f ≥ 0, then M (f ) ≤ C2 f ∗ for some constant C2 . Thus if f ≥ 0 and f ∈ H1 , then f = 0. 2. If f ∈ H1 , then RN f dx = 0. Proof. Let φ ∈ C ∞ (RN ) with suppφ ⊂ B1 (θ) and φ(θ) = 1. ∀x ∈ RN , ∀s ≥ |x|, one deﬁnes h(y) = cφ(s−1 x − 2y), where c is the normalized constant such that RN h(y)dy = 1. Then f ∗ (x) = sup |(ht ∗ f )(x)| t>0

x−y ≥ (2s)−N | h f (y)dy| 2s RN

6y7 φ = (2s)−N c| f (y)dy| , s RN since the last integral tends to RN f (y)dy as s → ∞. If RN f (y)dy = 0, then we would have lim inf |x|→∞ f ∗ (x)|x|N > 0, which contradicts f ∗ ∈ L1 (RN ). 3. If f ∈ L∞ (RN ) with compact suppf and RN f (x)dx = 0, then f ∈ H1 . Proof. Assume suppf ⊂ BR (θ). Since

[ht (x − y) − ht (θ)]f (y)dy |(ht ∗ f )(x)| = N R 1 = (x − y) · ∇ht (s(x − y))dsf (y)dy RN 0

≤C t−(N +1) dyf ∞ , |x−y|≤t, |y|≤R

therefore f ∗ (x) ≤ CR|x|−N −1 f ∞ . The conclusion follows. 1

The dual space of H is the BMO space, which is deﬁned as follows: ∀f ∈ L1loc (RN ), ∀x ∈ RN , ∀r > 0, let

1 f r (x) = f (y)dy , |Br (x)| Br (x) and deﬁne

4.5 Other Function Spaces

269

Deﬁnition 4.5.13 (BMO space) f BMO = sup sup x∈RN r>0

1 |Br (x)|

|f (y) − f r (x)|dy . Br (x)

BMO(RN ) is the space of functions such that f BMO < ∞. The quantity f BMO is a semi-norm; in fact, f BMO = 0 if and only if f = constant. After modulo constants, BMO is a Banach space. There is an equivalent BMO semi-norm:

1 |f (y) − c|dy . f = sup sup inf x∈RN r>0 c |Br (x)| Br (x) In fact, on one hand it is obvious f ≤ f BMO , and on the other hand, fBr (x) |f (y) − f r (x)|dy ≤ fBr (x) |f (y) − c|dy + |f r (x) − c| ≤ 2f , where f denotes the average. The following simple properties are easily seen from the deﬁnition. 1. L∞ ⊂ BMO ⊂ L1loc . 2. The function log|x| ∈ BMO. In fact, the scaling transformation: ∀λ > 0, f (x) → f (λx) maps BMO functions to BMO functions, and preserves the semi-norm. Under the scaling, log|x| is changed by adding a constant. The veriﬁcation is then reduced to

|log|x||dx ≤ C, or |log|x| − log |x0 ||dx ≤ C , B

B

where B is unit ball centered at x0 , The ﬁrst inequality is true for |x0 | ≤ 1 and the second for |x0 | ≥ 1. These hold by elementary calculation. 3. If f ∈ BMO,then |f | ∈ BMO, and then f± ∈ BMO. 1 denote the average; one has Let fΩ = |Ω| Ω fBr (x) ||f (y)| − |c||dy ≤ fBr (x) |f (y) − c|dy , which implies that |f | ≤ f . In the case where (x, r) is not speciﬁed, we deﬁne Q = Br (x) and mQ (f ) = f r (x). 4. If f ∈ L1loc (RN ) and if there exists C > 0 such that mQ (f ) − essinfQ f ≤ C ∀ Q, then f ∈ BMO and f BMO ≤ 2C. Proof. Since Q [f (y) − mQ (f )]dy = 0, fQ |f (y) − mQ (f )|dy = 2fQ max[mQ (f ) − f (y), 0]dy ≤ 2fQ [mQ (f ) − essinfQ f ]dy ≤ 2C .

270

4 Minimization Methods

5. Let f ∈ L1loc (RN ) with f ≥ 0. If there exists a constant C > 0 such that mQ (f ) ≤ CessinfQ f, ∀ Q, then log f ∈ BMO. Proof. Taking logarithms and using Jessen’s inequality, we have mQ (log f ) = fQ log f dy ≤ log fQ f dy = log mQ (f ) ≤ log C + essinfQ log f .

The conclusion follows from property 4.

6. Assume f ∈ L1loc (RN ) and that M (f )(x) = ∞ a.e. ∀ δ ∈ (0, 1), set w = M (f )δ . Then there exists a constant C = Cδ > 0 independent of Q, such that mQ (w) ≤ CinfQ w, ∀ Q. Proof. Fixing Q, we may assume Q = BR (θ) without loss of generality. Decompose f = f1 + f2 , where f1 = χQ · f with Q = 2Q. Let wi = M (fi )δ , i = 1, 2. Since M (f ) ≤ M (f1 ) + M (f2 ), there is a constant c depending on δ, such that w ≤ c(w1 + w2 ). Now we apply Kolmogorov’s inequality to w1 : mQ (w1 ) = fQ M (f1 )δ (y)dy ≤

Cδ |Q|1−δ f1 δL1 |Q|

= 2N Cδ mQ (f )δ ≤ C1 infQ w ≤ C1 infQ w where C1 is independent of Q. Next we turn to estimating w2 . It remains to verify that there exists a constant C2 > 0 such that M (f2 )(x) ≤ C2 M (f2 )(y), ∀ x, y ∈ Q . Indeed, it implies that w2 (x) ≤ C2 infQ w2 ≤ C2 infQ w ∀x ∈ Q, then mQ (w2 ) ≤ C2 infQ w. Combining these two estimates together, we obtain mQ (w) ≤ c(mQ (w1 ) + mQ (w2 )) ≤ CinfQ w . Now we return to estimate M (f 2 ). Notice that for any r > 0, only when Br (x) ∩ (Q )c = Ø, i.e., r ≥ R, Br (x) |f2 (z)|dz can be nonzero. From fBr (x) |f2 (z)|dz ≤

|Br+R (y)| f |f2 (z)|dz ≤ 2N M (f2 )(y) , Br (x) Br+R (y)

it follows that M (f2 )(x) ≤ 2N M (f2 )(y).

Combining properties 5 and 6, we obtain: 7. If f ∈ L1loc (RN ) and M (f )(x) = ∞ a.e., then log M (f ) ∈ BMO and log M (f )BMO ≤ cN , a constant depending on N only.

4.5 Other Function Spaces

271

Deﬁnition 4.5.14 (VMO space) The space VMO(RN ) is the closure of C0∞ (RN ) in BMO. It is proved (see Stein [Ste 2]): that (VMO)∗ = H1 , (H1 )∗ = BMO, in the following sense, ∀f ∈ BMO, the following integral makes sense:

1 ∞ g=0 l(g) = f · gdx, ∀g ∈ Hα := g ∈ L | supp (g) is compact, and (see property 3 of H1 functions), and has a unique bounded extension to H1 : |l(g)| ≤ Cf BMO gH1 for some constant C > 0. Conversely, ∀l ∈ (H1 )∗ , ∃f ∈ BMO, such that l(g) = f · g ∀g ∈ H1α , with f BMO ≤ Cl(H1 )∗ . Since H1 is the dual of a separable Banach space VMO, the weak-* topology on H1 is well deﬁned, and then the Banach–Alaoglu theorem is applicable. Namely: If {fj } is a bounded sequence in H1 , then there is a subsequence {fj }, which converges in the distribution sense to f ∈ H1 , with f H1 ≤ lim inf j→∞ fj H1 . 4.5.3 Compensation Compactness If we intend to use the space H1 in variational problems, then the prices we have to pay are: (1) To verify the H1 boundedness of nonlinear quantities appearing in the integrand of the functional along a minimizing sequence, and (2) such nonlinear quantities are sequentially weakly ∗ l.s.c. There are few examples of typical nonlinear quantities, which are in the space L1 with some additional compensation conditions so that they fall into H1 and preserve the w∗ -continuity. Example 1. (div-curl) Suppose {En } ⊂ Lp (RN , RN ), divEn = 0 in the distri bution sense, and {Bn } ⊂ Lp (RN , RN ), curlBn = 0 in the distribution sense, where p ∈ (1, ∞), p1 + p1 = 1. Assume that both {En }, {Bn } are bounded in their own spaces respectively, then En · Bn is bounded in L1 (RN , R1 ). The compensation conditions divEn = 0, curlBn = 0 play a role in the weak*convergence. Example 2. (Jacobian) Suppose u ∈ Lqloc (RN , RN ), for some q ∈ (1, ∞), in which ∇u ∈ LN (RN , RN ×N ). Then det(∇u) ∈ L1 (RN , R1 ). The special structure of the Jacobian is also a kind of compensation condition. First we verify that the compensation condition implises the nonlinear quantities are indeed in H 1 , and then these nonlinear quantities are w∗ continuous along weakly convergent sequences.

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4 Minimization Methods

Theorem 4.5.15 Suppose that E ∈ Lp (RN , RN ), B ∈ Lp (RN , RN ), with p ∈ (1, ∞), p1 + p1 = 1. If curl B = 0 and divE = 0, then E · B ∈ H1 and ∃C > 0 a constant such that E · BH1 ≤ CEp Bp . Proof. Taking α, β satisfying α1 + β1 = 1 + nonnegative, with h = 0, we claim that

1 N,α

∈ (1, p), β ∈ (1, p ), ∀h ∈ C0∞ 1

1

|{ht ∗ (E · B)}(x)| ≤ C(fBt (x) |E|α ) α (fBt (x) |B|β ) β , 1 denotes the average. where fB = |B| B

Indeed, From curl B = 0, we have ω ∈ W 1,p (RN , R1 ) such that ∇ω = B. divE = 0 yields div(ωE) = ωdivE + E · ∇ω = E · B . By suitably choosing h such that supp(h) ⊂ B1 (θ), we have

1 x−y |ht ∗ (E · B)(x)| = ∇h E(y)ω(y)dy N +1 t t

1 x − y ∇h = E(y){ω − fBt (x) ω}dy N +1 t t α 1 α 1 1 α α ≤ C(fBt (x) |E| ) fBt (x) |ω − fBt (x) ω| t 1

1

≤ C(fBt (x) |E|α ) α (fBt (x) |∇ω|β ) β from the Sobolev–Poincar´e inequality. With the aid of the maximal function of f ∈ L1loc (Rn ):

1 |f (y)|dy . M (f )(x) = sup r>0 m(Br (x)) Br (x) The above inequality can be rewritten as 1

1

sup ht ∗ (E · B)(x) ≤ C(M (|E|α )(x)) α (M (|B|β )(x)) β . t>0

Again by Holder inequality and the Lp boundedness of the maximal function, we have

1 E · BH = sup |ht ∗ (E · B)(x)|dx t>0

≤C

α

p α

(M (|E| )) dx

p1

pβ 1 dx) p (M (|B| ) β

≤ CEp Bp .

4.5 Other Function Spaces

273

The second example can be reduced to the ﬁrst one, since ∂i u1 A1i = ∇u1 · σ , det(∇u) = i 1

2

where u = (u , u , . . . , u ), and σ = (A11 , A12 , . . . , A1N ) is the N -vector consisting of all N − 1 minors of det(∇u) with respect to the ﬁrst row. Deﬁne N E = σ and B = ∇u1 , then E ∈ L N −1 (RN , RN ), satisfying divE = 0, and B ∈ LN (RN , RN ), satisfying curlB = 0. According to Theorem 4.5.15, det(∇u) ∈ H1 and there exists a constant C > 0 such that det(∇u)H1 ≤ C∇uN LN . N

Apart from the w∗ compactness, compensation compactness theory identiﬁes classes of such nonlinear quantities, which are sequentially weakly ∗ continuous. Generally speaking, weak convergence may not preserve nonlinear quantities, e.g., if un u, vn v, then we cannot conclude that un vn uv. This can be seen from the example un = vn = sin(nπx) in L2 [0, 1]. Both un and vn 0, but un vn = 12 (1 − cos(2nπx)) 12 . However, luckily we have: Theorem 4.5.16 Suppose that {Ej }, {Bj } are two bounded sequences in L2 (RN ) satisfying div(Ej ) = 0 and curl(Bj ) = 0 in the distribution sense. If further, Ej E, Bj B in L2 (RN ), then Ej · Bj E · B in the distribution sense, and then Ej · Bj ∗ E · B in H1 . Proof. If B is curl free, we claim that there exists a distribution w such that ˆ ∇w = B. This can be shown by Fourier transformations. Let B(ξ) be the ˆk −ξk B ˆi = 0, ∀ i, k = Fourier transform of B, since curlB = 0 implies that ξi B N ˆ ξ B 1, 2, . . . , N . Let w be the Fourier inverse transform of 1|ξ|2k k . Obviously, ∇w = B. Moreover, if B has a bounded support: suppB ⊂ BR (θ)◦ , then we have w ∈ L2 and suppw ⊂ BR (θ)◦ . Now let wj be the distribution such that Bj = ∇wj . For a given φ ∈ C0∞ (RN ), we want to prove that

Ej · Bj φdx → E · Bφdx as j → ∞ . RN

RN

In this case, we may assume all the supports supp wj ⊂ BR (θ)◦ j = 1, 2, . . . , for some R > 0. Thus {wj } ⊂ H 1 (BR (θ)) and is bounded. Modulo a subsequence, we may assume wj → w in L2 (BR (θ)). Thus Ej · Bj , φ = div(wj Ej ) , φ

= − Ej , wj (∇φ)

→ − E , w(∇φ)

= div(wE) , φ = E · B , φ

where in taking limits, we used Ej E and wj → w in L2 (BR (θ)). Since C0∞ is dense in VMO space, Ej · Bj ∗ E · B in H1 .

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4 Minimization Methods

Remark 4.5.17 The above theorem has the following generalization: Let Ω ⊂ RN be an open domain. Assume that {Ej }, {Bj } are two bounded sequences in L2 (Ω, RN ) such that 1. {div Ej } lies in a compact subset of W −1,2 (Ω), 2. {curl Bj } lies in a compact subset of W −1,2 (Ω, MN ×N ). If further, Ej E and Bj B in L2 (Ω, RN ), then Ej · Bj E · B in the distribution sense. Similarly, one has: Theorem 4.5.18 Assume that {uj } is bounded in W 1,N (RN , RN ), and uj u. Then det(∇uj ) det(∇u) in the distribution sense, so is div(∇uj ) ∗ div(∇u) in H1 (Rn ). Proof. Again we use the previous special decomposition: ∂k u1 A1k = ∇u1 · σ = ∇(u1 · σ) , det(∇u) = k

where σ = (A11 , A12 , . . . , A1N ). Since divσ = 0, for ∀φ ∈ C0∞ (RN ), one has det(∇uj ) , φ = − u1j · σj , ∇φ → − u1 · σ , ∇φ = det(∇u) , φ . One can deﬁne a Hardy space on a domain Ω in RN . A distribution f on Ω is said to be in H1loc (Ω) if for each compact set K ⊂ Ω, there is an > 0 so that

( sup |ht ∗ f (x)|)dx < ∞ , K 0 0, we deﬁne gR (y) = y if 0 ≤ y ≤ R, = 2R − y if R ≤ y ≤ 2R, = 0 if y > 2R or y < 0. Obviously, gR ∈ C0 (R1 ), and |yχ[0,R] (y)| ≤ gR (y) ≤ |y|. According to Remark 4.4.10, we have

uj dx ≤ lim sup gR (y)dνj (x, y) lim sup {uj ≤R}

j→∞

Ω×R1

j→∞

= Ω

R1

Ω

R1

gR (y)dνx (y)dx

≤

|y|dνx (y)dx < ∞ .

We choose sequences nj ≥ 2j , kj ≥ j such that

lim sup vj dx ≤ |y|dνx (y)dx, j→∞

{vj ≤nj }

Ω

where vj = ukj . ∀k ∈ N , let Ek =

+

j≥k {vj

(4.51)

R1

> nj }. We shall verify:

1. |Ek | → 0 as k → ∞. 2. For ∀k ∈ N , {vj χFk } is uniformly absolutely continuous in j, where Fk = Ω \ Ek . By deﬁnition we have the estimates: |Ek | ≤

|{vj > nj }| ≤ sup vj L1 j

j≥k

Thus (1) is veriﬁed. We claim that

1 1 ≤ k−1 sup vj L1 . nj 2 j j≥k

lim sup lim sup R→∞

j→∞

Fk ∩{vj >R}

vj dx = 0 .

(4.52)

If it is proved, then for ∀ > 0, let R0 > 0, j0 > 0 be such that Fk ∩{vj >R0 } vj < N measurable set U with |U | < 2R 0 , we have 2 as j > j0 . For any L

U ∩Fk

vj dx ≤

Fk ∩{vj >R0 }

vj dx +

U ∩Fk ∩{vj ≤R0 }

vj dx ≤

+ R0 |U | < , 2

for ∀j ≥ j0 , i.e., vj is uniformly absolutely continuous on Fk ∀k.

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4 Minimization Methods

Now we return to proving (4.52). Since for ∀ Borel sets E ∈ B(Ω), ∀φ ∈ C0 (R1 ),

φ(vj (x))dx = φ(y)dνx (y)dx , lim j→∞

E∩{vj ≤nj }

E

R1

Choosing φj ↑ |y|, it follows that

vj (x)dx ≥ lim inf j→∞

E∩{vj ≤nj }

E

R1

|y|dνx (y)dx ,

(4.53)

because for j ≥ k, Fk ⊂ {vj ≤ nj }, Fk ∪ ({vj ≤ nj } ∩ Ek ) = {vj ≤ nj }. Combining (4.51) and (4.53) with E = Ek , we have

lim sup vj (x)dx ≤ |y|dνx (y)dx . (4.54) j→∞

Fk

Fk

R1

Thus,

lim sup lim sup R→∞

Fk ∩{vj >R}

j→∞

−

= lim sup lim sup R→∞

vj (x)dx

j→∞

Fk

Fk ∩{vj ≤R}

vj (x)dx

|y|dνx (y)dx − ≤ lim sup lim sup j→∞ R→∞ Fk R1 Fk

= lim sup (|y| − gR (y))dνx (y)dx = 0 . R→∞

Fk

gR (vj (x) dx

R1

Deﬁnition 4.5.20 Let {uj } ⊂ L1 (Ω) be bounded; we say that uj converges to u ∈ L1 (Ω) in the sense of the bitting lemma, if for all > 0 there exists a measurable subset E ⊂ Ω such that |E| < and uj u in L1 (Ω\E). What is the relationship between H1 −w∗ convergence and the convergence in the sense of the bitting lemma? Lemma 4.5.21 Let {uj } ∗ u in H1 (RN ). For each R > 0, let v ∈ L1 (BR (θ)) be such that there exists a subsequence nj for which unj converges to v in the sense of the bitting lemma in BR (θ), then u = v a.e. on BR (θ). Proof. We want to show that for ∀φ ∈ C0∞ (RN ), uφdx = vφdx. Assume suppφ ⊂ BR (θ) for suitable R > 0, vj = unj ∀j. By the assumption, for all > 0 there exists E such that |E| < and vj v on L1 (Ω\E). For all λ > 0 deﬁne wλ = (1 + λ log M (χE ))+ , where M (f ) is the maximal function of f . Provided by the propositions (3) and (7) of BMO, wλ ∈ BMO, and log M (χE ) ≤ cN , and then wλ BMO ≤ Cλ, where C is a constant independent of λ and .

4.5 Other Function Spaces

277

Since χE ≤ M (χE ) ≤ 1 a.e., it follows χE ≤ wλ ≤ 1 a.e. According to the proposition (4) of the maximal function, we have 1 1 1 1 |{wλ > 0}| = M (χE ) > exp − ≤ C exp λ χE L ≤ Cε exp λ , λ and then

unj φdx = RN

vφdx

RN

+ RN

unj φwλ dx +

RN

(unj − v)φ(1 − wλ )dx −

RN

vφwλ dx

= I1 + I2 + I3 + I4 , where

I1 =

vφdx, RN

I2 ≤ unj H1 wλ φBMO ≤ Cwλ φBMO ,

(unj − v)(1 − wλ )φdx → 0, as j → ∞, I3 = Ω\E

I4 ≤ |v||φ|dx ≤ φL∞ |v| . {wλ >0}

BR (θ)∩{wλ >0}

The limit of I3 is due to the assumption that uj converges weakly to v in L1 (Ω\E). We turn to estimate I2 . ∀r > 0, ∀x ∈ RN , let Q = Br (x), we have

1 fQ dz |φ(z) − φ(y)|dy fQ |φwλ − fQ φwλ |dz ≤ φL∞ wλ BMO + |Q| {wλ >0} C(λ + R|{wλ > 0}|) if r < R , ≤ C(λ + R1N |{wλ > 0}|) if r ≥ R . ∀η > 0 ﬁxing λ > 0 such that Cλ < η2 at ﬁrst, and then we choose > 0 so small such that the summation of I4 and C(R + R−N )|{wλ > 0}| < η2 , we obtain the desired conclusion. Combining Lemmas 4.5.19 and 4.5.21, we have: Theorem 4.5.22 Assume that uj ∗ u in H1 (RN ). Then for all R > 0 there exists a subsequence {unj } converging to u on BR (θ) in the sense of the bitting lemma. As an application to the calculus of variations, we study the polyconvex functionals in Sect. 4.3.3. Recall that a function f : Ω × M n×n → R1 is called polyconvex, if f (x, A) = g(x, T (A)), where g : Ω×Rτn → R1 is a Caratheodory

278

4 Minimization Methods

function, for a.e. x ∈ Ω, ξ → g(x, ξ) is convex, T = {adjs }n1 : M n×n → Rτn , and 2 n n! . τn = s!(n − s)! 1 Theorem 4.5.23 Let Ω be an open bounded domain in Rn ; f (x, A) = g(x, T (A)) is polyconvex, satisfying C|A|n ≤ g(x, T (A)) ∀A ∈ M n×n , C > 0 . Then the functional

f (x, ∇u)dx

J(u) = Ω

has a minimizer in W01,n (Ω, Rn ). Proof. Let c = inf u∈W 1,n J(u), and {uj } be a minimizing sequence, i.e., 0 J(uj ) → c. By the coercive condition, ∇uj Ln (Ω,M n×n ) is bounded, and then uj W 1,n (Ω,Rn ) is bounded. Modulo a subsequence, we have uj n

n!

2

u in W 1,n (Ω, Rn ), adjs (∇uj ) adjs (∇u), in L s (Ω, R( s!(n−s)! ) ), 1 ≤ s < n. Since we can extend uj to be zero outside Ω, provided by Theorem 4.5.18, we have det(∇uj ) ∗ det(∇u) in H1 (Rn ). For ∀ > 0, we ﬁnd δ > 0 such that if |E| < δ then E g(x, T (u(x)))dx < . According to Theorem 4.5.22, there exists a measurable set E ⊂ Ω such that |E| < δ and det(∇(uj )) det(∇u) in L1 (Ω\E). Obviously, we also 2 n! have adjs (∇uj ) adjs (∇u), 1 ≤ s < n in L1 (Ω\E, R( s!(n−s)! ) ). Since a.e., x ∈ Ω, ξ → g(x, ξ) is convex, we have

g(x, T (∇uj (x)))dx ≥ lim inf g(x, T (∇uj (x)))dx lim inf j→∞ j→∞ Ω Ω\E

≥ g(x, T (∇u(x)))dx Ω\E

≥ g(x, T (∇u(x)))dx − . Ω

Since > 0 is arbitrary, we obtain

g(x, T (∇uj (x)))dx ≥ g(x, T (∇u(x)))dx . lim inf j→∞

Ω

Thus u ∈ W01,n (Ω, Rn ) is a minimizer.

Ω

Comparing this result with Theorem 4.3.12, the growth condition for polyconvex functions is dropped. It is useful in the elasticity theory. Another advantage in using Hardy space is on regularity. As we mentioned above, the Calderon–Zygmund theory can be applied to the Hardy space. Once some nonlinear expression of the minimizer is proved in H1 , the Euler

4.6 Free Discontinuous Problems

279

Lagrange equation is applied to obtain an a priori estimate, and gain better regularity of the solution, cf. Evans [Ev 2], Helein [Hel 1]. The compensation compactness method has been successfully applied to elasticity and hyperbolic systems by Ball [Bal 1], Di Perna [Di], Tartar [Tar 1] etc.

4.6 Free Discontinuous Problems 4.6.1 Γ-convergence Γ-convergence has been studied extensively by De Giorgi’s school. The idea is to use a family of functionals J , depending on a parameter and approaching the given functional J in a variational problem, such that the minimizers v of J have a limit point v which is a minimizer of J. Deﬁnition 4.6.1 Let X be a metric space and let Fn : X → [0, ∞] be a sequence of functionals. We say that Fn Γ-converges to F on X as n → ∞, written as Γ − limn→∞ Fn = F , if (1) ∀u ∈ X, ∀{un } such that un → u in X one has lim inf n→∞ Fn (un ) ≥ F (u), (2) ∀u ∈ X, ∃{un } such that un → u in X and lim supn→∞ Fn (un ) ≤ F (u). The importance of the notion lies in the following theorems. Theorem 4.6.2 If F = Γ − limn→∞ Fn , then F is l.s.c. Proof. We prove by contradiction. If F is not l.s.c., then ∃xn → x, such that F (x) > limn→∞ F (xn ). However, by deﬁnition, ∀n, ∃{xnm } ⊂ X, xnm → xn , satisfying Fn (xnm ) → F (xn ). We may assume that limn→∞ F (xn ) > −∞ and F (x) < ∞. Let δ = 14 (F (x) − limn→∞ F (xn )) > 0, then, ∀n ∈ N , ∃mn ∈ N such that |Fn (xnmn ) − F (xn )| < δ, mn → ∞, and xnmn → x .

(4.55)

On one hand, by Γ-convergence, F (x) ≤ lim inf Fn (xnmn ) . n→∞

On the other hand, by the deﬁnition of δ, Fn (xnmn ) > F (x) − δ for n large, we have (4.56) F (xn ) < F (x) − 3δ . It follows that Fn (xnmn ) − F (xn ) > 2δ, which contradicts (4.55).

Theorem 4.6.3 Suppose that Γ − limn→∞ Fn = F and that vn minimizes Fn over X ∀ n ∈ N . If v is a limiting point of vn , then v is a minimizer of F and F (v) = lim inf n→∞ Fn (vn ).

280

4 Minimization Methods

Proof. By (2) of the deﬁnition of Γ-convergence, for every u ∈ X, there exists {un } ⊂ X such that un → u, and lim Fn (un ) = F (u) .

n→∞

Since vn is a minimizer of Fn , we have Fn (vn ) ≤ Fn (un ). Therefore (1) of the deﬁnition of Γ-convergence yields F (u) ≥ lim sup Fn (un ) ≥ lim inf Fn (vn ) ≥ F (v) , n→∞

n→∞

(4.57)

as vn → v in X. This proves the theorem.

We note that Γ-convergence is diﬀerent from pointwise convergence. Example 4.6.4

then

⎧ 1 ⎪ ⎨1 if x ≥ n , Fn (x) = nx if x ∈ [− n1 , n1 ] , ⎪ ⎩ −1 if x ≤ − n1 1, if x > 0 , Γ − lim Fn (x) = n→∞ −1 if x ≤ 0 .

But pointwisely, Fn (x) → sgn(x) . The diﬀerence is at x = 0. Example 4.6.5 Γ − limn→∞ sin nx = −1. 4.6.2 A Phase Transition Problem We present here an example due to Cahn and Hilliard on phase transitions showing how Γ-convergence is applied to variational problems. In a container Ω ⊂ R3 ﬁlled with two immiscible and incompressible ﬂuids, the two ﬂuids arrange themselves in order to minimize the area of the interface which separates these two phases. Let u be a function, which takes the value 0 on the set occupied by the ﬁrst ﬂuid, and the value 1 on the set occupied by the second. Let V = u be the total volume of the second ﬂuid. Obviously one has 0 < V < m(Ω). Let S(u) be the interface, i.e., the singular set of u. The equilibrium conﬁguration is obtained by minimizing the energy F (u) := σH2 (S(u)), where σ is the surface tension between the two ﬂuids, and H2 is the two dimensional Hausdorﬀ measure. On the macroscopic level, we allow for a mixture of two ﬂuids; let u : Ω → [0, 1] be the average density of the second ﬂuid, and let W be a double-well potential, i.e., a continuous positive function that vanishes only at 0 and 1.

4.6 Free Discontinuous Problems

281

Cahn and Hilliard established a model: ∀ > 0, we consider a minimizer u of the following energy functional:

1 2 |∇u| + W (u), with V = u. (4.58) E (u) = Ω Ω Ω Under the volume constraint u cannot be a constant; but as → 0, for a.e. x ∈ Ω, u (x) must tend to either 0 or 1. Does u tend to a minimizer of F ? The following Modica–Mortola theorem gives an answer to this question. Before going to the proof, we need a few lemmas. Lemma 4.6.6 Suppose that um → u in a metric space (X, d) satisfying lim supm→∞ F (um ) ≤ F (u), and that ∀m, ∃um,n → um as n → ∞ such that lim supn→∞ Fn (um,n ) ≤ F (um ). Then ∃mn → ∞, such that for vn = umn n we have vn → u and lim supn→∞ Fn (vn ) ≤ F (u). 1 Proof. We may assume d(u, um ) < m . Deﬁne 1 1 mn = max m |, d(umn , u) < , Fm (umn ) ≤ F (um ) + . m m

It is easy to verify that mn → ∞. We set vn = umn n , then vn satisﬁes the requirements. The following lemma is just the one-dimensional version of the general theorem. Lemma 4.6.7

2 1 1 [|γ (t)| + W (γ(t))]dt | γ ∈ C (R ), γ(−∞) = 0, γ(∞) = 1 min R1 1

=2

W (t)dt .

0

Proof. We write the functional to be minimized as J(γ), and the right-hand integral as σ. According to the Cauchy–Schwarz inequality,

1

W (γ(t))γ (t)dt = 2 W (s)ds = σ . J(γ) ≥ 2 R1

0

√ The equality holds if and only if γ = W (γ). Since W is continuous and positive between the two zeroes 0 and 1, the ﬁrst-order equation has a global solution γ on R1 , with γ(−∞) = 0, γ(∞) = 1.

Let U ⊂ Rn be an open set with nonempty boundary ∂U . One deﬁnes the signed distance function as follows: dist(x, ∂U ), if x ∈ U, d(x) = −dist(x, ∂U ), if x ∈ / U. Thus, d is Lipschitzian continuous with Lipschitz constant 1.

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4 Minimization Methods

Lemma 4.6.8 Assume that ∂U ∈ C 2 is bounded. ∀t ∈ R1 we set Σt = {x ∈ Rn | d(x) = t}. Then ∃ > 0 such that for |s| < , Σs is a hypersurface of C 2 class and lims→0 Hn−1 (Σs ) = Hn−1 (∂U ). Moreover, |∇d(x)| = 1 . Proof. ∀x0 ∈ ∂U , let n(x0 ) be the unit outer normal vector of ∂U at x0 , and let Tx0 (∂U ) be the tangent hyperplane at x0 . Rotate the coordinates such that the ﬁrst n−1 coordinate axes lie on Tx0 (∂U ) and the xn coordinate axis points in the direction −n(x0 ). In a neighborhood V (x0 ) of x0 , ∂U can be represented by xn = f (x ) where x = (x1 , . . . , xn−1 ) with f (x0 ) = 0, ∇f (x0 ) = θ. There exists a tubular neighborhood Ξ of ∂U , such that ∀x ∈ Ξ, there is a unique representation ξ = π(x) ∈ ∂U such that x = π(x) − n(x)d(x), where n(x) = n(π(x)), and that ξ is the unique point satisfying |x − ξ| = d(x) is a neighborhood of x. There exists > 0 such that Σs ⊂ Ξ, ∀|s| < . We claim that: |∇d(x)| = 1, and ∇d(x) = −n(ξ) . Thus ∀x ∈ Σs , s = d(x), we have a local representation: x = F (x , s) := (x , f (x )) − n(x , f (x ))s. By the use of the implicit function theorem, one can verify that F is invertible as > 0 is small. This implies that d(x) = d(ξ − sn(ξ)) = s. Therefore, ∇d(x)n(ξ) = −1. Since d has Lipschitz constant 1, the claim is proved. Using the Gauss–Green formula, we have

div(∇d(x)) = ∇d(x)n(x)dHn−1 + ∇d(x)n(x)dHn−1 . 0 0, let E (u) = Ω |∇u|2 + 1 Ω W (u) if u ∈ W 1,2 ∩ X , F (u) := (4.59) +∞, otherwise , and

σDχE (Ω) if u = χE , F (u) = +∞, otherwise ,

(4.60)

1 where σ = 2 0 W (t)dt, and E is a set of ﬁnite perimetrics. Then Γ − lim F = F in X, and every sequence of minimizers of F is subconvergent in X to a minimizer of F.

4.6 Free Discontinuous Problems

283

Proof. 1. We claim that ∀u ∈ X and for every un → u in X with n → 0, if ∃M > 0 such that

1 2 W (un ) ≤ M , n |∇un | + n then W (u) ≤ lim inf n→∞ W (un ) = 0, provided by Fatou’s lemma. From the double well assumption on W u = χE for some Borel set E. 2. We begin with the veriﬁcation of (1) of the deﬁnition of Γ-convergence. We may assume without loss of generality that lim inf →0 F (u ) is bounded. According to step 1, u = χE for some Borel set E. From the Cauchy–Schwarz inequality and the l.s.c. of the total variations, we have

1 2 |∇u | + W (u ) dx lim inf F (u ) = lim inf →0 →0 Ω

≥ lim inf 2 W (u )|∇u |dx →0

Ω |∇(Ψ(u ))|dx = lim inf →0

Ω

= lim inf DΨ(u )(Ω) →0

≥ DΨ(u)(Ω) , √ where Ψ is the primitive of 2 W satisfying Ψ(0) = 0. One has Ψ(u) = Ψ(χE ) = Ψ(1)χE = σχE . Thus, DΨ(u) = F (u), and then (1) is veriﬁed. 3. We now verify (2) of the deﬁnition of Γ-convergence. By the deﬁnition of F , it is suﬃcient to verify (2) for u = χE . Suitably modifying the approximation theorem of BV functions (Theorem 4.5.7), we ﬁnd approximate functions of the form: j χΩi , with uj − uL1 → 0 , uj = i

Ωji , i

= 1, 2, . . . , j = 1, 2, . . . , are disjoint bounded open sets with where smooth boundary ∂Ωji . On account of Lemma 4.6.6, the veriﬁcation is reduced to verifying (2) for characteristic functions of open sets with bounded smooth boundaries. Let u = χΩ , let d be the signed distance function of ∂Ω, and let γ be a minimizer obtained in Lemma 4.6.7. ∀ > 0, we deﬁne u (x) = γ( d(x) ). From Lemma 4.6.8 and the co-area formula, we have

2 d 1 γ +W γ d F (u ) = dx Ω

1 g(t)[γ (t/)2 + W (γ(t/))]dt = R1

g(t)[γ (t)2 + W (γ(t))]dt . = R1

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4 Minimization Methods

where g(t) = Hn−1 (Σt ). Lemma 4.6.8 yields g(t) → Hn−1 (∂Ω) as → 0. Thus

[|γ (t)|2 + W (γ(t))]dt = F (u) . lim sup F (u ) ≤ DχE (Ω) R1

4. Finally we verify that if u is a minimizer of F , then {u } is precompact. In fact, one may assume that F (u ) is bounded, and then by step 1, so is DΨ(u )(Ω). Thus |Ψ(u )BV = Ψ(u )L1 + DΨ(u )(Ω) is bounded. According to the compactness property of BV functions, it is precompact in L1 (Ω). Since Ψ admits a continuous inverse, u is precompact in L1 . 4.6.3 Segmentation and Mumford–Shah Problem In image segmentation, we want to detect the edge of an image from a picture. Given a bounded domain Ω ⊂ R2 and an image represented by a function g ∈ L2 (Ω), ﬁnd a closed set K ⊂ Ω with ﬁnite one-dimensional Hausdorﬀ measure H1 (K), and a function u ∈ H 1 (Ω\K), which minimize the following cost functional:

|∇u|2 dx + µ |u − g|2 dx + λH1 (K) , E(K, u) = Ω\K

Ω\K

where λ, µ > 0 are parameters. This is the Mumford–Shah problem. The original image g is thereby estimated by u which is smooth in Ω except for the discontinuous set K (in fact, in H 1 (Ω\K)), and the latter is the estimated edge. This is a variational problem with unknown (K, u). Obviously, the diﬃculty lies in K, which is an object we have never met. Tentatively, suppose K is given. Since E is quadratic in u, so is convex and coercive, we obtain a minimizer u(K) with mΩ\K := E(K, u(K)). The problem is then reduced to minimizing the functional: J(K) = mΩ\K + λH1 (K) .

(4.61)

Since the discontinuous set K is only assumed to be Hausdorﬀ measurable, the problem is more complicated and requires further and deeper prerequisites. For pedagogical purposes, we follow Nordstr¨ om [No] and restrict ourselves to assuming that only the union of ﬁnitely many C 1 -curves γ is the admissible candidate for K. Given ξ ∈ C 1 ([0, 1], R2 ) the image of ξ is a C 1 -curve (may be self intersection). Thus the union of N − C 1 -curves γ is deﬁned by ξ = (ξ1 , . . . , ξN ) ∈ C 1 ([0, 1], R2 )N , with N 5 γ = ξ[0, 1] := ξn [0, 1] . n=1

4.6 Free Discontinuous Problems

285

Let us recall the notion of a domain with minimally smooth boundary (cf. Stein [Ste 1]). An open set O ⊂ Rn is said to have a minimally smooth boundary if ∃ε > 0, ∃ an integer N , a positive M > 0, and a sequence of open sets U1 , U2 , . . . , such that (i) If x ∈ ∂O, then Bε (x) ⊂ Ui for some i, (ii) No point of Rn is contained in more than N of Ui ’s, (iii) ∀i, ∃ a special Lipschitz domain Di whose bound does not exceed M so that Ui ∩ O = Ui ∩ Di . For example, if ∂O ∈ C 1 , then ∂O is minimally smooth. The introduction of this notion is due to the following extension theorem: Theorem 4.6.10 Let O ⊂ Rn be a domain with minimally smooth boundary. Then ∃ a linear operator T mapping functions on O to functions on Rn such that (1) T u|O = u, (2) T u W l,p (Rn ) Cl,p u W l,p (O) for some constant Cl,p , ∀l ∈ N , p ∈ [1, ∞]. Deﬁnition 4.6.11 An N − C 1 -curve γ is said to be an admissible image segmentation of Ω if ∀ connected components G of Ω\γ, ∀ε > 0, ∃ a domain with minimally smooth boundary Oε such that (1) Oε ⊂ G, (2) m(G − Oε ) < ε. We denote by EN the set of all N − C 1 -curve admissible image segmentations of Ω endowed with the C 1 ([0, 1], R2 )N topology. Thus for ∀γ ∈ EN ,

1

˙ |ξ(t)|dt :=

H1 (γ) = 0

N

n=1

1

|ξ˙n (t)|dt,

where γ = ξ[0, 1] =

0

N 5

ξn [0, 1] ,

n=1

i.e., the length of the piecewise C 1 -curve γ. Now, let us make some preparations. To simplify the notations, we assume λ = µ = 1. For any bounded open domain G, we deﬁne

2 |∇u| + |u − g|2 on H 1 (G) . IG (u) = G

G

It is known that IG has an unique minimizer uG . Let mG = IG (uG ) =

min

u∈H 1 (G)

IG .

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4 Minimization Methods

By simple calculations,

|g|2 − g · uG .

mG = G

Moreover, we have the following properties: (uG ) = 0, i.e., (1) IG

∇uG ∇ϕ + (uG − g)ϕ = 0 ∀ϕ ∈ H 1 (G) .

(2)

G

G

|∇uG | + |uG | 2

2

G

|g| . 2

G

Lemma 4.6.12 Let G be a connected component of Ω\γ where γ is an admissible image segmentation of Ω. Then for ∀ε > 0, ∃ a domain with minimally smooth boundary O such that |mO − mG | < ε .

Proof. By the deﬁnition of the admissibility for γ, ∀ > 0, ∃ a domain with minimally smooth boundary O ⊂ O ⊂ G such that m(G\O) < . Letting uO be the minimizer of IO , we have

(|g|2 − guG ) − (|g|2 − guO ) mG − mO = O

G 2 (|g| − guG ) + g(uO − uG ) . = O

G\O

Noticing that

∇uG ∇! uO + uG u !O G\O G\O

= g! uO − guG O

G

= g! uO + g(uO − uG ) ,

G\O

where u !O = T uO , we have

g(uO − uG ) = O

G\O

O

(∇uG ∇! uO + uG u !O ) −

g! uO . G\O

Applying the Cauchy–Schwarz inequality, the ﬁrst integral uO H 1 . By (2) and Theorem 4.6.10 ∃C0 > 0 such that uG H 1 (G\O) ! ! uO H 1 ≤ C0 uO H 1 (O) ≤ C0 gL2 (O) ≤ C0 gL2 (G) .

≤

4.6 Free Discontinuous Problems

Therefore ∃C > 0 a constant such that

287

12

|mG − mO | C g L2 (G)

|g| + 2

uG 2H 1 (G\O)

.

G\O

Since m(G\O) can be suﬃciently small, the conclusion follows.

Lemma 4.6.13 The function J restricted on EN is l.s.c. Proof. The second term H1 (γ) of J is trivially continuous in C 1 -topology, so is l.s.c. It is suﬃcient to verify the l.s.c. of mΩ\γ with respect to γ, i.e., for ∀γ0 ∈ EN , ∀ε > 0, ∃δ > 0 such that dist (γ, γ0 ) < δ implies that mΩ\γ > mΩ\γ0 − ε. Let E be the collection of all connected components of Ω\γ0 , then E is countable, say E = {Gi }∞ 1 . From ∞

i=1

|g|2 =

|g|2 g 2L2 (Ω) , Ω\γ

Gi

there exists K > 0 such that

∞ i=K+1

|g|2

0 be such that dist (Oi , ∂Gi ) δ > 0; we choose a 2δ neighborhood K + Oi ⊂ Ω\γ as dist (γ, γ0 ) < 2δ , and we have of γ0 , then i=1

mΩ\γ = IΩ\γ (uΩ\γ )

K

IOi (uΩ\γ )

i=1

K

mOi

i=1

−ε + mG i . 2 i=1 K

+K Let u(x) = uGi as x ∈ Gi , i = 1, 2, . . . , K and u(x) = 0 as x ∈ E\ i=1 Gi , +K +∞ then Ω\(γ0 ∪ i=1 Gi ) = i=K+1 Gi , we obtain from inequality (4.62) mΩ\γ0 ≤ IΩ\γ0 (u) =

K i=1

IGi (uGi ) + IΩ\(γ0 ∪+ k

i=1

Gi ) (0)

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4 Minimization Methods

≤

K

mG i +

i=1

ε . 2

This proves that mΩ\γ > mΩ\γ0 − ε . Now for ∀N ∈ N , ∀α ∈ (0, 1), ∀ρ > 0, ∀ω > 0, let us deﬁne CN (α, ρ, ω) =

γ = ξ[0, 1] ∈ EN | ξ C 1 +

1

˙ − ξ(s)| ˙ |ξ(t) |t − s|α s,t∈[0,1],s=t sup

˙ dt ≥ ω ξ(t)

≤ ρ, 0

endowed with the C 1 [0, 1]N topology, and let C = {Ø} ∪

∞ 5

CN (α, ρ, ω) .

N =1

We arrive at: Theorem 4.6.14 ∃γ0 ∈ C, ∃u0 ∈ H 1 (Ω\γ0 ), such that E(γ0 , u0 ) = Min(γ,u)∈(C×H 1 (Ω\γ)) E(γ, u) .

Proof. The lower bound ω > 0 forces us to minimize E on CN (α, ρ, ω) for some ﬁnite N , and the latter is a closed compact set in C 1 [0, 1]N . Our conclusion follows from the general principle of calculus of variations. The simpliﬁed model of Mumford–Shah problem is restricted to the space of possible discontinuous sets consisting of those corresponding to line drawings. The Mumford–Shah functional has been studied by De Giorgi and Ambrosio [DG 2], [Amb], [DA]. In the variational problem, the working space is an SBV space, a subspace of BV (Ω), which consists of those BV functions in which the Cantor part of their derivatives vanishes. The distributional derivative of a BV function u can be decomposed into three parts: Du = Da u + Dj u + Dc u, where Da , Dj , and Dc are the absolutely continuous part, jump part, and the Cantor part, resp. The advantage of the subspace SBV (Ω) is in the characterization of its w∗ -compactness. The reader is referred to Ambrosio [Amb]. Mumford and Shah conjectured that if (K, u) is an optimal essential pair of J, then K is locally in Ω the union of ﬁnitely many C 1,1 embedded curves, see Morel [Mor].

4.7 Concentration Compactness

289

4.7 Concentration Compactness The loss of compactness breaks down the standard variational techniques. However, most problems arising in geometry (prescribing scalar curvature problem, Yamabe problem, minimal surfaces, ﬁxed points and intersections in symplectic geometry etc.) and in physics (N -body problem, Yang–Mills equation, nonlinear Schr¨ odinger equations, gravity theory etc.) are in this realm. Semilinear elliptic PDEs in Rn and those with the critical Sobolev exponent are two typical examples on this topic. They have been studied extensively in the last two decades. If Ω ⊂ Rn is a bounded domain, then a bounded sequence {uj } in H 1 (Ω) has a weakly convergent subsequence, and then a strongly convergent subsequence in Lq (Ω) as q ∈ [2, 2∗ ), according to the compactness of the embedding H 1 (Ω) → Lq (Ω). This argument is frequently used. (See Example 3 in Sect. 4.2.) However, when Ω is unbounded, the argument does not work. This can be seen by the following example: Let Ω = Rn , {xj } be a sequence that tends to inﬁnity. Given a nonzero u0 ∈ C0∞ , let uj (x) = u0 (x − xj ). Then ||uj ||H 1 = ||u0 ||H 1 and |uj |q = |u0 |q . Therefore, uj (x) → 0 a.e., and then uj 0 in H 1 (Rn ), but uj cannot Lq converge to 0. The reason is that the associated measures

|uj |2 ∀ Lebesgue measurable set E ⊂ Rn (4.63) µj (E) = E

leak out at inﬁnity. 4.7.1 Concentration Function Relating to a measure µ, P. Levy introduced the concentration function Q(r) = sup µ(Br (x)), r ≥ 0 . x∈Rn

We have the following: Lemma 4.7.1 Suppose that {uj } ⊂ H 1 (Rn ) is a bounded sequence. If ∃ R > 0 such that limj→+∞ Qj (R) = 0, where Qj is the concentration functions with respect to µj deﬁned in equation (4.63), then there exists a subsequence {ujk } such that ujk → 0 in Lq (Rn ), ∀ q ∈ (2, 2∗ ). Proof. Due to the interpolation inequality, we have: θ ∗ ||u||Lq (BR (x)) ≤ ||u||1−θ L2 (BR (x)) ||u||L2 (BR (x))

∀ x ∈ Rn , where θ =

(q−2)n 2q .

4 In the case where θq ≥ 2, we have q ≥ 2 + and a constant C > 0 such n that

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4 Minimization Methods

(1−θ)q

2 |u|q ≤ C||u||L2 (BR (x)) ||u||θq−2 H 1 (BR (x)) uH 1 (BR (x)) BR (x)

(1−θ)q θq−2 ≤ C( sup ||u||L2 (BR (x)) )||u||H 1 (Rn ) (|u|2 + |∇u|2 ) x∈Rn

BR (x)

from the Sobolev embedding theorem. We ﬁnd a locally ﬁnite covering, i.e., ∃ {xk } and ∃ ∈ N such that ∀ x ∈ Rn , ∃ at most balls of BR (xk ) to which x belongs. Then

|u|q ≤ |u|q Rn

BR (xk )

k

≤ C||u||θq H 1 (Rn )

(1−θ)q

sup ||u||L2 (BR (x))

.

x∈Rn

By the assumptions that uj H 1 is bounded and modulo a subsequence supx∈Rn BR (x) |uj |2 → 0, it follows that Rn |uj |q → 0. 4 In the case where θq < 2, we have 2 < q < 2 + := q0 . Since θq0 = 2, n ||ujk ||Lq0 (Rn ) → 0. Letting 1q = λ2 + (1−λ) q0 , we obtain, by interpolation, that →0. ||ujk ||Lq (Rn ) ≤ |ujk |λ2 |ujk |1−λ q0 The lemma shows that the concentration function plays an important role in the compactness argument. It will be seen by the following: Corollary 4.7.2 (Strauss) Let Hr1 (Rn ) be the subspace of H 1 (Rn ) consisting of radial symmetric functions. The embedding Hr1 (Rn ) → Lp (Rn ), 2 < p < 2∗ , n ≥ 2 is compact. Proof. ∀x ∈ Rn , ∀R > 0, let m(x, R) be the largest number of disjoint balls with radius R and the centers lie on the same sphere with radius |x| centered at θ. It is easily seen that m(x, R) → ∞ as x → ∞. By deﬁnition ∀u ∈ L2 (Rn ), ∀r > 0,

|u|2 ≤ m(x, r)−1 u2L2 .

Br (x)

If {uj } is a bounded sequence in Hr1 (Rn ), then ∀ > 0, ∃R > 0 such that

|uj |2 | |x| ≥ R

sup

0, ∃ R > 0, ∃ {xj } ⊂ Rn such that µj (BR (xj )) ≥ 1−. 1 In fact, by deﬁnition, ∃ R0 > 0 s.t. Q(R0 ) > . Let xj ∈ Rn satisfying 2 1 Qj (R0 ) ≤ µj (BR0 (xj )) + . For ∈ (0, 12 ), we choose R1 > 0 such that j 1 Q(R1 ) > 1 − , and let yj satisfy Qj (R1 ) ≤ µj (BR1 (yj )) + . Then j µj (BR1 (yj )) + µj (BR0 (xj )) ≥ Qj (R1 ) + Qj (R0 ) −

2 > 1, for j large . j

Therefore BR1 (yj ) ∩ BR0 (xj ) = ∅, which implies BR1 (yj ) ⊂ B2R1 +R0 (xj ) and 1 1 µj (B2R1 +R0 (xj )) ≥ µj (BR1 (yj )) ≥ Qj (R1 ) − > 1 − − for j large. This j j is the conclusion. For λ ∈ (0, 1), ∀ > 0, ∃ R0 > 0, ∃ {xj } such that Qj (R0 ) ≥ µj (BR0 (xj )) ≥ λ − for j large. Also, we may ﬁnd a sequence Rj → ∞ such that Qj (R0 ) ≤ Qj (Rj ) < λ + for j large . Thus λ − < µj (BR0 (xj )) ≤ µj (BRj (xj )) ≤ λ + . Now for any given R > R0 , we may assume Rj > R for all j and let µ1j = µj |BR0 (xj ) and µ2j = µj |Rn \BRj (xj ) . Then 0 ≤ µ1j + µ2j ≤ µj , supp µ1j ⊂ BR0 (xj ), supp µ2j ⊂ Rn \ BRj (xj ) ⊂ Rn \ BR (xj ), and so |λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )| ≤ |λ − µj (BR0 (xj ))| + |λ − µj (BRj (xj )| < 2 . In summary, we have:

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4 Minimization Methods

Theorem 4.7.3 (Concentration compactness principle) Suppose that {µj } ⊂ M (Rn ) is a sequence of probability measures. Then one of the following three conclusions holds: 1. (Compactness) ∃ {xj } ⊂ Rn such that ∀ > 0, ∃ R > 0 with µj (BR (xj )) ≥ 1 − for all j. 2. (Vanishing) ∀ R > 0, limj→∞ (supx∈Rn µj (BR (x))) = 0. 3. (Dichotomy) ∃ λ ∈ (0, 1) such that ∀ > 0, ∃ R > 0 and ∃ {xj } with the property: Given R > R, there exist positive measures µ1j and µ2j such that 0 ≤ µ1j + µ2j ≤ µj , supp µ1j ⊂ BR (xj ), supp µ2j ⊂ Rn \ BR (xj ), and lim supj→∞ {|λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )|} ≤ . We shall present an example showing how this principle is applied. In contrast with Example 3 in Sect. 4.2, we consider the same equation but on the whole space Rn : Find a nontrivial solution u ∈ H 1 (Rn ) satisfying −u + u = |u|q−2 u, 2 < q < 2∗ . If we follow the steps in the proof of that example, then it can be stated as a minimizing problem for the functional:

J(u) = (|∇u|2 + |u|2 ) Rn

subject to the constraint:

M = u ∈ H 1 (Rn ) |

|u| = 1 . q

Rn

It is important to note that both J and M are translation invariant. From the translation invariance, we intend to prove that, for any minimizing sequence {uj }, after suitable translation, the new minimizing sequence {vj } is Lq subconvergent. Then the limit v0 ∈ M . Let Sq = inf J . M

In light of the concentration compactness principle, let µj = |uj |q dx. If one can exclude Cases 2 and 3, then ∃{xj } ⊂ Rn , ∀ > 0, ∃ R > 0, such that

1= |uj (x − xj )|q dx ≥ µj (BR (xj )) ≥ 1 − . Rn

Let vj (x) = uj (x − xj ). Again {vj } is a minimizing sequence, vj H 1 (Rn ) is bounded, after a subsequence vj v0 , and then vj → v0 strongly in Lq (BR (θ)).

|v0 (x)|q ≥ 1 − . Therefore,

Then we have 1 ≥ BR (0)

v0 ∈ M . Moreover, we have vj → v0 in Lq (Rn ) strongly.

Rn

|v0 |q = 1, i.e.,

4.7 Concentration Compactness

293

• To exclude the vanishing case: Suppose we had R > 0 such that

lim inf sup |uj |q = 0 .

j→∞ x∈Rn

|uj |2 = 0. According to the Lemma 4.7.1, uj → 0

Then lim inf sup

j→∞ x∈Rn

BR (x)

BR (x) ∗

in Lp (Rn ) ∀ p ∈ (2, 2 ). In particular, when we take p = q, it contradicts |uj |q = 1. • To exclude the dichotomy case: Suppose ∃λ ∈ (0, 1) such that ∀ > 0, ∃ R > 0, ∃ {xj }, and positive measures µ1j and µ2j such that 0 ≤ µ1j + µ2j ≤ µj , supp µ1j ⊂ BR (xj ), supp µ2j ⊂ Rn \ B2R (xj ), and limj→∞ {|λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )|} ≤ . Choosing j → 0, ∃ Rj > 0, after a subsequence, we have supp µ1j ⊂ BRj (xj ), supp µ2j ⊂ Rn \ B2Rj (xj ), and limj→∞ {|λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )|} = 0. In fact, we may assume Rj → ∞. Let φ ∈ C0∞ (B2 (θ)) such that φ = 1 x − xj ). We write vj = uj φj + uj (1 − φj ). Then in B1 (θ) and let φj (x) = φ( Rj J(uj ) = J(uj φj ) + J(uj (1 − φj )) + Bj , where Bj is the interaction term:

Bj = 2 ∇(uj φj )∇(uj (1 − φj )) + u2j φj (1 − φj ) . By simple estimation ∀ > 0, 1 Bj ≥ − − Ej , where

|uj ∇φj |2 ≤

Ej = 2 B2Rj (xj )\BRj (xj )

C Rj2

|uj |2 . B2Rj (xj )\BRj (xj )

Fixing > 0 arbitrarily small, since Ej → 0 as j → ∞, we have 1 J(uj ) ≥ Sq (|uj φj |2q + |uj (1 − φj )|2q ) − − Ej 2 2 1 ≥ Sq (µ1j (BRj (xj )) q + µ2j (Rn \ B2Rj (xj )) q ) − − Ej 2 2 q q ≥ Sq (λ + (1 − λ) ) − + o(1) . 2

2

But λ q + (1 − λ) q > 1 as λ ∈ (0, 1). This is a contradiction. Namely we obtain: Theorem 4.7.4 (P.L. Lions) For any minimizing sequence {uj } ⊂ H 1 (Rn ) of the functional J(u) = Rn (|∇u|2 + |u|2 ) subject to |u|q = 1 for q ∈ (2, 2∗ ),

294

4 Minimization Methods

there exists a sequence {xj } ⊂ Rn such that vj (x) = uj (x − xj ) Lq -strongly converges to a minimizer of J. Example 1. Assume a ∈ C(Rn ) satisﬁes a(x) → a∞ as |x| → ∞. One studies the nontrivial solution u ∈ H 1 (Rn ) of the equation: −∆u + a(x)u = |u|q−2 u, 2 < q < 2∗ . Deﬁne

I(u) = Rn

(|∇u|2 + a(x)|u|2 ), I∞ (u) =

Rn

(|∇u|2 + a∞ |u|2 ) ,

and M (u) := |u|qq =

|u|q . Rn

We deﬁne S = inf{I(u) | M (u) = 1} and S∞ = inf{I∞ (u) | M (u) = 1} , and conclude: If S < S∞ , then there is a nontrivial solution of the above equation. Before going to the proof, we need a lemma (see [BL]): Lemma 4.7.5 (Brezis–Lieb) Suppose that Ω ⊂ Rn and {uj } ⊂ Lp (Ω), p ∈ [1, ∞) If {uj } is bounded in Lp (Ω), and uj → u a.e., on Ω. Then lim (uj pp − uj − upp ) = upp .

j→∞

Proof. According to Fatou’s lemma, |u|pp ≤ lim inf j→∞ uj pp < ∞. We begin with an elementary inequality: ∀ > 0, there exists C > 0 such that ∀x, y ∈ R1 , ||x + y|p − |x|p | ≤ |x|p + C |y|p . Let vj, = (| |uj |p − |uj − u|p − |u|p | − |uj − u|p )+ , then vj, ≤ (1 + C )|u|p , and vj, → 0 a.e. Then we can use Lebesgue’s dominance theorem to conclude: Ω vj, → 0 as j → ∞. Since | |uj |p − |uj − u|p − |u|p | ≤ vj, + |uj − u|p ,

we have

| |uj |p − ||uj − u|p − |u|p | ≤ C ,

lim sup j→∞

where C = sup{uj −

upp

Ω

< ∞. Letting → 0, the lemma is proved.

4.7 Concentration Compactness

295

Now we turn to the proof of the conclusion in Example 1. In fact, by the previous discussion, after a subsequence, any minimizing sequence {uj } ⊂ H 1 (Rn ) ∩ M −1 (1) is weakly H 1 convergent to some u0 ∈ H 1 (Rn ), so it is suﬃcient to show that u0 ∈ M −1 (1). Let vj = uj − u0 , we have vj 0, in H 1 (Rn ) and vj → 0 in L2loc , and then vj → 0 a.e. Since we have: 1. I(uj ) = I(u0 ) + I(vj ) + o(1). 2. |uj |qq = |u0 |qq + |vj|qq + o(1) (from Lemma (4.7.6)) 3. a(x)|vj |2 = a∞ |vj |2 + o(1), (from the estimate

|a(x) − a∞ ||vj (x)|2 ≤ |a(x) − a∞ ||vj |2 + C |x|≥R

|x|≤R

|vj |2 ,

for suitably chosen R > 0, where C = 2 Max{a(x)}). It follows that I(vj ) = I∞ (vj ) + o(1). Let |u0 |qq = λ. Obviously λ ∈ [0, 1]; we are going to exclude the case that λ ∈ [0, 1). Suppose not, either λ ∈ (0, 1), we would have S ≥ S|u0 |2q + S∞ |vj |2q + o(1) 2

2

= Sλ q + S∞ (1 − λ) q + o(1) 2

2

≥ S(λ q + (1 − λ) q ) + o(1) > S , or, λ = 0, then S > S∞ . These are all impossible. 4.7.2 The Critical Sobolev Exponent and the Best Constants The following best constant plays an important role in many variational problems arising in geometry and analysis: |∇u|2 Rn , S= inf 2 u∈D 1,2 \{θ} ( n |u|2∗ ) 2∗ R

1 where D1,2 is the closure of C0∞ (Rn ) under the norm: u = ( Rn |∇u|2 ) 2 . We intend to ﬁgure out the precise value of S and reduce it to a variational problem. Let

|∇u|2 , I(u) = u2 := Rn

and ∗

M (u) = |u|22∗ =

Rn

∗

|u(x)|2 dx .

We study the minimization problem: Min{I(u) | M (u) = 1} .

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4 Minimization Methods

Both I and M are not only translation invariant, but also invariant under the scaling transformation: 6x7 n−2 Tθ : u(x) → θ− 2 u , ∀θ > 0 . θ By the same idea as in the previous subsection, one may use the concentration compactness principle to show that after translation and scaling, the ∗ minimizing sequence is again convergent in L2 (Rn ), and then a minimizer does exist. Let u0 be a minimizer; obviously |u0 | is also. Therefore we may assume the minimizer u0 is nonnegative. The Euler–Lagrange equation of the variational problem reads as −∆u = λ|u|2

∗

−2

u in Rn ,

(4.64)

where λ > 0 is the Lagrange multiplier. After suitably adjusting a factor one may assume λ = 1. By a moving plane argument due to Gidas, Ni and Nirenberg, [GNN] (see also Yanyan Li [Li 2]), u0 is radially symmetric: u0 (x) = 1 . Moreover, g is nonincreasing. g(|x|), for some nonnegative function on R+ Plugging g into equation (4.64), it becomes an ODE: −(rn−1 g ) = Srn−1 g 2

∗

−1

.

However, we shall present here a more direct proof due to Lieb. Lemma 4.7.6 If the problem Min{I(u) | M (u) = 1} possesses a minimizer, then it has a radially symmetric and nonincreasing minimizer u0 ; i.e., there is a nonincreasing function g deﬁned on R1+ such that u0 (x) = g(|x|). Proof. We may restrict ourselves to nonnegative minimizers, because |u| ≤ u, and |u|q = uq . Let u∗ be the Schwarz rearrangement of u (see Lemma 4.2.5), then u∗ is nonincreasing, radial symmetric and satisﬁes m(ut ) = m(u∗t ), ∀t ∈ R1 , where ut = {x ∈ Rn | u(x) ≥ t}. Thus,

∞

∞ q q |t| dm(ut ) = |t|q dm(u∗t ) = |u∗ |qq . |u|q = −∞

−∞

According to the Faber–Krahn inequality, it follows that

∗ 2 |∇u | ≤ |∇u|2 . Rn

Rn

Combining Corollary 4.7.2 with Lemma 4.7.6, one proves that the following equation possesses a positive solution in H 1 (Rn ) directly: −∆u + u = |u|q−2 u for 2 < q < 2∗ . Lemma 4.7.7 The minimizer of the problem in lemma 4.7.6 is achieved.

4.7 Concentration Compactness

297

Proof. 1. By changing variables, let F (αt) = eαt g(et ), where α = n2 − 1, and r = et ; we have

∞

∞

∞ ∗ 1 ∞ 2∗ F dt = g(r)2 rn−1 dr, α (F − F )2 dt = g (r)2 rn−1 dr . α −∞ 0 −∞ 0 ∞ 2 n−1 ∞ 2∗ n−1 Setting f (g) = 0 g (r) r dr, φ(g) = 0 g(r) r dr, and E0 = {g ∈ −1 1 1 (R+ ) | g ≥ 0, φ(g) = ωn−1 }, E0 = {g ∈ E0 | g is nonincreasing}, where Hloc n

ωn−1 =

nπ 2 Γ( n 2 +1)

is the area of the unit sphere S n−1 , we have:

Min{I(u) | M (u) = 1} ≤ Min{f (u) | u ∈ E0 } ≤ Min{f (u) | u ∈ E0 } ≤ Min{I(u) | M (u) = 1} . Our problem is reduced to ﬁnd: (4.65) Min{f (u) | u ∈ E0 } . ∞ ∗ ∞ 2. Let f1 (F ) = −∞ (F 2 + F 2 )dt, φ1 (F ) = −∞ F 2 dt, and E1 = {F ∈ −1 }. Then the problem (4.65) is equivalent to the probH 1 (R1 ) | φ1 (F ) = αωn−1 lem: Min{f1 (F ) | F ∈ E1 } . In fact, on the one hand, if F solves the latter, then F (t) → 0 as t → ±∞, One has

∞

∞ 2 (F − F ) dt = (F 2 + F 2 )dt . (4.66) −∞

−∞

On the other hand, functions with compact supports consist of a dense subset of E0 , and for g with compact support F (t) is zero as t becomes large. Moreover, F (t) ≤ g(0)et , and then F (t) → 0 as t → ±∞. Again equation (4.66) holds. 1 }. Again, by 3. Let E1 = {F ∈ E1 | F is even and nonincreasing on R+ symmetric rearrangement, we have Min{f1 (F ) | F ∈ E1 } = Min{f1 (F ) | F ∈ E1 } . Let {Fj } be a minimizing sequence in E1 , the sequence is weakly conver∗ α . gent to some F0 ∈ E1 . We are going to verify that |F0 |22∗ = ωn−1 ∗

Since Fj → F0 , a.e., it is suﬃcient to ﬁnd a dominant function in L2 (R1+ ). From Fj (t) → 0 as t → ∞, we have

∞

∞ 2 Fj (t) = −2 Fj Fj dt ≤ ( Fj2 + Fj2 )dt ≤ C . 0

t

Since Fj is nonincreasing on [0, ∞), and

t tFj (t)2 ≤ Fj2 ≤ C , 0 − 12

the function M (t) = Min{1, t (R1+ ).

∗

} is a dominant function for Fj in L2

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4 Minimization Methods

Now we can ﬁnd the minimizer explicitly. Following the above notations, we conclude that the function F0 satisﬁes the ordinary diﬀerential equation: ⎧ −2 2∗ −1 ⎪ , ⎨−F + F = Sα F F (0) = 0 , ⎪ ⎩ F (∞) = 0 . where S is the best constant, α = n2 − 1. The unique solution (see Lieb [Lieb], Aubin [Au 1] and Talenti [Tal]) reads as = α n(n − 2) sech(α−1 t) . F0 = 4S or

= g0 (r) = r−α F0 (α log r) =

n(n − 2) 1 S 1 + r2

n2 −1 .

By direct computation, we obtain S = n(n − 2)π

Γ(n) " # Γ n2

− n2 .

Therefore we have proved: Theorem 4.7.8 The functions u(x) =

(n(n − 2)θ) (θ2 + |x − y|2 )2

n−2 4

, ∀θ > 0, ∀ y ∈ Rn

(4.67)

are minimizers of the problem S = Min{u22 | |u|2∗ = 1}. In fact, all minimizers for S are of the form (4.67), see Caﬀarelli, Gidas, and Spruck [CGS]. A simple proof via the moving plane method [GNN] was given by W. Chen and C. Li, Yanyan Li, see [ChL 2], [Li 2]. Example 2. Find a nontrivial solution u ∈ H01 (Ω) of the following equation, where Ω ⊂ Rn is a bounded domain: −∆u − λu = |u|2

∗

−2

u in Ω .

(4.68)

where we assume λ < λ1 , the ﬁrst eigenvalue of −∆. Again, the minimization problem Min{Iλ (u) | |u|2∗ = 1} is considered, where Iλ (u) = u2 − λ|u|22 . Let Sλ (Ω) = inf {Iλ (u) | u ∈ H01 (Ω), |u|2∗ = 1} .

4.7 Concentration Compactness

299

Lemma 4.7.9 S0 (Ω) = S . Proof. 1. For Ω1 ⊂ Ω2 , by deﬁnition, S0 (Ω2 ) ≤ S0 (Ω1 ). 2. For ∀R > 0, let Ωj = BjR (θ), j = 1, 2. We claim that S0 (Ω1 ) = S0 (Ω2 ) This is due to the scaling invariance of the functional S0 (Ω). 3. By the translation invariance, we conclude that for all Ω, S0 (Ω) = S0 (B1 (θ)). This proves the lemma. Following the steps in Example 1, there exists a minimizing sequence {uj } with |uj |2∗ = 1 and uj u0 in H01 (Ω). Let vj = uj − u0 , we have. 1. Iλ (uj ) = Iλ (u0 ) + Iλ (vj ) + o(1). ∗ ∗ ∗ 2. |uj |22∗ = |u0 |22∗ + |vj |22∗ + o(1). ∗

Let |u0 |22∗ = µ ∈ [0, 1]. It remains to verify that µ = 1. We shall exclude the cases µ = 0, and µ ∈ (0, 1). In fact, if µ ∈ (0, 1), then Sλ ≥ Sλ (|u0 |22∗ + |vj |22∗ ) + o(1) 2

2

= Sλ (µ 2∗ + (1 − µ) 2∗ ) + o(1) > Sλ + o(1) . This is impossible. If µ = 0, then u0 = θ, i.e., uj θ in H01 (Ω), and then |uj |2 → 0. Thus Iλ (uj ) = |∇uj |22 + o(1) 2

≥ S|uj |22∗∗ + o(1) = S + o(1) . Now, we need the following: Lemma 4.7.10 (Brezis–Nirenberg [BN 1] For n ≥ 4, ∀λ > 0, Sλ (Ω) :=

inf

u∈H01 (Ω)\{θ}

Iλ (u) <S. |u|22∗

(4.69)

Proof. One may assume θ ∈ Ω. Let ϕ ∈ C0∞ (Ω), and ϕ(x) = 1 in a small neighborhood of θ. For ε > 0, we deﬁne vε (x) = Then ∇vε (x) =

(ε2

(ε2

ϕ(x) . n + |x|2 ) 2 −1

∇ϕ(x) (n − 2)ϕx − 2 n n . −1 2 2 + |x| ) (ε + |x|2 ) 2

We have constants K1 , K2 , K3 > 0 such that

300

4 Minimization Methods

vε 22

= (n − 2)

2 Ω

|x|2 dx + O(1) = (n − 2)2 (ε2 + |x|2 )n

Rn

|x|2 dx + O(1) (ε2 + |x|2 )n

K1 = n−2 + O(1) , ε

∗ |ϕ|2 dx dx K2 2∗ = + O(1) = n + O(1) , |v|2∗ = 2 + |x|2 )n 2 2 n (ε (ε + |x| ) ε Ω Rn and

|vε |22 = Ω

|ϕ|2 dx = (ε2 + |x|2 )n−2

as n ≥ 5, and

Rn

(ε2

dx K3 + O(1) = n−4 + O(1) , 2 n−2 + |x| ) ε

|vε |22 = K3 | log ε| + O(1) ,

as n = 4. In summary, we have

Sλ ≤

⎧ K1 −λK3 ε2 ⎪ if n ≥ 5 , ⎪ 1− 2 ⎨ n K2

K1 −λK3 ε2 log |ε| ⎪ ⎪ if n = 4 . 1 ⎩ 2 K2

Since K3 > 0 and

K1

1− 2 n

K2

= S for small ε > 0, Sλ < S.

Provided by Lemma 4.7.10, the vanishing case is excluded. We have proved the existence of a nontrivial solution. Remark 4.7.11 In fact, the restriction λ < λ1 is not necessary. Provided by minimax methods, which we shall study in the next subsection, one can remove this restriction. Also the case n = 3 has been discussed, see Brezis–Nirenberg [BN 1]. Remark 4.7.12 (Yamabe problem) Let (M, g0 ) be a connected compact ndimensional Riemannian manifold. One asks: Does there exist a metric g pointwise conformal to g0 such that the scalar curvature R with respect to g is a constant? 4

Set g = u n−2 g0 , where u ∈ C ∞ (M ) with u > 0. It is reduced to the following PDE: ∗ 4(n − 1) ∆g0 u + Rg0 u = Ru2 −1 , − n−2 where ∆g0 is the Laplacian Beltrami operator, and Rg0 is the scalar curvature with respect to g0 . Then it is transferred to the variational problem: n−2 |∇g0 u|2 + 4(n−1) R g0 u 2 M . λ(M ) = min 2 u∈H 1 (M ) ( M |u|2∗ ) 2∗

4.8 Minimax Methods

301

This is a problem very similar to Example 2. Again, λ(M ) ≤ S. If λ(M ) < S, then the problem is solvable by the previous method. However, it is easily seen that if (M, g0 ) = (S n , gˆ0 ), where gˆ0 is the canonical metric, then λ(S n ) = S. The Yamabe problem has been solved by Aubin [Au 1] and Schoen [Sco]. They proved: Theorem 4.7.13 For n ≥ 3, if (M, g0 ) is a compact connected Riemannian manifold, which is not conformally equivalent to (S n , gˆ0 ), then λ(M ) < S. Then Yamabe problem is solvable.

4.8 Minimax Methods 4.8.1 Ekeland Variational Principle It is well known that the direct method does not work in the lack of compactness (i.e., the coerciveness). Without coerciveness, only approximate minimizers can be found. Let us recall the Ekeland variational principle, which we have derived in Chap. 2 as an equivalence of Caristi ﬁxed-point theorem. Due to the importance of this principle, we present here a direct proof. Theorem 4.8.1 (Ekeland) Let (X, d) be a complete metric space, and let / + ∞. If f is bounded from below and l.s.c., and f : X → R1 ∪ {+∞}, but f ≡ if ∃ ε > 0, ∃xε ∈ X satisfying f (xε ) < inf X f + ε. Then ∃ yε ∈ X such that 1. f (yε ) f (xε ), 2. d(xε , yε ) 1, 3. f (x) > f (yε ) − εd(yε , x),

∀x = yε .

Proof. Note that the required yε is the minimum of the function f (x) + εd(yε , x), while the new function contains yε itself. Accordingly, we deﬁne a sequence approaching yε . Choose u0 = xε . Suppose that un is already chosen. Set Sn = {w ∈ X f (w) f (un ) − εd(w, un )} . Obviously Sn = ∅. We choose un+1 ∈ Sn satisfying 1 f (un+1 ) − inf f f (un ) − inf f , Sn Sn 2

(4.70)

n = 0, 1, 2, . . . . We want to show that {un } is a Cauchy sequence. Indeed, εd(um , un ) f (un ) − f (um ),

∀m > n .

(4.71)

Since f is bounded below, f (un ) − f (um ) → 0 as m, n → ∞, it follows un → u∗ ∈ X. From the l.s.c. of f and (4.70), we have

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4 Minimization Methods

f (u∗ ) lim f (un ) lim inf f . n→∞ Sn

n→∞

(4.72)

We are going to verify that yε = u∗ satisﬁes the conclusions 1–3. In fact, since from (4.71) f (un ) is nonincreasing, conclusion 1 is trivially true. Again by (4.71), we have εd(xε , yε ) = εd(u0 , u∗ ) f (xε ) − f (u∗ ) f (xε ) − inf f Sn

0, ∃ yε ∈ X such that f (x) > f (yε ) − εd(x, yε ), ∀x = yε . Let us introduce the following notion, which plays an important role in the calculus of variations in the large. Deﬁnition 4.8.3 Let X be a Banach space, and f ∈ C 1 (X, R1 ). If any se quence {xj }∞ 1 ⊂ X along which f (xj ) → c and f (xj ) → θ implies a convergent subsequence. Then f is said to satisfy the (P S)c condition. If f satisﬁes the (P S)c condition, for ∀c, then it is said to satisfy the (P S) condition. Let us return to the minimizing problem: Corollary 4.8.4 Let X be a Banach space, and let f ∈ C 1 (X, R1 ) be bounded from below. If (P S)c holds with c = inf f , then f possesses a minimum. X

Proof. According to Theorem 4.8.1, ∀n 0 ∃ xn ∈ X satisfying f (x) > f (xn ) − n1 x − xn f (xn ) < c + n1 which implies that f (xn ) → c and f (xn ) → θ. Therefore we obtain a convergent subsequence xnj → x∗ , according to (P S)c . So is f (x∗ ) = c = inf f . X

4.8 Minimax Methods

303

4.8.2 Minimax Principle In this section, we are concerned with those critical points that are saddle points but not minima. The following geometric intuition suggests a minimax consideration. A valley is surrounded by mountains. Starting from a point x1 on the ground outside these mountains, we intend to get into a place x0 in the valley. A path l one would like to take, is a path along which the highest point is lower than those at neighboring paths. The highest point on this path is indeed a saddle point of the height function. Theorem 4.8.5 (Ambrosetti–Rabinowitz [AR]) Let X be a Banach space, / Ω. and f ∈ C 1 (X, R1 ). Suppose that Ω ⊂ X is an open set, x0 ∈ Ω, and x1 ∈ Set Γ = {l ∈ C([0, 1], X) l(i) = xi , i = 0, 1} , and c = inf max f ◦ l(t) . l∈Γ t∈[0,1]

If 1. α = inf f (x) > max{f (x0 ), f (x1 )}, x∈∂Ω

2. (P S)c holds for f ; then c α is a critical point. We shall not prove this theorem at this moment, but introduce a more general notion. Deﬁnition 4.8.6 Let X be a Banach space. Let Q ⊂ X be a compact manifold with boundary ∂Q and let S ⊂ X be a closed subset of X. ∂Q is said linking with S, if 1. ∂Q ∩ S = ∅, 2. ∀ϕ : Q → X continuous with ϕ|∂Ω = id |∂Ω , we have ϕ(Q) ∩ S = ∅. Example 4.8.7 (Mountain pass) Let Ω, x0 , x1 be as in Theorem 4.8.5. Set Q = the segment {λx0 + (1 − λ)x1 | λ ∈ [0, 1] }, and S = ∂Ω. Then, ∂Q = {x0 , x1 } and S link. Example 4.8.8 Let X1 be a ﬁnite-dimensional linear subspace of the Banach space X, and let X2 be its complement: X = X1 + X2 . Let Q = BR ∩ X1 , S = X2 , where BR is the ball with radius R > 0 centered at θ. Then ∂Q and S link. Indeed, ∀ϕ : Q → X continuous with ϕ∂Q = id ∂Ω , we want to show: ϕ(Q) ∩ X2 = ∅.

304

4 Minimization Methods

Let P be the projection onto X1 . It is suﬃcient to show that P ◦ϕ : Q → X1 has a zero. Obviously deg(P ◦ ϕ, Q, θ) = deg(id, Q, θ) = 1 . The conclusion follows from the Brouwer degree theory. Theorem 4.8.9 Let X be a Banach space, and let f ∈ C 1 (X, R1 ). Assume that Q ⊂ X is a compact manifold with boundary ∂Q which links with a closed subset S ⊂ X. Set Γ = {ϕ ∈ C(Q, X) ϕ|∂Q = id |∂Q } , and c = inf max f ◦ ϕ(ξ) . ϕ∈Γ ξ∈Q

If ∃ α < β such that sup f (x) α < β inf f (x) , x∈S

x∈∂Q

and if (P S)c holds, then c ( β) is a critical value. Proof. Let d be the distance on C(Q, X). Then (Γ, d) is a metric space. Let J(ϕ) = max f ◦ ϕ(ξ) . ξ∈Q

Invoke the assumption that ∂Q and S link, J inf f . Moreover, J is locally S

Lipschitzian, since J(ϕ1 ) − J(ϕ2 ) max[f ◦ ϕ1 (ξ) − f ◦ ϕ2 (ξ)] ξ∈Q

max f (θϕ1 (ξ) + (1 − θ)ϕ2 (ξ))d(ϕ1 , ϕ2 ) . θ∈[0,1] ξ∈Q

It follows that

J(ϕ1 ) − J(ϕ2 ) Cd(ϕ1 , ϕ2 ) ,

where C is a constant depending on ϕ1 , ϕ2 . We apply the Ekeland variational principle to J, and obtain a sequence {ϕn } ⊂ Γ satisfying 1 , n

(4.74)

1 d(ϕ, ϕn ) , n

(4.75)

c J(ϕn ) < c + and J(ϕ) J(ϕn ) − n = 1, 2, 3, . . . . Set

M(ϕ) = {ξ ∈ Q f ◦ ϕ(ξ) = J(ϕ)} .

4.8 Minimax Methods

305

Obviously M(ϕ) is compact. We claim that M(ϕ) ⊂ int(Q). Indeed, since ∂Q and S link, if ∃ ξ0 ∈ M(ϕ) ∩ ∂Q, then f ◦ ϕ(ξ0 ) = max f ◦ ϕ(ξ) inf f β . S

ξ∈Q

But, f ◦ ϕ(ξ0 ) = f (ξ0 ) sup f (x) α . x∈∂Q

This is a contradiction. Set Γ0 = {ψ ∈ C(Q, X) ψ|∂Q = θ} . It is a linear closed subspace of X. Let · be the norm of C(Q, X). ∀h ∈ Γ0 , with h = 1, ∀λj ↓ 0, ∀ξj ∈ M(ϕn + λj h), we have λ−1 j [ f ◦ (ϕn + λj h)(ξj ) − f ◦ ϕn (ξj ) ] −

1 , n

(4.76)

from (4.75). Since {ξj } ⊂ Q, we obtain a convergent subsequence ξj → ηn∗ ∈ M(ϕn ), which depends on ϕn , λj and h. After taking limits, we have f ◦ ϕn (ηn∗ ), h(ηn∗ ) −

1 . n

(4.77)

We want to show that ∃ ηn ∈ M(ϕn ) such that f ◦ ϕn (ηn ), u −

1 , n

(4.78)

∀u ∈ X with u = 1. If not, ∀η ∈ M(ϕn ), ∃ vη ∈ X with vη = 1, satisfying f ◦ ϕn (η), vη < −

1 , n

then there exists a neighborhood of η, Oη ⊂ int(Q) such that 1 f ◦ ϕn (ξ), vη < − , n

∀ξ ∈ Oη .

Since M(ϕn )+is compact, there is a ﬁnite covering. Let m be the least number m of covering: i=1 Oηi ⊃ M(ϕn ). We obtain the associate {vηi }m 1 , vηi = 1 satisfying 1 ∀ξ ∈ Oηi , f ◦ ϕn (ξ), vηi < − n i = 1, 2, . . . , m. Construct a partition of unity subject to {Oηi }m 1 : 0 i 1, sup i ⊂ Oηi , i = 1, . . . , m, and m i=1

i (ξ) ≡ 1,

∀ξ ∈ M(ϕn ) .

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4 Minimization Methods

We set v = v(ξ) =

m

i (ξ)vηi .

i=1

Thus v ∈ Γ0 and v 1. But, ∃ ξ ∗ ∈ M(ϕn ) such that there is only one i0 satisfying ξ ∗ ∈ Oηi0 . Therefore v = 1. We obtain 1 f ◦ ϕn (ξ), v(ξ) < − , ∀ξ ∈ M(ϕn ) . n This contradicts (4.77). Thus (4.78) holds. Setting xn = ϕn (ηn ), we have f (xn ) → θ . Combining with (4.74), we have also f (xn ) → c . The (P S)c condition implies that c is a critical value.

The above proof is based on Shi [Shi] and Aubin and Ekeland [AE]. The following example due to Brezis and Nirenberg [BN 3] asserts that the (P S)c condition is crucial in Theorem 4.5.5. Example. The function ϕ(x, y) = x2 + (1 − x)3 y 2 deﬁned on R2 does have a mountain surround (0, 0): ϕ(x, y) ≥ c > 0 on the circle x2 + y 2 = 14 , ϕ(0, 0) = 0, and ϕ(4, 1) = −11. But by direct computation it has only one critical point: (0, 0). 4.8.3 Applications The mountain pass lemma and the related minimax principles are widely used in the study of diﬀerential equations. We are satisﬁed to give few examples. Example 1. We study the following forced resonance problem. Given h ∈ L1 ([0, π]) satisfying

π h(t) sin tdt = 0 .

(4.79)

0

Assume that g ∈ C(R1 ) is T -periodic, T > 0, and satisﬁes:

T g(t)dt = 0 .

(4.80)

0

Find a solution of the nonlinear BVP: x ¨(t) + x(t) = h(t) + g(x(t)) t ∈ (0, π) x(0) = x(π) = 0 . First we introduce a generalized Riemann-Lebesgue lemma.

(4.81)

4.8 Minimax Methods

307

Lemma 4.8.10 Under the above assumptions of g, if un → u in C([0, π]), αn → ∞, u ∈ C 1 ([0, π]) and k ∈ L1 ([0, π]), then

π g(un (t) + αn sin t)k(t)dt → 0 . lim n→∞

0

Proof. Since the sequence is bounded in C([0, π]), and since g(un (t) + αn sin nt) − g(u(t) + αn sin t) uniformly converges to 0, it is suﬃcient to prove that

π g(u(t) + αn sin t) · χE (t)dt = 0 ,

lim

n→∞

0

∀E = (a, b) ⊂ (0, π), where χE is the characteristic function of E. We may assume αn → +∞. We write vn (t) = αn−1 u(t) + sin t, and let G(t) be a primitive of g. Then G is T -periodic, according to (4.80). At ﬁrst, we assume π2 ∈ E. Then for n large, s = vn (t) is strictly increasing (decreasing), if b < π2 (or a > π2 resp.). We have t = vn−1 (s). Let δ = dist (E, π2 ), ∃ a constant Cδ depending on δ > 0, such that 1 Cδ a.e. By changing variables, |v (v −1 (s))| n

n

b vn (b) ds g(αn vn (t))dt = g(αn s) −1 a vn (a) vn (vn (s))

Cδ |G(αn vn (b)) − G(αn vn (a))| . αn

(4.82)

Since G is continuous and periodic, the RHS of the above inequality tends to zero as n → ∞. Second, we consider the case π2 ∈ E. Since g is bounded, provided by the absolute continuity of the integral of k, ∀ε > 0, ∃δ > 0 such that

π/2+δ ε (4.83) k(t)g(αn vn (t))dt < . π/2−δ 2 Combining (4.82) with (4.83), the conclusion follows.

Let us reformulate (4.81) in a variational version. Again let G be a primiT tive of g satisfying 0 G(t)dt = 0. Setting X = H01 ([0, π]),

π

π 1 2 2 (u˙ − u ) + hu dt, N (u) = I(u) = G(u(t))dt , 2 0 0 and J(u) = I(u) + N (u) . The functionals I and J are not coercive. In fact, letting un (t) = n sin t, we have I(un ) = 0, and from |G(t)| C, a constant, we have |N (un )| Cπ, and then |J(un )| Cπ.

308

4 Minimization Methods

Now, let us decompose X into H1 ⊕ span {e1 }, where e1 = sin t, and 1 H1 = e⊥ 1 , i.e., ∀u ∈ H0 ([0, π]) one has the orthogonal decomposition: u = u1 + αe1 ,

where

π

u(t) sin tdt .

α= 0

Then we have: (1) I is s.w.l.s.c. and coercive on H1 . Moreover, from (4.79), I(u) = I(u1 ). (2) N is bounded and weakly continuous on X, and according to Lemma 4.8.10, N (u1 + αe1 ) → 0 as α → ∞, ∀u1 ∈ H1 . (3) J is bounded from below on X and coercive on H1 . Let

m∗ = inf J(u) . u∈X

Lemma 4.8.11 If m∗ is not a minimum of J, then ∃{u1j } ⊂ H1 and αj → ∞ such that J(u1j + αj e1 ) → m∗ . Moreover, u1j u1 in H1 , which is a solution of x ¨ + x = h, and m∗ = I(u1 ) = inf I(u1 ), u1 ∈ H1 . Proof. 1. We choose a minimizing sequence {uj } of J, and decompose it on H1 ⊕ span {e1 }: uj = u1j + αj e1 . We claim that αj → ∞. Indeed, invoke the coerciveness of I on H1 and I(u1j ) = I(uj ) = J(uj ) − N (uj ) < m∗ + 1 + Cπ , {u1j } is bounded in H1 . Thus after a subsequence u1j u1 in H1 , which implies u1j → u1 in C([0, π]). If {αj } is bounded, then after a subsequence αj → α, we have N (u1j + αj e1 ) → N (u) , where u = u1 + αe1 . We have lim inf I(uj ) = lim I(u1j ) I(u1 ) = I(u) , j→∞

j→∞

and m∗ = lim J(uj ) = lim (I(uj ) + N (uj )) I(u) + N (u) = J(u) . j→∞

j→∞

Therefore, u ¯ is a minimizer of J. This is a contradiction. The claim is proved. 2. ∀α ∈ R1 , ∀u1 ∈ H1 , one has m∗ ≤ J(u1 + αe1 ) = I(u1 ) + N (u1 + αe1 ) .

4.8 Minimax Methods

309

Letting α → ∞, provided by Lemma 4.8.10, N (u1 + αe1 ) → 0, thus m∗ I(u1 ) = I(¯ u1 ),

∀u1 ∈ H1 .

(4.84)

But to the minimizing sequence {uj } we have I(u1 ) lim I(u1j ) j→∞

= lim (J(uj ) − N (u1j + αj e)) j→∞

= lim J(uj ) = m∗ . j→∞

(4.85)

Combining (4.84) and (4.85), we conclude that m∗ = I(u1 ) is the minimum of I on H1 , and u1 is the associated minimizer, which satisﬁes I (u1 ) = 0, i.e., u1 is a solution of x ¨+x=h, x(0) = x(π) = 0 . Lemma 4.8.12 ∀c > m∗ , J satisﬁes (P S)c . Proof. Assume that {uj } is a (P S)c sequence of J, i.e., J(uj ) → c and J (uj ) → θ in X . From the coerciveness of I on H1 , and I(u1j ) = J(uj ) − N (u1j + αj e1 ) c + 1 + cπ , after a subsequence, we have u1j u1 in H1 . We are going to verify that after a subsequence αj → α. For otherwise, αj → ∞, we would have g(uj ) ∗ θ in L∞ ([0, π]), according to Lemma 4.8.10. d 2 ) + 1 has a compact inverse K on Noticing that the diﬀerential operator ( dt H1 , from J (uj ) → θ, we have u1j = Kh + KP g(uj ) + o(1) ,

(4.86)

where P is the orthogonal projection from X onto H1 . Thus u1j → Kh = u1 in H1 strongly. It follows from 2 in the proof of Lemma 4.8.11, c = lim J(uj ) j→∞

= lim (I(u1j ) + N (u1j + αj e1 )) j→∞

= lim I(u1j ) j→∞

= I(u1 ) = m∗ .

310

4 Minimization Methods

This is a contradiction. Therefore αj → α. Substituting in (4.86), we obtain a convergent sequence: uj = u1j + αj e1 → Kh + KP g(u1 + αe1 ) + αe1 . Conclusion. The problem (4.81) has at least one solution. Proof. If m∗ is attainable at u∗ , then u∗ is a minimizer, which is a solution of the Euler Lagrange equation (4.81) for J. Otherwise, let b = inf J(u1 ) . u1 ∈H1

Note that J|H1 is s.w.l.s.c. and coercive, ∃u∗1 ∈ H1 such that b = J(u∗1 ). Therefore b > m∗ . Since lim J(u1 + αe1 ) = I(u1 ) = m∗ ,

α→±∞

we may choose a > 0 large enough such that J(u1 ± ae1 ) < b .

Deﬁne Γ=

l ∈ C([0, 1], X)| l

1 1 ± 2 2

= u1 ± ae1

,

and c=

inf

max J(l(t)) .

l∈Γ t∈[0,1]

Since for ∀l ∈ Γ, l[0, 1] ∩ H1 = Ø, we have c b > m∗ . Thus (P S)c holds. According to the mountain pass theorem there is a critical point u∗ with J(u∗ ) = c. Again u∗ solves (4.81). Example 2. We present here an application to the superlinear elliptic boundary problem, see Ambrosetti and Rabinowitz [AR]. Let Ω ⊂ Rn be a bounded domain with smooth boundary. Given a function f ∈ C(Ω × R1 ) satisfying the following conditions: n+2 ) (when n ≥ 3) [F1 ] There exist positive constants C1 , C2 and α ∈ (1, n−2 such that |f (x, t)| ≤ C1 + C2 |t|α . [F2 ] ∃θ ∈ (0, 12 ), ∃ a constant M > 0 such that

t

F (x, s)ds ≤ θtf (x, t), as |t| ≥ M .

0 < F (x, t) := 0

4.8 Minimax Methods

311

[F3 ] lim sup t→0

f (x, t) < λ1 , uniformly in x ∈ Ω . t

[F4 ] f (x, t) > λ1 , uniformly in x ∈ Ω . t where λ1 is the ﬁrst eigenvalue of the Laplacian −∆ on Ω with Dirichlet boundary conditions. lim inf t→∞

Theorem 4.8.13 Under the assumptions F1 , F2 , F3 , F4 , the problem −∆u = f (x, u), x ∈ Ω , (4.87) u = 0, x ∈ ∂Ω , has a nontrivial solution. Proof. We consider the functional on H01 (Ω):

1 |∇u|2 − F (x, u) dx . J(u) = Ω 2

(4.88)

From the growth condition F1 , J is C 1 . Noticing J(θ) = 0, we claim that there is a mountain surrounding θ and / Bρ (θ) with J(u0 ) = 0. Indeed, from F3 , we have ∃α > 0, ∃ρ > 0 and u0 ∈ > 0, δ > 0 such that f (x, t) ≤ λ1 − , as 0 < |t| < δ , t which implies that F (x, t) ≤

1 (λ1 − )t2 as |t| < δ . 2

Combining with F1 , we have a constant C3 such that F (x, t) ≤

1 (λ1 − )t2 + C3 |t|p , 2

where p = α + 1, thus

1 F (x, u(x))dx ≤ u2H 1 + C4 upLp . 1− 2 λ 1 Ω This proves that J(u) ≥

u2H 1 − C4 upLp . 2λ1

Therefore ∃α > 0, ∃ρ > 0 such that J|∂Bρ (θ) ≥ α.

312

4 Minimization Methods

Now, we take u0 = sφ1 , where φ1 > 0 is the normalized ﬁrst eigenfunction, and s > 0 is to be determined. Setting

λ1 2 s − F (x, sφ1 (x))dx , g(s) = J(sφ1 ) = 2 Ω From F4 , g(s) → −∞ as s → +∞. Therefore there exists s0 > 0 satisfying g(s0 ) = 0, set s = s0 , u0 is as required. It remains to verify the Palais-Smale condition. Suppose that a sequence {uj } ⊂ H01 (Ω) satisﬁes J(uj ) → c, J (uj ) → θ; we shall show that it possesses a convergent subsequence. Firstly, {uj } is bounded. In fact, ∃C3 > 0 such that

1 −C3 ≤ uj 2H 1 − F (x, uj (x))dx ≤ C3 . 2 Ω According to F2 ,

uj (x)f (x, uj (x))dx ≤ − F (x, uj (x))dx + C4 . , −θ Ω

it follows that

Ω

1 − θ uj 2H 1 + θJ (uj )(uj ) ≤ C5 . 2

Since J (uj ) → θ, this proves that {uj } is bounded in H01 (Ω). Next, we show the existence of a convergent subsequence. Noticing that 2n , from the Sobolev embedding theorem H 1 (Ω) → Lp (Ω) is bounded, p < n−2 so that uj Lp is bounded, and then after a subsequence, {f (x, uj (x))} is weakly convergent in Lp (Ω), p1 + p1 = 1. However, (−∆)−1 : Lp (Ω) → H01 (Ω) is compact. Therefore, (−∆)−1 f (·, uj ) is strongly convergent in H01 (Ω). Since J (uj ) = uj − (−∆)−1 f (·, uj ) → θ ,

uj converges strongly.

It is interesting to compare the assumptions F1 to F4 with those in Sect. 3.6, Example 4. They are very similar. But the diﬀerences lie as follows: The partial diﬀerential operator in the cited example is not necessarily a divergent form, in which case the variational method cannot be applied; while in the above example, the solution is not necessarily positive. Moreover, one may even change the assumption F3 (see below); in these cases the a priori estimate, which the degree theoretic argument relies on, either fails or holds under other assumptions. Example 3. (Ni [Ni]) We make the conditions [F3 ] [F4 ]

f (x, t) = o(t) as t → 0 uniformlyin x ∈ Ω . tf (x, t) ≥ 0, ∀(x, t) ∈ Ω × R1 .

4.8 Minimax Methods

313

Theorem 4.8.14 Under the assumptions F1 , F2 , F3 , F4 , assume a ∈ C(Ω) is positive. Then the following equation possesses a nontrivial solution: −∆u(x) = a(x)u(x) + f (x, u(x)), x ∈ Ω , (4.89) u(x) = 0. x ∈ ∂Ω . Proof. We consider the functional on H01 (Ω):

1 [|∇u|2 − au2 ] − F (x, u) dx , J(u) = 2 Ω For which the Euler–Lagrange equation is (4.89). In the same manner we verify the Palais–Smale condition. We claim that there are linking sets, which separate values of J. Indeed, let 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ 1 < λk+1 ≤ · · · be the eigenvalues and {φ1 , . . . , φk , φk+1 , . . .} be the eigenfunctions of the eigenvalue problem: −∆u = λa · u, u ∈ H01 (Ω) . Taking Ek = span{φ1 , . . . , φk }, we have the orthogonal decomposition: ˆ E. H01 (Ω) = Ek ˆ we have On E,

|∇u|2 dx ≥ λk+1 au2 dx , Ω

therefore

From F3 ,

Ω

1 1 2 J(u) ≥ |∇u| − F (x, u) dx . 1− 2 λk+1 Ω

F (x, u(x))dx = o(u2 ) as u → 0 . Ω

Therefore there exist ρ > 0, α > 0 such that ˆ. J(u) ≥ α as u ∈ ∂Bρ (θ) ∩ E From F4 , F (x, t) ≥ 0 ∀(x, t) ∈ Ω × R1 , it implies that

1 J(u) ≤ [|∇u|2 − a(x)u2 ]dx ≤ 0, ∀u ∈ Ek . 2 Ω From F2 , we have

1

F (x, t) ≥ C1 |t| θ − C2 ,

where C1 , C2 are positive constants. Since θ < 12 ,

1 2 au dx − C1 |u| θ dx + C2 |Ω| → −∞ , J(u) ≤ (λk+1 − 1) Ω

Ω

314

4 Minimization Methods

as u ∈ Ek+1 := span{φ1 , . . . , φk , φk+1 } and uH 1 → ∞. There exists R > ρ such that J(u) ≤ 0 as u ∈ ∂BR (θ) ∩ Ek+1 . ˆ and ∂Q = (BR (θ) ∩ Ek ) ∪ (∂BR (θ) ∩ {(x1 , . . . , Setting S = ∂Bρ (θ) ∩ E xk+1 ) ∈ Ek+1 | xk+1 ≥ 0}). Then S and ∂Q link, and they separate values of J. Theorem 4.8.9 yields the existence of a critical value c ≥ α > 0, which implies the existence of a nontrivial solution of equation (4.89).

5 Topological and Variational Methods

This chapter is devoted to the critical point theory and Conley index theory. A typical problem from geometry in the calculus of variations is to ﬁnd geodesics between two points q0 , q1 on a given Riemannian manifold (M, g). A path on M connecting q0 and q1 is denoted by γ : [0, 1] → M with γ(i) = qi , i = 0, 1. Let N = {γ ∈ C 1−0 ([0, 1], M ) | γ(i) = qi , i = 0, 1} , and deﬁne the energy functional:

I(γ) = 0

1

dγ 2 dt , dt

where · is the scalar product induced by the metric g, or in the local coordinates: dγ 2 dγ dγ j , = Σi,j gij (γ(t)) dt dt dt and γ = (γ 1 , . . . , γ p ), p = dim M. Then the Euler–Lagrange equation reads as the geodesic equation: 2 d dγ j dγ k , i = 1, 2, . . . , p . γ i = Σj,k Γijk (γ) dt dt dt 2

Geometrically, a curve satisfying the geodesic equation has the feature that tangents along it are parallel. Generally speaking, there are many geodesics joining q0 and q1 , which are critical points of the energy functional. However, besides those minimal in length, how do we reach the others? Inﬂuenced by the pioneering work of Birkhoﬀ (1917) [Bi] on closed geodesics, two global methods have appeared: The minimax method and the Morse theory. Both are based on topological arguments. The minimax method, as an outgrowth of the max-min characterization of the eigenvalues of Laplacian with Dirichlet data, was successfully developed

316

5 Topological and Variational Methods

by Ljusternik and Schnirelmann (1934)[LjS 1]. The method provided a proof of the existence of at least three closed geodesics on a closed surface with genus zero, see Ljusternik and Schnirelmann [Lj, LjS 1], Ballmann [Bm], Klinenberg [Kl]. In the 1950s Krasnosel’ski [Kr 1] introduced the notion of genus, which is a topological index related to the group Z2 , in the study of the nonlinear eigenvalue problem for a class of integral equations. An important step towards the recent development of critical point theory is due to Palais (1966) [Pa 1, Pa 2]. Two major contributions are: (1) The Ljusternik Schnirelmann theory was extended to inﬁnite-dimensional manifolds. (2) Homotopically stable families were used in the minimax principle. The reviva of the study of the minimax method began with the works of Ambrosetti and Rabinowitz (1974)[AR] and Rabinowitz [Ra 3]. The ﬁrst paper deals with functionals unbounded from below, and provides many applications in the study of nonlinear elliptic equations, Hamiltonian systems, and problems from geometry and mathematical physics. The second paper provides a proof of the existence of a closed orbit of an autonomous Hamiltonian system on a star-shaped hypersurface in R2n (independently, see Weinstein [Wei 1]). This is the ﬁrst global result on this problem and is a breakthrough to the Weinstein conjecture. In parallell, in the 1930s, Morse developed a theory which reveals the relationship between the critical points of a nondegenerate function and the topology of the underlying compact manifold. Although the nondegeneracy condition and the compactness assumption in his theory do not meet in most variational problems, he succeeded in applying his theory in the study of closed geodesics and unstable minimal surfaces, see Morse [Mo], Milnor [Mi 1], Bott[Bo 1], Morse and Tompkins [MT], Schiﬀman [Sch]. Moreover, Morse theory became a basic tool in computing the homology of compact manifolds. The work of Smale on the solution of the Poincar´e conjecture for n ≥ 5 pushes the development of the theory to a new peak, see Smale [Sm 1], Milnor [Mi 3]. In the 1950s and 1960s Rothe [Ro 1–3], Palais [Pa 3], and Smale [Sm 2] extended the Morse theory to inﬁnite-dimensional manifolds by using the Palais–Smale condition. Later, Marino and Prodi [MP 1, MP 2] and Gromoll and Meyer [GM] endeavored in weakening the restriction of the nondegeneracy. In applying Morse theory to diﬀerential equations, most functionals are unbounded from below, and the nondegenerate assumption should be removed; to this end, critical groups for isolated critical points are introduced and the relative homology is involved, by which the interconnection with the minimax principle has been revealed, see K. C. Chang [Ch 4,5,7]. Conley extended the Morse theory to ﬂows on a compact space without a variational structure. The Conley index for isolated invariant sets with respect to the ﬂow is a homotopy invariant. A Morse decomposition is also extended. With the aid of Conley’s theory, Floer established his homology theory, which is an important tool in the symplectic geometry.

5.1 Morse Theory

317

In this chapter we shall introduce these theories as tools in the study of multiple solutions for diﬀerential equations. Morse theory is introduced in Sect. 5.1, we shall present it in a way comparing with the Leray–Schauder degree. After the basic theorems, we shall focus on the computation of critical groups for isolated critical points, in particular, those obtained by various minimax methods. Section 5.2 is on the minimax principles, which include the Ljusternik– Schnirelmann category theory, cap length estimates, Krasnosel’ski’s genus and other index theories. Various extensions can be found in Rabinowitz [Ra 4], [Ra 5]. The central result is the multiplicity theorem. Section 5.3 is deals with an application to the Weinstein conjecture, see Viterbo [Vi 1]. We return to the prescribed Gauss curvature problem on S 2 in Sect. 5.4. We shall apply the Morse theory with boundary conditions to attack the problem. We introduce the deﬁnitions of the isolated invariant set, the index pair, and the Conley index in metric space without compactness. The fundamental properties of the Conley index, in particular, the homotopy invariance and the Morse decomposition are studied. Examples and the relationship with the Morse theory are also presented.

5.1 Morse Theory 5.1.1 Introduction Morse revealed a deep relation between the critical points of any nondegenerate function and the topology of the underlying compact manifold M . Let f ∈ C 1 (M ). ∀a ∈ R1 , the set fa = {x ∈ M |f (x) ≤ a} is called a level set. We denote the set of all critical points by K. Here the terminology “critical point” is coincident with that for general diﬀerential mappings, which was introduced in Sect. 1.3.5, i.e., K = {x ∈ M | f (x) = θ}. A real value {c} is called critical if f −1 (c) ∩ K = ∅; otherwise, it is called regular. According to the Sard–Smale theorem, the set of critical values is of the ﬁrst category. A C 2 function f is called nondegenerate if it has only nondegenerate critical points. A critical point p is called nondegenerate if the Hessian f (p) at this point has a bounded inverse (“boundedness” is assumed if M is an inﬁnite-dimensional Hilbert–Riemannian manifold). As an application of the Sard–Smale theorem, the set of nondegenerate functions is dense in C 2 (M ). To a nondegenerate critical point p, we call the dimension of the subspace of negative eigenvectors of f (p) the Morse index, and denote it by ind(f (p)). The basic idea in the Morse theory is to relate the local behavior of a nondegenerate critical point with the variations of the topological structures of the level sets fa . More precisely, if f −1 ∩ K = {p} and ind(f (p)) = j, then fc+ ∼ fc− ∪ B j ,

318

5 Topological and Variational Methods

where B j is the j-ball, for > 0 small, “∪” means “attached by” and ∼ means homotopical equivalence. Since M is compact, the nondegenerate function f can only have ﬁnitely many critical points {p1 , . . . , pl }. Then the space M is homotopically equivalent to the space by gluing ﬁnitely many balls with dimensions {ind(f (pi ))|i = 1, . . . , l} according to their levels. The above consideration is based on the following Morse lemma. We assign a Riemannian structure M . Lemma 5.1.1 (Morse lemma) Suppose that f ∈ C 2 (M, R1 ) and that p is a nondegenerate critical point; then there exists a neighborhood U of p and a local diﬀeomorphism Φ : U → Tp (M ) with Φ(p) = θ, such that 1 f ◦ Φ−1 (ξ) = f (p) + (f (p)ξ, ξ)p 2

∀ξ ∈ Φ(U ) ,

where (, )p is the Riemannian structure at p. From the homology point of view, for a pair of topological spaces (A, B) with B ⊂ A, let Hq (A, B, G) be the qth relative homology group with coeﬃcient group G, q = 0, 1, 2, . . . . They are homotopically invariant. According to the Morse lemma, after a linear homeomorphism, roughly speaking, we may assume that f is a quadratic function in a neighborhood B (θ) of θ in Tp (M ) of the form: 1 f (x) = c + ( x+ 2 − x− 2 ) , 2 where x = x+ + x− ∈ H± , dimH− = j, and Rn = H+ ⊕ H− is an orthogonal decomposition, where n = dimM . To characterize homologically the nondegenerate critical point p with Morse index j, we have Hq (fc+ , fc− , G) = Hq (B j , G) = δqj G

∀q ∈ N .

Since p satisﬁes f (p) = θ, it is a nondegenerate zero of the mapping f : M −→ T ∗ (M ), the index of the Brouwer degree at p is well deﬁned and

i(f , p) = sgn det (f (p)) = (−1)j =

n

(−1)q rankHq (B j , G) .

q=0

If further, p is the only critical point in f −1 [c − , c + ], then as a local characterization of f at p, Hq (fc+ , fc− , G), q = 0, 1, . . . , n, is better than the Brouwer index. We are encouraged by using the homotopy invariance {Hq (fb , fa ; G)}nq=0 (for a < b) as a replacement for deg(f , f −1 (a, b), θ). Unfortunately there is no additivity (but only the subadditivity, see later). In the degree theory, the excision property and then the Kronecker existence,

5.1 Morse Theory

319

which makes it useful in the study of ﬁxed points, are based on the additivity; while for relative homology groups, the excision property is related to a deformation argument. In this context, we shall establish the counterpart of the Kronecker existence theorem by deformation as follows: Theorem 5.1.2 (Nontrivial interval theorem) If ∃q ∈ N and ∃a < b such that Hq (fb , fa ; G) is nontrivial, then K ∩ f −1 [a, b] = ∅. One proves the theorem by contradiction, i.e., if there is no critical value in the interval [a, b], then fb ∼ fa . In fact, a homotopy between fb and fa can be easily constructed by the negative gradient ﬂow: x(t) ˙ = −f (x(t)) ∀x ∈ f −1 [a, b] . Noticing that along the ﬂow line f is strictly decreasing and that f −1 [a, b] is compact, the proof is left to readers as an exercise. We shall present Morse theory in this book as a topological tool in the study of the existence and the multiplicity of solutions of certain nonlinear diﬀerential equations with variational structures. To our purpose the two assumptions: the compactness of manifolds and the nondegeneracy of functions, are too restricted. In dealing with inﬁnitedimensional manifolds, a certain kind of compactness is assumed on the function f , e.g., the Palais–Smale condition. Under this condition, deformation theorems are derived, which implies the noncritical interval theorem. Let us use singular relative homology groups with an Abelian coeﬃcient group G, H∗ (X, Y ; G) to describe the topological diﬀerence between the topological spaces X and Y , with Y ⊂ X. Namely, if H∗ (fb , fa ; G) is not trivial, then K ∩ f −1 [a, b] = ∅. This is the main result in Sect. 5.1.2. Instead of the nondegeneracy, we study a certain kind of isolatedness of critical points. A series of critical groups is introduced to replace the Morse index in describing the local behavior of isolated critical points. The basic properties and computations of critical groups are studied, e.g., the shifting theorem and that for mountain pass points. The local theory is described in Sect. 5.1.3. The Morse relation, which is the subadditivity of topological invariances Hq (fb , fa ; G), q = 0, 1, 2, . . . , links, on one hand, the global invariants: H∗ (fb , fa ; G), ∗ = 0, 1, . . . , and on the other hand, the local invariants: the critical groups for isolated critical points in f −1 [a, b]. It is applied to the estimation of the critical groups for isolated critical points obtained by minimax principles. It is also used to set up the relationship between the Leray– Schauder index and the critical groups for an isolated zero of a potential compact vector ﬁeld, by which, we see that for a potential compact vector ﬁeld, critical groups provide more information than the Leray–Schauder index. This is the contents of Sect. 5.1.4. 5.1.2 Deformation Theorem The following terminologies are used:

320

5 Topological and Variational Methods

Let X be a topological space. A deformation of X is a continuous map η : X × [0, 1] → X such that η(·, 0) = id. Deﬁnition 5.1.3 (Deformation retract) Given a topological pair Y ⊂ X. A continuous map r : X → Y is called a deformation retract, if r ◦ i = idY and i ◦ r ∼ idX , where i : Y → X is the injection. In this case Y is called a deformation retraction of X. Deﬁnition 5.1.4 (Strong deformation retract) A deformation retract r is called a strong deformation retract, if there exists a deformation η : X × [0, 1] → X, such that η(·, t)|Y = idY ∀t ∈ [0, 1] and η(·, 1) = i ◦ r. Then Y is called a strong deformation retraction of X. According to the homotopy invariance of the relative singular homology group H∗ (X, Y ; G) (where G is the coeﬃcient group), if Y is a strong deformation retraction of X, then H∗ (X, Y ; G) = 0. In the following, M is assumed to be a Banach–Finsler manifold with a Finsler structure · . But for pedagogical reasons, all of the following theorems are proved only on Banach spaces. Readers who are familiar with the background material on inﬁnite-dimensional manifolds may complete the proofs themselves or may refer to [Ch 1]. As we have seen in Sect. 5.1.1, the deformation is constructed by negative gradient ﬂow on Riemannian manifolds. However, if the C 1 function f is deﬁned on a Banach space M , f (x) ∈ M ∗ , the dual space of M , then the gradient ﬂow does not make sense. We introduce: Deﬁnition 5.1.5 (Pseudo-gradient vector ﬁeld) Let M be a Banach–Finsler manifold and f ∈ C 1 (M, R1 ). Let K be the critical set of f . A vector ﬁeld X over M is called a pseudo-gradient if (1) X(p) < 2|f (p)| and 1 := M \K, (2) f (p), X(p) > |f (p)|2 , ∀p ∈ M where , is the duality between T ∗ (M ) and T (M ), · and | · | stand for the Finsler structures on T (M ) and T ∗ (M ), respectively. Our main results of this section are the following: Theorem 5.1.6 (Noncritical interval theorem) If f ∈ C 1 (M, R1 ) satisﬁes (PS)c ∀c ∈ [a, b] and if K ∩ f −1 [a, b] = ∅ , then fa is a strong deformation retraction of fb . As a direct consequence, Theorem 5.1.2 is extended to Banach–Finsler manifolds for C 1 functions satisfying the (P S)c condition ∀c ∈ [a, b]. Theorem 5.1.7 (Second deformation theorem) If f ∈ C 1 (M, R1 ) satisﬁes (PS)c ∀c ∈ [a, b] and M is C 2 . If K ∩ f −1 (a, b] = ∅ and the connected components of K ∩ f −1 (a) are only isolated points, then fa is a strong deformation retraction of fb . We need the following lemmas:

5.1 Morse Theory

321

Lemma 5.1.8 (The existence of a pseudo-gradient vector ﬁeld) ∀f ∈ C 1 (M, R1 ), there exists a continuous pseudo-gradient vector ﬁeld of f on 1 = M \K. M 1, ∃ξ(p0 ) ∈ Tp (M ) such that ξ(p ) = 1 and f (p ), ξ(p ) > Proof. ∀p0 ∈ M 0 0 0 0 2 3 3 |f (p0 )|. Setting Xp0 = 2 |f (p0 )|ξ(p0 ), we have Xp0 < 2|f (p0 )|, and f (p0 ), Xp0 > |f (p)|2

∀p ∈ Vp0 .

1 is metrizable, it is paracompact. There is a locally ﬁnite partition of Since M 1. Let unity {ηβ |β ∈ B}, with supp ηβ ⊂ Vp0 for some p0 = p0 (β) ∈ M X(p) =

ηβ (p)Xp0 (β) .

β∈B

This is a required pseudo-gradient vector ﬁeld.

Lemma 5.1.9 If f satisﬁes (PS)c ∀c ∈ [a, b], and if K ∩ f −1 [a, b] = ∅, then ∃, δ > 0 such that |f (x)| ≥

∀x ∈ f −1 [a − δ, b + δ] .

Proof. If not, then there exists a sequence {xn } ⊂ f −1 [a − n1 , b + n1 ] such that |f (xn )| < n1 . We ﬁnd a subsequence {xn } such that f (xn ) → c ∈ [a, b] and f (xn ) → θ. According to (PS)c , we ﬁnd x∗ ∈ f −1 (c) ∩ K. This is a contradiction. Proof. (Proof of the nontrivial interval theorem) The deformation is constructed by the ﬂow, which deforms the level sets: σ(t) ˙ = −X(σ(t)) σ(0) = x0 ∈ f −1 [a, b] . 1. It is a decreasing ﬂow: d f (σ(t)) = f (σ(t)), σ(t)

˙ ≤ −|f (σ(t))|2 . dt 2. The ﬂow does not stop until it arrives at fa . Indeed, if the maximal existence time for the initial data x0 is Tx0 and f (σ(Tx0 )−0) > a, from Lemma 5.1.9:

t f (σ(ι)), σ(ι) dι ˙ < −2 t ∀ t < Tx0 , a − b ≤ f (σ(t)) − f (x0 ) = 0

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5 Topological and Variational Methods

then Tx0

0 such that lim f ◦ σ(t, x) = a. t→Tx −0

1. We claim that

lim σ(t, x) does exist and then

t→Tx −0

f (σ(Tx − 0, x)) = a . Indeed, (PS)a implies that Ka is compact. Either one of the following cases holds: (a) inf dist(σ(t, x), Ka ) > 0 , t∈[0,Tx )

(b) inf dist(σ(t, x), Ka ) = 0 . t∈[0,Tx )

In case (a), again by (PS)c ∀c ∈ [a, b], ∃α > 0 such that inf t∈[0,Tx )

f (σ(t, x)) ≥ α .

5.1 Morse Theory

323

Thus

dist(σ(t2 , x), σ(t1 , x)) ≤

t2

t1 t2

≤

t1

Since Tx is ﬁnite,

dσ dt dt dt ≤ X(σ(t, x))

t2

t1

|t2 − t1 | dt ≤ . f (σ(t, x)) α

lim σ(t, x) = z.

t→Tx −0

In case (b), we shall prove that ∃z ∈ Ka such that lim σ(t, x) = z .

t→Tx −0

First, we claim that lim dist(σ(t, x), Ka ) = 0 .

t→Tx −0

If not, ∃0 > 0, ∃ti → Tx − 0, such that dist(σ(ti , x), Ka ) ≥ 0 . By the assumption (b), ∃ti → Tx − 0 such that lim dist(σ(ti , x), Ka ) = 0 .

i→∞

Thus we have two sequences t∗i < t∗∗ i both converging to Tx such that 0 , 2 ∗∗ dist(σ(ti , x), Ka ) = 0 , dist(σ(t∗i , x), Ka ) =

and

σ(t, x) ∈ (Ka ) \(Ka )◦0 ∀t ∈ [t∗i , t∗∗ i ], 0

2

where (Ka )δ denotes the δ-neighborhood of Ka . Again by (PS)c ∀c ∈ [a, b], we have inf

∗∗ t∈[t∗ i ,ti ]

f (σ(t, x)) ≥ α > 0 .

Therefore 0 ∗ ≤ dist(σ(t∗∗ i , x), σ(ti , x)) 2

t∗∗ i dσ dt ≤ ∗ dt tt 1 ≤ |t∗∗ − t∗i | → 0 . α i

(5.1)

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5 Topological and Variational Methods

This is a contradiction, (5.1) is proved. It follows that lim f (σ(t, x)) = θ , t→Tx −0

from the compactness of Ka . The (PS)a condition then implies that the limit set A of the orbit {σ(t, x)|t ∈ [0, Tx )} is nonempty and that, for each sequence ti → tx−0 , there ti , x) is convergent. exists a subsequence ! ti such that σ(! Next we prove that A is a compact connected subset of Ka , and then a single point, according to the assumption that the connected components of Ka are isolated points. The compactness of A is obvious. We only want to prove the connectedness. If not, ∃ open subsets O and O such that O ∩ O = ø, A = (O ∩ A) ∪ (O ∩ A) , and O ∩ A = ø, O ∩ A = ø. Choosing z ∈ O ∩ A, z ∈ O ∩ A, ∃ti → Tx − 0, ti → Tx − 0 satisfying σ(ti , x) → z, σ(ti , x) → z . For large i, we have σ(ti , x) ∈ O, σ(ti , x) ∈ O , so ∃t∗i ∈ [ti , ti ] (or [ti , ti ]) such that σ(t∗i , x) ∈ O ∪ O . Denote the limit of σ(t∗i , x) by z ∗ . (It exists because of (PS)a .) So z ∗ ∈ A, but z ∗ ∈ O ∪ O . This is a contradiction. We conclude that A = {z} ⊂ Ka . In both cases, claim 1 is proved. 2. We shall prove the continuity of the function Tx . As in the proof of Theorem 5.1.6, if σ(Tx0 − 0, x0 ) ∈ Ka , then the function Tx is continuous at x0 . So we restrict ourselves to the case z = σ(Tx0 −0, x0 ) ∈ Ka . If Tx is not continuous at such a x0 , then ∃0 > 0, ∃xn → x0 such that |Txn − Tx0 | ≥ 0 , so there exists a subsequence either Txn ≤ Tx0 − 0 , or Txn ≥ Tx0 + 0 . Since

f ◦ σ(Tx − , x) − f ◦ σ(t, x) =

Tx −

f (σ(t, x))dt

t

1 ≤ − (Tx − − t) , 4

5.1 Morse Theory

325

we have

1 f ◦ σ(t, x) ≥ a + (Tx − − t) . 4 But for any ﬁxed > 0, according to the continuous dependence of the initial data ODE, dist(σ(Tx0 − , xn ), σ(Tx0 − , x0 )) → 0 . If Txn ≥ Tx0 + 0 , then we have f ◦ σ(Tx0 − , x − 0) = lim f ◦ σ(Tx0 − , xn ) n→∞ 1 ≥ lim a + (Txn − Tx0 + ) n→∞ 4 1 ≥ a + (0 + ) . 4 Letting → 0, we obtain a≥a+

0 . 4

This is a contradiction. Similarly, we prove that Txn ≤ Tx0 − 0 is impossible. Therefore, Tx is continuous. 3. Finally, we deﬁne the deformation retract as follows: ⎧ if (t, x) ∈ [0, 1] × fa , ⎨x if (t, x) ∈ [0, 1) × (fb \(fa ∪ Kb )) , η(t, x) = σ(Tx t, x) ⎩ σ(Tx − 0, x) if (t, x) ∈ {1} × (fb \(fa ∪ Kb )) . Claim. Only the continuity of η has to be veriﬁed. Four cases are distinguished: (a) (b) (c) (d)

◦

(t, x) ∈ [0, 1] × f a , (t, x) ∈ [0, 1) × (f −1 (a, b]\Kb ), (t, x) ∈ {1} × (f −1 (a, b]\Kb ), (t, x) ∈ [0, 1] × f −1 (a).

Only cases (c) and (d) have to be veriﬁed. Since their proofs are similar, we only give the veriﬁcation for (c). If η is discontinuous at (1, x0 ), then ∃ > 0, ∃tn → Tx0 − 0, and ∃xn → x0 such that dist(σ(tn , xn ), σ(Tx0 − 0, x0 )) ≥ . Let z = σ(Tx0 − 0, x0 )(∈ Ka ), and let ◦

F1 = {z}, F2 = (M \B (z)) ∩ Ka . Both F1 and F2 are compact subsets of Ka . Provided by the assumption of Ka , and Lemma 3.5.2, we have compact subsets K1 , K2 ⊂ Ka such that

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5 Topological and Variational Methods

K1 ∩ K2 = ∅, Fi ⊂ Ki , i = 1, 2, and K1 ∪ K2 = Ka . Obviously, we may take ◦

◦

K1 ⊂ B (z). Let N = K2 ∪ (M \B (z)), then α = dist(N, K1 ) > 0 . The continuity of the ﬂow as well as of the arrival time Tx implies that ∃δ > 0 such that α dist(σ(t, x0 ), σ(Tx0 − 0, x0 )) < 8 if t ∈ [Tx0 − δ, Tx0 ], and ∃δ1 > 0 such that Tx > Tx0 − δ if x ∈ Bδ (x0 ). For t ∈ [Tx0 − δ, Tx0 ) ∩ [Tx0 − δ, Tx ), x ∈ Bδ1 (x0 ), ∃t ∈ (0, δ) such that α dist(σ(t, x), σ(t, x0 )) < . 8 In summary, for such a t, and for any x ∈ Bδt (x0 ), α neighborhood of K1 . σ(t, x) ∈ (K1 ) α4 , the 4 For large n, xn ∈ Bδt (x0 ), let tn be such a t, satisfying σ(tn , xn ) ∈ (K1 ) α4 . Reducing δ, and repeating the above procedure, we obtain subsequences {xn , tn , tn } such that ⎧ ⎨ tn , tn → Tx0 − 0 σ(tn , xn ) ∈ (K1 ) α4 ⎩ σ(tn , xn ) ∈ B (z) . We may assume tn < tn ; then we have tn , tn such that tn ≤ tn < tn ≤ tn and σ(tn , xn ) ∈ ∂[(K1 ) α4 ] σ(tn , xn ) ∈ ∂[(N ) α4 ] σ(t, xn ) ∈ (K1 ) α4 ∪ (N ) α4 ∀t ∈ [tn , tn ] , n = 1, 2, . . . . According to (PS)c , ∀c ∈ [a, b] β = inf { f (x) |x ∈ f −1 [a, b)\((K1 ) α4 ∪ (N ) α4 )} > 0 . Therefore α ≤ dist(σ(tn , xn ), σ(tn , xn )) ≤ 2

tn

tn

1 dσ dt ≤ |tn − tn | → 0 . dt β

This is a contradiction. The proof is completed.

5.1 Morse Theory

327

Remark 5.1.10 A ﬂow on a metric space M is a continuous map η : M × R1 → M possessing the following properties: (i) η(x, 0) = x ∀x ∈ M , (ii) η(η(x, t1 ), t2 ) = η(x, t1 + t2 )

∀t1 , t2 ∈ R1 , ∀x ∈ M .

A set A ⊂ M is said to be (positively) invariant with respect to η, if η(x, t) ∈ A

1 ∀x ∈ A, ∀t ∈ R1 (∀t ∈ R+ ).

! = ∪ η(A, t) (A+ = ∪ η(A, t)) is the By deﬁnition, for any subset A, A 1 1 t∈R

t∈R+

smallest η- (positively) invariant set containing A. With these terminologies, it is easy to extend Theorems 5.1.6 and 5.1.7 to pseudo-gradient ﬂow invariant sets. Namely, let η be a pseudo-gradient ﬂow for f ∈ C 1 (M, R1 ) satisfying (PS)c ∀c ∈ [a, b], and let A be a η-positively invariant set. If K ∩ f −1 (a, b] ∩ A = ∅, and the connected components of K ∩ f −1 (a) ∩ A are only isolated points, then fa ∩ A is a strong deformation retraction of fb ∩ A. 5.1.3 Critical Groups Deﬁnition 5.1.11 Let f be a C 1 function deﬁned on M , let p be an isolated critical point f , and let c = f (p). Cq (f, p) = Hq (fc ∩ U, (fc \{p}) ∩ U ; G) is called the qth critical group of f at p, q = 0, 1, 2, . . . , where U is an isolated neighborhood of p, i.e., K ∩ U = {p}. According to the excision property of the singular homology theory, the critical groups are well deﬁned, i.e., they do not depend on the special choice of U . From the deﬁnition, we have: Example 1. If p is an isolated minimum point of f , then Cq (f, p) = δq0 · G . Under some additional conditions, the converse of the above statement is true. Namely: If f ∈ C 2−0 (M, R1 ) satisﬁes (PS), and if p is an isolated critical point, which is not a local minimum point, then C0 (f, p) = 0 . See [Ch 5].

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5 Topological and Variational Methods

Example 2. If M is n-dimensional, and p is an isolated local maximum point of f , then Cq (f, p) = δqn · G . In the following, we assume that M is a Hilbert–Riemannian manifold. The Morse lemma, which is a special case of the following more general decomposition theorem – the splitting theorem – proved later, is a cornerstone in studying the local behavior of a nondegenerate critical point. Deﬁnition 5.1.12 Let p be a nondegenerate critical point of f , we call the dimension of the negative space corresponding to the spectral decomposition of f (p), the Morse index of p, and denote it by ind(f, p). Example 3. Suppose that f ∈ C 2 (M, R1 ) and p is a nondegenerate critical point of f with Morse index j; then Cq (f, p) = δqj · G . Proof. According to the Morse lemma, after a linear homeomorphism, we may assume that f is a quadratic function on a Hilbert space H of the form: f (x) =

1 ( x+ 2 − x− 2 ) , 2

where x = x+ + x− , x± ∈ H± , and H = H+ ⊕ H− is an orthogonal decomposition. Let U = B be the -ball centered at θ: B ∩ f0 = {x ∈ B | x+ ≤ x− } . Deﬁne η(x, t) = x− + tx+

∀(x, t) ∈ (B ∩ f0 ) × [0, 1] .

It is a strong deformation retract from (B ∩ f0 , B ∩ (f0 \{θ})) to (H− ∩ B , (H− \{θ}) ∩ B ). Thus Cq (f, q) ∼ = Hq (H− ∩ B , (H− \{θ}) ∩ B ) ∼ = Hq (B j , S j−1 ) = δqj · G , if j < +∞. Nevertheless, for j = +∞, since S ∞ is contractible, again, we have Cq (f, p) ∼ =0. The conclusion is proved.

Theorem 5.1.13 (Splitting) Suppose that U is a neighborhood of θ in a Hilbert space H, and that f ∈ C 2 (U, R1 ). Assume that θ is the only critical point of f , and that A = f (θ) is a Fredholm operator. Then there exist a ball B ⊂ U centered at θ, an origin-preserving local homeomorphism ϕ deﬁned on B, and a C 1 map h : B ∩ N → N ⊥ , where N = ker(A) such that

5.1 Morse Theory

f ◦ ϕ(z + y) =

1 (Az, z) + f (y + h(y)) 2

329

∀x ∈ B ,

where y ∈ N and z ∈ N ⊥ , with x = y + z. Proof. • Decomposing H into N ⊕ N ⊥ , let P be the orthogonal projection onto N ⊥ . We solve the equation for ﬁxing y ∈ B ∩ N : P f (y + z) = θ1

(θ1 and θ2 are the origins in N ⊥ and N resp.) .

Since f (θ1 + θ2 ) = θ1 , and f (θ1 + θ2 ) = A, by the implicit function theorem, there exist a ball B and h : B ∩ N → N ⊥ satisfying P f (y + h(y)) = θ1

∀y ∈ B ∩ N .

• Setting u = z − h(y), and letting F (u, y) = f (z + y) − f (h(y) + y) , F2 (u) = 12 (Au, u) , we have F (θ1 , y) = 0 Fu (θ1 , y) = P f (y + h(y)) = θ1 Fu (θ1 , θ2 ) = P f (θ) = A|N ⊥

, , .

Deﬁne ξ : (u, y) → u0 ∈ F2−1 ◦ F (u, y) ∩ {η(u, t)||t| F (u, y) > F2 (η(u, t)) holds for = 2 u ≤ t ≤ u . 1− 1− C We conclude the existence and the uniqueness of the function t(u, y), with = 2 |t(u, y)| ≤ 1 − 1 − u . C Remark 5.1.14 In the case N = {0}, the Morse lemma is a consequence of the above theorem. It remains to verify that ϕ is a diﬀeomophism. We omit the veriﬁcation, in fact, a weaker version is suﬃcient to prove Example 3. Remark 5.1.15 In the study of nonlinear PDEs, sometimes we work on Banach spaces rather than Hilbert spaces. We note that the above theorem can be applied under the following assumptions: Let X be a Banach space, embedded continuously into a Hilbert space H as a dense linear subspace. Assume: (A1) There are Banach spaces as follows: X = Xn

Xn−1 K

...

Yn−1

G

K

G

...

X0 = H

X1

Y1

Fig. 5.2.

(A2) f ∈ C 2 (H, R1 ) has a derivative of the form f (x) = x − K · G(x) ,

K

G Y0

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5 Topological and Variational Methods

where K : Yi → Xi+1 is linear and continuous, and G : Xi → Yi is continuous, i = 0, 1, . . . , n − 1. Let A = I − K · G (θ), and let H∗ , ∗ = −, 0, +, be the negative, null, and positive spaces resp., according to the spectral decomposition of A. (A3) H0 ⊂ X. A map is called regular, if it maps X into X. We claim that the splitting theorem remains true for the Banach space X if assumptions (A1 ), (A2 ) and (A3 ) hold. To this end, we continue to use the notations ϕ, ξ, η, h and B of Theorem 5.1.13, and prove that ϕ is regular. It is suﬃcient to show that the map ψ : (u, y) −→ (ξ(u, y), y) is regular, and that Im h ⊂ X. Indeed, the ﬂow η is a reparametrization of the following ﬂow ζ: ˙ ζ(s) = −ζ(s) + K · G (θ)ζ(s) ζ(0) = u . Or equivalently, −t

t

ζ(t) = e u + K · G (θ)

et−s ζ(s)ds . 0

By a bootstrap procedure, we have ζ ∈ X whenever u ∈ X ∩ B. This shows the regularity of ψ. Again, as an element in H, z = h(y) satisﬁes the equation: P f (y + z) = θ , i.e., h(y) = −P K · G(y + h(y)) + (I − P )h(y) . Noticing that (I − P )H = H0 ⊂ X, P is regular. By the same reason, P Xi ⊂ Xi , i = 0, 1, . . . , n. After a bootstrap iteration, it follows that Im h ⊂ X. Theorem 5.1.16 Under the assumptions of the splitting theorem, (A1), (A2) and (A3), we have Cq (f, θ) = Cq (f |X , θ) .

Proof. We use ϕ and B as in the splitting theorem, and choose U = ϕ(B). Deﬁne a deformation η : U × [0, 1] → U as follows: η(x, t) = ϕ(tx+ + x− ) , where x+ + x− = ϕ−1 (x), and x+ ∈ H+ , x− ∈ H− ⊕ H0 . Then η is a deformation retract fc ∩ U → fc ∩ U− , where U− = ϕ(B ∩ (H− ⊕ H0 )). Therefore C∗ (f, θ) = H∗ (fc ∩ U, fc ∩ U \{θ})

5.1 Morse Theory

333

= H∗ (fc ∩ U− , fc ∩ U− \{θ}) , where c = f (θ). Since ϕ is regular, we also have C∗ (fˆ, θ) = H∗ (fˆc ∩ U− , fˆc ∩ U− \{θ}) , where fˆ = f |X . However, fc ∩ U− = fˆc ∩ U− from (A3). This proves our conclusion. As a consequence of the splitting theorem, we have: Theorem 5.1.17 (Shifting) Assume that the Morse index of f at an isolated critical point p is j < +∞. Under the assumptions of the splitting theorem, we have Cq (f, p) = Cq−j (f!, θ) , where f!(y) = f (y + h(y)) as in the splitting theorem. Proof. We may assume that f is of the form f (z + y) = f1 (z) + f (y + h(y)), where f1 (z) = 12 ( z+ 2 − z− 2 ), z = z+ +z− , (z+ , z− ) ∈ H+ ⊕H− = N ⊥ , and y ∈ N , where the notations are inherited from the splitting theorem. Since dimN < ∞, it is easy to construct a new function f2 such that f2 equals f (y + h(y)) in a neighborhood of θ in N , and f2 satisﬁes the (PS) condition. Let Ui be an isolated neighborhood of θi for fi in the space Yi , i = 1, 2, where Y1 = N ⊥ and Y2 = N . Suppose Ui ⊂ fi−1 [−, ], i = 1, 2, for some !i ∩ f −1 [−, ], then Wi is again an isolated neighborhood > 0. Let Wi = U i !i is the smallest η-invariant set containing Ui , i = 1, 2. of θi for fi , where U We have a strong deformation retract ψ deforming (f2 )ε to (f2 )0 , according to the second deformation theorem. One constructs a deformation as follows: ϕ(z, y, t) = z− + (1 − t)z+ + ψ(y, t) . Accordingly, C∗ (f, p) ∼ = H∗ (f0 ∩ (W1 × W2 ), (f0 ∩ (W1 × W2 )\{(θ1 , θ2 )}); G) ∼ = H∗ (((f1 )0 ) ∩ W1 ) × ((f2 )0 ∩ W2 ) , ((f1 )0 ) ∩ W1 ) × ((f2 )0 ∩ W2 )\{(θ1 , θ2 )}; G) ∼ = H∗ ((f1 )0 ∩ W1 , (f1 )0 ∩ W1 \{θ1 }; G) ⊗H∗ ((f2 )0 ∩ W2 , (f2 )0 ∩ W2 \{θ2 }; G) ∼ C∗ (f1 , θ1 ) ⊗ C∗ (f2 , θ2 ) = Provided by the Kunneth formula, where we omit the veriﬁcation of the pair ((((f1 )0 ∩ W1 )\{θ1 }) × ((f2 )0 ∩ W2 ), ((f1 )0 ∩ W1 ) × ((f2 )0 ∩ W2 )\{θ2 }) is an excision couple. By the use of Example 3, we obtain Cq (f, p) = Cq−j (f2 , θ) . Since f2 (y) = f (y + h(y)) = f!(y), this proves our conclusion.

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5 Topological and Variational Methods

Corollary 5.1.18 Under the assumption of the shifting theorem with dimN = k, if p is: (1) a local minimum point of f!, then Cq (f, p) = δqj · G; (2) a local maximum point of f!, then Cq (f, p) = δq(j+k) · G; (3) neither a local minimum point nor a local maximum point, then Cq (f, p) = 0 for q ≤ j and q ≥ j + k . Deﬁnition 5.1.19 (Mountain pass point) An isolated critical point p of f is called a mountain pass point if C1 (f, p) = 0. Theorem 5.1.20 Suppose that f ∈ C 2 (M, R1 ) has a mountain pass point p, and that f (p) is a Fredholm operator satisfying the condition: (Φ) f (p) ≥ 0 and 0 ∈ σ(f (p)) =⇒ dim ker f (p) = 1 . Then Cq (f, p) = G · δq1 .

Proof. Let j = ind(f (p)). If p is nondegenerate, then by Example 3: Cq (f, p) = δqj ·G; we have j = 1. The conclusion follows. Otherwise, from the shifting theorem, Cq (f, p) = Cq−j (f!, p); we obtain j ≤ 1. In the case where j = 1, C0 (f!, θ) = 0. θ is a local minimum of f , from Example 1, Cq (f, p) = Cq−1 (f!, θ) = δq1 · G. In the case where j = 0, C1 (f!, θ) = 0. Now, dim ker (f (p)) = 1, θ is a local maximum. From Example 2, Cq (f, p) = Cq (f!, θ) = δq1 · G. The proof is complete. 5.1.4 Global Theory For a function f ∈ C 1 (M, R1 ) satisfying the (PS)c ∀c ∈ [a, b], where a, b are regular values, we know from the nontrivial interval theorem that the nontriviality of H∗ (fb , fa ; G) implies the existence of a critical point in f −1 (a, b). In this subsection we shall show that the topological invariance H∗ (fb , fa ; G) possesses the homotopy invariance and the subadditivity as follows. Theorem 5.1.21 Suppose that {fσ ∈ C 1 (M, R1 )|σ ∈ [0, 1]} is a family of functions satisfying the (PS) condition. Assume that a(σ) and b(σ) are two continuous functions deﬁned on [0, 1] with a(σ) < b(σ), and that both a(σ) and b(σ) are regular values of fσ , ∀σ ∈ [0, 1]. Assume that σ → fσ is continuous in C 1 (M ) topology. Then the homology group H∗ ((fσ )b(σ) , (fσ )a(σ) ; G) is independent of σ.

5.1 Morse Theory

335

Proof. For simplifying notations, ∀σ0 , σ1 ∈ [0, 1], we write fσi , a(σi ), b(σi ), (a(σi ), b(σi )) as fi , ai , bi and Ki , respectively, i = 0, 1. and K(fσi ) ∩ fσ−1 i As |σ1 − σ0 | is small, we have c < d such that f0 (K0 ) ⊂ (c, d) ⊂ [c, d] ⊂ (a0 , b0 ) ∩ (a1 , b1 ) . The ﬁrst inclusion is due to the (PS) condition for f0 . By the continuity of f0 , we have δ > 0 such that f0 ((K0 )δ ) ⊂ (c, d) , where (K0 )δ is the δ-neighborhood of K0 . Again by the (PS) condition of f0 , ∃ = (δ) > 0 such that f0 (x) ≥ ∀x ∈ f0−1 [a0 , b0 ]\(K0 )δ . As |σ1 − σ0 | is small, we have K1 ⊂ (K0 )δ , f1 ((K0 )δ ) ⊂ (c, d) and f1−1 [c, d] ⊂ f0−1 (a, b). Thus fi (Kj ) ⊂ (c, d), i, j = 0, 1. Again as |σ1 − σ0 | is small, we can construct a pseudo-gradient vector ﬁeld for f1 , which coincides with that of f0 in ((f1 )b1 ∩ (f0 )b0 )\(K0 )δ . Thus the pseudo-gradient ﬂow deﬁnes strong deformation retracts ((f1 )b1 , (f0 )c ∩ (f1 )c ) → ((f0 )d ∩(f1 )d , (f0 )c ∩(f1 )c ) and (f1 )c → (f0 )c ∩(f1 )c , according to the nontrivial interval theorem. In the same manner we have strong deformation retracts ((f0 )b0 , (f0 )c ∩ (f1 )c ) → ((f0 )d ∩ (f1 )d , (f0 )c ∩ (f1 )c ) and (f0 )c → (f0 )c ∩ (f1 )c . Therefore the exactness of the homology sequence yields: H∗ ((f0 )b0 , (f0 )c ; G) = H∗ ((f1 )b1 , (f1 )c ; G) . Again by the nontrivial interval theorem, there are strong deformation retracts (f0 )c → (f0 )a0 , (f1 )c → (f1 )a1 . We arrive at H∗ ((f0 )b0 , (f0 )a0 ; G) = H∗ ((f1 )b1 , (f1 )a1 ; G) .

Remark 5.1.22 One may use other homology theories instead, e.g., the sin¯ ∗ (A, B; F ), gular cohomology H ∗ (A, B; F ) or Alexander–Spanier cohomology H with a coeﬃcient ﬁeld F , etc. Since they share most common properties but have only a few diﬀerences, we use special homology theory for special problems. The homotopy invariance H∗ (fb , fa ; G) can be localized on a pseudogradient ﬂow invariant set. For details see Chang and Ghoussoub [CG]. We present here a special case, which will be used later. Let f ∈ C 1 (M, R1 ) satisfy the (PS) condition, and let a < b be regular values. Since K ∩ f −1 [a, b] is compact, it has a bounded neighborhood U ⊂ ! be the η-invariant set for a pseudo-gradient ﬂow η for f . f −1 [a, b]. Let U ! ∩ f −1 [a, b] is bounded. Lemma 5.1.23 The set W = U

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5 Topological and Variational Methods

Proof. From the (PS) condition, we have > 0 such that f (x) ≥

∀x ∈ f −1 [a, b]\U .

∀y ∈ W \U , there exist x ∈ ∂U and t > 0 (or t < 0) such that y = η(x, t) and ! \U ∀s > 0 (∀s < 0 resp.). Let X be the pseudo-gradient vector η(x, s) ∈ U ﬁeld with respect to η. Thus

t 2 f (η(x, s)) 2 ds t≤ 0

t f (η(x, s)), X(η(x, s)) ds ≤ 0

t d f (η(x, s))ds = f (x) − f (y) ≤ b − a =− ds 0 as t > 0. We obtain the estimate 0 < t ≤ b−a 2 ; similarly, for t < 0, by which, we have the estimate

t y − x = η(x, t) − x ≤ η(x, ˙ s) ds 0

≤2 t

t

f (η(x, s)) 2 ds

1/2

0

≤ 2(t(b − a))1/2 ≤

2(b − a) .

Therefore W is bounded.

Since the regular set is open, we have regular values c, d such that [a, b] ⊂ ! ∩ f −1 [c, d]. (c, d). Let W = U Corollary 5.1.24 With the above notations, the singular homology groups ! , fa ∩ U ! ; G) are invariant under small C 1 (W ) perturbation of f , H∗ (fb ∩ U i.e., if g − f C 1 (W ) is small and both f and g satisfy the (PS) condition, then ! , fa ∩ U ! ; G) = H∗ (gb ∩ U ! , ga ∩ U ! ; G) . H∗ (fb ∩ U

Corollary 5.1.25 With the (PS) condition, the critical groups C∗ (f, p) for an isolated critical point p are invariant under C 1 topology on any bounded neighborhood of p. Now we turn to the subadditivity. Let S be an integer-valued function on certain pairs of spaces. S is called subadditive if S(X, Z) ≤ S(X, Y ) + S(Y, Z), whenever Z ⊂ Y ⊂ X .

5.1 Morse Theory

337

Example. ∀q ∈ N, rank Hq (X, Y ; G) is subadditive. In fact, by examining the exact sequence: j∗

i

∂

∗ Hq (X, Y ; G) →∗ Hq−1 (Y, Z; G) → · · · · · · → Hq (Y, Z; G) → Hq (X, Z; G) →

we obtain rank Hq (X, Z; G) = rank Im i∗ + rank ker i∗ = rank Im i∗ + rank Im j∗ ≤ rank Hq (X, Y ; G) + rank Hq (Y, Z; G) . Applying this to the triple (fc , fb , fa ) with regular values c > b > a, we have rank Hq (fc , fa ; G) ≤ rank Hq (fb , fa ; G) + rank Hq (fc , fb ; G) . If further, we write q (X, Y ) = rank Im i∗ , q (x, z) = rank Im j∗ and q−1 (Y, Z) = rank Im ∂∗ , then rank Hq (X, Y ) = q (X, Y ) + q−1 (Y, Z) , rank Hq (X, Z) = q (X, Z) + q (X, Y ) , rank Hq (Y, Z) = q (Y, Z) + q (X, Z) . Summing them up, we obtain P (t; X, Y ) + P (t; Y, Z) = P (t; X, Z) + (1 + t)Q(t; Y, Z) , where P (t; X, Y ) =

∞

tq rank Hq (X, Y ; G), and Q(t; Y, Z) =

q=0

∞

tq q (Y, Z)

q=0

are formal series with nonnegative coeﬃcients. If there is a multiple X0 ⊂ X1 ⊂ X2 ⊂ · · · ⊂ Xn , we have n

P (t, Xj , Xj−1 ) = P (t, Xn , X0 ) + (1 + t)Q(t) .

(5.2)

j=1

where Q is a formal series with nonnegative coeﬃcients. Thus, ∀ t ≥ 0, P (t; X, Y ) is subadditive, and P (−1, X, Y ) is additive. We turn to establishing the relationship between the singular relative homology groups H∗ (fb , fa , G) and the critical groups C∗ (f, p). This is the Morse relation. Suppose that f ∈ C 1 (M, R1 ) has only isolated critical values, and that each of them corresponds to a ﬁnite number of critical points; say · · · < c−2 < c−1 < c0 < c1 < c2 < · · ·

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5 Topological and Variational Methods

are critical values with i K ∩ f −1 (ci ) = {zji }m j=1 , i = 0, ±1, ±2, . . . .

One chooses 0 < i < max {ci+1 − ci , ci − ci−1 }, i = 0, ±1, ±2, . . . . Deﬁnition 5.1.26 For a pair of regular values a < b, we call Mq (a, b) = rank Hq (fci +i , fci −i ; G) aδ, and x0 ∈ H will be determined later. Let β = inf { f (x) x ∈ Bδ \Bδ/2 } , The (PS) condition of f yields β > 0. We choose x0 ∈ H such that 0 0 fc+ would be a strong deformation retract of H. Since m is the minimum, fm− = ∅, it follows that χ(fc+ , fm− ) = χ(H) = 1. According to the Morse equality, we would have 1 + (−1)j = 1 .

This is a contradiction.

Example 1. Let Ω ⊂ Rn be a bounded domain with smooth boundary; we study the problem: −u = g(u) in Ω , (5.5) u=0 on ∂Ω . Let λ1 < λ2 ≤ · · · be the eigenvalues of − with the Dirichlet condition, and let ci , i = 1, 2, 3 be various constants. Assume (g1) g ∈ C 1 (R1 ) with g(0) = 0. (g2) |g (t)| ≤

c1 (1 + |t|p−1 ), p < no restriction

n+2 n−2

t (g3) G(t) := 0 g(s)ds ≤ c2 |t|2 + c3 , where c2 < λ1 /2. (g4) ∃i ≥ 1 such that λi < g (0) ≤ λi+1 .

if n ≥ 3 , if n ≤ 2 .

Theorem 5.1.36 The problem (5.5) under assumptions (g1)–(g4) has at least three distinct solutions.

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5 Topological and Variational Methods

Proof. Deﬁne

J(u) = Ω

1 |∇u|2 − G(u) on H01 (Ω) . 2

According to (g1) and (g2), J is C 2 . (g3) implies that J is bounded below. g(0) = 0 means that θ is a critical point of J. The veriﬁcation of the (PS) condition follows from (g2) and (g3). In fact, (g3) implies that a (PS) sequence must be bounded in H 1 , and (g2) implies that |g(t)| ≤ c4 (1+|t|p ), the compact embedding is applied. Since J (θ) = id − g (0)(−∆)−1 , (g4) implies that θ is nondegenerate with ﬁnite index ind(J (θ)) = i ≥ 1. 2. Bifurcation We know from Theorem 3.5.1 that on a Banach space, for a compact vector ﬁeld with parameter λ : F (x, λ) = x − λT x + N (x, λ), where T is compact, N (x, λ) = ◦( x ) uniformly in λ, the bifurcation point occurs at eigenvalues λ with odd multiplicity. Generally speaking, the odd multiplicity condition cannot be removed. However, if ∀ λ ∈ R1 , F (·, λ) is a potential operator, i.e., the diﬀerential of a certain functional with a parameter λ, we have: Theorem 5.1.37 Suppose that H is a Hilbert space, and that f ∈ C 2 (H, R1 ) satisﬁes f (θ) = θ. If λ0 is an isolated eigenvalue of the self-adjoint operator f (θ) with ﬁnite multiplicity, then (θ, λ0 ) is a bifurcation point of the equation: F (u, λ) = f (u) − λu = θ .

(5.6)

Let f (u) = 12 (Lu, u) + g(u), where g(u) = o(u2 ), then L = f (θ), and f (u) = Lu + G(u), with G(u) = g (u), G(u) = ◦( u ) as u → 0. The proof depends upon the Lyapunov–Schmidt reduction. Let X = ker (L − λ0 I), with dim X = n; and let P, P ⊥ be the orthogonal projections onto X and X ⊥ , respectively. Then (5.6) is equivalent to a pair of equations

λ0 x + P G(x + x⊥ ) = λx , Lx⊥ + P ⊥ G(x + x⊥ ) = λx⊥ ,

(5.7) (5.8)

where u = x + x⊥ , x ∈ X, x⊥ ∈ X ⊥ . According to the IFT, equation (5.8) is uniquely solvable in a small bounded neighborhood O of (λ0 , θ) ∈ R1 × X, say x⊥ = ϕ(λ, x) for (λ, x) ∈ O, where ϕ ∈ C 1 (O, X ⊥ ). Substituting x⊥ = ϕ(λ, x) into (5.7), we have (5.9) λ0 x + P G(x + ϕ(λ, x)) = λx , which is again an Euler–Lagrange equation on the ﬁnite-dimensional space X. Indeed, let

5.1 Morse Theory

345

λ ( x 2 + ϕ(λ, x) 2 ) 2 λ 1 1 = (λ0 − λ) x 2 + (Lϕ, ϕ) − ϕ 2 +g(x + ϕ) . 2 2 2

Jλ (x) = f (x + ϕ(λ, x)) −

It is easy to verify that (5.9) is the Euler–Lagrange equation for Jλ , and that ϕ(λ, x) = ◦( x ) as x → θ. The problem is reduced to ﬁnding the critical points of Jλ near x = θ for ﬁxed λ near λ0 , where Jλ ∈ C 1 (Ω1 , R1 ), Ω1 is a neighborhood of θ in X. If (θ, λ0 ) were not a bifurcation point, then there would be a small neighborhood Ω1 × (λ0 − δ, λ0 + δ), δ > 0 such that θ is the unique critical point in Ω1 for Jλ as |λ − λ0 | < δ. However, as λ < λ0 , θ is a local minimum point, Cq (Jλ , θ) = δq0 G, but as λ > λ0 , θ is a local maximum point, Cq (Jλ , θ) = δqn G. This contradicts the invariance of critical groups under C 1 -perturbation on bounded neighborhoods of θ. Note: From the degree point of view, there is no diﬀerence between λ > λ0 and λ < λ0 , except n is odd. Compare with Krasnoselski’s theorem (Theorem 3.5.1). 3. Superlinear Elliptic Problem We turn to the problem studied previously in Sect. 3.6 and Sect. 4.8. Example 2.

−u = g(x, u) in Ω , u = 0 on ∂Ω ,

(5.10)

where Ω ⊂ Rn is a bounded domain with smooth boundary. Assume: n+2 (g1) |g(x, t)| ≤ C(1 + |t|α ) α < n−2 if n ≥ 3. (g2) ∃θ > 2, ∃M > 0 such that 0 < θG(x, t) ≤ tg(x, t) ∀x ∈ Ω, for |t| ≥ M , t where G(x, t) = 0 g(x, s)ds. (g3) g ∈ C 1 (Ω × R1 ) with g(x, 0) = gt (x, 0) = 0.

Deﬁne J(u) =

Ω

1 |∇u|2 − G(x, u(x)) dx . 2

Lemma 5.1.38 (Z.Q. Wang) Under the assumptions of (g1), (g2) and (g3), there exists a constant A > 0, such that Ja S ∞ , the unit sphere in H01 (Ω) for −a > A. Proof. By (g2 ), we have a constant C > 0 such that G(x, t) ≥ C(|t|θ − 1) ∀t, |t| ≥ M .

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5 Topological and Variational Methods

Thus ∀u ∈ S ∞ ,

J(tu) → −∞ as t → +∞ .

First we want to show that ∃A > 0 such that ∀a < −A, if J(tu) ≤ a, then d dt J(tu) < 0. In fact, set |g(x, t)| + 1 . A = 2M |Ω| max (x,t)∈Ω×[−M,M ]

If J(tu) =

2

t 2

−

Ω

G(x, tu(x))dx ≤ a, then

d J(tu) = (J (tu), u) dt

= t− u(x) · g(x, tu(x))dx Ω

2 1 ≤ G(x, tu(x))dx − tu(x) · g(x, tu(x))dx + a t 2 Ω Ω

1 1 2 ≤ − tu(x)g(x, tu(x))dx + (A − 1) + a t θ 2 |tu(x)|≥M

1 1 2 θ θ − ≤ |t| |u(x)| dx − 1 < 0 , Cθ t θ 2 |tu(x)|≥M as a < −A. The implicit function theorem is employed to obtain a unique T (u) ∈ C(S ∞ , R1 ) such that J(T (u)u) = a ∀u ∈ S ∞ . Next, we claim that T (u) possesses a positive lower bound > 0. In fact, 2 by (g3), g(x, 0) = gt (x, 0) = 0, J(tu) = t2 − ◦(t2 ) ∀u ∈ S ∞ as |t| is small. The conclusion follows. Finally, let us deﬁne a deformation retract η : [0, 1] × (H\B (θ)) → H\B (θ), where H = H01 (Ω) and B (θ) is the -ball with center θ, by η(s, u) = (1 − s)u + sT (u)u ∀u ∈ H\B (θ) . This proves that H\B (θ) Ja , i.e., Ja S ∞ .

Indeed, J satisﬁes the (PS) condition (see e.g., Sect. 4.8.3, Example 2). It is easy to give a diﬀerent proof of the conclusion we obtained previously, i.e., there is a nontrivial solution of (5.10). Indeed, θ is a critical point of J, which is an isolated minimum. Therefore Cq (J, θ) = δq0 G. It is known that S ∞ is contractible. The pair (H01 (Ω), Ja ) must be trivial, i.e., Hq (H01 (Ω), Ja ; G) = 0 ∀q .

5.2 Minimax Principles (Revisited)

347

If there were no other critical points, then the Morse relation would be 1 = 0. This is impossible. In fact, from Sect. 4.8, Example 2, we have already known that there is a critical point via the mountain pass lemma, we shall see in the next section, equation (5.10) possesses at least three nontrivial solutions.

5.2 Minimax Principles (Revisited) The minimax method was initiated by G. Birkhoﬀ in his pioneering work on closed geodesics. It was successfully developed by Ljusternik and Schnirelmann in proving that on a closed surface with genus zero there are at least three closed geodesics. The method has been well developed by M. A. Krasnoselski, P. S. Palais, A. Ambrosetti and P. Rabinowitz etc. 5.2.1 A Minimax Principle We have studied saddle points by the minimax method in Sect. 4.8. In this section we shall introduce a general minimax principle based on the deformation theorem, which contains various types of this method as special cases. Deﬁnition 5.2.1 For a given function f ∈ C(M, R1 ) satisfying the (PS) condition on a Banach–Finsler manifold M , for a given a ∈ R1 , let Φa (f ) = {φ = η(1, ·) | η ∈ C([0, 1] × M, M ) satisfy η(t, x) = x, ∀ (t, x) ∈ ({0} × M ) ∪ ([0, 1] × fa )}. A family of subsets F of M is called Φa (f )-invariant if ∀A ∈ F φ(A) ∈ F, whenever φ = Φa (f ). Example 1. (Linking) Let X be a Banach space, let Q ⊂ X be a compact manifold with boundary ∂Q and let S ⊂ X be a closed subset of X. Assume ∂Q and S link (see Deﬁnition 4.8.6). Let f ∈ C 1 (X, R1 ) satisfy the (PS) condition. Assume a = max∂Q f < inf S f . We set the family F = {ψ(Q)|ψ ∈ C(Q, X) with ψ|∂Q = id∂Q }. Then F is Φa (f )-invariant. In fact, for A = ψ(Q), one has φ(A) = φ ◦ ψ(Q), ∀φ ∈ Φa (f ), and then φ ◦ ψ|∂Q = id|∂Q , so is φ(A) ∈ F. Example 2. (Homology class) Let f ∈ C 1 (M, R1 ) satisfy the (PS) condition. For a pair of real numbers a < b, if [σ] ∈ Hq (fb , fa ; G) is a nontrivial q-relative homology class for some q ∈ N. Set F = {|σ||σ ∈ [σ] is a singular q-closed chain in (fb , fa ) with coeﬃcient group G} , where |σ| is the support of σ. Then F is Φa (f )-invariant. In fact, let φ = η(1, ·) where η ∈ C([0, 1] × M, M ) satisﬁes η(t, x) = x ∀ (t, x) ∈ ({0} × M ) ∪ ([0, 1] × fa ), then ∀σ ∈ [σ], φ(|σ|) = |φ(σ)|, and φ(σ) ∈ [σ].

348

5 Topological and Variational Methods

Theorem 5.2.2 (Minimax principle) Suppose that M is a smooth Banach– Finsler manifold and that f ∈ C 1 (M, R1 ) satisﬁes the (PS) condition. Let F be a Φa (f )-invariant family of subsets of M for some a ∈ R1 . Set c = inf sup f (x) . A∈F x∈A

(5.11)

If (1) c is ﬁnite, (2) a < c, then c is a critical value of f . Proof. It is proved by contradiction. If c is not a critical value, then there exists > 0 such that a < c − , f −1 [c − , c + ] ∩ K = ∅, provided by the (PS) condition. Taking A0 ∈ F such that sup f (x) < c + i.e., A0 ⊂ fc+ . x∈A0

According to the nontrivial interval theorem, ∃ a strong deformation retraction η : [0, 1] × M → M such that η(0, ·) = id, η(t, ·)|fc− = idfc− ∀ t ∈ [0, 1], and η(1, fc+ ) ⊂ fc− . Thus η(t, x) = x ∀ (t, x) ∈ ({0} × M ) ∪ ([0, 1] × fa ), and then φ := η(1, ·) ∈ Φa (f ). Since F is Φa (f )-invariant, it follows that inf sup f (x) < c − .

A∈F x∈A

This is a contradiction.

Thus, in Example 1, Since Q is compact, c < ∞. From the S and ∂Q link, we have c > a. Theorem 5.2.2 implies Theorem 4.8.9. Also, applying Theorem 5.2.2 to Example 2, c = inf sup f (x)

(5.12)

σ∈[σ]x∈|σ|

is a critical value of f . In fact, obviously c is ﬁnite. We verify that c is a critical value, if not, c ≤ a. ∀ > 0, ∃σ ∈ [σ], with |σ| ⊂ fc+ , according to the second deformation theorem, there exists η deforming fc+ to fc . Since η(1, σ) ∈ [σ] and |η(σ)| ⊂ fa , [σ] is trivial in (fb , fa ). A contradiction. What can we say about the critical point with critical value c deﬁned by equation (5.12)? Theorem 5.2.3 Suppose that M is a smooth Banach–Finsler manifold, and that f ∈ C 1 (M, R1 ) satisﬁes (PS)d ∀d ∈ [a, b], where a, b are regular values. If [σ] is a nontrivial class in Hq (fb , fa ; G), and c is deﬁned by (5.12). Then c ∈ (a, b) and K ∩ f −1 (c) = ∅. Moreover, if c is isolated and K ∩ f −1 (c) consists of isolated critical points, then ∃p ∈ K ∩ f −1 (c) such that Cq (f, p) = 0 . Proof. The ﬁrst part of the conclusion has been proved. Since c is an isolated critical value, i.e., ∃ > 0 such that f −1 [c − , c + ] ∩ K = K ∩ f −1 (c) and a < c − .

5.2 Minimax Principles (Revisited)

349

It is suﬃcient to show Hq (fc− , fc+ ; G) = 0, for then by Theorem 5.1.27, there must be a p ∈ K ∩ f −1 (c) such that Cq (f, p) = 0. Now we prove it by contradiction. If Hq (fc+ , fc− ; G) = 0, then by the long exact sequence j∗

i

∗ Hq (fc+ , fa ; G) −→ Hq (fc+ , fc− ; G) −→ · · · , · · · −→ Hq (fc− , fa ; G) −→

we have Im i∗ = Hq (fc+ , fa ; G). This shows that the class [σ] is again non trivial in Hq (fc− , fa ; G). But this contradicts the deﬁnition of c. Combining Theorem 5.2.3 with the shifting theorem, we have: Corollary 5.2.4 Let M be a smooth Hilbert–Riemannian manifold. Suppose that f ∈ C 2 (M, R1 ) satisﬁes the assumptions of Theorem 5.2.3, and that f (x) is a Fredholm operator ∀x ∈ K ∩ f −1 (c). Then ∃p ∈ K ∩ f −1 (c) such that ind(f (p)) ≤ q ≤ ind(f (p)) + dim ker f (p). Return to the mountain pass point, we have: Corollary 5.2.5 Suppose that M is a smooth Hilbert–Riemannian manifold and that f ∈ C 1 (M, R1 ). If ∃p0 , p1 ∈ M such that c = inf sup f ◦ l(t) > max {f (p0 ), f (p1 )} , l∈Γt∈[0,1]

where Γ = {l ∈ C([0, 1], M )|l(i) = pi , i = 0, 1}. Assume (PS)c . If K ∩ fc consists of isolated points, then there exists a mountain pass point p ∈ K ∩ f −1 (c) (see Deﬁnition 5.1.19) i.e., C1 (f, p) = 0 .

(5.13)

5.2.2 Category and Ljusternik–Schnirelmann Multiplicity Theorem The notion of category was introduced by Ljusternik and Schnirelmann. It is a topological invariant used in the estimate of the lower bound of the number of critical points. Deﬁnition 5.2.6 Let M be a topological space, A ⊂ M be a closed subset. Set catM (A) = inf{m ∈ N ∪ {+∞} | ∃m contractible closed subsets of M : F1 , F2 , . . . , Fm such that A ⊂ ∪m i=1 Fi } . A set F is called contractible (in M ) if ∃η : [0, 1] × M → M such that η(0, ·) = idM and η(1, F ) = one-point set. Example 1. If C is a closed convex set in a Banach space X, then catX (C) = 1. In fact, pick any p ∈ C, and set

350

5 Topological and Variational Methods

η(t, x) = (1 − t)x + tp . Thus C is contractible. Example 2. Let S n be the unit ball in Rn+1 . Then catS n (S n ) = 2. It is easy to see that catS n (S n ) ≤ 2, because the upper and the lower n = {(x , xn+1 ) ∈ Rn × R1 | x 2 +x2n+1 = 1, ±xn+1 ≥ hemispheres S± 0} are contractible. On the other hand, S n is not contractible in itself, i.e., catS n (S n ) > 1. This can be shown by contradiction. Suppose there were a continuous η : [0, 1]×S n → S n such that η(0, ·) = idS n and η(1, S n ) = p ∈ S n . Let A : B1n+1 (θ) → S n be deﬁned by: −p if x = θ , A(x) = x −η(1− x , x ) if x ∈ B1n+1 (θ)\{θ} , then A would be continuous. According to the Brouwer ﬁxed-point theorem, ∃p0 ∈ S n such that A(p0 ) = p0 , But by the deﬁnition of A, A(p0 ) = −p0 , thus p0 = θ. This is a contradiction. Example 3. S ∞ is contractible, so is catS ∞ (S ∞ ) = 1. (see Sect. 3.3).

CatT 2 (T 2) = 3 Fig. 5.3.

Example 4. Let T 2 be the two-dimensional torus, i.e., S 1 × S 1 . Then catT 2 (T 2 ) = 3. This can be shown by Fig. 5.3. The following fundamental properties for the category hold: catM (A) = 0 ⇔ A = ∅. (Monotonicity) A ⊂ B ⇒ catM (A) ≤ catM (B). (Subadditivity) catM (A ∪ B) ≤ catM (A) + catM (B). (Deformation nondecreasing) If η : [0, 1] × M → M is continuous such that η(0, ·) = idM , then catM (A) ≤ catM (η(1, A)). (5) (Continuity) If A is compact, then there is a closed neighborhood N of A such that A ⊂ int(N ) and catM (A) = catM (N ). (6) (Normality) capM ({p}) = 1 ∀p ∈ M .

(1) (2) (3) (4)

5.2 Minimax Principles (Revisited)

351

In fact properties (1), (2), (3) and (6) follow from the deﬁnition directly. We shall prove (4) and (5) below. Proof. Proof of (4). Let B = η(1, A) and let catM (B) = m, i.e., ∃ closed −1 (1, Fi ), contractible sets F1 , . . . , Fm of M such that B ⊂ ∪m i=1 Fi . Let Gi = η then Gi is closed and contractible, because ∃ξ : [0, 1] × M → M such that ξ(0, . . .) = idM and ξ(1, Fi ) = pi ∈ M . Now let ϕ = ξ ◦ η : [0, 1] × M → M , we have ϕ(0, ·) = idM and ϕ(1, Gi ) = pi , i = 1, 2, . . . , m. Since A ⊂ ∪m i=1 Gi , property (4) is proved. In order to verify (5), we need a result on continuous extensions due to Hanner [Hann] and Palais [Pa 3]: Let X, Y be metrizable Banach manifolds, and let A ⊂ X be a closed subset. Then for any continuous map φ : ({0} × X) ∪ ([0, 1] × A) → Y , there is a continuous extension φ! : [0, 1] × X → Y . Proof. Since A is compact, catM (A) < ∞, i.e., ∃ closed contractible sets F1 , . . . , Fm such that A ⊂ ∪m i=1 Fi . Since ∃ϕi :: [0, 1] × M → M such that ϕi (0, . . .) = idM and ϕi (1, Fi ) = pi ∈ M , we apply the above continu!i : [0, 1] × M → M such that ous extension theorem to φi and obtain ϕ ϕ˜i |({θ}×M )∪([0,1]×Fi ) = ϕi |({θ}×M )∪([0,1]×Fi ) , i = 1, 2, . . . , m. Let Vi be a !−1 closed contractible neighborhood of pi , then Ui = ϕ i (1, Vi ) is closed and −1 m contractible, because ϕ˜i (0, Vi ) = Vi . Thus N = ∪i=1 Ui is a closed neighbor hood of A, we obtain m = catM (A) ≤ catM (N ) ≤ m. As a consequence of the additivity and normality, we have (7) If catM (A) = m, then A ≥ m, i.e., there are at least m distinct points in A. The main theorem of this subsection is the following multiplicity theorem: Theorem 5.2.7 (Ljusternik–Schnirelman theorem) Let M be a smooth Banach–Finsler manifold. Suppose that f ∈ C 1 (M, R1 ) is a function bounded from below, satisfying the (PS) condition. Then f has a least catM (M ) critical points. This theorem is based on the following version of the deformation theorem: Theorem 5.2.8 Let M be a smooth Banach–Finsler manifold. Suppose f ∈ C 1 (M, R1 ). Let N ⊂ N be two closed neighborhoods satisfying 7 δ, δ > 0 . 8 Suppose that there are positive constants b and , such that dist(N , ∂N ) ≥

df (x) ≥ b ∀x ∈ fc+ \(fc− ∪ N ) , 1 2 1 0 < < 3 min δb , δb . 4 8 Then for any 0 < < 2 , there exists η ∈ C([0, 1] × M, M ) satisfying

352

(1) (2) (3) (4) (5)

5 Topological and Variational Methods

η(0, ·) = id, η(t, ·)|Cf −1 [c−,c+] = id|Cf −1 [c−,c+] , η(t, ·) : M → M is a homeomorphism ∀t ∈ [0, 1], η(1, fc+ \N ) ⊂ fc− , f ◦ η(t, x) is nonincreasing in t ∀(t, x) ∈ [0, 1] × M .

Proof. Deﬁne a smooth function: 0 for s ∈ [c − , c + ] , p(s) = 1 for s ∈ [c − , c + ] , with 0 ≤ p(s) ≤ 1. Let A = M \(N ) δ , and B = N be two closed subsets. Let 8

g(x) =

dist(x, B) . dist(x, A) + dist(x, B)

We see 0 ≤ g ≤ 1, g = 0 on N and g = 1 outside (N ) δ . Deﬁne 8

q(s) =

1 1 s

0≤s≤1, s≥1.

Letting X be a pseudo-gradient vector ﬁeld of f , we deﬁne V (x) = −g(x)p(f (x))q( X(x) )X(x) . Thus V ∈ C 1−0 , and V (x) ≤ 1. Consider the ODE: σ(t) ˙ = V (σ(t)) σ(0) = x0 ∀x0 ∈ M . Since V is bounded, the global existence and uniqueness of the ﬂow σ(t) on R1 are known. Let η(t, x) = σ(t), with σ(0) = x . Then η ∈ C([0, 1] × M, M ) satisﬁes (1), (2), (3) and (5). It remains to verify (4). From (5), it is suﬃcient to verify (4) for x ∈ fc+ \(fc− ∪ N ). More precisely, we shall prove: 3 δ, x ≤ c − ∀x ∈ fc+ \(fc− ∪ N ) . f ◦η 4 Indeed, if not, ∃x ∈ f −1 [c − , c + ]\N such that 3 c − < f ◦ η(t, x) ≤ c + ∀t ∈ 0, δ , 4 it follows that p(f ◦ η(t, x)) = 1 .

5.2 Minimax Principles (Revisited)

353

Noticing that η(t, x) − η(0, x) ≤ t, we have dist(η(t, x), (N ) δ ) ≥ dist(η(0, x), (N ) δ ) − t 8 8 7 1 3 − − > δ=0, 8 8 4 so g ◦ η(t, x) = 1. Now, d f ◦ η(t, x) = f (η(t, x)), η(t, ˙ x)

dt = −q( X(η(t, x)) ) f (η(t, x)), X(η(t, x))

≤ −q( X(η(t, x)) ) f (η(t, x)) 2 . If X(η(t, x)) ≤ 1, then we have d f ◦ η(t, x) ≤ − f (η(t, x)) 2 ≤ −b2 . dt Otherwise, d f ◦ η(t, x) ≤ − f (η(t, x)) 2 / X(η(t, x)) dt 1 1 ≤ − f (η(t, x)) < − b . 2 2 In summary,

d 2 1 f ◦ η(t, x) ≤ − min b , b . dt 2

Thus, 3 2 1 f ◦ η(t, x) ≤ f ◦ η(0, x) − δ min b , b 4 2 1 2 1 ≤ c + − 3δ min b , b > 0 such that (1)–(5) hold. With the aid of category, we introduce families of subsets of M . Let Fk = {A| closed subset of M with catM (A) ≥ k}, ∀k ∈ N .

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5 Topological and Variational Methods

If f ∈ C 1 (M, R1 ) is bounded from below, then Fk is Φa (f )-invariant with a < inf M f , provided by the deformation nondecreasing property (4) of the category. According to minimax principle ck = inf sup f (x) A∈Fk x∈A

k = 1, 2, . . . ,

(5.14)

are critical values whenever f satisﬁes the (PS) condition. Since Fk+1 ⊂ Fk , we have ck ≤ ck+1 , k = 1, 2, . . .. Generally, we do not have ck < ck+1 . However, in order to get the multiplicity of critical points, we need: Lemma 5.2.10 Let M be a smooth Banach–Finsler manifold. Suppose that f ∈ C 1 (M, R1 ) is bounded from below and satisﬁes the (PS)c condition, where c = ck+1 = · · · = ck+m , are deﬁned by (5.14), then catM (Kc ) ≥ m. Proof. Since Kc is compact, provided by the continuity of the category, we have a closed neighborhood N of Kc such that catM (N ) = catM (Kc ). From the deﬁnition of c, ∀ > 0 ∃ a closed subset A ⊂ fc+ such that catM (A ) ≥ k + m. Applying ﬁrst deformation lemma, ∃η : [0, 1] × M → M satisfying (1)–(5) in Theorem 5.2.8. Let φ = η(1, ·), then ◦

◦

φ(A \N ) ⊂ φ(fc+ \N ) ⊂ fc− . It follows that ◦

k + m ≤ catM (A ) ≤ catM (A\N ) + catM (N ) ◦

≤ catM (φ(A\N )) + catM (Kc ) ≤ catM (fc− ) + catM (Kc ) ≤ k + catM (Kc ) . Therefore catM (Kc ) ≥ m . Proof. Proof of Theorem 5.2.7: Deﬁne ck , k = 1, 2, · · · catM (M ); by (5.14), we have c1 ≤ c2 ≤ . . . ≤ ck ≤ . . .. According to Lemma 5.2.10, the multiplicity of c = ck is dominated by catM (Kc ), and then by the number of critical points in Kc . Combining with the fact that distinct critical values correspond to distinct critical points, the conclusion is proved. 5.2.3 Cap Product It is natural to ask: If we have two singular homology classes [τ1 ], [τ2 ] ∈ H∗ (fb , fa ; G), both nontrivial, and if c1 , c2 are deﬁned in the same way as

5.2 Minimax Principles (Revisited)

355

in (5.12), are there two distinct critical points? Generally speaking, no, but with the aid of the notion of the cap product, the answer is yes if [τ1 ] and [τ2 ] are related by a cap product. Noticing that there is a ring structure on the cohomology groups. let X be a topological space and Cq (X, G) be the set of all singular q-chains in X with coeﬃcient-group G. Let C q (X, G) be the set of all singular q-cochains, i.e., Hom(Cq (X, G), G). It is a module, and then the duality [,] is a bilinear form over Cq (X, G) × C q (X, G). For a topological pair (X, Y ), let C q (X, Y ; G) = {c ∈ C q (X, G)|[σ, c] = 0 ∀σ ∈ Cq (Y, G)} . Under the duality, the coboundary operator δq : C q−1 (X, G) → C q (X, G) is deﬁned to be the dual of the boundary operator ∂q : Cq (X, G) → Cq−1 (X, G): [∂q σ, c] = [σ, δq c]

∀σ ∈ Cq (X, G), ∀c ∈ C q−1 (X, G) .

The qth singular cohomology group H q (X, G) = ker δq+1 /Imδq . q A cup product structure on the graded singular cochains ⊕∞ q=0 C (X, G) is deﬁned as follows: [σ, c ∪ d] = [σ ◦ λq , c] · [σ ◦ µq , d] ∀c ∈ C p (X, G), ∀d ∈ C q (X, G) and ∀σ ∈ Cp+q (X, G), where λp : p → p p p+q and µq : q → p+q read as: λp ( i=0 xi ei ) = i=0 xi ei , and q q µq ( i=0 xi ei ) = i=0 xi ei+p , and j is the standard j-simplex. It is well deﬁned on H ∗ (X, G), and makes the latter a graded algebra. The cup product is bilinear, associative, and possesses the unit element, i.e., the 0-cochain 1, which is deﬁned by [x, 1] = e, the unit element of G, ∀x ∈ X. The cap product is deﬁned to be the dual operator of the cup product. For a topological pair (X, Y ), one deﬁnes ∩ : Cp+q (X, Y ; G) × C p (X, G) → Cq (X, Y ; G) by [σ ∩c, d] = [σ, c∪d] ∀σ ∈ Cp+q (X, Y ; G), ∀c ∈ C p (X; G), ∀d ∈ C q (X, Y ; G). Since one has ∂(σ ∩ c) = (−1)p (∂σ ∩ c − σ ∩ δc)

∀σ ∈ Cp+q (X, Y ; G) ∀c ∈ C p (X, G) ,

the cap product is also induced to relative singular homology groups: ∩ : Hp+q (X, Y ; G) × H p (X, G) → Hq (X, Y ; G) . Deﬁnition 5.2.11 Let (X, Y ) be a topological pair. [τ1 ], [τ2 ] ∈ H∗ (X, Y ; G) are nontrivial classes. [τ1 ] is called subordinate to [τ2 ] if ∃ω ∈ H ∗ (X; G), dim ω > 0 such that [τ1 ] = [τ2 ] ∩ ω , where ∩ is the cap product. In this case we write [τ1 ] < [τ2 ].

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5 Topological and Variational Methods

Lemma 5.2.12 Let B ⊂ X be a contractible set, and [ω] ∈ H q (X; G) with q > 0. Then ∃ˆ ω ∈ [ω] such that |ˆ ω | ∩ B = ∅. Proof. Since B is contractible, there is a closed contractible neighborhood N of B (property (5) of the category). For all closed singular q-chains σ, by adding a q + 1 singular chain τ with |τ | ⊂ N and |σ| ∩ N ⊂ |∂τ |, after subdivision, we ﬁnd a closed singular q-chain σN satisfying: (1) σ − σN = ∂τ, τ ∈ Cq+1 (X, G) and |τ | ⊂ N , (2) |σN | ∩ B = ∅.

σ

σN

τ B

N Fig. 5.4.

For any ω ∈ [ω], deﬁne ω ˆ by [σ, ω ˆ ] = [σN , ω] , then we have: (1) ω ˆ ∈ [ω]. Because [σ, ω ˆ ] = [σN , ω] = [σ, ω] − [∂τ, ω] = [σ, ω]

∀σ ∈ Cq (X, G) .

(2) |ˆ ω | ∩ B = ∅. Because ∀σ ∈ Cq (X, G) with |σ| ⊂ B, we have |σN | = ∅. Therefore [σ, ω ˆ ] = [σN , ω] = 0. Theorem 5.2.13 Let M be a smooth Banach–Finsler manifold. Suppose that f ∈ C 1 (M, R1 ) with regular values a < b. Assume that [τ1 ] < [τ2 ] < · · · < [τm ] are nontrivial homology classes in H∗ (fb , fa ; G). Let

5.2 Minimax Principles (Revisited)

357

ci = inf sup f (x), i = 1, 2, . . . , m . τi ∈[τi ]x∈|τi |

If c = c1 = c2 = · · · = cm , and if f satisﬁes the (P S)c condition, then we have catM (Kc ) ≥ m .

Proof. Since Kc is compact, we may choose neighborhoods N ⊂ N ⊂ N of Kc satisfying the following conditions: (1) catM (N ) = catM (Kc ); (2) there exist constants 1 < < < min {b − c, c − a} and η ∈ C(M, M ) such that η|fc− = idfc− η ∼ id , and

η(fc+ \N ) ⊂ fc− ,

according to the ﬁrst deformation theorem. We prove our theorem by contradiction. If catM (Kc ) ≤ m − 1, then there m are (m − 1) contractible closed sets {Bj }m 2 that cover N : ∪j=2 Bj ⊃ N . Since [τ1 ] < [τ2 ] < · · · < [τm ], there exist ω2 , ω3 , . . . , ωm ∈ H ∗ (fb , G), with dim ωj > 0 such that [τj−1 ] = [τj ] ∩ ωj ,

j = 2, 3, . . . , m .

ωj | ∩ Bj = Provided by Lemma 5.2.12, we always may choose ω ˆ j ∈ ωj , with |ˆ ∅, j = 2, 3, . . . , m. Again we choose τ ∈ [τm ] with |τ | ⊂ fc+ . Subdividing τ into τ = σ1 + σ2 + · · · + σm such that |σ1 | ⊂ fc+ \N , and |σj | ⊂ Bj , j = 2, 3, . . . , m , one has ω2 ∪ · · · ∪ ω ˆm) τ = τ ∩ (ˆ = σ1 ∩ (ˆ ω2 ∪ · · · ∪ ω ˆm) . Hence |τ | ⊂ fc+ \N and η(|τ |) ⊂ fc− . However, τ ∈ [τ1 ]; therefore, η(τ ) ∈ [τ1 ], which implies that c1 ≤ c − . This is a contradiction. Deﬁnition 5.2.14 For a pair of topological spaces (X, Y ), let L(X, Y ; G) = sup{l ∈ Z+ | ∃ nontrivial classes [τ1 ], [τ2 ], . . . , [τl ] ∈

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5 Topological and Variational Methods

H∗ (X, Y ; G) such that [τ1 ] < · · · < [τl ]} . L(X, Y ; G) measures the length of the chain of subordinate nontrivial singular homology classes. It is directly related to the notion of cup length: CL(X, Y ; G) = sup{l ∈ Z+ | ∃c0 ∈ H ∗ (X, Y ; G), ∃c1 , . . . , cl ∈ H ∗ (X; G) such that dim(ci ) > 0, i = 1, 2, . . . , l, and c0 ∪ · · · ∪ cl = 0} . It is known from algebraic topology that catX (X) ≥ CL(X, ∅; G), i.e., the cup length provides a lower bound estimate for the category; we have catP n (P n ) = n + 1, catT n (T n ) = n + 1, where Pn is the n- dimensional real projective space, and T n is the n-torus. 5.2.4 Index Theorem Symmetric functions, or more generally, a function, which is invariant under some G-group action, may have more critical points. This is due to the fact that the underlying space quotient by the G-group action has more complicated topology. In this subsection we shall deal with this problem. Let us recall the proof of the Ljusterik–Schnirelmann multiplicity theorem, in which only the fundamental properties (1)–(6) of the category were used. We are inspired to extend an abstract theory based on these properties for G-group action invariant functions. Let M be a Banach–Finsler manifold with a compact group action G. Let Σ be the set of all G-invariant closed subsets of M , and H be the set of all G-equivariant continuous mappings from M into itself, i.e., h ∈ H if and only if h ∈ C(M, M ) and h ◦ g = g ◦ h ∀g ∈ G. Deﬁnition 5.2.15 An index (Σ, H, i) with respect to G is deﬁned by i : Σ → N ∪ {+∞}, which satisﬁes: i(A) = 0 ⇔ A = ∅, ∀A ∈ Σ. (Monotonicity) A ⊂ B ⇒ i(A) ≤ i(B), ∀A, B ∈ Σ. (Subadditivity) i(A ∪ B) ≤ i(A) + i(B), ∀A, B ∈ Σ. (Deformation nondecreasing) If η : [0, 1] × M → M satisﬁes η(t, ·) ∈ H ∀t ∈ [0, 1], and η(0, ·) = idM , then i(A) ≤ i(η(1, A)), ∀A ∈ Σ. (5) (Continuity) If A ∈ Σ is compact, then there is a neighborhood N ∈ Σ of A such that A ⊂ int(N ) and i(A) = i(N ). (6) (Normality) i([p]) = 1 ∀p ∈ FixG where [p] = {g · p| g ∈ G} and FixG = {x ∈ M | g · x = x ∀g ∈ G} is the ﬁxed-point set of G, if G = {e}.

(1) (2) (3) (4)

Example 1. If G = {e}, then category (Σ, H, catM ) is an index. Example 2. Recall the genus deﬁned in Sect. 3.3, where M is a Banach space, G = Z2 = {I, −I}, i.e., Ix = x, (−I)x = −x ∀x ∈ M ,. Thus Σ = the set of all closed symmetric subset, H is the set of odd continuous mappings.

5.2 Minimax Principles (Revisited)

Let

359

⎧ if A = ∅ , ⎨0 γ(A) = inf{k ∈ N| ∃ an odd map φ ∈ C(A, Rk \{θ})} , ⎩ +∞ if no such odd map .

Then genus (Σ, H, γ) is an index with respect to Z2 . (Veriﬁcations) By deﬁnition, (1) and (2) are trivial. (3) (Subadditivity) Set γ(A) = n, γ(B) = m; we may assume that both n, m < ∞. This means that ∃ϕ : A → Rn \{θ} ∃ψ : B → Rm \{θ} both are odd and continuous. By Tietze’s theorem ∃ϕ ! ∈ C(M, Rn ), ∃ψ! ∈ C(M, Rm ) ! B = ψ. Without loss of generality, we may assume that such that ϕ| ! A = ϕ, ψ| ! ! ϕ ! and ψ are odd. Deﬁne f (x) = (ϕ(x), ! ψ(x)), then f ∈ C(M, Rn+m ) is odd n+m \{θ}. This implies that γ(A ∪ B) ≤ γ(A) + γ(B). and f (A ∪ B) ⊂ R (4) (Deformation nondecreasing) Suppose η : [0, 1]×M → M is continuous and odd, and satisﬁes η(0, ·) = id with γ(η(1, A)) = n, i.e., ∃ϕ : η(1, A) → Rn \{θ} is continuous and odd. Let ψ = ϕ ◦ η(1, ·) then ψ : A → Rn \{θ} is continuous and odd. Therefore γ(A) ≤ n. (5) (Continuity) Let γ(A) = n. From (2), with no loss of generality, we may assume n < +∞. There exists an odd continuous ϕ : A → Rn \{θ}. By Tietze’s theorem, there exists an odd continuous mapping ϕ ! : M → Rn , with ϕ| ! A = ϕ. Since A is compact and θ ∈ ϕ(A), ! ∃δ > 0 such that θ ∈ ϕ(N ! ), where N = Aδ , the closure of the δ-neighborhood of A, i.e., ϕ ! : N → Rn \{θ} . Combining with the monotonicity, we obtain n = γ(A) ≤ γ(N ) ≤ n (6) (Normality) Note that FixG = {θ}, if p = θ, then [p] = {p, −p}. We deﬁne φ(±p) = ±1; it follows that γ([p]) = 1. Moreover, applying the Borsuk–Ulam theorem in Sect. 3.2, we have computed γ(S n−1 ) = n. Example 3. (S 1 -index) Let M be a Banach space, S 1 = {eiθ | θ ∈ [0, 2π]} be a compact Lie group. Let G be the isometry representation of S 1 , i.e., the group consists of homomorphisms T : S 1 → L(M, M ), i.e., T (eiθ ) are isometric operators, satisfying T (eiθ ) · T (eiϕ ) = T (ei(θ+ϕ) ) ∀θ, ϕ ∈ [0, 2π]. Set Σ = the set of all G − invariant closed subsets of M . H = {h ∈ C(M, M ), T (eiθ ) ◦ h = h ◦ T (eiθ ) ∀θ ∈ [0, 2π]} . ∀A ∈ Σ, deﬁne

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5 Topological and Variational Methods

⎧ 0 if A = ∅, ⎪ ⎪ ⎨ inf{k ∈ N| ∃φ ∈ C(A, Ck \{θ}), ∃n ∈ N satisfying γ(A) = φ(T (eiθ )x) = einθ φ(x) ∀(x, eiθ ) ∈ A × S 1 }, ⎪ ⎪ ⎩ +∞ if there is no such φ, where Ck is k-dimensional unitary space (complex k space). We shall verify that (Σ, H, γ) is an index with respect to S 1 . By deﬁnition, (1) and (2) are trivial. Before going on to verify other fundamental properties, we need: Lemma 5.2.16 For any A ∈ Σ, if φ ∈ C(A, Ck \{θ}) satisﬁes φ(T (eiθ )x) = einθ φ(x) ∀x ∈ A for some n ∈ N , ! A = φ and then ∃φ! : M → Ck satisfying φ| ! ! (eiθ )x) = einθ φ(x) ∀(x, eiθ ) ∈ M × S 1 . φ(T ˆ A = φ. Setting Proof. By Tietze’s theorem, ∃φˆ ∈ C(M, Ck ) satisfying φ| 1 ! φ(x) = 2π

2π

ˆ (eiϕ )x)dϕ , e−inϕ φ(T

0

we have inθ ! (eiθ )x) = e φ(T 2π

2π

ˆ (ei(θ+ϕ) )x)dϕ = einθ φ(x) ! e−in(θ+ϕ) φ(T ,

0

and ∀x ∈ A, by the assumption. 1 ! φ(x) = 2π

2π

φ(x)dx = φ(x) . 0

(Veriﬁcations) (3) (Subadditivity) ∀A, B ∈ Σ, let γ(A) = n, γ(B) = m. We may assume n, m < ∞. There exist ϕ : A → Cn \{θ}, ψ : B → Cm \{θ} and k, l ∈ N such that ϕ(T (eiθ )x) = eikθ ϕ(x) and ψ(T (eiθ )y) = eilθ ψ(y) ∀(x, y, eiθ ) ∈ A × B × S 1 . Deﬁne ! k) , f (x) = (ϕ(x) ! l , ψ(x) where the power map z → z p is deﬁned to be z = (z1 , . . . , zk ) → (z1p , . . . , zkp ). Thus f : A ∪ B → Cn+m is continuous and f (z) = θ as z ∈ A ∪ B. Moreover, ! (eiθ )x)k ) f (T (eiθ )x) = (ϕ(T ! (eiθ )x)l , ψ(T ! k) = eilkθ (ϕ(x) ! l , ψ(x) = eilkθ f (x) .

5.2 Minimax Principles (Revisited)

361

Therefore γ(A ∪ B) ≤ γ(A) + γ(B) . (4) (Deformation nondecreasing) Assume A ∈ Σ, η ∈ [0, 1] × M → M is continuous and satisfying η(0, ·) = id, η(t, ·) ∈ H, and γ(η(1, A)) = k. By deﬁnition, ∃ϕ : η(1, A) → Ck \{θ} continuous and satisfying ϕ(T (eiθ )η(1, x)) = einθ ϕ(η(1, x)) for some n ∈ N, ∀(x, eiθ ) ∈ A × S 1 . Now let ψ = ϕ ◦ (η(1, ·)). Then ψ : A → Ck \{θ} and ψ(T (eiθ )x) = ϕ(η(1, T (eiθ )x)) = ϕ(T (eiθ )η(1, x)) = einθ ϕ(η(1, x)) = einθ ψ(x) . Therefore γ(A) ≤ k. (5) (Continuity) Assume that A ∈ Σ is compact and γ(A) = k. Provided ˜ (eiθ )x) = by Lemma 5.2.16, ∃ a continuous φ! : M → Ck ∃n ∈ N satisfying φ(T ˜ ! A = φ and φ : A → Ck \{θ}. φ| einθ φ(x), Setting V = φ!−1 (Ck \{θ}), then V is an open neighborhood of A. Since A is compact, ∃δ > 0 such that the closure of the δ-neighborhood of A, N ⊂ V . Therefore φ! : N → Ck \{θ} . It follows that γ(N ) ≤ k. (6) (Normality) Let p ∈ FixG , i.e., ∃θ ∈ [0, 2π] such that T (eiθ )p = p. Since [p] = {T (eiθ )p| θ ∈ [0, 2π]} ∈ Σ and is not empty, γ([p]) ≥ 1. There exists θ0 > 0, which is the minimal period of T (eiθ )p, i.e., T (eiθ0 )p = p but T (eiθ )p = p ∀θ ∈ (0, θ0 ). n = 2π θ0 must be an integer. We deﬁne φ : [p] = {T (eiθ )p| θ ∈ [0, θ0 ]} → C1 \{θ} as follows: φ : T (eiθ )p → einθ . It is continuous, and φ(T (eiϕ )T (eiθ )p) = φ(T (ei(θ+ϕ) p)) = ein(θ+ϕ) = einϕ φ(T (eiθ )p) . Therefore γ([p]) = 1. We have proved that γ so deﬁned is an index. In particular, if M is a Hilbert space, and G is an unitary representation of S 1 , we have the following Borsuk–Ulam-type theorem for S 1 equivariant maps. Theorem 5.2.17 Under the above assumptions, suppose that V 2k is a 2kdimensional S 1 -invariant subspace, which is isomorphic to Ck . If V 2k ∩FixG = {θ}, then γ(V 2k ∩ ∂B1 (θ)) = k.

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5 Topological and Variational Methods

Proof. According to Stone’s representation theorem for one-parameter unitary groups, one may diagonalize {T (eiθ )| θ ∈ [0, 2π]} simultaneously: T (eiθ ) = diag{eiλ1 θ , . . . , eiλk θ } . Since T (eiθ ) is 2π-periodic, λj ∈ Z, j = 1, 2, . . . , k. By the assumption that there is no ﬁxed point of {T (eiθ )| θ ∈ [0, 2π]} in Ck \{θ}, therefore λj = 0, j = 1, 2, . . . , k. Firstly, we show γ(V 2k ∩ ∂B1 (θ)) ≤ k by constructing a continuous map ∧

∧ λ

φ : z = (z1 , . . . , zk ) → (z1λ1 , . . . , zk k ), where ∧ = |λ1 . . . λk |. Obviously, φ : Ck ∩ ∂B1 (θ) → Ck \{θ} continuously and φ(T (eiθ )z) = ei∧θ φ(z), ∀z ∈ Ck ∩ ∂B1 (θ). Next we are going to show that γ(Ck ∩ ∂B1 (θ)) ≥ k. Suppose γ(Ck ∩ ∂B1 (θ)) = k1 < k, i.e., ∃ a continuous φ : Ck ∩ ∂B1 (θ) → Ck1 \{θ}, and ∃n ∈ N such that φ(T (eiθ )z) = einθ φ(z). From Lemma 5.2.16, ! Ck ∩∂B (θ) = φ. Accordthere is an continuous extension φ! : Ck → Ck1 with φ| 1 ing to Theorem 3.3.10, ! B1 (θ), θ) = deg(φ, B1 (θ), θ) = 0 . deg(φ, Thus for y ∈ Ck with small y , one has ! B1 (θ), y) = 0 . deg(φ, In particular, we take y ∈ Ck \Ck1 , then φ!−1 (y)∩B1 (θ) = ∅, which contradicts ! k ) ⊂ Ck1 . φ(C The Ljusternik–Schnirelmann multiplicity theorem is extended as follows: Theorem 5.2.18 (Multiplicity theorem) Let M be a smooth Finsher–Banach manifold with a compact Lie group action G. Let (Σ, H, i) be an index with respect to G. Suppose that f ∈ C 1 (M, R1 ) satisﬁes (PS)c conditions, where c = ck+1 = · · · = ck+m is ﬁnite, and cn = inf sup f (x)

n = 1, 2, . . .

i(A)≥nx∈A

Then i(Kc ) ≥ m. Proof. The proof follows step by step that of Lemma 5.2.10. The only thing to be worked out is that the deformation constructed in Corollary 5.2.9 is G-equivariant. In fact, let η : [0,1] × M → M be the deformation obtained in Theorem 5.2.8. Then ηg (t, z) = G g −1 η(t, g · z)dµ is the G-equivariant deformation, where dµ is the right invariant Haar measure on compact continuous group G. We omit the veriﬁcations.

5.2 Minimax Principles (Revisited)

363

5.2.5 Applications Example 1. (An existence of three nontrivial solutions) Return to Sect. 5.1.5, Example 2, where we proved the existence of a nontrivial solution for the superlinear elliptic BVP both at inﬁnity and at zero. In fact, it was also studied in Sect. 4.8, Example 2, where we showed the existence of a mountain pass solution p. After Corollary 5.2.5, we know that C1 (f, p) = 0; if further, the condition (Φ) in Theorem 5.1.20 is fulﬁlled, then we have Cq (f, p) = δq1 G. We assume condition (Φ) at this moment, and conclude: Statement 5.2.19 Under assumptions (g1) (g2) and (g3) of Sect. 5.1.5 Example 2, equation (5.10) possesses at least three nontrivial solutions. Proof. We continue to use the notations in Sect. 5.1.5 Example 2. 1. From (g3), 1 (5.15) J(u) = u 2 + ◦ ( u 2 ) . 2 Therefore θ is a local minimum, and Cq (J, θ) = δq0 G. 2. We claim the existence of two nontrivial solutions. Let us deﬁne g(x, t) t ≥ 0 g+ (x, t) = 0 t≤0,

and J+ (u) =

Ω

1 2 |∇u| − G+ (x, u(x)) dx , 2

where

G+ (x, t) =

t

g+ (x, s)ds . 0

Similarly, one veriﬁes that J+ ∈ C 2 (H01 (Ω), R1 ) satisﬁes (PS), and that J+ (tϕ1 ) → −∞ as t → +∞ , where ϕ1 > 0 is the ﬁrst eigenvector of − with 0-Dirichlet data. On the other hand, ∃δ > 0 such that J+ |∂Bδ (θ) ≥

1 2 δ , 4

From (5.15). The mountain pass lemma is applied to obtain a critical point u+ ∈ H01 (Ω), with critical value c+ > 0, which satisﬁes −u+ = g+ (x, u+ ) u+ |∂Ω = 0 . By using the maximum principle, u+ ≥ 0, so it is again a critical point of J. Analogously, we deﬁne

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5 Topological and Variational Methods

g− (x, t) =

g(x, t) 0

t≤0 t>0,

and obtain a critical point u− ≤ 0, with critical value c− > 0. Since (Φ) is assumed, Cq (J± , u± ) = δq1 G. According to Theorem 5.1.16, we have ! Cq (J± , u± ) = Cq (J1 ± , u± ) = Cq (J, u± ) = Cq (J, u± ) , where J! = J|C01 . Therefore Cq (J, u± ) = δq1 G . 3. Suppose that there were no other critical points of J. The Morse type numbers over the pair (H01 (Ω), Ja ) would be M0 = 1, M1 = 2, Mq = 0, q ≥ 2 , but the Betti numbers read as βq = 0 ∀q = 0, 1, 2, . . . From Lemma 5.1.38: Hq (H01 (Ω), Ja ) Hq (H01 (Ω), S ∞ ) 0. This contradicts the Morse relation.

Now we return to verifying the condition (Φ): J (u) ≥ 0 and 0 ∈ σ(J (u)) imply dim ker(J (u)) = 1. It is based on the following: Theorem 5.2.20 (Manes–Micheletti) Suppose that m ∈ C(Ω) satisﬁes sup{m(x)| x ∈ Ω} > 0. Then the equation −u = λmu in Ω u|∂Ω = 0 admits a smallest positive eigenvalue λ1 , associated with a positive eigenfunction. Moreover, dim ker(− − λ1 m·) = 1 . Continuation of the proof of Statement 5.2.19 (Veriﬁcation of the condition (Φ)) Now

|∇v|2 − mv 2 ∀v ∈ H01 (Ω) , (5.16) (J (u)v, v) = Ω

where m(x) = g (x, u(x)). Assume that the right-hand side of equation (5.16) is nonnegative and that ∃φ ∈ H01 (Ω)\{θ} satisﬁes the equation −v − m · v = 0 .

(5.17)

5.2 Minimax Principles (Revisited)

365

We want to show that all solutions of equation (5.17) are multiples of φ. After Theorem 5.2.20, it is suﬃcient to show that λ = 1 is the smallest positive eigenvalue of (5.18) −v = λmv v ∈ H01 (Ω) . On the one hand, |∇φ|2 |∇v|2 inf ≤ =1, λ1 := mv 2 mφ2 v∈H01 \{θ} from Ω mφ2 = Ω |∇φ|2 = 0. On the other hand, provided by the nonnegativeness of J (u), |∇v|2 λ = Ω ≥1 mv 2 Ω for every λ ∈ σ(J (u)). Therefore the smallest positive eigenvalue λ1 of equation (5.18) equals 1. Condition (Φ) is veriﬁed. We turn to the proof of Theorem 5.2.20. Proof. 1. Let T : u → (−)−1 m · u be the self-adjoint compact operator on H01 (Ω). According to Courant’s max-min characterization, ± Ω mu2 −1 , n = 0, 1, 2, . . . ±λ±n = sup inf 2 V ∈Fn u∈V Ω |∇u| are eigenvalues of T , where Fn is the family of all n-dimensional linear subspaces of H01 (Ω). Thus, ±λ±1 ≤ ±λ±2 ≤ · · · , if they exist. 2. Under our assumption, λ1 > 0. Let ω be an eigenfunction. We claim that ω does not change sign. Otherwise, let ω+ = ω ∨ 0 and ω− = ω ∧ 0; they are not zero, then

2 2 2 mω = mω+ + mω− = α+ + α− , Ω Ω Ω

|∇ω|2 = |∇ω+ |2 + |∇ω− |2 = β+ + β− , Ω

Ω

Ω

2 > 0 and β± = Ω |∇ω± |2 . where α± = Ω mω± α+ +α− α+ +α− − − or αβ+ < max{ αβ++ , αβ− }. However, the latter Either β+ +β− = β+ = αβ− + +β− α+ +α− −1 case cannot occur, because λ1 = β+ +β− = supu Ω mu2 / Ω |∇u|2 . In the former case, both ω± are eigenfunctions with respect to λ1 and ±m · ω± ≥ 0. From the maximum principle, we obtain ±ω± (x) > 0 a.e. x ∈ Ω. This is impossible. 3. We show that dim ker(− − λ1 m·) = 1. Suppose that there are ω1 and ω2 ∈ H01 (Ω) satisfying

−ωi = λ1 mωi

i = 1, 2 .

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5 Topological and Variational Methods

Then by step 2, ∀α ∈ R1 , ω1 + αω2 does not change sign. Let A+ = {α ∈ R1 | ω1 + αω2 ≥ 0} and A− = {α ∈ R1 | ω1 + αω2 ≤ 0}. Then both A+ and A− are nonempty closed subsets and A+ ∪ A− = R1 . Let α0 ∈ A+ ∩ A− , then ω1 + α0 ω2 = 0, i.e., ω1 and ω2 are colinear. This proves dim ker(− − λ1 m·) = 1. Example 2. (Spectrum of the p-Laplacian) Let Ω ⊂ Rn be a bounded open domain. ∀p ∈ (1, ∞) we deﬁne the pLaplacian operator to be p u = div(|∇u|p−2 ∇u)

∀u ∈ W01,p (Ω) .

In particular, 2 is the Laplacian. λ is called an eigenvalue of −p if there exists a nontrivial weak solution u ∈ W01,p (Ω) of the equation: −p u = λ|u|p−2 u i.e.,

in Ω,

(5.19)

|∇u|p−2 ∇u∇ϕ = λ Ω

|u|p−2 u · ϕ

∀ϕ ∈ W01,p (Ω) ,

Ω

where p1 + p1 = 1. This is reduced to a constraint variational problem. Set

1 1 p |u| and g(u) = |∇u|p . f (u) = p Ω p Ω Let M = g −1 (p−1 ). Itis a nonempty closed submanifold of W01,p (Ω). In fact, ∀u ∈ M, g (u), u = Ω |∇u|p = 1, which implies that g (u) = θ. Let f! = f |M . We have

|u|p (p u) ∈ W −1,p (Ω) , df!(u) = |u|p−2 u + Ω

which satisﬁes df!(u), u = 0 ∀u ∈ M . Set J0 = (−2 )−1/2 and J1 w = |w|p −2 w/ w pp −2 , we have J0 w p = w −1,p , J0 w 1,p = w p , and J1 w p = w p .

Thus J = J0 ◦ J1 ◦ J0 : W −1,p (Ω) → W 1,p (Ω) is well deﬁned and satisﬁes: (1) J is continuous, (2) Jw, w = w 2−1,p ∀w ∈ W −1,p (Ω), (3) Jw 1,p = w −1,p ∀w ∈ W −1,p (Ω).

5.2 Minimax Principles (Revisited)

367

An odd pseudo-gradient vector ﬁeld for f! is deﬁned by X(u) = J · df!(u) − λu ∈ W 1,p (Ω) , where λ = J · df!(u), −p u . We claim that X(u) ∈ Tu M . ˙ = In fact, let s : (−1, +1) → M be a C 1 -curve with s(0) = u and s(0) X(u), then d g(s(t))|t=0 = X(u), g (u) = X(u), −p u = 0 . dt But we have X(u), df!(u) = J · df!(u), df!(u) = df!(u) 2−1,p . From

X(u) 1,p ≤ df!(u) −1,p + |λ| u 1,p ,

and |λ| ≤ J · df!(u) 1,p −p u −1,p ≤ df!(u) −1,p u 1,p = df!(u) −1,p , it follows

X(u) 1,p ≤ 2 df!(u) −1,p .

Thus X(u) is a pseudo-gradient vector ﬁeld for f!. Now we verify the (PS)c condition for f!, as c > 0. Suppose that {uj } is a sequence on M such that df!(uj ) → θ and f!(uj ) → c > 0. {uj } is bounded in W01,p (Ω), and then after a subsequence uj u0 ∈ W01,p (Ω). From df!(uj ) → θ (W −1,p (Ω)), and f!(uj ) → c one has −p uj → (pc)−1 |u0 |p−2 u0

W −1,p (Ω) .

From Sect. 2.6, Example 4, −p is a monotone operator with a continuous inverse: (−p )−1 : W −1,p (Ω) → W01,p (Ω), therefore uj strongly converges in W01,p (Ω). We notice that M is the unit sphere in W01,p (Ω), and that the function ! f is even. Quotient by a Z2 -group, M/Z2 is homeomorphic to the inﬁnitedimensional projective space P ∞ . Applying the Ljusternik–Schnirelmann theorem, we have: Statement 5.2.21 For 1 < p < ∞, the equation (5.19) has inﬁnitely many 1 × (W01,p (Ω)\{θ}), m = 1, 2, . . .. Moreover, pairs of solutions (λm , ±um ) ∈ R+ let σp be the spectrum of the p-Laplacian, i.e., the set of all λ associated with nontrivial solutions. Then σp is unbounded.

368

5 Topological and Variational Methods

Proof. Since f! satisﬁes (PS)c , c > 0, we apply the Ljusternik–Schnirelmann theorem to −f!, i.e., let ck =

inf sup − f!(u)

k = 1, 2, . . .

γ(A)≥ku∈A

where A is a closed symmetric (with resp. to θ) subset of M and γ is the genus. All these ck are negativet critical values with ck ≤ ck+1 . We have λk = − c1k , k = 1, 2, . . .. We shall prove that σp is unbounded by contradiction. If not, then there exists > 0 such that ck < −. ∀k large. Let us choose a sequence of linear subspaces {Ej }∞ 1 , such that 1,p Ej ⊂ Ej+1 , dim Ej = j, and ∪∞ j=1 Ej = W0 (Ω) .

Set dk =

inf

sup

γ(A)≥ku∈A∩E ⊥

− f!(u) ,

k−1

Ek⊥

where denotes a complementary linear subspace of Ek . We need the following intersection result: Lemma 5.2.22 Let X be a Banach space and E ⊂ X be a j < ∞ dimensional linear subspace. If A is a symmetric closed set with γ(A) > j then A∩E ⊥ = ∅, where E ⊥ is any complementary subspace of E. the projection Proof. We prove it by contradiction. If A ∩ E ⊥ = ∅, one deﬁnes P : X → E, according to the direct sum composition: X = E E ⊥ . Deﬁne 1 1 a function ρ ∈ C(R+ , R+ ) to be ρ (t) = as t ≤ , and ρ (t) = t as t ≥ for > 0 and an odd deformation: η(t, x) = (1 − t)x + t

Px . ρ ( P x )

From A ∩ E ⊥ = ∅, we have > 0 such that P (A) ∩ B (θ) = ∅. Since η(1, x) =

Px , η(1, A) ⊂ E ∩ ∂B1 (θ) . ρ ( P x )

Provided by the monotonicity and the deformation nondecreasing, γ(A) ≤ γ(η(1, A)) ≤ γ(E ∩ ∂B1 (θ)) = j . This is a contradiction.

(Continuing the proof of Statement 5.2.21) According to Lemma 5.2.22, if ⊥ = ∅, therefore dk is well deﬁned, and dk ≤ ck , k = γ(A) ≥ k, then A ∩ Ek−1 1, 2, . . . .

5.2 Minimax Principles (Revisited)

369

We suppose ck < − ∀k large, then there exists a closed symmetric subset ⊥ such that −f!(uk ) < −, i.e., Ak ∈ , with γ(Ak ) ≥ k, and ∃uk ∈ Ak ∩ Ek−1 f!(uk ) > ∀k large. However, uk ∈ M implies that after a subsequence uk v in W01,p (Ω), and then uk → v in Lp (Ω), in particular, f!(uk ) → f (v), as k → ∞. But ⊥ , which implies uk θ in W01,p (Ω), and f (θ) = 0. This contradicts uk ∈ Ek−1 f!(uk ) > . Example 3. Let Ω ⊂ Rn be a bounded domain with smooth boundary. We consider the problem: −u(x) = λg(x, u(x)) in Ω (5.20) u(x) = 0 on ∂Ω where g ∈ C(Ω × R1 ) is odd in t, and satisﬁes: (1) ∃t0 > 0 such that g(x, t0 ) ≤ 0 ∀x ∈ Ω (2) ∃a ∈ C(Ω), a > 0, such that g(x, t) − a(x)t = ◦(t) as t → 0 uniformly in x ∈ Ω. One has the following abstract result: Theorem 5.2.23 Let X be a Banach space, and f ∈ C 1 (X, R1 ) be an even function satisfying the (PS) condition. Assume a < b and either f (θ) < a or f (θ) > b. If further, (1) there are an m-dimensional linear subspace E and ρ > 0 such that sup

f (x) ≤ b ,

x∈E∩∂Bρ (θ)

(2) there is a j-dimensional linear subspace F such that inf f (x) > a ,

x∈F ⊥

where F ⊥ is a complementary space of F , (3) m > j, then f has at least m − j pairs of distinct critical points. Proof. Let cn =

inf sup f (x)

n = 1, 2, . . .

γ(A)≥nx∈A

1. From condition (1) E ∩ ∂Bρ (θ) ⊂ fb , by the monotonicity of genus γ(fb ) ≥ γ(E ∩ ∂Bρ (θ)) = m. We obtain cm ≤ b. 2. We verify that cn ≥ a as n > j. If it is not true, then ∃n0 > j such that cn0 < a, i.e., ∃ a closed symmetric set A with γ(A) ≥ n0 such that

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5 Topological and Variational Methods

f (x) < a ∀x ∈ A. Condition (2) implies that f (x) > a ∀x ∈ F ⊥ . But according to Lemma 5.2.22, A ∩ F ⊥ = ∅. This is a contradiction. Combining steps 1 and 2, we obtain critical values in [a, b] cj+1 ≤ · · · ≤ cm . From the multiplicity theorem, there are at least m − j pairs of critical points. Remark 5.2.24 Another proof of Theorem 5.2.23 based on the estimation of cup length can be found in [Ch 5]. Now we return to equation (5.20). First, we replace g by the function g(x, t) t ≤ t0 gˆ(x, t) = g(x, t0 ) t > t0 . From the maximum principle, we know that if u ∈ H01 (Ω) is a solution of the equation: −u(x) = λˆ g (x, u(x)) in Ω (5.21) u(x) = 0 on ∂Ω , then u(x) < t0 ∀x ∈ Ω, and then it is a solution of equation (5.20). Let us write p(x, t) = gˆ(x, t) − a(x)t t and P (x, t) = 0 p(x, s)ds. Consider the functional with a parameter λ:

Jλ (u) = Ω

1 |∇u|2 − λ 2

au2 + P (x, u(x)) dx . 2

Obviously, ∀λ ∈ R1 , Jλ is bounded from below. Since a > 0, the eigenvalue problem: −v = µa · v

v ∈ H01 (Ω)

has eigenvalues λ1 < λ2 ≤ · · · ≤ λk ≤ · · · . If λ > λk then there exists ρ > 0 such that Jλ |Ek ∩∂Bρ (θ) < 0, where Ek is the subspace spanned by eigenfunctions with eigenvalues µ ≤ λk . Omitting the veriﬁcation of the (PS) condition, we apply Theorem 5.2.23 and obtain: Theorem 5.2.25 Under assumptions (1) and (2) of Example (3), (5.20) possesses at least k distinct pairs of solutions for λ > λk .

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

371

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture For H ∈ C 1 (Rn × Rn , R1 ), the following ODE system: x˙ = −Hp (x, p) p˙ = Hx (x, p)

(5.22)

∀ (x, p) ∈ Rn × Rn , is called a Hamiltonian system, and H is called a Hamiltonian function. Sometimes we use the notations: z = (x, p) ,

and J=

0 I

−I 0

,

where I is the n × n unit matrix. then (5.22) has a simple form: −J z˙ = H (z) , or

z˙ = JH (z) .

(5.23)

J grad is called a symplectic gradient. For the Hamiltonian system two types of periodic solution problems are often examined. 1. For a given period T , ﬁnd a solution z satisfying (5.23) and z(t) = z(t+T ). 2. For a given energy surface, i.e., a real number c, ﬁnd a periodic solution z of (5.23) lying on the given energy surface, i.e., H(z(t)) = c. The reason why we can set up the type 2 problem is that Hamiltonian systems are conservative: d H(z(t)) = (H (z(t)), z(t)) ˙ dt = (H (z(t)), JH (z(t))) = 0 ,

(5.24)

i.e., along a trajectory t → z(t), the Hamiltonian function is a constant. Note: the periodic trajectory z on Σ = H −1 (c) is virtually independent of the behavior of the function H outside Σ , that is, we have: Lemma 5.3.1 Let H, H1 ∈ C 1 (R2n , R1 ), c ∈ R1 . Assume that Σ = H −1 (c) is a compact manifold, If H|Σ = H1 |Σ , then z˙ = JH (z) and z˙ = JH1 (z) has the same periodic trajectories on Σ.

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5 Topological and Variational Methods

Proof. By the hypotheses, ∃ µ ∈ C(Σ, R1 ) satisfying H (z) = µ(z)H1 (z),

∀z ∈ Σ .

Since µ(z) = 0 and Σ is compact, there are constants m, M > 0, m µ(z) M,

∀z ∈ Σ .

(5.25)

Let z be an integral curve of H on Σ, deﬁne z1 (t) = z ◦ s(t) , where s : R1 → R1 will be determined later. First we solve the equation: 1 σ(t) ˙ = (µ◦z)(σ(t)) (5.26) σ(0) = 0 . Since the functions µ, z are all bounded and continuous, (5.26) has a solution on every open interval in R1 . Let T be the period of z, since σ(t) → +∞ as t → +∞, there exists τ > 0 such that σ(τ ) = T . Let t t s(t) = T +σ t− τ , τ τ where [t] denotes the largest integer less than t, then s ∈ C 1 (R1 , R1 ), s(0) = 0. It remains to verify the diﬀerentiability at t = jτ, j = 0, ±1, ±2, . . . . In fact, for j = 1, 1 1 1 = = (µ ◦ z)(σ(0)) (µ ◦ z)(0) (µ ◦ z)(T ) = σ(τ ˙ − 0) = s (τ − 0)

˙ = s (τ + 0) = σ(0)

Similarly, we prove this for j = 1. Hence, s : R1 → R1 is a diﬀeomorphism and satisﬁes 1 s(t) ˙ = (µ◦z)(s(t)) , s(0) = 0 . It follows that

˙ z(t) ˙ ◦ s(t) = JH1 (z1 (t)) , z˙1 (t) = s(t)

and z1 (0) = z(0) ∈ Σ, hence z1 is an integral curve of H1 on Σ. The converse can be proved in the same way. Thus a given hypersurface Σ determines the orbits of every Hamiltonian vector ﬁeld J grad H with Σ as regular energy surface H −1 (c) for some constant c. In this sense, one can ask the question: Which hypersurfaces carry a periodic orbit?

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

373

5.3.1 Hamiltonian Operator Let S 1 be the unit circle {eit t ∈ [ 0, 2π ]} and V = Rn × Rn . Deﬁne the real Hilbert space L2 (S 1 , V ) as follows: z ∈ L2 (S 1 , V ) means that z is deﬁned on S 1 with range in V and |z|2 is integrable, the inner product is deﬁned by z, w =

1 2π

2n

S1

∀z, w ∈ L2 (S 1 , V ) .

zj (t)wj (t)dt,

j=1

Similarly deﬁne the Sobolev space H 1 (S 1 , V ) with the inner product 1 z, w = 2π

2n

S 1 j=1

[zj (t)wj (t) + z˙j (t)w˙ j (t)] dt .

Let J=

0 In

−In 0

,

where In is the n × n identity matrix. Consider the linear operator A : z → d z in L2 (S 1 , V ) with domain D(A) = H 1 (S 1 , V ). −J dt The real space V is isomorphic to the complex linear space Cn = Rn +iRn . The isomorphism is deﬁned as follows: Let e1 , . . . , e2n be an orthonormal basis in V , and let φj = ej + iej+n , j = 1, 2, . . . , n . 2n

ˆ = They form a basis in Cn . Let z = j=1 zj ej be corresponding to z n j=1 (zj − izj+n )φj . Deﬁne the inner product [ zˆ, w ˆ ] = Re

n

(zj − izj+n )(wj − iwj+n )

j=1

=

n 2n (zj wj + zj+n wj+n ) = zj wj . j=1

j=1

Then the correspondence z → zˆ is linear and inner product preserving, and then {φj j = 1, . . . , n} is an orthogonal basis in Cn . The real Hilbert space L2 (S 1 , V ) is isomorphic and homeomorphic to the complex Hilbert space L2 (S 1 , Cn ) with inner product deﬁned by

1 [ zˆ(t), w(t) ˆ ]dt = z, w , 2π S 1 then z → zˆ is an inner product preserving linear isomorphism of L2 (S 1 , V ) → L2 (S 1 , Cn ). The space L2 (S 1 , Cn ) has an orthonormal basis: {e−imt φj j = 1, 2, . . . , n; m = 0, ±1, ±2, . . . } .

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5 Topological and Variational Methods

Every zˆ ∈ L2 (S 1 , Cn ) has the Fourier expansion +∞ n −imt zˆ = cjm e φj . m=−∞

j=1

One has

+∞ n

|cjm |2 = z2 =

j=1 m=−∞

1 2π

[ zˆ, zˆ ]dt . S1

Hence zˆ ↔ c = {cjm } is an isometric isomorphism of L2 (S 1 , Cn ) → (l2 )n =

n :

l2 .

j=1 n

According to the Fourier expansion, zˆ ∈ H 1 (S 1 , C ) if and only if +∞ n

(1 + |m|)2 |cjm |2 < +∞ .

j=1 m=−∞

Note that

d −imt (e φj ) = me−imt φj ; dt it is easy to verify that A is a self-adjoint operator with domain D(A) = H 1 (S 1 , Cn ), and then {e−imt φj } is just the diagonalized orthonormal basis. Let M (m) = span{e−imt φ1 , . . . , e−imt φn }, m ∈ Z , −J

then L2 (S 1 , Cn ) = ⊕m∈Z M (m) ; this is the spectral decomposition or A. 5.3.2 Periodic Solutions Theorem 5.3.2 Suppose that H ∈ C 2 (Rn × Rn , R1 ) satisﬁes: (1) H(θ) = 0, H (θ) = θ, H (θ) = 0, H(z) 0. (2) H(z) = γ2 |z|2 , for |z| R, R > 0, γ ∈ (1, 2). Then the Hamiltonian system (5.22) possesses a nontrivial 2π-periodic solution. d Proof. Deﬁne G(z) = H(z) − γ2 |z|2 . Set A = −J dt ,

H(z(t))dt , ψ(z) = S1

and

G(z(t))dt ;

g(z) = S1 n

then ψ ∈ C 2 (H, R1 ), (H = L2 (S 1 , C )) possesses the following properties:

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

(1) ψ(θ) = 0, ψ (θ) = θ, ψ (θ) = 0, ψ(x) 0 (2) ∃ b, M > 0 such that ∀x ∈ H

375

∀x ∈ H.

g (x)L(H) M , γ ψ(x) x2H − b , 2 ψ (x) = γx + g (x) . 1

1

Denote the Sobolev space H 2 (S 1 , Cn ) = D(|A| 2 ) by E, and let E ∗ denote the subspace associated with the ∗ eigenvalues of E, where ∗ = +, 0, −. Let P ∗ be the orthogonal projections onto E ∗ , ∗ = +, 0, −. We deﬁne a bilinear form on E: 1 1 ((z, w)) = (|A| 2 z, |A| 2 w) , where ( , ) is the inner product on H. The system (5.22) is rewritten as Az = ψ (z), and equivalently,

z ∈ D(A) ,

(5.27)

(A − γ)z = g (z) .

We introduce a functional on E: Φ(x) =

1 (((P + − P − )x, x)) − ψ(x) . 2

Obviously, if z ∈ D(A) is a critical point of Φ, then z solves (5.27). We choose an arbitrary element e in M (1) with norm one, and set Q = ((E − ⊕ E 0 ) ∩ BR (θ)) × [0, R]e S = E + ∩ ∂Bε (θ)

for R > 0 large, and

for ε > 0 small .

Since ψ(x) = o(x2H ) as xH → 0, we have β > 0 such that inf Φ S

ε2 + o(ε2 ) β . 2

We turn out to estimate the values of Φ on ∂Q: Φ(x) −ψ(x) 0 ∀x = x− + x0 ∈ E − ⊕ E 0 , 1 1 γ Φ(x− + x0 + se) − x− 2 + (1 − γ)s2 + b − x0 2 0 , 2 2 2 as x− + x0 + se ∈ ∂Q, s > 0 and R large enough. In summary Φ|∂Q 0. Since both S and Q are inﬁnite dimensional, and the intersection of ∂Q and S has not been discussed previously, we shall use the Galerkin method instead. Let Ek = ⊕|j |k M (j), and Φk = Φ|Ek . Then Sk and ∂Qk link, where Sk = S ∩ Ek , Qk = Q ∩ Ek , ∀k. We verify the (P S) condition for Φk . Denote by Πk the orthogonal projection onto Ek .

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5 Topological and Variational Methods

Let {xj } ⊂ Ek be a sequence along which Φk (xj ) → θ. Then (A − γ)xj = Πk g (xj ) + o(1) , i.e.,

xj = (A − γ)−1 (Πk g (xj ) + o(1)) .

Since g (xj ) is bounded, {xj } is bounded, and then {xj } is subconvergent. Applying the minimax theorem for linkings, we have a critical point zk ∈ Ek satisfying (A − γ)zj = Πk g (zk ) , Φ(zk ) β > 0 . It remains to prove that {zk } is subconvergent in E. Since we have zk = (A − γ)−1 Πk g (zk ) , and that the linear map (A − γ)−1 : L2 → H 1 → E is compact, {zk } has a convergent subsequence with limit zˆ. Thus Φ(ˆ z ) β > 0 and zˆ = (A − z ), which implies zˆ ∈ D(A) and that zˆ is a nontrivial solution of γ)−1 g (ˆ (5.27). The proof is complete. 5.3.3 Weinstein Conjecture As to the given energy question, the breakthrough is due to Rabinowitz and Weinstein. In 1978, Rabinowitz [Ra 3] proved that a strongly star-shaped hypersurface (Weinstein [Wei 1] proved for a convex hypersurface) carries at least one periodic orbit. Nevertheless, the star-shapedness, and then the convexity, is not invariant under canonical diﬀeomorphism. Weinstein [Wei 2] called for a symplectic invariant version. He made the following conjecture: Every compact smooth hypersurface of contact type carries at least a periodic orbit. A hypersurface Σ is said to be of contact type if there exists a vector ﬁeld X in a neighborhood of Σ, such that (1) X Σ. (2) The ﬂow φt , generated by X (i.e., (dφt )J(dφt )T = et J, ∀t.

d t dt φ

= X ◦ φt and φ0 = id) satisﬁes

A matrix A satisfying AJAT = µJ, where µ is a nonzero constant, is called a canonical matrix with multiple µ. A diﬀeomorphism φ : R2n → R2n satisfying (dφ)J(dφ)T = µJ, is called a canonical diﬀeomorphism with multiple µ. Remark 5.3.3 In the deﬁnition of contact type, the condition (2) is equivalent to LX ω = ω, where ω is the sympectic form ω(u, v) = Ju, v , ∀u, v ∈ R2n , · is the inner product in R2n , and LX is the Lie-derivative of the 2-form.

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

377

Lemma 5.3.4 Suppose that φ is a canonical diﬀeomorphism with multiple µ, and that z4 is a solution of the Hamiltonian system z˙ = JH (z) . 4 Let H(u) = µH(z), where u = φ(z). Then u ˆ = φ(ˆ z ) is a solution of 4 (u) . u˙ = J H

Proof. By changing variables: u˙ = dφ(z) · z˙ = (dφ(z)) J(∇z H) 1 T 4 = (dφ(z)) J (dφ(z)) ∇u H µ 7 6 1 T 4 = (dφ(z)) · J · (dφ(z)) · ∇u H µ 4 (u) . = JH Example 5.3.5 A hypersurface Σ is called strongly star-shaped, if there exists a point x0 ∈ R2n , such that x − x0 , n(x) > 0

∀x ∈ Σ ,

where n(x) is the unit outward normal vector on Σ. A strongly star-shaped hypersurface is of contact type. Indeed, without loss of generality, we may assume x0 = θ. Let the vector ﬁeld 1 X(x) = x . 2 The strong star-shapedness implies that X Σ; the ﬂow φt generated by X is deﬁned to be φt (x) = et/2 x , i.e., φt (x) = et/2 id. Therefore, (dφt )J(dφt )T = et J . Now we turn to the Weinstein conjecture. By deﬁnition, for a contact-type hypersurface Σ, we have@a+family Σε = φε (Σ), ε ∈ (−1, 1). & % of hypersurfaces 2n Σε |ε| < δ has two components: The interior Let 0 < δ < 1, then R B, which with no with loss of generality contains & exterior + %the origin, and the ε ∈ (−1, −δ) ⊂ B. A, which is unbounded. Also one may assume Σ ε + Let d = diam {Σε | ε ∈ (−1, 1)}. We deﬁne d < r < 2d, γ2 r2 < b < r2 , γ = 3 ∞ ∞ 2 , and functions f ∈ C (−1, 1), g ∈ C (0, ∞) satisfying

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5 Topological and Variational Methods

0, s ∈ ( −1, −δ ] , f (s) = b, s ∈ [ δ, 1) , b, s ≤ r, g(s) = γ 2 2 s , s ≥ R, where R > r , g(s) ≥ f (s) > 0

γ 2 s for s > r , 2 0 < g (s) < γs

for s ∈ (−δ, δ)

sr.

Then we deﬁne a Hamiltonian function: ⎧ 0 z∈B, ⎪ ⎪ ⎪ ⎨f (ε) z ∈ Σε |ε| δ , H(z) = ⎪ b z ∈ A and |z| r , ⎪ ⎪ ⎩ g(|z|) |z| > r . The function H satisﬁes assumptions (1) and (2) of Theorem 5.3.2. According to Theorem 5.3.2, there is a nontrivial 2π-periodic solution zˆ for this Hamiltonian system, with

d 1 H(ˆ z (t))dt β for some β > 0 . −J zˆ, zˆ − Φ(ˆ z) = 2 dt S1 Lemma 5.3.6 The solution zˆ lies on Σε for some ε ∈ (−δ, δ). Proof. 1. If |ˆ z (0)| > r, then |ˆ z (t)| = |ˆ z (0)|, because the energy surface of H at zˆ(0) is a sphere. Thus

2π 1 g (|ˆ z (0)|) |ˆ z (0)|2 · 2π − g(|ˆ z (t)|)dt 2 0 = γπ|ˆ z (0)|2 − γπ|ˆ z (0)|2 = 0 .

Φ(ˆ z) =

This is impossible. + Σε , then H(ˆ z (0)) = b or 0. In either case, 2. If |ˆ z (0)| r but zˆ(0) ∈ / |ε| 0 is a necessary condition for the existence of a solution. However, in this section we always assume that K(x) > 0 ∀x ∈ S 2 . We introduce

1 [|∇u|2 + 2u] , S(u) = 4π S 2 and J(u) = S(u) − log

1 4π

Ke2u . S2

The Euler–Lagrange equation for J reads as 4πKe2u on S 2 . u = 1 − 2u Ke 2 S

(5.29)

point of J if and only Noticing that J(u+c) = J(u) ∀c ∈ R1 , u0 is a critical 1 2u0 Ke . Once we obtain if u solves (5.28), where u = u0 −c, and c = 12 log 4π S2 a critical point of J, after adjusting a constant, we solve equation (5.29). In order to avoid the indetermination caused by the translation invariance of J, we add a constraint. Let

e2u = 4π , X = u ∈ H 1 (S 2 )| S2

and consider the functional J on X. The Euler–Lagrange equation is again (5.29). 5.4.1 The Conformal Group and the Best Constant In this section, we ﬁrst analyze the special case K ≡ 1, and ﬁnd out all the minimizers. Now, we introduce the following stereographic projection: ˆ = C ∪ {∞} π : S2 → C x1 + ix2 π(x1 , x2 , x3 ) = z = ∀(x1 , x2 , x3 ) ∈ S 2 ⊂ R3 . 1 − x3 This is a conformal diﬀeomorphism with (π −1 )∗ g0 =

4 dzdz . (1 + |z|2 )2

5.4 Prescribing Gaussian Curvature Problem on S 2

381

Let Conf(S 2 ) be the conformal group consisting of all conformal diﬀeoobius group morphisms from S 2 into itself. Conf(S 2 ) is isomorphic to the M¨ SL(2, C). A subgroup D of Conf(S 2 ) is deﬁned as follows: for any Q ∈ S 2 , we choose an orthonormal frame {e1 , e2 , e3 }, such that Q is the north pole, i.e., Q = e3 , by which we deﬁne a stereographic projection πQ , ∀t ∈ [1, ∞), let ˆ φQ,t = π −1 ◦ τt ◦ πQ , and let τt z = tz ∀z ∈ C, Q D = {φQ,t | (Q, t) ∈ S 2 × [1, ∞)} . Since φQ,1 = id, and φ−Q,t = φQ,t−1 , D is a submanifold and is parameterized to be B 3 (S 2 × [1, ∞))/(S 2 × {1}) under the correspondence: φQ,t → t−1 t Q. For any Riemannian metric g on S 2 , ∀φ ∈ Conf(S 2 ), the function ψ=

1 log det(dφ) 2

(5.30)

satisﬁes φ∗ g0 = e2ψ g0 . The conformal group action on the function v reads as: (5.31) φ∗ v := vφ = v ◦ φ + ψ . We deﬁne the conformal Laplacian as follows: Lg ϕ = −g ϕ + Kg

∀ϕ ∈ C ∞ (S 2 ) ,

(5.32)

where g is the Laplacian–Beltrami operator and Kg is the Gaussian curvature with respect to g, respectively. Thus, for g = e2u g0 , we have Lg ϕ = −g ϕ + Kg = e2ϕ Ke2ϕ g −2u

=e

−2u

=e

·e

2(u+ϕ)

(5.33) Ke2(u+ϕ) g0

Lg0 (u + ϕ) .

Lemma 5.4.1 Lg0 vφ = det(dφ)(Lg0 v) ◦ φ

(5.34)

Proof. (Lg0 v) ◦ φ = Le2ψ g0 (v ◦ φ) = e−2ψ Lg0 (v ◦ φ + ψ) = det(dφ)−1 Lg0 vφ ,

provided by (5.33) and (5.31).

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5 Topological and Variational Methods

Lemma 5.4.2 All solutions of the equation: −ψ + 1 = e2ψ on S 2

(5.35)

are of the form (5.30). Proof. Since Lg0 ψ = −ψ + 1, plugging ψ = oφ into (5.34), (5.35) follows from Lemma (5.4.1). Conversely, if ψ is a solution of (5.35), then Kg = 1 for g = e2ψ g0 . According to the isometric theorem in Riemannian geometry, there exists an isometry φ : (S 2 , g) → (S 2 , g0 ) satisfying φ∗ g0 = g, i.e., φ ∈ Conf(S 2 ). By (5.30), ψ has the form 12 log det(dφ). 1 Lemma 5.4.3 Both the functionals S and T (u) = 4π e2u are conformal S2 invariant. Furthermore, S(ψ) = 0 ∀ψ of form (5.30). Proof. 1. Obviously, ∀φ ∈ Conf(S 2 ), ∀v ∈ H 1 (S 2 ),

1 1 2(v◦φ) T (vφ ) = e det(dφ) = e2v = T (v) . 4π S 2 4π S 2 2. We prove that S(ψ) = 0 ∀ψ of the form (5.30). For any φ0 ∈ Conf(S 2 ), we introduce a diﬀerentiable path φt ∈ Conf(S 2 ), d ψt |t=0 . Then by step 1, with φt |t=0 = φ0 . Let ψt = 12 log det(dφt ), and w = dt

d 1 0 = T (oφt )|t=0 = e2ψ0 w , (5.36) dt 2π S 2 and

d 1 S(ψt )|t=0 = [∇ψ0 · ∇w + w] dt 2π S 2

1 = [−ψ0 + 1]w 2π S 2

1 e2ψ0 w = 0 . = 2π S 2

Since the path φt is arbitrary, we have S(ψ) = const. In particular, let φ0 = id, then ψ0 = 0 and S(ψ) = 0. 3. Finally, we turn to proving the conformal invariant of S. ∀φ ∈ Conf(S 2 ),

1 |∇(v ◦ φ)|2 + 2 [∇(v ◦ φ)∇ψ + v ◦ φ] + S(ψ) S(vφ ) = 4π 2 S2 S

1 |∇v|2 + 2 (Lg0 ψ)v ◦ φ = 4π 2 S2 S

1 = |∇v|2 + 2 v ◦ φe2ψ 4π S2 S2 = S(v) .

5.4 Prescribing Gaussian Curvature Problem on S 2

383

In the sequel of this subsection we are going to prove that S(u) ≥ 0 ∀u ∈ X, and then, all the minimizers of S on X are of the form (5.30). Firstly, we give a necessary condition for the solvability of (5.28). Namely: Theorem 5.4.4 (Kazdan–Warner) If u is solution of (5.28) then

∇K, ∇xi e2u = 0 i = 1, 2, 3 .

(5.37)

S2

Proof. By Lemma 5.4.3, ∀φQ,t ∈ D, J(uφQ,t ) = S(u) − log

1 4π

S2

2u K ◦ φ−1 . Q,t e

(5.38)

If u is a solution of (5.28), then u is a critical point of J, therefore ; < d d J(uφQ,t )|t=1 = J (u), uφQ,t |t=1 = 0 . dt dt By (5.38), d 1 J(uφQ,t )|t=1 = 2u dt K ◦ φ−1 Q,t e S2

S2

∇K, ∇(x, Q) e2u .

We choose Q = e1 , e2 , and e3 , respectively; (5.37) follows.

For any u ∈ H 1 (S 2 ), we deﬁne the mass center of u to be

−1

e2u · x1 e2u , x2 e2u , x3 e2u . P (u) = S2

S2

S2

S2

Obviously, P (u) ∈ B 3 . We want to show that ∀u ∈ H 1 (S 2 ), after a conformal transform, the mass center can be moved to the origin, i.e., there exists (Q, t) ∈ S 2 × [1, ∞) such that P (u ◦ φQ,t ) = θ, or equivalently, the nonlinear system:

xi e2uφθ,t = 0, i = 1, 2, 3 , (5.39) S2

is solvable. We solve the nonlinear system by IFT. Let F : H 1 (S 2 ) × B 3 → B 3 be the map

x ◦ φ−Q,t e2u , (5.40) F (u, ξ) = S2

where ξ = s(t)Q, and s(t) =

1 − t−1 1 − t−2 ln t

is a diﬀeomorphism: [1, ∞) → [0, 1). We need some estimate for ∂ξ F .

t ∈ [1, 2] , for t large ,

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5 Topological and Variational Methods

Lemma 5.4.5 Assume {Q(t)} ⊂ S 2 , Q(t) → Q0 as t → +∞. Then

x ◦ φQ(t),t e2u → Q0 e2u as t → +∞ S2

(5.41)

S2

uniformly in {u ∈ H 1 (S 2 )| u ≤ c} ∀c > 0. Moreover, e2ψQ,t → 4πδ(−Q) as t → +∞ .

(5.42)

Proof. Since the mapping H 1 (S 2 ) → L1 (S 2 ) : w → e2w is compact, it is suﬃcient to prove (5.41) for ﬁxed u. Noticing that x ◦ φQ,t = D−1 {2t(x − (Q · x)Q) + [t2 (1 + Q · x) − (1 − Q · x)]Q} , where D = t2 (1 + Q · x) + (1 − Q · x) , we have x ◦ φQ(t),t → Q0 except x = −Q0 . Our conclusion follows from the Lebesgue dominance theorem.

Lemma 5.4.6 For any compact subset C ⊂ H 1 (S 2 ) and any constants b > 1, the matrix ∂ξ F (u, ξ) has a uniformly bounded inverse ∀u ∈ C, t ≤ b, where ξ = s(t)Q solves the equation F (u, ξ) = θ. Proof. We write x ! = x ◦ φQ,t , and compute 1 (Q − (! x · Q)! x), t (t2 − 1)2 t2 − 1 ∂e x (e − (! x · e)! x) + [(! x · e)Q − (! x · Q)e] , != 2t 2t

∂t x !=

∀e ∈ TQ (S 2 ). One chooses an orthonormal frame {e1 , e2 } in TQ (S 2 ); the 2 2 −1 −1 matrix ∂ξ F (u, ξ) is then an array of the three vectors ( t 2t g1 , t 2t g2 , ts1(t) g3 ), where

1 (ej − (ˆ x · ej )ˆ x)e2u j = 1, 2, 3 , gi = 4π S 2 x ˆ = x ◦ φ−Q,t and e3 = Q, because ξ is a solution. Let A be the matrix consisting of (g1 , g2 , g3 ). It is symmetric and positive 3 deﬁnite. Indeed, ∀λ = i=1 λi ei with |λ| = 1, 3 λi gi · λ (Aλ) · λ = i=1

= 1−

1 4π

S2

(ˆ x · λ)2 e2u .

5.4 Prescribing Gaussian Curvature Problem on S 2

385

Fixing u, ∃δ = δ(u) > 0 such that

1 (ˆ x · λ)2 e2u ≤ 1 − δ 4π S 2 uniformly in (Q, t) ∈ S 2 × [1, b]. Since C is compact, ∃δ0 > 0 such that

1 (ˆ x · λ)2 e2u ≤ 1 − δ0 , ∀(u, ξ) ∈ C × (S 2 × [1, b]) . 4π S 2 i.e., (Aλ) · λ ≥ δ0 . Thus ∃δ1 > 0 such that | det (A)| ≥ δ1 , and then | det (∂ξ F (u, ξ))| ≥

(t2 − 1)2 δ1 . 4t3 s (t)

The lemma is proved.

We are ready to solve (5.40) by the continuity method. According to Lemma 5.4.6, equation (5.39) is locally solvable for any solution (u0 , ξ0 ) of (5.39) from the implicit function theorem. One chooses a starting point (u0 , ξ0 ), satisfying F (u0 , ξ0 ) = θ. For any u ∈ H 1 (S 2 ), we consider a path u(τ ) = 12 log(τ e2u + (1 − τ )e2u0 ) connecting u0 and u, and study the equation F (u(τ ), ξ(τ )) = θ for τ ∈ [0, 1]. Let [0, τ0 ) be the maximal existence interval. If τ0 = 1, then (5.39) is solvable for such a u. Otherwise, there must be a sequence τj → τ0 such that tj = t(τj ) → +∞. Because, if not, t(τ ) is bounded on [0, τ0 ), and then by Lemma 5.4.6, (∂ξ F (u(τ ), ξ(τ )))−1 is uniformly bounded. The solution is extendible beyond τ0 . This contradicts the maximality of τ0 . After subtracting a subsequence from {τj }, again denoted by {τj }, we have Qj = Q(τj ) → Q0 ∈ S 2 . By Lemma 5.4.5, θ = F (u(τj ), ξ(τj ))

2u(τj ) x ◦ φ−Qj ,tj e → −Q0 = S2

e2u(τ0 ) .

S2

But this is impossible. Thus we have completed the proof. Theorem 5.4.7 ∀u ∈ H 1 (S 2 ), there exists (Q, t) ∈ S 2 × [1, ∞) satisfying (5.39). Now let us deﬁne

1 e2u . I(u) = S(u) − log 4π S 2 Obviously, I(u) = S(u) ∀u ∈ X. We shall prove:

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5 Topological and Variational Methods

Theorem 5.4.8 I(u) ≥ 0 ∀u ∈ H 1 (S 2 ), and the equality holds if and only if u = 12 log det(dφ) + c, ∀φ ∈ Conf(S 2 ) ∀c ∈ R1 . Thus all the minimizers of S on X are of the form (5.30). Proof. First we prove the existence of a minimizer of I on Y = {u ∈ H 1 (S 2 )| P (u) = θ, S 2 u = 0}. According to Aubin’s inequality Sect. 4.2, ∀u ∈ Y .

1 1 2 [|∇u| + 2u] − log e2u I(u) = 4π S 2 4π S 2 1 1 − ≥ u 2 − log C , 4π 8π − i.e., I is coercive on Y . Since Y is weakly closed in H 1 (S 2 ), we conclude that there exists u0 ∈ Y satisfying I(u0 ) = min{I(u)| u ∈ Y }. The Euler–Lagrange equation reads as −1

1 2u0 e e2u0 = µ · xe2u0 , for some µ ∈ R3 . −u0 + 1 − 4π S 2 2u 1 Let c = 12 log 4π e 0 and v = u0 − c, then −v + 1 = (1 + µ · xe2c )e2v . Applying the necessary condition in Theorem 5.4.4, we have

µe2u0 = θ , S2

i.e., µ = θ. Hence v is a solution of (5.35), it must be of form (5.30). However, P (v) = θ; it follows that v = θ, or u0 = const. Therefore I(u0 ) = 0 and then I(u) ≥ 0 ∀u ∈ Y . Since I is translation invariant: I(u+c) = I(u) ∀c ∈ R1 , and also conformal invariant: I(uφ ) = I(u) ∀φ ∈ Conf(S 2 ), our conclusion follows from Theorem 5.4.7. Returning to equation (5.28), in which K ≡ 1, all solutions are also the minimizers of the functional I. We may even write down these solutions explicitly: 1 − α2 1 (5.43) ψ = log 2 (1 + αQ · x)2 2

2 2 where α = 1−t 1+t2 , for (Q, t) ∈ S × [1, ∞). Although Conf(S ) is a group with 1 six parameters, the representation ψQ,t = 2 log det(dφQ,t ) only has three, i.e., it is totally determined by the subgroup D.

Corollary 5.4.9 (Onofri) ∀u ∈ H 1 (S 2 ), we have

1 1 2u 2 e ≤ exp (|∇u| + 2u) . 4π S 2 4π S 2 see also Hong [Hon 1].

5.4 Prescribing Gaussian Curvature Problem on S 2

387

5.4.2 The Palais–Smale Sequence From Theorem 5.4.8, we see that J is bounded below. We claim that if K = const, then the inﬁmum of J is not achieved. Indeed, let P0 ∈ S 2 be the maximum of K. On one hand, we have J(u) ≥ − log K(P0 ), provided by Corollary 5.4.9. On the other hand,

1 1 2ψ−P0 ,t Ke = − log Ke2ψ−P0 ,t . J(ψ−P0 ,t ) = S(ψ−P0 ,t ) − log 4π S 2 4π S 2 Applying (5.42), we obtain J(ψ−P0 ,t ) → − log K(P0 ), i.e., inf J = − log K(P0 ). If u0 were a minimum of J, then we would have J(u0 ) = − log K(P0 ). Deﬁne

1 ˆ J(u) = S(u) − log K(P0 )e2u , 4π S 2 then min Jˆ = − log K(P0 ), which is achieved by u = ψQ,t + c ∀c ∈ R1 , ∀(Q, t) ∈ S 2 × [1, ∞). Then ˆ 0 ) ≤ J(u0 ) = − log K(P0 ) , − log K(P0 ) ≤ J(u ˆ Therefore, u0 = ψQ,t + c for some which implies that u0 is a minimum of J. (Q, t) ∈ S 2 × [1, ∞) and c ∈ R1 . But this is impossible. The above argument proved indirectly that the PS condition does not hold for J (Ekeland variational principle). Now, we shall analyze the (PS) sequence carefully, and then reveal the reason why the (PS) condition breaks down. Firstly, we continue the proof in Theorem 5.4.7, and obtain: Theorem 5.4.10 Let P (u) be the mass center of u, and let X0 = {u ∈ X| P (u) = θ} , then

◦

X∼ = X0 × B 3 , where ∼ = means diﬀeomorphism. Proof. After Theorem 5.4.7, it remains to prove that the solution of equation (5.39) is unique. Suppose not, then we may assume that ξi = s(ti )Qi satisﬁes (5.39), i = 0, 1. Deﬁne wi = uφQi ,ti , i = 0, 1; we pick a path (Q(τ ), t(τ )) connecting (Q0 , t0 ) and (Q1 , t1 ), and let w(τ ) = 12 log(τ e2w1 + (1 − τ )e2w0 ) ∀ τ ∈ [0, 1]. Set , u(τ ) = w(τ )φ−1 Q(τ ),t(τ )

then F (u(τ ), ξ(τ )) =

S2

x ◦ φ−Q(t),t(τ ) e2u(τ ) = θ .

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5 Topological and Variational Methods

We introduce a new parameter σ, and deﬁne for (σ, τ ) ∈ [0, 1]2 , u(σ, τ ) =

1 log[(1 − σ)e2u(τ ) + σe2u ] 2

and we solve the equation: F (u(σ, τ ), ξ(σ, τ )) = θ . By the same argument as used in Theorem 5.4.8, we have solutions (Q(σ, τ ), t(σ, τ )) depending on σ, τ continuously, and satisfying Q(0, τ ) = Q(τ ), t(0, τ ) = t(τ ) , Q(σ, i) = Qi , t(σ, i) = ti , i = 0, 1, and F (u, ξ(1, τ )) = θ . This contradicts the local uniqueness of the solution of (5.39), which follows from the implicit function theorem. The smoothness of (Q, t) depending on u follows directly. Theorem 5.4.10 provides a parameterization of X : X → X0 × (S 2 × [1, ∞)/S 2 × {1}), u → (w, Q, t). Theorem 5.4.11 Suppose that {uj = (wj , Qj , tj )} is a Palais–Smale sequence of J on X, then either it is subconvergent or tj → +∞. Proof. Since

J(uj ) = S(wj ) − log

K ◦ φQj ,tj e2wj → c ,

and 4πK(! xj )e2wj + λj e2wj + (µj · x)e2wj + ◦(1) , −wj + 1 = K(! xj )e2wj

(5.44)

where (λj , µj ) ∈ R1 × R3 , if {tj } is bounded, then after a subsequence, x !j = ! = x ◦ φQ,t for some (Q, t) ∈ S 2 × [1, ∞). x ◦ φQj ,tj → x It follows that S(wj ) and λj are both bounded. Since − K(! xj )xe2wj + ◦(1) , ∧(wj )µj = K(! xj )e2wj

where ∧(wj ) =

1 4π

S2

xi xk e2wj

, 3×3

and ∧(wj ) is uniformly positive deﬁnite, {µj } is bounded. Moreover, the embedding w → e2w from H 1 to Lp , p > 1, is compact; we may use (5.44) to conclude that {wj } is subconvergent.

5.4 Prescribing Gaussian Curvature Problem on S 2

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5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 From Theorem 5.4.11, we know that the Palais–Smale Condition for J breaks down at t = +∞. We are inspired with courage to compactify the open ball ◦

B 3 . By simple estimates, we obtain the limit of the integral:

1 K ◦ φQ,t e2w = K(Q) + ◦(1) as t → ∞ . 4π S 2 Let us extend J to be J! deﬁned on the manifold with boundary M = X0 × B where ∂M = X0 × S 2 : J(u) u∈X ! J(u) = S(w) − log K(Q) (w, Q) ∈ X0 × S 2 .

3

If J! were C 1 on M , then Remark 5.1.31 would be applicable directly to this problem. Unfortunately, this is not true. We are forced to ﬁgure out the asymptotic expansions for the partial derivatives for J. Namely we have v ∈ Tw (X0 ) such that Lemma 5.4.12 ∃b > 0, cb > 0 and ∃! 1

S (w), v! ≥ cb S 2 (w) v! , if S(w) ≤ b. Furthermore, ∀ µ, δ > 0, ∃N = N (δ, b, µ), ∃T = T (δ, b, µ) such that ∂e K(Q) | ≤ N t−1+µ log t ∀e ∈ TQ (S 2 ) , K(Q) 1 1 2K(Q) | ≤ N (|∇K(Q)|t log− 2 tS 2 (w) |∂s J − K(Q) |∂e J +

+ (log t)−1 + t2 (log t)−1 S 2 (w)) . if K(Q) ≥ δ and t ≥ T . The proof is referred to Chang and Liu [CL 2]. Combining Lemma 5.4.12 with the partition of unity, we obtain: Theorem 5.4.13 Assume K > 0. If K = 0 whenever ∇K(x) = θ, then 3 there exists a function f deﬁned on X0 × B possessing the same critical set as J, and ∃ a neighborhood U of X0 × S 2 , such that on U f (w, sQ) = S(w) − log K(Q) −

2K(Q) (1 − s) , K(Q)

where s : [1, ∞) → [0, 1) is a diﬀeomorphism with s(t) = 1 − t−2 log t for t large.

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5 Topological and Variational Methods

Furthermore, f satisﬁes the (PS) condition. Now we are ready to apply the Morse theory under general boundary 3 conditions to f on X0 × B . For given a < b such that e−a , e−b are regular values of K, we assume: (I) K has only a ﬁnite number of critical points with values in the interval [e−b , e−a ]. Set Ω = {x ∈ S 2 | K(x) < 0}, CR0 (a, b) = {x ∈ Ω| K(x) ∈ (e−b , e−a ), x is a local maximum of K}, CR1 (a, b) = {x ∈ Ω| K(x) ∈ (e−b , e−a ), x is a saddle point of K} . Theorem 5.4.14 Under assumption (I), assume (II) K(x) = 0 whenever ∇K(x) = θ . If J has only isolated critical points, then the following Morse inequalities hold: (mq + µq − βq )tq = (1 + t)Q(t) , where Q is a formal series with nonnegative coeﬃcients, and mq = qth Morse type number of J in J −1 [a, b] , βq = rank Hq (Jb , Ja ) , 0 q≥2 µg = CRq (a, b) q = 0, 1 .

Proof. It is equivalent to study the Morse inequalities for f .We notice that f is of separate variables, and that S(w) has no critical point except the minimum θ on X0 , from Lemma 5.4.12. Moreover, f possesses the same critical set as J and equals J in a neighborhood of the critical set. Now the restriction fˆ on the boundary X0 × S 2 reads as S(w) − log K(Q), which has only isolated critical points. Since f (w, sQ) = S(w) − log K(Q) − 2

K(Q) (1 − s) K(Q)

near the boundary, and the normal n(w, Q) = (θ, Q) ∈ Tw (X0 ) × R3 , we see that Σ− = {(w, Q) ∈ X0 × S 2 | f (u), n(u) ≤ 0} = {(w, Q) ∈ X0 × S 2 | Q ∈ Ω ∩ K −1 [e−b , e−a ]} .

5.4 Prescribing Gaussian Curvature Problem on S 2

391

It remains to compute the critical groups of the critical points of − log K on Ω. Now ∀Q ∈ CR0 (a, b), rankCq (− log K, Q) = δq0 , and ∀Q ∈ CR1 (a, b) rankCq (− log K, Q) = δq1 , from (II). We claim that rankC2 (− log K, Q) = 0 ∀Q ∈ Ω. If not, Q must be a local minimum of K, and then K(Q) ≥ 0. This is a contradiction. Obviously we have rankCq (− log K, Q) = 0 ∀q > 2 . On the other hand, θ is the minimum of S on X0 ; it follows that rankCq (S, θ) = δq0 ∀q . Thus the Morse type numbers of fˆ read as 0 ∀q ≥ 2 µq = CRq (a, b) q = 0, 1 .

The proof is complete. Corollary 5.4.15 Under assumptions (I) and (II). Let p = local maxima of K in Ω , q = saddle points of K in Ω . If p = q + 1, then (5.28) admits a solution.

¯ 3 is conProof. By taking e−b < min K and e−a > max K. Since X0 × B tractible, the conclusion follows from the Morse inequalities. Even assumption (I) can be dropped. Namely, Theorem 5.4.16 Assume (II) and if deg(Ω, ∇K, θ) = 1, then (5.28) admits a solution. ! satisfying: Proof. First, we perturb K to K ! > 0 possesses only nondegenerate critical points. (1) K ! (2) For a given neighborhood U of the critical set V of K, K(x) = K(x) ∀x ∈ U. ! C 2 is small. (3) K − K

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5 Topological and Variational Methods

! is constructed as follows: Find a cover ∪l B (xi ) ⊂ U of V, ∀i, we pull K i=1 back the linear function on B (xi ), and obtain a function φa,i (x) such that ∇φa,i (x) = a ∀x ∈ B (xi ), where a is any given vector in R2 . Let {ρi }l1 be the partition of unity of V with suppρi ⊂ B (xi ), and let ! K(x) = K(x) − φa,i (x)ρi (x) . By Sard’s theorem, a can be chosen as small as we wish such that (1) and (3) hold. Second, as in Theorem 5.4.13, we can construct a functional f! similar to f satisfying: ! and (1) In a neighborhood O of X0 × S 2 , f! is similar to f with respect to K, there are no critical points there. (2) Outside O, f! = f = J. ! Now, let p, q be the numbers of local maxima and saddle points of K 2 ! ! in Ω = {x ∈ S | K(x) < 0}, respectively. By homotopy invariance of the Brouwer degree: ! ∇K, ! θ) = p − q . 1 = det(Ω, ∇K, θ) = deg(Ω, This proves the existence of a critical point of f! so does for J.

Remark 5.4.17 Other suﬃcient conditions were obtained by Chen and Ding [CD]. Chang and Yang [CY 1, 2], Han [Han], and Hong [Hon 2]. All these results are contained in Theorem 5.4.14, see [CL 2]. Remark 5.4.18 The prescribing scalar curvature problem on S n is a similar problem. For n = 3, it was due to Bahri and Coron [BC] and later, Schoen and Zhang [SZ]. For general n, see Li [Li], and Chen and Lin [ChL].

5.5 Conley Index Theory The Conley index is another topological tool in nonlinear analysis. It can be seen as an extension of both the Leray–Schauder degree and the critical groups. In fact, the Conley index theory is an extension of the Morse theory. It is based on ﬂows without a variational structure. In contrast with the degree theory and the Morse theory, Conley’s theory studies the invariant sets of a ﬂow rather than either the ﬁxed points for a compact vector ﬁeld or the critical points for diﬀerentiable functions. Let (X, d) be a metric space, and let ϕ : R1 × X → X be a ﬂow, i.e., it is continuous and satisﬁes: (1) ϕ(0, x) = x ∀x ∈ X, (2) ϕ(t, ϕ(s, x)) = ϕ(t + s, x)

x ∈ X, ∀t, s ∈ R1 .

5.5 Conley Index Theory

393

Deﬁnition 5.5.1 A set S ⊂ X is called invariant with respect to the ﬂow ϕ, if ϕ(R1 , S) = S, i.e., ∀x ∈ S, ∀t ∈ R1 , ϕ(t, x) ∈ S. Example 1. Let X be a Banach space, and let K : X → X be compact. Let ϕ be the ﬂow derived by the compact vector ﬁeld f = id − K: ϕ(t, ˙ x) = f (ϕ(t, x)) ∀(t, x) ∈ R1 × X , ϕ(0, x) = x , then any subset of the ﬁxed-point set of K is an invariant set for the ﬂow ϕ. Example 2. Let M be a smooth Banach–Finsler manifold and let f ∈ C 1 (M, R1 ) satisfy the (PS) condition. If V is a pseudo-gradient vector ﬁeld of f and ϕ is the associated ﬂow: ϕ(t, ˙ x) = −g(ϕ(t, x))

V (ϕ(t, x)) with ϕ(0, x) = x , V (ϕ(t, x))

(5.45)

where g(x) = Min{d(x, K), 1}, and K is the critical set of f . Then any subset of the critical set of f is an invariant set of the ﬂow ϕ. Deﬁnition 5.5.2 We call the triple (M, f, ϕ) a pseudo-gradient ﬂow if M is a smooth Banach–Finsler manifold, f ∈ C 1 (M, R1 ) satisﬁes the (PS) condition, and ϕ is deﬁned by equation (5.45.) 5.5.1 Isolated Invariant Set Invariant sets can be extremely complicated. It is known from dynamical systems that there are chaotic dynamics and fractal structure. As we have learned in Chap. 3 and in Sects. 5.1 and 5.2, we single out isolated invariant sets as the object of our study. Before going on, we introduce some notions from dynamical systems. 1 ∪ {+∞}, for any subset A of X, we write ∀T ∈ R+ GT (A) = ∩ ϕ(t, A), and ΓT (A) = {x ∈ GT (A)| ϕ([0, T ], x) ∩ ∂A = ∅} . |t|≤T

In particular, we write I(A) = G∞ (A). It is the maximal invariant subset of A. By deﬁnition, we have the following properties: (1) GT (A) = GT (A), GT1 (A) ⊂ GT2 (A), if T1 ≥ T2 , A1 ⊂ A2 ⇒ GT (A1 ) ⊂ GT (A2 ), GT (A) is closed, GT1 +T2 (A) = GT2 (GT1 (A)). (2) If GT (A) ⊂ int(A), then G2T (A) ⊂ int(GT (A)).

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5 Topological and Variational Methods

Proof. Suppose not, then ∃y ∈ G2T (A) ∩ ∂GT (A), i.e., ∃yn → y with ϕ([−T, T ], yn ) ⊂ A, or ∃tn ∈ [−T, T ], ϕ(tn , yn ) ∈ A. There exists a subsequence {tn } of {tn } such that tn → t ∈ [−T, T ] and ϕ(t, y) ∈ ∂A. But y ∈ G2T (A), i.e., ϕ([−2T, 2T ], y) ∈ A, which implies that ϕ(t, y) ∈ GT (A) = GT (A) ⊂ int(A), a contradiction. Now, we introduce a family of closed sets: Σ = Σ(ϕ) = {A ⊂ X| A is closed, and ∃ T > 0 such that GT (A) ⊂ int(A)} . Deﬁnition 5.5.3 A neighborhood U of an invariant set S for the ﬂow ϕ is called isolating if U ∈ Σ and I(U ) = S. An invariant set S is called an isolated invariant set if there exists an isolating neighborhood. Remark 5.5.4 If U is compact and if I(U ) = S ⊂ int(U ), then U ∈ Σ, i.e., ∃T > 0 such that GT (U ) ⊂ int(U ). In fact, if not, i.e., Gn (U ) ⊂ int(U ) ∀n ∈ N, we have yn ∈ Gn (U )\int(U ), i.e., ϕ([−n, n], yn ) ⊂ U but yn ∈ int(U ). Since U is compact, there is a subsequence yni → y0 ∈ I(U ), but y0 ∈ int(U ). A contradiction. Moreover, we have: (3) If U ∈ Σ, then ∃T1 > 0 such that GT1 (U ) ∈ Σ, and ∀t ∈ R1 , ϕ(t, U ) ∈ Σ. Proof. It is known that both GT1 (U ) and ϕ(t, U ) are closed. Since ∃T > 0 such that GT (U ) ⊂ int(U ). Taking T1 = T , from (1) and (2), we have GT (GT1 (U )) = G2T (U ) ⊂ int(GT (U )) = int(GT1 (U )). Also GT (ϕ(t, U )) = ∩ ϕ(t+s, U ) = ϕ(t, GT (U )) ⊂ ϕ(t, int(U )) = int(ϕ(t, U )) . |s|≤T

(4) If U ∈ Σ, then ΓT (U ) ⊂ ∂GT (U ) is closed. Proof. If {xn } ∈ ΓT (U ) and xn → x, then ∃tn ∈ [0, T ] such that ϕ(tn , xn ) ∈ ∂U , after a subsequence, we have tn → t ∈ [0, T ] and ϕ(t, x) ∈ ∂U . Thus x ∈ ΓT (U ). This shows that ΓT (U ) is closed. Next, ∀x ∈ ΓT (U ), we verify x ∈ ∂GT (U ). By deﬁnition, ∃t¯ ∈ [0, T ], ∃yn ∈ U such that yn → ϕ(t, x). Let xn = ϕ(−t, yn ), we have xn → x and / int(GT (U)), therefore xn ∈ GT (U ). Since GT (U ) is closed, we have x ∈ x ∈ ∂GT (U ). The following terminologies are taken from dynamical systems: Deﬁnition 5.5.5 ∀x ∈ X, the set ω(x) = ∩ ϕ([t, ∞), x) t>0

is called the ω-limit set of x, and the set

5.5 Conley Index Theory

395

ω ∗ (x) = ∩ ϕ((−∞, −t], x) t>0

is called the ω ∗ -limit set of x. Given a subset S ⊂ X, the set [S] = {x ∈ X| ω(x) ∪ ω ∗ (x) ⊂ S} is called the invariant hull of S. By deﬁnition, ω(x) = ω(ϕ(t, x)), ω ∗ = ω ∗ (ϕ(t, x)) ∀t ∈ R1 , and then [S] is invariant. Moreover, if S is invariant, then S ⊂ [S]. Lemma 5.5.6 Let (M, f, ϕ) be a pseudo-gradient ﬂow. Then for any x ∈ M the set ω(x) is compact and is a subset of Kc for some critical value c, where Kc = K ∩ f −1 (c). The same holds true for the set ω ∗ (x). Proof. Since the function is nonincreasing along the pseudo-gradient ﬂow, ﬁrst we show that ω(x) is on one level, say ω(x) ⊂ f −1 (c) for some c ∈ R1 . Indeed, if not, then there exist tn , tn ↑ ∞ such that ϕ(tn , x) → y and ϕ(tn , x) → y with f (y) < f (y ). We may always assume that tn > tn , which means that f (y ) = limf (ϕ(tn , x)) ≤ limf (ϕ(tn , x)) = f (y) . n

n

This is a contradiction. Next, we prove that ω(x) ⊂ K. Indeed, if ∃y ∈ ω(x)\K, then we choose regular values a < b such that both f (x) and f (y) are in (a, b). Since Kab = K ∩ f −1 [a, b] is compact, there exists r > 0 such that Br (y) ∩ (Kab )r = ∅. By deﬁnition, there exists tn → +∞ such that xn = ϕ(tn , x) → y. From the (PS) condition, there exists δ > 0 such that f (x) ≥ δ for all x ∈ f −1 [a, b]\(Kab )r , We claim that there exists tn → +∞ such that xn = ϕ(tn , x) ∈ ∂(Kab )r . If not, ∃T > 0 such that ϕ([T, ∞), x) ∩ (Kab )r = ∅, then we would have f (y) = limf (xn ) = lim inf f (ϕ(t, x)) ≤ a , n

t→+∞

This is impossible. Now we choose tn → +∞ with tn < tn such that xn = ϕ(tn , x) ∈ Br (y), ϕ([tn , tn ], x) ∩ (Kab )r = ∅ . It follows that f (xn ) − f (xn ) ≥ δxn − xn ≥ δdist(Br (y), (Kab )r ) . Again, this is impossible. The lemma is proved.

Deﬁnition 5.5.7 A subset W of X is said to have a mean value property (MVP) with respect to the ﬂow ϕ, if ∀x ∈ X, ∀t1 < t2 , ϕ(ti , x) ∈ W, i = 1, 2 implies that ϕ([t1 , t2 ], x) ⊂ W . We now introduce the following key concept:

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5 Topological and Variational Methods

Deﬁnition 5.5.8 (Dynamically isolated critical set) Let (M, f, ϕ) be a pseudo-gradient ﬂow. A subset S of the critical set K is said to be a dynamically isolated set if there exist a closed neighborhood O of S and regular values α < β of f such that O ⊂ f −1 [α, β] , and

! ∩ K ∩ f −1 [α, β] = S , cl(O)

! = ∪ ϕ(t, O). We shall then say that (O, α, β) is an isolating triplet where O 1 t∈R

for S. Lemma 5.5.9 Let (M, f, ϕ) be a pseudo-gradient ﬂow and K be the critical set of f . If U is a closed (MVP) neighborhood of S satisfying U ∩ K = S, then [S] = I(U ). If further, there exist real numbers α < β and a (MVP) closed set W , satisfying U ⊂ W ⊂ f −1 [α, β] and W ∩ K = S, then ∃T > 0 such that GT (W ) ⊂ int(U). Proof. 1. [S] = I(U ) “⊂” ∀x ∈ [S], by deﬁnition ω(x) ∪ ω ∗ (x) ⊂ S, then ∃t± n → ±∞ such that ± + ϕ(tn , x) ∈ U , and then ϕ([t− n , tn ], x) ⊂ U . Since n is arbitrary, it follows that ϕ(t, x) ∈ U ∀t ∈ R1 , i.e., x ∈ I(U ). “⊃” ∀x ∈ I(U ), ϕ(t, x) ∈ U ∀t ∈ R1 . Since U is closed, ω(x) ∪ ω ∗ (x) ⊂ U . From Lemma 5.5.6, ω(x) ∪ ω ∗ (x) ⊂ K, therefore ω(x) ∪ ω ∗ (x) ⊂ U ∩ K = S, i.e., x ∈ [S]. 2. GT (W ) ⊂ int(U ). From the (PS) condition, ∃δ ∈ (0, 1) such that dist(x, K) ≥ δ and f (x) ≥ δ ∀x ∈ W \int(U ). Set T > δ −2 (β − α). We shall prove that ∀x ∈ int(U ) ∃t ∈ [−T, T ] such that ϕ(t, x) ∈ W . It is divided into three cases: (a) x ∈ W . By taking t = 0, it is done. ). If ϕ([−T, T ], x) ⊂ W , then (b) x ∈ W \int(U

f (ϕ(−T, x)) − f (ϕ(T, x)) =

T

−T

f (ϕ(s, x)), ϕ(s, ˙ x) ds

≥ 2T δ 2 > 2(β − α) . The contradiction shows ϕ([−T, T ], x) ⊂ W . )\int(U )) ∩ W = int(U ) ∩ W \int(U ). Since either x ∈ (c) x ∈ (int(U ∪ ϕ(t, int(U )) or x ∈ ∪ ϕ(t, int(U )). t>0

t 0 small. Again, we would have β − α ≥ δ 2 (t2 − t1 ) so that t2 < T . Therefore ϕ([−T, T ], x) ⊂ W . Similarly for the second case. Theorem 5.5.10 Let (M, f, ϕ) be a pseudo-gradient ﬂow. If (O, α, β) is an isolating triplet for a dynamically isolated critical set S for f , then [S] is an isolated invariant set. ! ∩ Moreover, any closed MVP neighborhood U of [S], satisfying U ⊂ cl(O) f −1 [α, β], is an isolating neighborhood for [S], and U ∈ Σ. ! ∩ f −1 [α, β] ∈ Σ, is an isolated Proof. 1. Applying Lemma 5.5.9, W = cl(O) invariant neighborhood of [S] . 2. To prove that U ∈ Σ is an isolating neighborhood for [S], it is suﬃcient to show that [S] = I(U ) ⊂ GT (U ) ⊂ int(U ) for some T > 0. Since [S] = I([S]) ⊂ I(U ) ⊂ I(W ) = [S] , ! ∩ f −1 [α, β], we obtain [S] = I(U ). where W = cl(O) Again by Lemma 5.5.9, we have GT (U ) ⊂ GT (W ) ⊂ int(U ).

Example 1. If c is an isolated critical value, i.e., Kc = K ∩ f −1 (c) = ∅, and there is no critical point on the levels in [c − , c + ]\{c} for some > 0, then the set Kc is a dynamically isolated critical set. Example 2. If x0 is an isolated critical point of f , then S = {x0 } is a dynamically isolated critical set. 5.5.2 Index Pair and Conley Index As we have seen in the Morse theory, isolated neighborhoods only are not enough in characterizing the isolated critical points, the local dynamic behavior provides the necessary information. This leads us to: Deﬁnition 5.5.11 Let (N, L) be a pair of subspaces of X. A subset L of N is called positively invariant in N with respect to the ﬂow ϕ, if x ∈ L and ϕ([0, t], x) ⊂ N imply ϕ([0, t], x) ⊂ L . It is called an exit of N , if ∀x ∈ N, ∃t1 > 0 such that ϕ(t1 , x) ∈ N , implies ∃t0 ∈ [0, t1 ) such that ϕ([0, t0 ], x) ⊂ N and ϕ(t0 , x) ∈ L. Example. Let (M, f, φ) be a pseudo-gradient ﬂow, and let α < β < γ. Let N = f −1 [α, γ] and L = f −1 [α, β]. L is positively invariant in N , and also an exit set of N . To an isolated invariant set S, we introduce: Deﬁnition 5.5.12 For U ∈ Σ, let (N, L) be a pair of closed subsets of U with L ⊂ N . It is called an index pair relative to U if: (1) N \L ∈ Σ,

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(2) L is positively invariant in N , (3) L is an exit set of N , (4) N \L ⊂ U and ∃T > 0 such that GT (U ) ⊂ N \L. According to the deﬁnition S := I(U ) = I(N \L) is an isolated invariant set, and both U and N \L are isolating neighborhoods of S. Example. Let (M, f, ϕ) be a pseudo-gradient ﬂow. If (O, α, β) and (O , α , β ) are two isolating triplets for a dynamically isolated critical set S for f , with O ⊂ O, [α , β ] ⊂ [α, β], then ∃T > 0 such that N = GT (W ), L = ϕ(−T, W− ) ! ∩ 1 ) ∩ f −1 [α , β ], where W = cl(O) is an index pair relative to U = cl(O −1 −1 ! f [α, β] and W− = cl(O) ∩ f (α). Moreover, both U and N are isolating neighborhoods of [S]. Note that N \L = N and int(N \L) = int(N ). In order to verify the conclusion, we need: 1. ∀T > 0, N = GT (W ) is a closed MVP neighborhood of [S]. It is suﬃcient to verify that [S] ⊂ int(N ). From Lemma 5.5.9, we have [S] ⊂ N . If the conclusion is not true, then ∃x ∈ [S] ∩ ∂N , i.e., ∃xn ∈ GT (W ) with xn → x. This means that there are tn ∈ [−T, T ], such that ϕ(tn , xn ) ∈ W . After a subsequence, we have tn → t ∈ [−T, T ] and then ϕ(t, x) ∈ int(W ). But x ∈ [S] implies that ϕ(t, x) ∈ [S] ⊂ int(W ), provided by Lemma 5.5.9. This is a contradiction. Now, we are going to verify conditions (1)–(4). 2. Applying Lemma 5.5.9 to U = N , ∃T0 > 0 such that GT +T0 (W ) ⊂ GT0 (W ) ⊂ int(N ) . Since GT +T0 (W ) = GT0 (N ). This shows that N \L ∈ Σ, (1) is veriﬁed. 3. Again applying Lemma 5.5.9, ∃T > 0 such that N = GT (W ) ⊂ int(U ) ⊂ U . Moreover, GT (U ) ⊂ GT (W ) = N = N \L . Poperty (4) is veriﬁed. 4. Since W− is an exit set of W , L is an exit set of N . Obviously, L is positively invariant in N . This completes the veriﬁcation. From Lemma 5.5.9, both U and N are isolating neighborhoods of [S]. For a system without variational structure, does there exist an index pair relative to any set U ∈ Σ? We have: Theorem 5.5.13 (Existence of an index pair) Let ϕ be a ﬂow on a metric space X. ∀U ∈ Σ, (GT (U ), ΓT (U )) is an index pair relative to U , where T > 0 is assumed such that GT (U ) ⊂ int(U ). Proof. From the properties (1) and (4), both GT (U ) and ΓT (U ) are closed. We shall verify the four conditions in Deﬁnition 5.5.11 successively.

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(1) By property (4), int(GT (U )\ΓT (U )) = int(GT (U )). Applying property (2), GT (GT (U )\ΓT (U )) ⊂ G2T (U ) ⊂ int(GT (U )) . Thus GT (U )\ΓT (U ) ∈ Σ. (2) ΓT (U ) is positively invariant in GT (U ), i.e., if x ∈ ΓT (U ) and ϕ([0, T1 ], x) ∈ GT (U ), then ϕ([0, T1 ], x) ⊂ ΓT (U ). Suppose not, then ∃t ∈ [0, T1 ] such that ϕ(t, x) ∈ ΓT (U ). Let t∗ = inf{s ∈ [0, T1 ]| ϕ(s, x) ∈ ΓT (U )}. Since ΓT (U ) is closed, y = ϕ(t∗ , x) ∈ ΓT (U ) and ∃n → +0 such that ϕ(t∗ + n , x) ∈ ΓT (U ). Thus ϕ([0, T ], y) ∩ ∂U = ∅ and ϕ([n , T ], y) ∩ ∂U = ∅; it follows that y ∈ ∂U . But y ∈ GT (U ) ⊂ int(U ). This is a contradiction. (3) ΓT (U ) is an exit set of GT (U ), i.e., if x ∈ GT (U ) and if ∃t1 > 0 such that ϕ(t1 , x) ∈ GT (U ), then ∃t0 ∈ [0, t1 ) such that ϕ([0, t0 ], x) ⊂ GT (U ) and y = ϕ(t0 , x) ∈ ΓT (U ). Let us deﬁne t0 = inf{s > 0| ϕ([s − T, s + T ], x) ⊂ U } = inf{s > 0| ϕ(s, x) ∈ GT (U )} ; we have t0 ∈ [0, t1 ]. Since GT (U ) is closed, ϕ([0, t0 ], x) ⊂ GT (U ), therefore t0 < t1 . Deﬁning y = ϕ(t0 , x), we have ϕ(T, y) = ϕ(t0 + T, x) ∈ ∂U , therefore y ∈ ΓT (U ). (4) From property (4), GT (U )\ΓT (U ) = GT (U ) ⊂ U . For T1 > T , we obtain GT1 (U ) ⊂ GT (U ) = GT (U )\ΓT (u) . A topological invariant is introduced to describe the index pair (N, L) relative to an isolating neighborhood U . Conley called the homotopy type h(U ) = [N/L] the invariant. In comparing with that in the Morse theory, we prefer to replace it by the relative homology groups. However, in order to match Conley’s deﬁnition, Alexander–Spanier cohomology is more suitable, because it possesses a special excision property not shared by singular cohomology theory. ∗ For a topological pair (X, A) and a coeﬁcient ﬁeld F , H (X, A; F ) stands for Alexander–Spanier cohomology. The following excision property holds. Suppose that X and Y are paracompact Hausdorﬀ spaces, and that A and B are closed in X and Y respectively. If X\A is homeomorphic to Y \B. Then ∗ ∗ H (X, A; F ) ∼ = H (Y, B; F ). Deﬁnition 5.5.14 (Conley index) For a given ﬂow ϕ and a given U ∈ Σ, let (N, L) be an index pair with respect to (U, ϕ). We call

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5 Topological and Variational Methods ∗

h(U ) = h(U, ϕ) = H (N, L; F ) the Conley index for U (or for the isolated invariant set S = I(U )) with respect to the ﬂow ϕ. In contrast with equation (5.3), we also write h(t, U ) =

∞

¯ q (N, L; F ) . tq H

q=0

In the following we omit ϕ in the notation h, if there is no ambiguity. In order to show that the Conley index is well-deﬁned, one has to verify that if (N1 , L1 ) and (N2 , L2 ) are two index pairs relative to U , then ∗ ∗ H (N1 , L1 ; F ) ∼ = H (N2 , L2 ; F ). In fact by deﬁnition, ∃T > 0 such that GT (N1 \L1 ) ⊂ GT (U ) ⊂ N2 \L2 and GT (N2 \L2 ) ⊂ GT (U ) ⊂ N1 \L1 . Because Ni \Li ∈ Σ, i = 1, 2, for suﬃcient large T > 0, we have GT (N1 \L1 ) ⊂ int(N2 \L2 ) and GT (N2 \L2 ) ⊂ int(N1 \L1 ) .

(5.46)

Lemma 5.5.15 If (N1 , L1 ) and (N2 , L2 ) are two index pairs relative to U ∈ Σ, and if T > 0 such that (5.46) holds, then the map f : [T, ∞) × N1 /L1 → N2 /L2 deﬁned by ϕ(3t, x) if ϕ([0, 2t], x) ⊂ N1 \L1 and ϕ([t, 3t], x) ⊂ N2 \L2 , f (t, [x]) = otherwise [L2 ] is continuous. Proof. There are three cases in logic: (1) ϕ([t, 3t], x) ⊂ N2 \L2 , (2) ϕ([0, 2t], x) ⊂ N1 \L1 , (3) ϕ([t, 3t], x) ⊂ N2 \L2 and ϕ([0, 2t], x) ⊂ N1 \L1 . In case (1), by the continuity of the ﬂow, there exists a neighborhood W of (t, x) such that ϕ([s, 3s], y) ⊂ N2 \L2 ∀(s, y) ∈ W then f (s, [y]) = [L2 ] = f (t, [x]), i.e., f is continuous at (t, [x]). Similarly for case (2). In case (3), we consider two possibilities: (a) ϕ([t, 3t], x) ∩ L2 = ∅. Since ϕ([t, 3t], x) ⊂ N2 \L2 , it follows that ϕ(2t, x) ∈ Gt (N2 \L2 ) ⊂ int(N1 \L1 ) ⊂ N1 \L1 . Again from ϕ([0, 2t], x) ⊂ N1 \L1 and the positive invariance of L1 in N1 , it follows that ϕ([0, 2t], x) ⊂ N1 \L1 . Therefore f (t, [x]) = ϕ(3t, x) ∈ N2 \L2 . For any open neighborhood U of ϕ(3t, x) in N2 \L2 , by the continuity of the ﬂow ϕ, ∃ a neighborhood W in N1 /L1 of (t, x) such that ϕ([0, 2s], y) ⊂ N1 \L1 , ϕ([s, 3s], y) ⊂ N2 \L2 and ϕ(3s, y) ∈ U ∀(s, y) ∈ W . Thus f (s, [y]) = ϕ(3s, y) ∈ U as (s, y) ∈ W with y ∈ N1 , i.e., f is continuous at this point (t, x).

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(b) ϕ([t, 3t], x) ∩ L2 = ∅. Since L2 is an exit set of N2 , ϕ(3t, x) ∈ L2 . Let [U ] be any neighborhood of [L2 ] in N2 \L2 , deﬁne V = {x ∈ N2 \L2 | x ∈ [U ]} ∪ (X\N2 ) ∪ L2 , then V is a neighborhood of L2 in X, and [U ] = (V ∩ (N2 \L2 )) ∪ [L2 ]. By the continuity of the ﬂow ϕ, ∃ a neighborhood W of (t, x) such that ϕ(3s, y) ∈ V ∀(s, y) ∈ W . Thus f (s, [y]) ∈ {[ϕ(3s, y)], [L2 ]} ⊂ (V ∩ (N2 \L2 )) ∪ [L2 ] ⊂ [U ] , ∀(s, y) ∈ W with y ∈ N1 . Again f is continuous at this point. The proof is complete. Theorem 5.5.16 If (N1 , L1 ) and (N2 , L2 ) are two index pairs relative to ∗ ∗ U ∈ Σ, then H (N1 , L1 ; F ) ∼ = H (N2 , L2 ; F ). Proof. According to Lemma 5.5.15, there are continuous functions: f : [T, ∞)× N1 /L1 → N2 /L2 and g : [T, ∞) × N2 /L2 → N1 /L1 deﬁned by ϕ(3t, x) if ϕ([0, 2t], x) ⊂ N1 \L1 and ϕ([t, 3t], x) ⊂ N2 \L2 , f (t, [x]) = otherwise , [L2 ] and

ϕ(3t, x) g(t, [x]) = [L1 ]

if ϕ([0, 2t], x) ⊂ N2 \L2 and ϕ([t, 3t], x) ⊂ N1 \L1 , otherwise ,

respectively. One deﬁnes ηi : [0, T ] × Ni /Li → Ni /Li ϕ(6t, x) if ϕ([0, 6t], x) ⊂ Ni \Li , ηi (t, [x]) = otherwise , [Li ] i = 1, 2. Again they are continuous and satisfy η1 (T, [x]) = g(T, f (T, [x])), η2 (T, [x]) = f (T, g(T, [x])) and ηi (0, [x]) = idNi /Li i = 1, 2. This shows that N1 /L1 and N2 /L2 have the same homotopy type. It follows that ∗

∗

H (N1 /L1 , [L1 ]; F ) ∼ = H (N2 /L2 , [L2 ]; F ) . By the special excision property, we have ∗ ∗ H (Ni , Li ; F ) ∼ = H (Ni /Li , [Li ]; F ), i = 1, 2 .

Combining them together, we obtain ∗ ∗ H (N1 , L1 ; F ) ∼ = H (N2 , L2 ; F ) .

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Example 1. (Hyperbolic ﬁxed point) Recall the hyperbolic system and the Hartman–Grobman theorem in Chap. 1. Let f : Rn → Rn be C 1 . Assume that θ is a hyperbolic ﬁxed point with k-dimensional unstable manifold, i.e., A = eL with L = f (θ) satisfying σ(A) ∩ S 1 = ∅. Then we have the decomposition Rn = E u ⊕ E s where E u and E s are invariant subspaces of A, on which the eigenvalues of Au = A|E u lie outside the unit circle, and those of As = A|E s lie inside the unit circle with dim E u = k. Thus L = Lu ⊕ Ls , where Lu = L|E u and Ls = L|E s . The real parts of all eigenvalue of Lu are greater than zero, while those of Ls are less than zero. According to the Hartman–Grobman theorem the ﬂow in a neighborhood of θ for the ODE x˙ = f (x) is topologically equivalent to the ﬂow in a neighborhood of x˙ = Lx, i.e., ϕ(t, x) = eLt x. An isolating neighborhood of the origin is given by B k × B n−k and the exit set by S k−1 × B n−k . Let U = N = B k × B n−k and L = S k−1 × B n−k . It follows that h(t, U ) = tk . Example 2. An invariant manifold S is an invariant set with manifold structure. It is called normally hyperbolic, if the tangent bundle over Rn restricted on S has the bundle decomposition: (TRn )S = E u ⊕ E s with u s Eϕ(t,x) = T ϕ(t, ·)Exu , Eϕ(t,x) = T ϕ(t, ·)Exs

∀x ∈ S, ∀t ∈ R1 .

According to Thom’s isomorphism theorem, see [BT],[Mi 3],[Sp], we have the following result: Let S be a normally hyperbolic invariant manifold. Let E u be the vector bundle over S deﬁned by the local unstable manifold of S. If E u is a rank k orientable bundle, then h(U ) = H

∗+k

(S; F ) ,

for U ∈ Σ with I(U ) = S. Thus, if S is a hyperbolic periodic orbit with an oriented unstable manifold of dimension k + 1, then h(t, U ) = tk + tk+1 . Example 3. (Critical groups) Let (M, f, φ) be a pseudo-gradient ﬂow, and let {O, α, β} be an isolating triplet for a dynamically isolated critical set S of f . Then ∗ ! ∩ fβ , O ! ∩ fα ; F ) . h(O) = H (O In particular, if S is an isolated critical point p, then h(O) = C∗ (f, p) , where C∗ (f, p) is the critical groups of f at S. The Conley index possesses many important properties: Wazewski’s principle, the continuation and the Morse–Smale decomposition, which are the

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403

counterparts of the Kronecker existence, the homotopy invariance and the (sub) additivity of the degree theory. First, we introduce a type of compactness condition on U ∈ Σ. Deﬁnition 5.5.17 (Condition(B)) U ∈ Σ is said to satisfy Condition (B) if ∀ closed neighborhoods W of I(U ) there exists T > 0 such that GT (U ) ⊂ W . In particular, if U ∈ Σ satisﬁes Condition (B) and I(U ) = ∅, then ∃T > 0 such that GT (U ) = ∅. It is known that if U ∈ Σ is compact, then U satisﬁes Condition (B) (see Remark 5.5.4). We investigate other suﬃcient conditions of Condition (B): Rubakowski [Ry] introduced: Condition (A) U ∈ Σ is said to satisfy Condition (A) if ∀xn ∈ U, ∀tn → +∞, ϕ([0, tn ], xn ) ⊂ U implies that ϕ(tn , xn ) is subconvergent. Condition (A) ⇒ Condition (B). Indeed, if not, ∃W , a closed neighborhood of I(U ) such that Gn (U ) ⊂ W ∀n, i.e., ∃yn ∈ Gn (U )\W . Let xn = ϕ(−n, yn ), then ϕ([0, n], xn ) ⊂ U . By Condition (A), yn = ϕ(n, xn ) subconvergent to y0 ∈ U , and then y0 ∈ I(U ). But W is closed, therefore y0 ∈ int(W ). This is a contradiction. The isolating neighborhoods of the examples of dynamically isolated critical sets after Theorem 5.5.10 satisfy Condition (B). More precisely, let (M, f, ϕ) be a pseudo-gradient ﬂow, and let c be an isolated critical value. If Kc = K ∩ f −1 (c) is isolated (or, p ∈ Kc is isolated), then the set ˜ ∩ f −1 [c − , c + ] is an isolating neighborhood of Kc (or p resp.) W = cl(O) satisfying Condition (B), if > 0 is small so that f −1 [c − , c + ] ∩ K = Kc , and O is a neighborhood of Kc (or p resp.) containing only critical points in Kc (or p resp.). From Lemma 5.5.9, it is suﬃcient to show that any neighborhood V of p, containing p as the only critical point, contains a (MVP) neighborhood U of p. In fact, let BR (p) ⊂ V for some R > 0, we choose r = R2 , O = Br (p), ˜ ∩ f −1 (c − δ, c + δ) for small δ > 0. By standard estimates, and U = cl(O) U ⊂ BR (p) ⊂ V . The following Wazewski principle holds: Theorem 5.5.18 Let U ∈ Σ satisfy Condition (B). If h(U ) = 0 then I(U ) = ∅. ∗

Proof. We prove that I(U ) = ∅ implies H (N, L; F ) = 0 for all index pairs (N, L) relative to U . According to Theorem 5.5.13, it is suﬃcient to verify that for a special index pair (N, L) = (GT (U ), ΓT (U )). Condition (B) yields the existence of large T > 0 such that GT (U ) = ∅. The conclusion follows trivially.

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Next, we turn to the continuation. A special form of the homotopy invariance of the topological degree asserts that the degree deg(f, Ω, p) is continuous with respect to the vector ﬁeld f . In Conley index theory, the vector ﬁeld f is replaced by the ﬂow ϕ, so we should study: Under what condition on U ∈ Σ, and in which sense of the variance between ﬂows ϕ and ψ, does one have h(U, ϕ) = h(U, ψ)? First we notice: Lemma 5.5.19 Assume that U ∈ Σ(ϕ) satisﬁes Condition (B) and that I(U ) is compact. Then U satisﬁes the condition: Σ0 (ϕ) : ∃T, δ > 0 such that Nδ (GT (U )) ⊂ int(U ) , where Nδ (A) is the δ-neighborhood of the set A. Moreover, ∃T, δ > 0 such that Nδ (GT (U )) ⊂ GT /2 (U ), Nδ (GT /2 (U )) ⊂ int(U ) .

(5.47)

Proof. 1. Let = d(∂U, I(U )) where d(·, ·) is the distance between two sets induced by the metric d. Since I(U ) is compact, > 0. Let V = N/2 (I(U )); by Condition (B), one has T > 0 such that GT (U ) ⊂ V . Let δ = /2; the ﬁrst conclusion follows. 2. To prove the second conclusion, we claim that ∂GT (U ) ∩ I(U ) = ∅. Indeed, if not, ∃x0 ∈ ∂GT (U ) ∩ I(U ). From x0 ∈ ∂GT (U ), it implies that ∃t0 ∈ [−T, T ] such that ϕ(t0 , x0 ) ∈ ∂U . But from x0 ∈ I(U ), we have ϕ([t0 − T, t0 + T ], x0 ) ⊂ U , then ϕ(t0 , x0 ) ∈ GT (U ) ⊂ int(U). This is a contradiction. Then 1 := d(∂GT (U ), I(U )) > 0; we repeat the proof in step 1, and obtain T1 > 0 such that GT1 (U ) ⊂ V := N 21 (I(U )). Thus, N 21 (GT1 ) ⊂ GT (U ). One deﬁnes the Hausdorﬀ metric between two sets U and V as follows: dH (U, V ) = sup d(x, V ) + sup d(U, x) . x∈U

x∈V

Lemma 5.5.20 For U ∈ Σ0 (ϕ), there exist T, > 0 such that if a ﬂow ψ satisﬁes (5.48) d(ϕ(t, x), ψ(t, x)) ≤ ∀(t, x) ∈ [−T, T ] × Nδ (U ) and a closed set V satisﬁes dH (U, V ) ≤ , then U, V ∈ Σ0 (ψ) and V ∈ Σ0 (ϕ).

(5.49)

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405

Proof. By deﬁnition, one has T, δ > 0 such that Nδ (GTϕ (U )) ⊂ int(U ) . Letting ≤ δ/4, we have N (GTψ (U )) ⊂ N2 (GTϕ (U )) ⊂ int(U ) , i.e., U ∈ Σ0 (ψ), and N2 (GTψ (V )) ⊂ N4 (GTϕ (U )) ⊂ int(U ) . Thus dH (N (GTψ (V )), ∂V ) ≥ ; it follows that N (GTψ (V )) ⊂ int(V ) i.e., V ∈ Σ0 (ψ). Similarly, we prove that V ∈ Σ0 (ϕ). Theorem 5.5.21 (Continuation) For U ∈ Σ0 (ϕ), ∃T, > 0 such that h(U, ϕ) = h(V, ψ) , whenever (ψ, V ) satisﬁes (5.48) and (5.49), and that both ϕ and ψ are uniformly continuous on [−T, T ] × N2δ (U ), namely, d(φ(t, x), φ(t, y))

0, and show (N, L) = (GT , ΓT ), (N ! T /Γ !T , that g2 ◦ g1 and g1 ◦ g2 are homotopic to identity maps on GT /ΓT and G respectively. Let us deﬁne for λ ∈ [0, 1] the map [ϕ(2T, ξ−λ (T, x))] ϕ([0, 2T ], ξ−λ (T, x)) ⊂ GT H(λ, [x]) = [ΓT ] otherwise . Note: It is equivalent to replace ϕ([0, 2T ], ξ−λ (T, x)) ⊂ GT by ϕ(2T, ξ−λ (T, x)) ∈ GT on the right-hand side of the deﬁnition. Since H(1, [x]) = g2 ◦ g1 and [ϕ(2T, x)] ϕ([0, 2T ], x) ⊂ GT H(0, [x]) = [ΓT ] otherwise , the latter is homotopic to id|GT /ΓT as shown in Theorem 5.5.16. It remains to verify that the deﬁnition is well-deﬁned and that H(λ, [x]) is continuous. We shall show that x ∈ ΓT ⇒ H(λ, x) = [ΓT ] i.e., ϕ([0, 2T ], ξ−λ (T, x)) ⊂ T G . From x ∈ ΓT , by step 1, ξ−λ (T, x) ∈ N (ΓT ), i.e., ∃x ∈ ΓT such that d(ξ−λ (T, x), x) < . Following the argument in 3, ϕ([ 12 T, 32 T ], x) ∩ ∂U = ∅, we have ϕ([0, 2T ], ξ−λ (T, x)) ⊂ GT . As to the continuity, we deﬁne h2 : GT /ΓT → GT /ΓT by [ϕ(T, x)] ϕ(T, x) ∈ GT h2 ([x]) = [ΓT ] otherwise . Since ϕ(T, x) ∈ GT can be replaced by ϕ([0, T ], x) ⊂ GT , the continuity of h2 has been proved in Theorem 5.5.16, and from ϕ([0, T ], x) ⊂ GT ⊂ int(U ), it implies x ∈ ΓT , h2 is well deﬁned. By deﬁnition H(λ, [x]) = h2 ◦ h1 ◦ ξ−λ (T, x) , therefore H is continuous on [0, T ] × GT /ΓT . 6. Finally, we consider the case U = V , but with ϕ ! = ϕ, and write GT (U ) = T T T T Gϕ (U ), G (V ) = Gϕ (V ). Since Nδ (G (U )) ⊂ int(U ), Nδ (GT (V )) ⊂ int(V ), if one takes 0 < < δ/3 and dH (U, V ) < , we have N (GT (U )) ⊂ V and N (GT (V )) ⊂ U .

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5 Topological and Variational Methods

! , L) ! be index pairs related to U and V resp., then Let (N, L) and (N ! \L. ! ∃T1 , T2 > 0 such that GT1 (N \L) ⊂ GT1 (U ) ⊂ V , and GT2 (V ) ⊂ N

! \L. ! Similarly one has GT1 +T2 (N ! \L) ! ⊂ N \L. By Thus GT1 +T2 (N \L) ⊂ N ∗ ∗ ! , L; ! F ), i.e., H (N the proof of Theorem 5.5.16, we obtain H (N, L; F ) ∼ = h(U, ϕ) = h(V, ϕ). Combining steps 5 and 6, we have h(U, ϕ) = h(V, ϕ) = h(V, ψ) .

The proof is complete. 5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension

In the study of the dynamics on invariant sets the Morse–Smale decomposition is an extension of the Morse relation. Before going on, let us introduce the notion of attractor–repeller pairs. On a metric space (X, d) with ﬂow ϕ, ∀Y ⊂ X, deﬁne ω(Y ) = ∩ ϕ([t, ∞), Y ) and ω ∗ (Y ) = ∩ ϕ((−∞, −t], Y ) . t>0

t≥0

By deﬁnition, ω(Y ) is an invariant closed set. Deﬁnition 5.5.22 Let S be a compact invariant set for ϕ; a subset A ⊂ S is called an attractor in S, if there exists a neighborhood U of A such that ω(U ∩ S) = A. The dual repeller of A in S is deﬁned by A∗ = {x ∈ S| ω(x) ∩ A = ∅} . The pair (A, A∗ ) is called an attractor–repeller pair. The set C(A∗ , A, S) = {x ∈ S| ω(x) ⊂ A, ω ∗ (x) ⊂ A∗ } is called the set of connecting orbits from A∗ to A in S. The following properties hold. (1) A and A∗ are disjoint compact invariant sets. Proof. In the following, we use the notation U as in Deﬁnition 5.5.22. In fact, A = ω(U ∩ S) is invariant. Both ω(x) and A are invariant, so is A∗ . If ∃x ∈ A ∩ A∗ , then ω(x) ⊂ A. But by deﬁnition of A∗ , ω(x) ∩ A = ∅. Since S is compact, ω(x) = ∅. This is a contradiction. Therefore A ∩ A∗ = ∅. By deﬁnition, A is closed. It remains to verify the closedness of A∗ . If {xn } ⊂ A∗ with xn → x ∈ S, and if ω(x) ∩ A = ∅, then ϕ(t, x) ∈ U for some t > 0. Consequently ϕ(t, xn ) ∈ U for n large. Therefore ω(xn ) ⊂ ω(U ∩ S) = A. This is impossible. Therefore ω(x) ∩ A = ∅, i.e., x ∈ A∗ .

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(2) For any open neighborhood V of A, ∃T = T (V ) such that ∪ ϕ(t, U ∩ t≥T

S) ⊂ V . Proof. If not, ∃V ⊃ A open, ∃xn ∈ U ∩ S ∃tn → +∞ such that ϕ(tn , xn ) ∈ V . Since S is a compact invariant set, ϕ(tn , xn ) → y ∈ S\V . But y ∈ ω(U ∩ S) = A. This is a contradiction. (3) If B is a closed set disjoint from A, then ∀ > 0 ∃T = T > 0 such that d(x, A∗ ) < , whenever x ∈ S and t ≥ T such that ϕ(t, x) ∈ B. Proof. If not, ∃ > 0 ∃tn → ∞ ∃xn ∈ S such that ϕ(tn , xn ) ∈ B but d(xn , A∗ ) ≥ . Since S is compact and invariant, ∃x ∈ S such that xn → x. Then d(x, A∗ ) ≥ , which implies that x ∈ A∗ , i.e., ω(x) ∩ A = ∅. Thus for the neighborhood U of A; we have t1 ∈ R1 such that ϕ(t1 , x) ∈ U ∩ S and then ϕ(t1 , xn ) ∈ U ∩ S for n large. Let V = X\B, which is an open neighborhood of A, from (2), ∃T > 0 such that ϕ(t, xn ) ∈ V ∀t ≥ T . This contradicts ϕ(tn , xn ) ∈ B ∀n. For any x ∈ X, we denote the orbit passing through x by o(x) = {ϕ(t, x)| t ∈ R1 }. (4) ω(y) ∩ A∗ = ∅ ⇒ o(y) ⊂ A∗ . Proof. Let B be a closed neighborhood of A∗ such that A ∩ B = ∅. Since ω(y) ∩ A∗ = ∅, ∃tn → +∞ such that ϕ(tn , y) ∈ B. ∀t ∈ R1 , let z = ϕ(t, y); we have ϕ(tn − t, z) = ϕ(tn , y) ∈ B, thus from (3), ∀ > 0, we have d(z, A∗ ) < . Since > 0 is arbitrarily small, o(y) ⊂ A∗ . (5) ω ∗ (y) ∩ A = ∅ ⇒ o(y) ⊂ A. Proof. By the assumption that ∃tn → +∞ such that ϕ(−tn , y) ∈ U ∩ S, then ∀t ∈ R1 , t + tn ≥ 0 for n large, from ϕ(t, y) = ϕ(tn + t, ϕ(−tn , y)) we have ϕ(t, y) ∈ ω(U ∩ S), i.e., o(y) ⊂ A. Lemma 5.5.23 Let (A, A∗ ) be an attractor–repeller pair of a compact invariant set S, then S = A ∪ A∗ ∪ C(A∗ , A, S) . Proof. It is suﬃcient to prove that ∀x ∈ S\(A ∪ A∗ ), ω(x) ⊂ A, and ω ∗ (x) ⊂ A∗ . 1. ω(x) ⊂ A : ∀y ∈ ω(x), i.e., ∃tn → +∞ such that ϕ(tn , x) → y. Let B = S\U , where U is as in Deﬁnition 5.5.22, then either ∃n0 ∈ N such that z = ϕ(tn0 , x) ∈ U or ϕ(tn , x) ∈ B, ∀n. But the latter case is impossible, because from (3) we would have x ∈ A∗ . In the former case, y ∈ ω(S ∩U ) = A. 2. ω ∗ (x) ⊂ A∗ : ∀y ∈ ω ∗ (x), ∃tn → +∞ such that zn = ϕ(−tn , x) → y. Let B = {x}. Since {x} ∩ A = ∅, we have d(zn , A∗ ) → 0, provided by (3). Thus y ∈ A∗ .

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Combining the conclusions of Lemma 5.5.23 and properties (4) and (5), we see that ∀x ∈ C(A∗ , A, S), ω(x) ⊂ A and ω ∗ (x) ⊂ A∗ , and there are connecting orbits from A to A∗ but none from A∗ to A. Let us deﬁne the Morse decomposition for a compact invariant set. Deﬁnition 5.5.24 Let S be a compact invariant set of X with respect to the ﬂow ϕ. An ordered collection (M1 , . . . , Mn ) of invariant subsets Mj ⊂ S is called a Morse decomposition of S, if there exists an increasing sequence of attractors in S: ∅ = A0 ⊂ A1 ⊂ · · · ⊂ An = S such that Mj = Aj ∩ A∗j−1 , 1 ≤ j ≤ n. Example 1. An attractor–repeller pair (A, A∗ ) of a compact invariant set S is a Morse decomposition, where A0 = ∅, A1 = A, A2 = S. Example 2. Suppose f ∈ C 1 (M, R1 ), where M is a compact manifold. Assume that f −1 [a, b] ∩ K = {p1 , . . . , pn }, where a, b are regular values of f with f (pi ) ≤ f (pi+1 ), i = 1, . . . , n − 1. Then ({p1 }, . . . , {pn }) is a Morse decomposition of S = I(f −1 ([a, b])). In fact, by setting A0 = ∅, and Ai = {x ∈ S| ω ∗ (x) ⊂ {p1 , . . . , pi }}, i = 1, 2, . . . , n. We shall verify that this is an increasing sequence of attractors in S with A∗i = {x ∈ S| ω(x) ⊂ {pi+1 , . . . , pn }}, and then Ai ∩ A∗i−1 = {pi }. It is proved by induction. Let Sk = I(f −1 [a, ak ]), where ak ∈ (f (pk ), f (pk+1 )), k = 1, . . . , n − 1, and an = b. Thus Sn = S. We verify that Ai = {x ∈ Sk | ω ∗ (x) ⊂ {p1 , p2 , . . . , pi }}, i = 1, 2, . . . , k, is an increasing sequence of attractors in Sk , and A∗i = {x ∈ Sk | ω(x) ⊂ {pi+1 , . . . , pk }}. For n = 1. Obviously, A1 = {p1 } is an attractor in S1 = {p1 }, with A∗1 = ∅ and A∗0 = S1 . Thus A1 ∩ A∗0 = {p1 }. If the conclusion holds for n = k, i.e., Ai = {x ∈ Sk | ω ∗ (x) ⊂ {p1 , . . . , pi }}, i = 1, . . . , k, is an increasing sequence of attractors in Sk . For n = k + 1. It is easily seen that Sk is an attractor in Sk+1 . Indeed, let Uk = f −1 (a, ak ), then ω(Sk+1 ∩ Uk ) = Sk with Sk∗ = {pk+1 }. Then Ak = {x ∈ Sk+1 | ω ∗ (x) ⊂ {p1 , . . . , pk }} = Sk is an attractor in Sk+1 with A∗k = Sk∗ = {pk+1 }. In this case ∅ = A0 ⊂ A1 ⊂ . . . ⊂ Ak = Sk ⊂ Ak+1 = Sk+1 is an increasing sequence of attractors in S = Sk+1 . By easy computation, A∗i = {x ∈ Sk+1 | ω(x) ⊂ {pi+1 , . . . , pk+1 }} and then Ai ∩ A∗i−1 = {pi } . Our conclusion follows from the mathematical induction. By the deﬁnition of the Morse decomposition (M1 , . . . , Mn ) of S, we have the following conclusions: (1) {Mi | 1 ≤ i ≤ n} are pairwise disjoint.

5.5 Conley Index Theory

411

Indeed if i < j, then Mi ∩ Mj = Ai ∩ A∗i−1 ∩ Aj ∩ A∗j−1 ⊂ Ai ∩ A∗j−1 ⊂ Aj−1 ∩ A∗j−1 = ∅, from property (1). (2) ∀x ∈ S\( ∪ Mi ), ∃i < j +1 such that ω(x) ⊂ Mi and ω ∗ (x) ⊂ Mj+1 . 1≤i≤n

Indeed, set i = min{k ∈ N| ω(x) ⊂ Ak } and j = max{k ∈ N| ω ∗ (x) ⊂ A∗k } . From A0 = ∅ and An = S, we see 0 < i and j < n. Since ω(x) ⊂ Ai−1 , provided by the decomposition lemma, x ∈ A∗i−1 , and then by property (4), o(x) ⊂ A∗i−1 , in particular, ω(x) ⊂ A∗i−1 . By the same reason, ω ∗ (x) ⊂ Aj+1 , we have o(x) ⊂ Aj+1 and ω ∗ (x) ⊂ Aj+1 . We claim that i ≤ j + 1. For otherwise, j + 1 ≤ i − 1, then o(x) ⊂ Aj+1 ∩ A∗i−1 ⊂ Ai−1 ∩ A∗i−1 = ∅, a contradiction. For i = j + 1, we have / Mi . Therefore the only possibility o(x) ⊂ Ai ∩ A∗i−1 = Mi , it contradicts x ∈ is i < j + 1. In this case we have ω(x) ⊂ Ai ∩ A∗i−1 = Mi and ω ∗ (x) ∈ Aj+1 ∩ A∗j = Mj+1 . Comparing with Example 2, the following conclusion is easily veriﬁed: (3) Aj = {x ∈ S| ω(x) ∪ ω ∗ (x) ⊂ M1 ∪ . . . ∪ Mj } and A∗j = {x ∈ S| ω(x) ∪ ω ∗ (x) ⊂ Mj+1 ∪ . . . ∪ Mn }, j = 1, 2, . . . , n. (4) If (M1 , . . . , Mn ) is a Morse decomposition of S, and if ∅ = A0 ⊂ A1 ⊂ . . . ⊂ An = S is the associated increasing sequence of attractors, then for 0 < i ≤ j < n, (M1 , . . . , Mi−1 , Mji , Mj+1 , . . . , Mn ) is again a Morse decomposition of S, where Mji = Aj ∩ A∗i−1 = {x ∈ S| ω(x) ∪ ω ∗ (x) ⊂ Mi ∪ . . . ∪ Mj }. (5) If S is an isolating invariant compact set, and if {M1 , . . . , Mn } is a Morse decomposition of S, then Mi , 1 ≤ i ≤ n are isolating. Proof. We assume that there exists a compact neighborhood U of S such that I(U ) = S ⊂ int(U ). According to (1), Mi and Mj are disjoint, ∀i = j, we have a neighborhood Ui ⊂ U of Mi such that Ui ∩ Mj = ∅ ∀j = i. We claim that I(Ui ) = Mi . “⊂” ∀x ∈ I(Ui ), o(x) ⊂ I(Ui ) ⊂ Ui ⊂ U . Therefore ω(x) ∪ ω ∗ (x) ⊂ Ui ∩ S. According to (2), ω(x) ∪ ω ∗ (x) ⊂ Mi and then x ∈ Mi . “⊃” Since Mi is an invariant set in Ui , Mi ⊂ I(Ui ). For a compact isolating invariant set S, let U ∈ Σ be its compact isolated neighborhood, i.e., I(U ) = S ⊂ int(U ). We deﬁne 1 , x) ⊂ U, ω(x) ⊂ Mj ∪ . . . ∪ Mn } and Ij+ = {x ∈ U | ϕ(R+ 1 , x) ⊂ U, ω ∗ (x) ⊂ M1 ∪ . . . ∪ Mj }, 1 ≤ j ≤ n . Ij− = {x ∈ U | ϕ(R−

(6) 1 I1+ ⊃ I2+ ⊃ . . . ⊃ In+ , ϕ(R+ , Ij+ ) = Ij+ , 1 ≤ j ≤ n , 1 I1− ⊃ I2− ⊃ . . . ⊃ In− , ϕ(R− , Ij− ) = Ij− , 1 ≤ j ≤ n .

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5 Topological and Variational Methods

Aj ⊂ Ij− , A∗j−1 ⊂ Ij+ , and then

Mji = Aj ∩ A∗i−1 = Ii+ ∩ Ij− ,

for i ≤ j. (7) Ij± , 1 ≤ j ≤ n are compact. Proof. It is suﬃcient to show that they are closed. For I1+ and In− it follows from the continuity of the ﬂow and the closedness of U . In the case where n = 2, it suﬃces to verify the closedness of I2+ and I1− . Let (M1 , M2 ) be the Morse decomposition, and let {xn } ⊂ I2+ and xn → x; we want to show that x ∈ I2+ . Since x ∈ I1+ , we have ω(x) ⊂ M1 ∪ M2 . It is suﬃcient to show that ω(x) ⊂ M1 . Suppose not, i.e., ω(x) ⊂ M1 . We choose open neighborhoods Vi of Mi , i = 1, 2 such that V 1 ∩ V 2 = ∅. Since ω(xn ) ⊂ M2 and xn → x, we have tn ≥ 0 and tn ≥ 0 such that ϕ([tn , ∞), xn ) ⊂ V2 and ϕ(tn , xn ) ∈ V1 . Thus ∃tn ∈ [tn , tn ] such that ϕ([tn , ∞), xn ) ⊂ U \V1 and ϕ(tn , xn ) ∈ U \(V1 ∪ V2 ). From the compactness of U , after a subsequence ϕ(tn , xn ) → y ∈ M1 ∪ M2 and ϕ([0, +∞), y) ⊂ U \V1 . From xn ∈ I1+ , we have ϕ(tn , xn ) ∈ I1+ ; it follows that y ∈ I1+ , and then ω(y) ⊂ M2 . If {tn } is bounded, then after a subsequence, tn → s; we have ω(ϕ(−s, y)) = ω(y) ⊂ M2 . But ϕ(−s, y) = x. This contradicts ω(x) ⊂ M1 . If {tn } is unbounded, without loss of generality may assume, tn → +∞. ∀t > 0 ∃n large such that ϕ([−t, 0], ϕ(tn , xn )) = ϕ([tn −t, tn ], xn ) ⊂ ϕ([0, ∞), x) ⊂ U ; it follows that ϕ([−t, 0], y) ⊂ U . Since t > 0 is arbitrary, 1 1 , y) ⊂ U . Combining with ϕ(R+ , y) ⊂ U \V1 , it follows that we have ϕ(R− o(y) ⊂ U , i.e., y ∈ I(U ) = S. Since (M1 , M2 ) is a Morse decomposition of S, and ω(y) ⊂ M2 , it follows that y ∈ M2 , again a contradiction. Similarly, I1− is closed. For n > 2, it can be reduced to the case where n = 2 by deﬁning M1 = Mj−1,1 and M2 = Mn,j . Thus (M1 , M2 ) is again a Morse decomposition of S from (4), since Ij+ = I2+ , where I2+ is that in (M1 , M2 ). Thus Ij+ is closed.

Similarly Ij− = Ij− , where M1 = Mj1 and M2 = Mn,j+1 . The proof is complete. (8) Let U ∈ Σ be compact. If Z ⊂ U is closed, and if ∀x ∈ Z ∃t > 0 such that ϕ(t, x) ∈ U , then the minimal positively invariant set in U which contains Z, deﬁned by P (Z, U ) = {x ∈ U | ∃t ≥ 0 such that ϕ([−t, 0], x) ⊂ U and ϕ(−t, x) ∈ Z} is closed. It follows directly from the continuity of the ﬂow and the closedness of U . Lemma 5.5.25 Suppose that U ∈ Σ is a compact isolating neighborhood with S = I(U ), and that (M1 , . . . , Mn ) is a Morse decomposition of S. Then for every open neighborhood V of Ij− , there exists a compact neighborhood Nj such that Nj ⊂ V and Nj is positively invariant in U .

5.5 Conley Index Theory

413

+ Proof. Since Ij+1 ∩ Ij− = ∅, we choose a compact neighborhood W ⊂ U of + − Ij+1 ∩ In such that W ∩ Ij− = ∅. 1. ∃t∗ > 0 such that ϕ([−t∗ , 0], x) ⊂ U \W ⇒ x ∈ V ∩ (U \W ). If not, 1 with tn → +∞ such that ϕ([−tn , 0], xn ) ⊂ U \W , ∃{xn } ⊂ U and {tn } ⊂ R+ but xn ∈ V ∩ (U \W ). Let x be a limit point of xn , then x ∈ V ∩ (U \W ) 1 , x) ⊂ U \W . Thus ω ∗ (x) ⊂ M1 ∪ . . . ∪ Mj . Therefore x ∈ Ij− ⊂ and ϕ(R− V ∩ (U \W ). A contradiction. 2. Deﬁne two disjoint subsets of Ij−

A = {x ∈ Ij− | ϕ([0, t∗ ], x) ⊂ U } and B = {x ∈ Ij− | ϕ([0, t∗ ], x) ⊂ U } , then ∀x ∈ A ∃δ = δ(x) > 0 such that ϕ([0, t∗ ], Bδ (x)) ⊂ V ∩(X\W ), from step 1. ∀x ∈ B ∃t = t(x) > 0 such that ϕ([0, t], x) ⊂ V ∩ (X\W ) and ϕ(t, x) ∈ U . This enables us to choose δ = δ(x) > 0 such that ϕ([0, t(x)], Bδ (x)) ⊂ V ∩ (X\W ) and ϕ(t(x), Bδ(x) (x)) ∩ U = ∅. Since Ij− is compact, ∃x1 , . . . , xk ∈ Ij− k

such that Ij− ⊂ ∪ Bδ(xi ) (xi ), and then we choose a compact neighborhood of i=1

k

Ij− , Z ⊂ ∪ Bδ(xi ) (xi ). i=1

3. We claim that P (Z, U ) ⊂ V ∩ (U \W ). ∀x ∈ P (Z, U ), by deﬁnition ∃t ≥ 0 such that ϕ([−t, 0], x) ⊂ U , and ϕ(−t, x) ∈ Z. Then by the deﬁnition of Z, ∃i ∈ [1, k], such that ϕ(−t, x) ∈ Bδ(xi ) (xi ). If x ∈ V ∩ (U \W ), there are two cases: Either xi ∈ A, then ϕ([−t, t∗ − t], x) ⊂ V ∩ (X\W ) which implies t > ∗ t , and then ∃t1 ∈ [0, t∗ − t] such that ϕ([−t, −t1 ], x) ⊂ V ∩ (X\W ) and y = ϕ(−t1 , x) ∈ V ∩ (X\W ). It implies that ϕ([−t∗ , 0], y) ⊂ U \W , but y ∈ V ∩ (X\W ). This contradicts step 1. Or xi ∈ B, then ϕ([−t, t(xi ) − t], x) ⊂ V ∩ (X\W ) and ϕ(t(xi ) − t, x) ∈ U . From ϕ([−t, 0], x) ⊂ U , we see that t(xi ) > t. But if x ∈ V ∩ (X\W ), we would have t(xi ) < t. Again this is a contradiction. We have proved that x ∈ V ∩ (X\W ) ∩ U = V ∩ (U \W ). 4. P (Z, U ) is compact. In fact, if {xn } ⊂ P (Z, U ) with xn → x, i.e., ∃tn ≥ 0 such that ϕ([−tn , 0], xn ) ⊂ U and ϕ(−tn , xn ) ∈ Z, then ϕ([−tn , 0], xn ) ⊂ P (Z, U ) ⊂ 1 , x) ⊂ V ∩ (U \W ). It follows V ∩ (U \W ). If {tn } is unbounded, then ϕ(R− ∗ that ω (x) ∈ Mi for some i ≤ j, and then x ∈ Ij− ⊂ P (Z, U ). Otherwise, {tn } is bounded; after a subsequence, tn → s ≥ 0, then ϕ([−s, 0], x) ⊂ U and ϕ(−s, x) ∈ Z; again x ∈ P (Z, U ). Now we set Nj = P (Z, U ). Combining steps 1 and 4 and (8), the conclusion follows. Inductively applying Lemma 5.5.25, we have the following: Theorem 5.5.26 Let S be an isolating invariant set with a compact isolated neighborhood U0 . If (M1 , . . . , Mn ) is a Morse decomposition of S, then there exists an increasing sequence of compact sets: N0 ⊂ N1 ⊂ . . . ⊂ Nn such that

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5 Topological and Variational Methods

(Nj , Ni−1 ) is an index pair for Mji ∀i ≤ j. In particular, (Nn , N0 ) is an index pair for S and (Nj , Nj−1 ) is an index pair for Mj , ∀j. Proof. According to Theorem 5.5.13, in combination with Remark 5.5.4, there exists an index pair (Nn , N0 ) for S. Setting U = Nn \N0 , we deﬁne compact sets Ij± j = 1, 2, . . . , n see (7), and Un = U ∩ Nn . Applying Lemma 5.5.25, we − have a compact neighborhood Un−1 ⊂ U of In−1 , which is positively invariant + in Nn , and Un−1 ∩ In = ∅. Deﬁne Nn−1 = Un−1 ∪ N0 . Inductively, one deﬁnes a compact neighborhood Uj ⊂ U of Ij− , which is positively invariant in Nj+1 + and Uj ∩ Ij+1 = ∅, and Nj = Uj ∪ N0 , j = n − 1, . . . , 1. Recalling (5.2) and (5.3), as a consequence of Theorem 5.5.26, we have the Morse relations for the Morse decomposition (M1 , . . . , Mn ), i.e., n

P (t; Nj , Nj−1 ) = P (t; Nn , N0 ) + (1 + t)Q(t) ,

j=1

where Q is a formal series with nonnegative coeﬃcients, and P (t; Nj , Nj−1 ) =

∞

tq

q

rank H (Nj , Nj−1 ; F ) ,

q=0

or, in the spirit of Theorem 5.1.29, n

h(t; Mj ) = h(t; U0 ) + (1 + t)Q(t) .

j=1

Let us return to Example 2, and assume that f is a Morse function, i.e., ∀p ∈ K, f (p) is invertible, or in other words, the gradient system x˙ = −f (x) is hyperbolic. ∀p ∈ K, let W s (p) = {x ∈ M | ω(x) = p} and W u (p) = {x ∈ M | ω ∗ (x) = p} be the stable and unstable manifolds at p, respectively. One has dim W s (p) = ind(f, p) = codimW u (p) . According to the Sard–Smale theorem, one may choose a generic Riemannian metric on M such that W s (p)W u (q) ∀p, q ∈ K. Such Morse functions are said to be of Morse–Smale type. Thus, if ind(f, q) − ind(f, p) = k, and let M (q, p) = W u (q) ∩ W s (p), dim M (q, p) = k . We have seen that if K = {p1 , . . . , pn }, then ({p1 }, . . . , {pn }) is a Morse decomposition of M , where p1 , . . . , pn are ordered by their values {f (pi )| i ≤ i ≤

5.5 Conley Index Theory

415

n}. However, there is another way to make a Morse decomposition according to their Morse indices. Setting Sk = {p ∈ K| ind(f, p) = k}, 0 ≤ k ≤ m = dim M , and letting Aj = {x ∈ M | ω(x) ∪ ω ∗ (x) ⊂ S0 ∪ . . . ∪ Sj } 0 ≤ j ≤ m , then (S0 , S1 , . . . , Sm ) is a Morse decomposition of M . According to Theorem 5.5.26, there exists an increasing sequence of compact sets: ∅ = N−1 ⊂ N0 ⊂ . . . ⊂ Nm such that (Nj , Nj−1 ) is an index pair of Sj , 0 ≤ j ≤ m. Noticing the following exact sequences, we have: (1) j is injective, ◦ j = 0, and (3) ∂k = j ◦ i. Thus ∂k−1 ◦ ∂k = 0. (2) ∂k−1

0 = Hk (Nk−1 , Nk−2 )

Hk (Nk+1 , Nk−2 )

ξ

Hk (Nk , Nk−2 ) i

...

Hk+1 (Nk+1 , Nk )

∂k

j

Hk (Nk , Nk−1 )

Hk (Nk+1 , Nk−1 )

...

∂k−1

Hk−1 (Nk−1 , Nk−2 )

Fig. 5.6.

With suitable geometric representation of {Hk (Nk , Nk−1 ; G)| k = 0, 1, . . . , m}, Floer used ∂ to establish the Floer homology. In order to extend the notion of Morse decomposition to noncompact isolated invariant sets, let us keep in mind the Morse relations for (PS) functions on Banach–Finsler manifolds, and introduce the following:

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5 Topological and Variational Methods

Deﬁnition 5.5.27 Let U1 , U2 ∈ Σ with int(U1 ) ∩ int(U2 ) = ∅. One deﬁnes an order U2 > U1 , if ∃T > 0 such that U1 ∩ GT (U1 ∪ U2 ) is positively invariant with respect to GT (U1 ∪ U2 ). If U2 > U1 and U1 > U2 , then we say that U1 and U2 are ϕ-disconnected. Otherwise, we say that they are ϕ-connected. Lemma 5.5.28 Let U1 , U2 ∈ Σ with U2 > U1 , and let U = U1 ∪ U2 . Then there exists closed subsets N0 ⊂ N1 ⊂ N2 such that (N2 , N0 ), (N2 , N1 ) and (N1 , N0 ) are index pairs for U, U2 and U1 , respectively. Proof. Taking T > 0 large enough such that (GT (U ), ΓT (U )) is an index pair for U , GT (U1 ) ⊂ int(U1 ) and U1 ∩GT (U ) is positively invariant with respect to GT (U ). Setting N0 = ΓT (U ), N1 = (U1 ∩ GT (U )) ∪ ΓT (U ) and N2 = GT (U ). Obviously (N2 , N0 ) is an index pair for U (Theorem 5.5.13). (1) In order to verify that (N1 , N0 ) is an index pair for U1 , it is suﬃcient to verify the four conditions in Deﬁnition 5.5.12 as follows: 1. GT (N1 \N0 ) = GT (U1 ∩ GT (U )) ⊂ GT (U1 ) ∩ G2T (U ) ⊂ int(U1 ∩ GT (U )) = int(N1 \N0 ) . 2. Since ΓT (U ) is positively invariant in GT (U ), and U1 ∩ GT (U ) is positively invariant in GT (U ), ΓT (U ) is positively invariant in N1 . 3. By the same reasoning, ΓT (U ) is the exit set of N1 . 4. N1 \N0 = (U1 ∩ GT (U )) ⊂ U1 , GT (U1 ) ⊂ U1 ∩ GT (U ) = N1 \N0 ⊂ N1 \N0 . (2) Similarly, we verify that (N2 , N1 ) is an index pair for U2 . Since N2 \N1 = GT (U )\U1 = GT (U ) ∩ U2 , the veriﬁcation of GT (N2 \N1 ) ⊂ int(N2 \N1 ) follows similarly to that of (N1 , N0 ). Moreover GT (U2 ) ⊂ GT (U ) ∩ U2 = N2 \N1 .

Other veriﬁcations are omitted.

Deﬁnition 5.5.29 Let U ∈ Σ, a family of subsets of U , {U1 , . . . , Un } is called a Morse decomposition of U , if (1) Ui ∈ Σ, i = 1, . . . , n, (2) U = ∪ Ui , 1≤i≤n

(3) int(Ui ) ∩ int(Uj ) = ∅ as i = j, (4) Ui+1 > ∪ Uj , for i = 1, 2, . . . , n − 1. 1≤j≤i

By observing the exactness of sequences q

q

q

→ H (N2 , N1 ; G) → H (N2 , N0 ; G) → H (N1 , N0 ; G) → . . . . and the proof of (5.3), we arrive at:

5.5 Conley Index Theory

417

Theorem 5.5.30 If {U1 , . . . , Un } is a Morse decomposition of U ∈ Σ, then there is a formal series Q with nonnegative coeﬃcients such that n j=1

h(t; Uj ) = h(t; U ) + (1 + t)Q(t) .

Notes

Chapter 1 Section 1.1 contains the basic diﬀerential calculus on Banach spaces; the material can be found in any nonlinear functional analysis book, for instance, Schwartz [Scw], Nirenberg [Ni 1], Deimling [De], Zeidler [Zei], Berger [Ber 1] etc. The discussion on Nemytscki operator can be found in Vainberg [Va], but the proof is much simpler than that in the reference. The presentations of the implicit function theorem and the inverse function theorem are standard. Interesting applications are scattered in the literature, for instance, Nirenberg [Ni 4], Kazdan [Ka], Chow, Hale [CH], Mawhin [Maw 3] etc. The continuity method is extensively used in the existence proof of diﬀerential equations. The global implicit function theorem is due to Hadamard [Ha], Caccioppoli [Cac 1]; extensions can be found in Browder [Bd 1] and Plastock [Pl]. However, the proof presented here is very diﬀerent. The application of the continuity method to an a priori bound as a simpler proof for semi-linear elliptic equations with quadratic growth was given by Amann and Crandall [AC]; the result for quasi-linear elliptic equations can be found in Ladyzenskaya and Ural’zeva [LU]. The Lyapunov–Schmidt reduction is extensively used in nonlinear problems. The application to the study of bifurcation problems introduced here is due to Crandall and Rabinowitz [CR 1], [CR 2]. The references on bifurcation theory are recommended to Chow and Hale [CH]. Sections 1.3.3–1.3.4 provide examples on gluing, which is an important technique in symplectic geometry. The material here is taken from Floer and Weinstein [FW] and Oh [Oh]. Further references are Floer [Fl 1] [Fl 2], Hofer and Zehnder [HZ 2]. Parallel to the implicit function theorem method, a gluing technique via the variational method was developed by Sere [Se]; it has been applied to the homoclinic orbits in Hamiltonian systems, multi-bump solutions, and multi-peak solutions for elliptic diﬀerential equations, see also Coti Zelati and Rabinowitz [CR], Li [Li 1], Gui [Gu] etc. Section 1.3.5 is another important technique in applying the implicit function theorem. The

420

Notes

notion of transversality is taken from diﬀerential geometry. Combining with the Sard theorem, it provides a method for proving various generic type results. The ﬁnite-dimensional form of the transversality theorem can be found in Guillemin and Podollak [GP]. The Sard–Smale theorem is taken from Smale [Sm 3]. The simplicity of eigenvalues of the Laplacian on generic domains is due to Uhlenbeck [Uh]. Section 1.4 is on the Nash–Moser technique. See Nash [Na 2] and Moser [Mos 1] [Mos 2]. KAM theory is due to Kolmogorov [Ko], Arnold [Ar 2] and Moser [Mo 3]. The presentation here can be found in Hormander [Hor 2]. Applications in diﬀerential geometry can be found in Hamilton [Ham]; a version setting up on Frechet spaces is given therein. Other versions can be found in Nirenberg [Ni 1], see also Zehnder [Ze]. For further reading on recent developments of KAM theory to partial diﬀerential equations, Bourgain [Bou], Kuksin [Kuk], Wayne [Way], and Poschel [Po] are recommended. Chapter 2 The order method is very diﬀerent from other methods in this book. In concrete problems, once the assumptions are met, the method is simple and powerful. Our discussions start with the Bourbaki–Kneser principle [Bo], [Kn], see also Tarski [Ta]. The Amann theorem [Am 1] is a version that is easy to apply. The sub- and super-solutions method is extensively applied in ODE and PDE whenever the maximum principle is applicable. However, the constructions of sub- and super-solutions require special knowledge and techniques. We are satisﬁed introducing the method by an example. The Caristi ﬁxed-point theorem [Ca] is among a few ﬁxed-point theorems without assuming the continuity of the nonlinear mappings, applications can be found in [Lie]. Also, an equivalent version of this theorem is the very important Ekeland variational principle [Ek 1]; various applications in diﬀerent branches of analysis can be found in [Ek 1] and de Figueiredo [dF]. There are lots of books on convex analysis; in Sect. 2.2, we only present very brieﬂy the necessary material for the sequel discussions. References can be found in Ekeland and Temam [ET], Aubin and Ekeland [AE] etc. Fixed points for nonexpansive mappings have been studied by Browder [Bd 2], Goebel [GK], etc. The importance of this class of mappings is that one of the ﬁxed points can be ﬁgured out by iteration methods. Many algorithms for ﬁnding feasible solutions for convex programming have been studied in recent years, see [B], [BB]. There are many ways to introduce the Schauder ﬁxed-point theorem and related topics. We use the KKM theorem [KKM] and Ky Fan’s inequality [FK 3] as the starting point. The Nash equilibrium [Na 3], the Von Neumann– Sion minimax theorem [VN] [Si], the Schauder ﬁxed-point theorem [Sc], the Schauder–Techonoﬀ theorem, the Ky Fan–Glicksberg theorem [FK 1], [FK 2], and the existence result of Hartman and Stampacchia on variational inequalitiy [HS] are direct consequences. The variational inequality [LS], [Bd 3], [Stm]

Notes

421

is another direction in convex analysis with many applications in free boundary problems from mathematical physics, see Duvaut and Lions [DL], Kinderlehrer and Stampacchia [KS], Friedman [Fr] etc. However, the approach based on the ﬁxed points of set-valued mappings is due to the author, see [Ch 1], [Ch 2]. The theory of monotone operators and pseudo-monotone operators attracted much attention in the 1960s and 70s. The works of Minty [Min], Browder [Bd 4], Hartman and Stampacchia [HS], H. Brezis [Br 1], etc. constitute the basic content of the theory. Again we use the version of the Ky Fan inequality due to Brezis, Nirenberg and Stampacchia [BNS] to derive the most important results on this topics. For applications to quasi-linear elliptic equations see Leray and Lions [LL], to boundary value problems in nonlinear partial diﬀerential equations see Lions [LJ 1], and to nonlinear semigroups of operators see Brezis [Br 1], Crandall and Ligget [CLi]. Chapter 3 The Brouwer degree is a topological invariant; roughly speaking, there are two approaches: algebraic (see for instance Spanier [Sp], Greenberg [GH]) and diﬀerential (Milnor [Mi 2]). The Leray–Schauder degree is its extension to compact vector ﬁelds [LS] on Banach space. The analytic presentation can be found in many books and lecture notes, for instance: J. Schwartz [Scw], Nirenberg [Ni 1], Rabinowitz [Ra 2], Guillemin and Podollak [GP], Zeidler [Zei] etc. The materials of Sects. 3.1–3.4 are taken from these references. An application of the transversality theorem to the proof for the Borsuk–Ulam theorem and to the computation of S 1 -invariant degree are due to Nirenberg [Ni 3]. Sections 3.5 and 3.6.3 are adapted from Rabinowitz [Ra 1], [Ra 2]. Section 3.6.2 is based on Dancer [Dan 1]. The material of Sects. 3.6.4 and 3.6.5 is taken from Amann [Am 2], Krasnosel’ski [Kr 2], de Figueiredo, Lions and Nussbaum [FLN], Chang [Ch 3], and Dancer [Dan 2]. The blowing up method in a priori estimates is a useful technique, see for instance, Gidas and Spruck [GS]. Section 7 contains various extensions of the Leray–Schauder degree; for α-set contraction mappings, see Darbo [Dar], Stuart and Toland [ST]; for condensing mappings, see Nussbaum [Nu], Sadovskii [Sa], set-valued mappings see Browder [Bd 5], Cellina [Ce], Ma [Ma], Chang [Ch 2]. There are other directions: Fredholm operators, see Elworthy and Tromba [ElT 1] [ElT 2], Nirenberg [Ni 4], Llyord [Ll], Fitzpatrick, Pejsachowicz and Rabier [FP], [FPR], [PR 1, PR 2]; coincidence degree theory, Mawhin [Maw 1], etc. Chapter 4 Section 4.1 is an introduction to the calculus of variations. The derivations of the Euler–Lagrange equation, the Legendre–Hadamard condition and the Ljusternik theorem on constraint variational problems can be found in any

422

Notes

standard textbook. For the dual variational principle, i.e., the Legendre– Fenchel transformation, see for instance, Arnold [Ar 1], Ekeland and Temam [ET] or Aubin and Ekeland [AE]. For the Hamiltonian systems, the second version of the dual variational principle, i.e., the Legendre transform to the Hamiltonian function of all variables if the latter is strictly convex, see Clarke and Ekeland [CE]. In Sect. 4.2, the direct method is a general principle in the calculus of variations. We introduce a few interesting examples showing how the principle works. Harmonic maps were introduced by Eells and Sampson [ES] in representing homotopy classes of mappings between two manifolds. Here we only touch on the existence of a weak solution. For m = 2 the harmonic map with minimal energy is smooth, see Morrey [Mo 2]; for m > 2, Schoen and Uhlenbeck [ScU 1], [ScU 2] proved that the singularity has at most a codimension 3 ﬁnite Hausforﬀ measure. A nonsmooth minimal energy harmonic map was given by Lin [Lin]. As to nonminimal energy harmonic maps, the smoothness for m = 2 was proved by Helein [Hel 1], and the existence of a nowhere continuous harmonic for m = 3 constructed by Riviere [Ri]. The example on constant mean curvature surface is taken from Hildebrandt [Hi 1]; a systematic introduction can be found in Struwe [St 2]. The prescribing scalar curvature problem can be found in Kazdan [Ka], Kazdan and Warner [KW 1], [KW 2]. The result for χ(M ) = 0 is due to Berger [Ber 2], that for real projective space P 2 is due to Moser [Mos 4]; however, we present an easy proof with the aid of Aubin’s inequality [Au 3]. Section 4.3 is from Morrey [Mo 1] [Mo 2], Dacorogna [Dac], Acerbi and Fusco [AF], Marcellini [Mar], Muller [Mul 1] [Mul 2] Sverak [Sv 1] [Sv 2], Zhang [Zh 1] and Ball [Bal 1]. The relaxation method is from Buttazzo [Bu] and Dacorogna [Dac]. For the two-well problem, Zhang [Zh 2] has obtained an explicit expression of the quasi-convex envelope for the square distance function. The Young measure was introduced by Young [Yo]; the presentation here is due to Ball [Bal 2], see also [Mul 1], Kinderlehrer and Pedregal [KP 1, KP 2]. In Chipot [Chi], there are examples on the computation of Young measures. There are few books in dealing with the BV space, see for instance, Giusti [Gi], Evans and Gariepy [EG], Zimer [Zi], Ambrosio [Amb] etc. Section 4.5.1 is adapted from these books. The Hardy space and BMO space are important parts in harmonic analysis, see Stein [Ste 1, Ste 2] and Stein–Weiss [SW]. They are applied to PDEs in compensated compactness (as seen in Sect. 4.5.2) and in regularity (for harmonic maps see Helein [Hel 2] Evans [Ev 2]. For the background material we recommend Brezis and Nirenberg [BN 3], and Semmes [Se] as references. The connection between Hardy space and the compensated compactness is revealed by Coifman, Lions, Meyer and Semmes [CLMS]. In the applications to the calculus of variations, the bitting theorem due to Chacon is required, see Brooks and Chacon [BC]. Compensated compactness is also an important

Notes

423

tool in nonlinear analysis; we refer the reader to Tartar [Tar 1, Tar 2], DiPerna [Di] and Evans [Ev 2]. We brieﬂy introduce the Γ-convergence in Sect. 4.6 by a result of Modica and Mortola [MM], see also Alberti [Al]. Readers who want to know more can read Dal Maso [D]. The Munford–Shah model [MS] in the segmentation of the image processing is an interesting subject; the existence proof was given by Ambrisio [Amb], De Giorgi and [DG 2], and De Giorgi and Ambrosio [GA]. More details can be found in Ambrosio Fusco and Pallara [AFP], Dal Maso, Morel and Solimini [DMS] and Morel and Solimini [MS]. As to the regularity of the edge curve, see Morel [Mor]. However, all of these results require deep knowledge on BV functions. A much simpliﬁed model presented here is from Nordstr¨ om [No]. Concentration compactness [LP] in combination with blowing up analysis [SU] is one of the most important techniques in nonlinear problems without compactness. Since interesting concrete problems require more background knowledge, they are out of the scope of this book. We present in Sect. 4.7 just an introduction of the idea: The concentration phenomenon and the role of the bubble (or best constant). Two examples on semilinear elliptic equations are studied: The subcritical exponent on Rn , see Coti Zelati [CZ], and the critical exponent on bounded domain, see Brezis and Nirenberg [BN 1]. Section 4.8 is the preliminary of the minimax method. The Palais–Smale condition [PS] and the mountain pass lemma due to Ambrosetti and Rabinowitz [AR] are introduced in accordance with the Ekeland minimizing principle. The proof was given by Shi [Shi] and Aubin and Ekeland [AE]. Examples are taken from Mawhin [Maw 2] and Ambrosetti and Rabinowitz [AR]. Chapter 5 Section 5.1: The most recommended book in introducing Morse theory on compact manifolds is Milnor [Mi 1]. The classics are Morse [Mo] and Morse and Cairns [MC]. The extension to inﬁnite-dimensional space can be found in Chang [Ch 4], [Ch 5]. See also Mawhin and Willem [MW]. The background material in this section can be found in Palais [Pa 1], Palais and Smale [PS], Rothe [Ro 1], [Ro 2], [Ro 3], Marino and Prodi [MP 1], [MP 2], Chang [Ch 7], [Ch 8], Wang [Wa 1], [Wa 2], Castro and Lazer [CaL]. The critical point theory has been extended to non-diﬀerentiable functionals, see Chang [Ch 6] for locally Lipschitzian functionals on Banach space, and Kartiel [Ka], Ioﬀe, Schwartzmann [IS], and Corvellec, De Giovanni, Marzocchi [CGM] for continuous functions on metric spaces. Section 5.2: There are several books on minimax methods, see for instance Rabinowitz [Ra 4], Struwe [St 1], Ghoussoub [GH], and Willem [Wi] etc. The background material can be found in Rabinowitz [Ra 4], [Ra 5], Palais [Pa 2], [Pa 3], Ni [Ni], Chang [Ch 5], Tian [Ti], Liu [Liu 1], [Liu 2], Viterbo [Vi 2], Solimini [So], Chang, Long and Zehnder [CYZ], Fournier and Willem [FoW].

424

Notes

The notion of category was discovered by Ljusternik and Schnirelmenn [LjS 1]. The genus was introduced by Krasnosel’ski in [Kr 1], while for general index theory see Fadell and Rabinowitz [Fr]. The geometric index for S 1 is taken from Benci [Be 1] and Nirenberg [Ni 3]; other extensions of the index theory can be found in Benci and Rabinowitz [BR 2]. Section 5.3: There are many papers on periodic solutions for Hamiltonian systems via variational methods. The results related to this section are Rabinowitz [Ra 3], Weinstein [We 1], [We 2], Viterbo [Vi 1], Hofer and Zehnder [HZ 2], Struwe [St 1]. Further development on the Weinstein conjecture can be found in Hofer and Viterbo [HV], Hofer [Ho 2] and Liu and Tian [LT 1]. The following books are recommended: Hofer and Zehnder [HZ 1], Ekeland [Ek 2], Mawhin and Willem [MW]. Other important topics on the periodic solutions include: • For Arnold conjecture on the symplectic ﬁxed points and Lagrangian intersections, see Conley and Zehnder [CZ 1], Floer [Fl 1], [Fl 2], [Fl 3], Liu and Tian, [LT 2], Fukaya Ono [FO]. • The number of the periodic orbits on compact convex hypersurface has been estimated by Long and Zhu [LZ]. • For the N-body problem, see Bahri and Rabinowitz [Br]. Section 5.4: The main result of this section is taken from Chang and Yang [CY 1], [CY 2], and Chang and Liu [CL 2]. See also Han [Han], Chen and Li [ChL 1], [ChL 2]. The related prescribing scalar curvature problem on S n has received extensive attention. For n = 3 see Bahri and Coron [BC], and for high dimension, see Schoen and Zhang [SZ], Bahri [Bar 1], [Bar 2], Li [Li 1], and Chen and Lin [ChL]. Section 5.5: Conley’s index theory was introduced by Conley [Co], in which isolating neighborhoods for isolated invariant sets are compact. The theory is completed by Conley and Zehnder [CZ 2], Salamon [Sal], Salamon and Zehnder [SZ]. There are many ways to extend the theory to inﬁnite-dimensional spaces, see Rybakowski [Ry], Rybakowski and Zehnder [RZ], Benci [Be 2] etc. The relationship between the Conley theory and the Morse theory can be found in Chang and Ghoussoub [CG]. For further reading see Michaikov [Mic] and Smoller [Smo].

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Abhyankar, S.S. Resolution of Singularities of Embedded Algebraic Surfaces 2nd enlarged ed. 1998 Alexandrov, A.D. Convex Polyhedra 2005 Andrievskii, V.V.; Blatt, H.-P. Discrepancy of Signed Measures and Polynomial Approximation 2002 Angell, T. S.; Kirsch, A. Optimization Methods in Electromagnetic Radiation 2004 Ara, P.; Mathieu, M. Local Multipliers of C*-Algebras 2003 Armitage, D.H.; Gardiner, S.J. Classical Potential Theory 2001 Arnold, L. Random Dynamical Systems corr. 2nd printing 2003 (1st ed. 1998) Arveson, W. Noncommutative Dynamics and E-Semigroups 2003 Aubin, T. Some Nonlinear Problems in Riemannian Geometry 1998 Auslender, A.; Teboulle M. Asymptotic Cones and Functions in Optimization and Variational Inequalities 2003 Bang-Jensen, J.; Gutin, G. Digraphs 2001 Baues, H.-J. Combinatorial Foundation of Homology and Homotopy 1999 Brown, K.S. Buildings 3rd printing 2000 (1st ed. 1998) Chang, K.-C. Methods in Nonlinear Analysis 2005 Cherry, W.; Ye, Z. Nevanlinna’s Theory of Value Distribution 2001 Ching, W.K. Iterative Methods for Queuing and Manufacturing Systems 2001 Crabb, M.C.; James, I.M. Fibrewise Homotopy Theory 1998 Chudinovich, I. Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes 2005 Dineen, S. Complex Analysis on Inﬁnite Dimensional Spaces 1999 Dugundji, J.; Granas, A. Fixed Point Theory 2003 Elstrodt, J.; Grunewald, F. Mennicke, J. Groups Acting on Hyperbolic Space 1998 Edmunds, D.E.; Evans, W.D. Hardy Operators, Function Spaces and Embeddings 2004 Engler, A.J.; Prestel, A. Valued Fields 2005 Fadell, E.R.; Husseini, S.Y. Geometry and Topology of Conﬁguration Spaces 2001 Fedorov, Y.N.; Kozlov, V.V. A Memoir on Integrable Systems 2001 Flenner, H.; O’Carroll, L. Vogel, W. Joins and Intersections 1999 Gelfand, S.I.; Manin, Y.I. Methods of Homological Algebra 2nd ed. 2003 Griess, R.L. Jr. Twelve Sporadic Groups 1998 Gras, G. Class Field Theory corr. 2nd printing 2005 Hida, H. p-Adic Automorphic Forms on Shimura Varieties 2004 Ischebeck, F.; Rao, R.A. Ideals and Reality 2005 Ivrii, V. Microlocal Analysis and Precise Spectral Asymptotics 1998 Jech, T. Set Theory (3rd revised edition 2002) Jorgenson, J.; Lang, S. Spherical Inversion on SLn (R) 2001 Kanamori, A. The Higher Inﬁnite corr. 2nd printing 2005 (2nd ed. 2003) Kanovei, V. Nonstandard Analysis, Axiomatically 2005 Khoshnevisan, D. Multiparameter Processes 2002 Koch, H. Galois Theory of p-Extensions 2002

Komornik, V. Fourier Series in Control Theory 2005 Kozlov, V.; Maz’ya, V. Differential Equations with Operator Coefﬁcients 1999 Landsman, N.P. Mathematical Topics between Classical & Quantum Mechanics 1998 Leach, J.A.; Needham, D.J. Matched Asymptotic Expansions in Reaction-Diffusion Theory 2004 Lebedev, L.P.; Vorovich, I.I. Functional Analysis in Mechanics 2002 Lemmermeyer, F. Reciprocity Laws: From Euler to Eisenstein 2000 Malle, G.; Matzat, B.H. Inverse Galois Theory 1999 Mardesic, S. Strong Shape and Homology 2000 Margulis, G.A. On Some Aspects of the Theory of Anosov Systems 2004 Murdock, J. Normal Forms and Unfoldings for Local Dynamical Systems 2002 Narkiewicz, W. Elementary and Analytic Theory of Algebraic Numbers 3rd ed. 2004 Narkiewicz, W. The Development of Prime Number Theory 2000 Parker, C.; Rowley, P. Symplectic Amalgams 2002 Peller, V. (Ed.) Hankel Operators and Their Applications 2003 Prestel, A.; Delzell, C.N. Positive Polynomials 2001 Puig, L. Blocks of Finite Groups 2002 Ranicki, A. High-dimensional Knot Theory 1998 Ribenboim, P. The Theory of Classical Valuations 1999 Rowe, E.G.P. Geometrical Physics in Minkowski Spacetime 2001 Rudyak, Y.B. On Thom Spectra, Orientability and Cobordism 1998 Ryan, R.A. Introduction to Tensor Products of Banach Spaces 2002 Saranen, J.; Vainikko, G. Periodic Integral and Pseudodifferential Equations with Numerical Approximation 2002 Schneider, P. Nonarchimedean Functional Analysis 2002 Serre, J-P. Complex Semisimple Lie Algebras 2001 (reprint of ﬁrst ed. 1987) Serre, J-P. Galois Cohomology corr. 2nd printing 2002 (1st ed. 1997) Serre, J-P. Local Algebra 2000 Serre, J-P. Trees corr. 2nd printing 2003 (1st ed. 1980) Smirnov, E. Hausdorff Spectra in Functional Analysis 2002 Springer, T.A. Veldkamp, F.D. Octonions, Jordan Algebras, and Exceptional Groups 2000 Sznitman, A.-S. Brownian Motion, Obstacles and Random Media 1998 Taira, K. Semigroups, Boundary Value Problems and Markov Processes 2003 Talagrand, M. The Generic Chaining 2005 Tauvel, P.; Yu, R.W.T. Lie Algebras and Algebraic Groups 2005 Tits, J.; Weiss, R.M. Moufang Polygons 2002 Uchiyama, A. Hardy Spaces on the Euclidean Space 2001 Üstünel, A.-S.; Zakai, M. Transformation of Measure on Wiener Space 2000 Vasconcelos, W. Integral Closure. Rees Algebras, Multiplicities, Algorithms 2005 Yang, Y. Solitons in Field Theory and Nonlinear Analysis 2001 Zieschang P.-H. Theory of Association Schemes 2005

Kung-Ching Chang

Methods in Nonlinear Analysis

ABC

Kung-Ching Chang School of Mathematical Sciences Peking University 100871 Beijing People’s Republic of China E-mail: [email protected]

Library of Congress Control Number: 2005931137 Mathematics Subject Classiﬁcation (2000): 47H00, 47J05, 47J07, 47J25, 47J30, 58-01, 58C15, 58E05, 49-01, 49J15, 49J35, 49J45, 49J53, 35-01 ISSN 1439-7382 ISBN-10 3-540-24133-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24133-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and TechBooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

SPIN: 11369295

41/TechBooks

543210

Preface

Nonlinear analysis is a new area that was born and has matured from abundant research developed in studying nonlinear problems. In the past thirty years, nonlinear analysis has undergone rapid growth; it has become part of the mainstream research ﬁelds in contemporary mathematical analysis. Many nonlinear analysis problems have their roots in geometry, astronomy, ﬂuid and elastic mechanics, physics, chemistry, biology, control theory, image processing and economics. The theories and methods in nonlinear analysis stem from many areas of mathematics: Ordinary diﬀerential equations, partial diﬀerential equations, the calculus of variations, dynamical systems, diﬀerential geometry, Lie groups, algebraic topology, linear and nonlinear functional analysis, measure theory, harmonic analysis, convex analysis, game theory, optimization theory, etc. Amidst solving these problems, many branches are intertwined, thereby advancing each other. The author has been oﬀering a course on nonlinear analysis to graduate students at Peking University and other universities every two or three years over the past two decades. Facing an enormous amount of material, vast numbers of references, diversities of disciplines, and tremendously diﬀerent backgrounds of students in the audience, the author is always concerned with how much an individual can truly learn, internalize and beneﬁt from a mere semester course in this subject. The author’s approach is to emphasize and to demonstrate the most fundamental principles and methods through important and interesting examples from various problems in diﬀerent branches of mathematics. However, there are technical diﬃculties: Not only do most interesting problems require background knowledge in other branches of mathematics, but also, in order to solve these problems, many details in argument and in computation should be included. In this case, we have to get around the real problem, and deal with a simpler one, such that the application of the method is understandable. The author does not always pursue each theory in its broadest generality; instead, he stresses the motivation, the success in applications and its limitations.

VI

Preface

The book is the result of many years of revision of the author’s lecture notes. Some of the more involved sections were originally used in seminars as introductory parts of some new subjects. However, due to their importance, the materials have been reorganized and supplemented, so that they may be more valuable to the readers. In addition, there are notes, remarks, and comments at the end of this book, where important references, recent progress and further reading are presented. The author is indebted to Prof. Wang Zhiqiang at Utah State University, Prof. Zhang Kewei at Sussex University and Prof. Zhou Shulin at Peking University for their careful reading and valuable comments on Chaps. 3, 4 and 5. Peking University September, 2003

Kung Ching Chang

Contents

1

Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Diﬀerential Calculus in Banach Spaces . . . . . . . . . . . . . . . . . . . . . 1.1.1 Frechet Derivatives and Gateaux Derivatives . . . . . . . . . . 1.1.2 Nemytscki Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 High-Order Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Implicit Function Theorem and Continuity Method . . . . . . . . . . 1.2.1 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Continuity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lyapunov–Schmidt Reduction and Bifurcation . . . . . . . . . . . . . . 1.3.1 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Lyapunov–Schmidt Reduction . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 A Perturbation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Hard Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Small Divisor Problem . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Nash–Moser Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 7 9 12 12 17 23 30 30 33 43 47 49 54 55 62

2

Fixed-Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1 Order Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.2 Convex Function and Its Subdiﬀerentials . . . . . . . . . . . . . . . . . . . 80 2.2.1 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2.2 Subdiﬀerentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.3 Convexity and Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.4 Nonexpansive Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.5 Monotone Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.6 Maximal Monotone Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

VIII

Contents

3

Degree Theory and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.1 The Notion of Topological Degree . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.2 Fundamental Properties and Calculations of Brouwer Degrees . 137 3.3 Applications of Brouwer Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.3.1 Brouwer Fixed-Point Theorem . . . . . . . . . . . . . . . . . . . . . . 148 3.3.2 The Borsuk-Ulam Theorem and Its Consequences . . . . . 148 3.3.3 Degrees for S 1 Equivariant Mappings . . . . . . . . . . . . . . . . 151 3.3.4 Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.4 Leray–Schauder Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 3.5 The Global Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.6.1 Degree Theory on Closed Convex Sets . . . . . . . . . . . . . . . 175 3.6.2 Positive Solutions and the Scaling Method . . . . . . . . . . . . 180 3.6.3 Krein–Rutman Theory for Positive Linear Operators . . . 185 3.6.4 Multiple Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.6.5 A Free Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3.6.6 Bridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 3.7 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.7.1 Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 3.7.2 Strict Set Contraction Mappings and Condensing Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 198 3.7.3 Fredholm Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4

Minimization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.1 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.1.1 Constraint Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.1.2 Euler–Lagrange Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.1.3 Dual Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.2 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.2.1 Fundamental Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement . . . . . . . . . . 223 4.3 Quasi-Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.3.1 Weak Continuity and Quasi-Convexity . . . . . . . . . . . . . . . 232 4.3.2 Morrey Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 4.3.3 Nonlinear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 4.4 Relaxation and Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.4.1 Relaxations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 4.4.2 Young Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4.5 Other Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.5.1 BV Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 4.5.2 Hardy Space and BMO Space . . . . . . . . . . . . . . . . . . . . . . . 266 4.5.3 Compensation Compactness . . . . . . . . . . . . . . . . . . . . . . . . 271 4.5.4 Applications to the Calculus of Variations . . . . . . . . . . . . 274

Contents

IX

4.6 Free Discontinuous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 4.6.1 Γ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 4.6.2 A Phase Transition Problem . . . . . . . . . . . . . . . . . . . . . . . . 280 4.6.3 Segmentation and Mumford–Shah Problem . . . . . . . . . . . 284 4.7 Concentration Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.7.1 Concentration Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.7.2 The Critical Sobolev Exponent and the Best Constants 295 4.8 Minimax Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 4.8.1 Ekeland Variational Principle . . . . . . . . . . . . . . . . . . . . . . . 301 4.8.2 Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.8.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5

Topological and Variational Methods . . . . . . . . . . . . . . . . . . . . . . 315 5.1 Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 5.1.2 Deformation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 5.1.3 Critical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 5.1.4 Global Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 5.1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 5.2 Minimax Principles (Revisited) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5.2.1 A Minimax Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 5.2.2 Category and Ljusternik–Schnirelmann Multiplicity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 5.2.3 Cap Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 5.2.4 Index Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 5.2.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 5.3.1 Hamiltonian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 5.3.2 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 5.3.3 Weinstein Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 5.4 Prescribing Gaussian Curvature Problem on S 2 . . . . . . . . . . . . . 380 5.4.1 The Conformal Group and the Best Constant . . . . . . . . . 380 5.4.2 The Palais–Smale Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 387 5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 5.5 Conley Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 5.5.1 Isolated Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 5.5.2 Index Pair and Conley Index . . . . . . . . . . . . . . . . . . . . . . . . 397 5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

1 Linearization

The ﬁrst and the easiest step in studying a nonlinear problem is to linearize it. That is, to approximate the initial nonlinear problem by a linear one. Nonlinear diﬀerential equations and nonlinear integral equations can be seen as nonlinear equations on certain function spaces. In dealing with their linearizations, we turn to the diﬀerential calculus in inﬁnite-dimensional spaces. The implicit function theorem for ﬁnite-dimensional space has been proved very useful in all diﬀerential theories: Ordinary diﬀerential equations, diﬀerential geometry, diﬀerential topology, Lie groups etc. In this chapter we shall see that its inﬁnite-dimensional version will also be useful in partial diﬀerential equations and other ﬁelds; in particular, in the local existence, in the stability, in the bifurcation, in the perturbation problem, and in the gluing technique etc. This is the contents of Sects. 1.2 and 1.3. Based on Newton iterations and the smoothing operators, the Nash–Moser iteration, which is motivated by the isometric embeddings of Riemannian manifolds into Euclidean spaces and the KAM theory, is now a very important tool in analysis. Limited in space and time, we restrict ourselves to introducing only the spirit of the method in Sect. 1.4.

1.1 Diﬀerential Calculus in Banach Spaces There are two kinds of derivatives in the diﬀerential calculus of several variables, the gradients and the directional derivatives. We shall extend these two to inﬁnite-dimensional spaces. Let X, Y and Z be Banach spaces, with norms · X , · Y , · Z , respectively. If there is no ambiguity, we omit the subscripts. Let U ⊂ X be an open set, and let f : U → Y be a map.

2

1 Linearization

1.1.1 Frechet Derivatives and Gateaux Derivatives Deﬁnition 1.1.1 (Fr´echet derivative) Let x0 ∈ U ; we say that f is Fr´echet diﬀerentiable (or F-diﬀerentiable) at x0 , if ∃A ⊂ L(X, Y ) such that f (x) − f (x0 ) − A(x − x0 ) Y = ◦( x − x0 X ) . Let f (x0 ) = A, and call it the Fr´echet (or F-) derivative of f at x0 . If f is F-diﬀerentiable at every point in U , and if x → f (x), as a mapping from U to L(X, Y ), is continuous at x0 , then we say that f is continuously diﬀerentiable at x0 . If f is continuously diﬀerentiable at each point in U , then we say that f is continuously diﬀerentiable on U , and denote it by f ∈ C 1 (U, Y ). Parallel to the diﬀerential calculus of several variables, by deﬁnition, we may prove the following: 1. If f is F-diﬀerentiable at x0 , then f (x0 ) is uniquely determined. 2. If f is F-diﬀerentiable at x0 , then f must be continuous at x0 . 3. (Chain rule) Assume that U ⊂ X, V ⊂ Y are open sets, and that f is F-diﬀerentiable at x0 , and g is F-diﬀerentiable at f (x0 ), where f

g

U −−−−→ V −−−−→ Z Then

(g ◦ f ) (x0 ) = g ◦ f (x0 ) · f (x0 ) .

Deﬁnition 1.1.2 (Gateaux derivative) Let x0 ∈ U ; we say that f is Gateaux diﬀerentiable (or G-diﬀerentiable) at x0 , if ∀h ∈ X, ∃ df (x0 , h) ⊂ Y , such that f (x0 + th) − f (x0 ) − tdf (x0 , h)Y = ◦(t) as t → 0 for all x0 + th ⊂ U . We call df (x0 , h) the Gateaux derivative (or G-derivative) of f at x0 . We have

d f (x0 + th) |t=0 = df (x0 , h) , dt if f is G-diﬀerentiable at x0 . By deﬁnition, we have the following properties: 1. If f is G-diﬀerentiable at x0 , then df (x0 , h) is uniquely determined. 2. df (x0 , th) = tdf (x0 , h) ∀t ∈ R1 . 3. If f is G-diﬀerentiable at x0 , then ∀h ∈ X, ∀y ∗ ∈ Y ∗ , the function ϕ(t) = y ∗ , f (x0 + th) is diﬀerentiable at t = 0, and ϕ (t) = y ∗ , df (x0 , h) . 4. Assume that f : U → Y is G-diﬀerentiable at each point in U , and that the segment {x0 + th | t ∈ [0, 1]} ⊂ U , then f (x0 + h) − f (x0 ) Y sup df (x0 + th, h) Y 0 0, ∃δ = δ(M, ε) > 0 such that | ϕu (x, ξ) − ϕu (x, ξ ) |< ε, ∀x ∈ Ω , as |ξ|, |ξ | M and |ξ − ξ | δ. We choose M = u0 + h , then for |t| < δ < 1, |ϕu (x, u0 (x) + tθ(x)h(x)) − ϕu (x, u0 (x))| < ε . It follows that df (u0 , h)(x) = ϕu (x, u0 (x))h(x). Noticing that the multiplication operator h → A(u)h = ϕu (x, u(x)) · h(x) is linear and continuous, and the mapping u → A(u) from C(Ω) into L(C(Ω), C(Ω)) is continuous, from Theorem 1.1.3, f is F-diﬀerentiable, and (f (u0 ) · v)(x) = ϕu (x, u0 (x)) · v(x) ∀v ∈ C(Ω) .

1.1 Diﬀerential Calculus in Banach Spaces

5

We investigate nonlinear diﬀerential operators on more general spaces. Let Ω ⊂ Rn be a bounded open set, and let m be a nonnegative integer, older space C m,γ (Ω)) is deﬁned to be the function γ ∈ (0, 1). C m (Ω) (and the H¨ m space consisting of C functions (with γ-H¨older continuous m-order partial derivatives). The norms are deﬁned as follows: |∂ α u(x)| , u C m = max x∈Ω

|α|≤m

and u C m,γ = u C m +

|∂ α u(x) − ∂ α u(y)| , |x − y|γ |α|=m

max x,y∈Ω

where α = (α1 , α2 , . . . , αn ) is a multi-index, |α| = α1 + α2 + · · · + αn , ∂ α = ∂xα11 ∂xα22 · · · ∂xαnn . We always denote by m∗ the number of the index set {α = (α1 , α2 , . . . , αn ) | |α| m}, and Dm u the set {∂ α u | |α| m}. ∗ Suppose that r is a nonnegative integer, and that ϕ ∈ C ∞ (Ω×Rr ). Deﬁne a diﬀerentiable operator of order r: f (u)(x) = ϕ(x, Dr u(x)) . Suppose m r, then f : C m (Ω) → C m−r (Ω) (and also C m,γ (Ω) → (Ω)) is F-diﬀerentiable. Furthermore C ϕα (x, Dr u0 (x)) · ∂ α h(x), ∀h ∈ C m (Ω) , (f (u0 )h)(x) = m−r,γ

|α|≤r

where ϕα is the partial derivative of ϕ with respect to the variable index α. The proof is similar to Example 3. ∗

Example 4. Suppose ϕ ∈ C ∞ (Ω × Rr ). Deﬁne

f (u) = ϕ(x, Dr u(x))dx ∀u ∈ C r (Ω) . Ω

Then f : C r (Ω) → R1 is F-diﬀerentiable. Furthermore

f (u0 ), h = ϕα (x, Dr u0 (x))∂ α h(x)dx

∀h ∈ C r (Ω) .

Ω |α|≤r

Proof. Use the chain rule: ϕ(·,D r u(·))

C r (Ω) −−−−−−−→ C(Ω) −−−Ω−→ R1 , and combine the results of Examples 1 and 3.

6

1 Linearization

In particular, the following functional occurs frequently in the calculus of variations (r = 1, r∗ = n + 1). Assume that ϕ(x, u, p) is a function of the form: ϕ(x, u, p) =

n 1 2 |p| + ai (x)pi + a0 (x)u , 2 i=1

where p = (p1 , p2 , . . . , pn ), and ai (x), i = 0, 1, . . . , n, are in C(Ω). Set

n 1 2 |∇u(x)| + f (u) = ai (x)∂xi u + a0 (x)u(x) dx , Ω 2 i=1 we have

∇u(x) · ∇h(x) +

f (u), h = Ω

n

ai (x)∂xi h(x) + a0 (x)h(x) dx

i=1

∀h ∈ C 1 (Ω) . Example 5. Let X be a Hilbert space, with inner product (, ). Find the Fderivative of the norm f (x) = x , as x = θ. Let F (x) = x 2 . Since t−1 ( x + th 2 − x 2 ) = 2(x, h) + t h 2 , we have dF (x, h) = 2(x, h). It is continuous for all x, therefore F is Fdiﬀerentiable, and F (x)h = 2(x, h) . 1

Since f = F 2 , by the chain rule F (x) = 2 x ·f (x) . As x = θ,

x ,h . f (x)h = x In the applications to PDE as well as to the calculus of variations, Sobolev spaces are frequently used. We should extend the above studies to nonlinear operators deﬁned on Sobolev spaces. ∀p 1, ∀ nonnegative integer m, let

W m,p (Ω) = {u ∈ Lp (Ω) | ∂ α u ∈ Lp (Ω) | |α| m} , where ∂ α u stands for the α-order generalized derivative of u, i.e., the derivative in the distribution sense. Deﬁne the norm ⎞ p1 ⎛ ∂ α u pLp (Ω) ⎠ . u W m,p = ⎝ |α≤m

The Banach space is called the Sobolev space of index {m, p}. W m,2 (Ω) is denoted by H m (Ω), and the closure of C0∞ (Ω) under this norm is denoted by H0m (Ω).

1.1 Diﬀerential Calculus in Banach Spaces

7

1.1.2 Nemytscki Operator On Sobolev spaces, we extend the composition operator u → ϕ(x, u(x)) such that ϕ may not be continuous in x. The class of operators is sometimes called Nemytski operators. Deﬁnition 1.1.4 Let (Ω, B , µ) be a measure space. We say that ϕ : Ω × RN → R1 is a Caratheodory function, if 1. ∀a.e.x ∈ Ω, ξ → ϕ(x, ξ) is continuous. 2. ∀ξ ∈ RN , x → ϕ(x, ξ) is µ-measurable. The motivation in introducing the Caratheodory function is to make the composition function measurable if u(x) is only measurable. Indeed, there exists a sequence of simple functions {un (x)}∞ , such that un (x) → u(x) a.e., ϕ(x, un (x)) is measurable according to (2). And from (1), ϕ(x, un (x)) → ϕ(x, u(x)) a.e., therefore ϕ(x, u(x)) is measurable. p

Theorem 1.1.5 Assume p1 , p2 1, a > 0 and b ∈ Ldµ2 (Ω). Suppose that ϕ is a Caratheodory function satisfying p1

|ϕ(x, ξ)| b(x) + a|ξ| p2 . Then f : u(x) → ϕ(x, u(x)) is a bounded and continuous mapping from p p Ldµ1 (Ω, RN ) to Ldµ2 (Ω, RN ). Proof. The boundedness follows from the Minkowski inequality: p1 p

f (u) p2 b p2 + a u p 2 , 1

where · p is the Lpdµ (Ω, RN ) norm. We turn to proving the continuity. It is p1 suﬃcient to prove that ∀{un }∞ 1 if un → u in L , then there is a subsequence p2 {uni } such that f (uni ) → f (u) in L . Indeed one can ﬁnd a subsequence {uni } of {un } which converges a.e. to u, along which uni − uni−1 p1 < 21i , i = 2, 3, . . .; therefore |uni (x)| Φ(x) := |un1 (x)| +

∞

|uni (x) − uni−1 (x)| .

i=2

Since Φ is measurable, and

|Φ(x)|p1 dµ

p1

1

Ω

un1 p1 +

∞ i=2

p

we conclude that Φ ∈ Ldµ1 (Ω). Noticing

uni − uni−1 p1 < +∞ ,

8

1 Linearization

f (uni ) = ϕ(x, uni (x)) → ϕ(x, u(x)) and

p1

a.e. ,

p

|f (uni )(x)| b(x) + a(Φ(x)) p2 ∈ Ldµ2 (Ω, RN ) , we have f (uni ) − f (u) p2 → 0, according to Lebesgue dominance theorem. This proves the continuity of f . Corollary 1.1.6 Let Ω ⊂ Rn be a smooth bounded domain, and let 1 ≤ ∗ p1 , p2 ≤ ∞. Suppose that ϕ : Ω × Rm → R is a Caratheodory function satisfying m αj |ξj | p2 , |ϕ(x, ξ0 , . . . , ξm )| b(x) + a j=0

where ξj is a #{α = (α1 , . . . , αn )| |α| = j}-vector, αj ( p1 − 1

m−j −1 , n )

a>

0, and b ∈ L (Ω). Then f (u)(x) = ϕ(x, D u(x)) deﬁnes a bounded and continuous map from W m,p1 (Ω) into Lp2 (Ω). p2

m

Corollary 1.1.7 Suppose that Ω ⊂ Rn and that ϕ : Ω×R1 → R1 and ϕξ (x, ξ) 2n are Caratheodory functions. If |ϕξ (x, ξ)| b(x) + a|ξ|r , where b ∈ L n+2 (Ω), n+2 (if n ≤ 2, then the restriction is not necessary), then the a > 0, and r = n−2 functional

ϕ(x, u(x))dx

f (u) = Ω

is F-diﬀerentiable on H 1 (Ω), with F-derivative

f (u), v = ϕξ (x, u(x)) · v(x)dx , Ω

where , is the inner product on H 1 (Ω). Proof. The Sobolev embedding theorem says that the injection i : H 1 (Ω) → 2n 2n L n−2 (Ω) is continuous, so is the dual map i∗ : L n+2 (Ω) → (H 1 (Ω))∗ . 2n 2n According to Theorem 1.1.5, ϕξ (·, ·) : L n−2 → L n+2 is continuous. Therefore the Gateaux derivative

ϕξ (x, u(x)) · v(x)dx ∀v ∈ H 1 (Ω) df (u, v) = Ω

is continuous from H 1 (Ω) to (H 1 (Ω))∗ . Applying Theorem 1.1.3, we conclude that f is F-diﬀerentiable on H 1 (Ω). The proof is complete. Corollary 1.1.8 In Corollary 1.1.6, the diﬀerential operator f (u(x)) = ϕ(x, Dm u(x)) from C l,γ (Ω) to C l−m,γ (Ω), l m, 0 γ < 1, is F-diﬀerentiable, with ϕα (x, Dm u(x))∂ α h(x) ∀h ∈ C l,γ (Ω) . (f (u0 )h)(x) = |α|≤m

1.1 Diﬀerential Calculus in Banach Spaces

9

1.1.3 High-Order Derivatives The second-order derivative of f at x0 is deﬁned to be the derivative of f (x) at x0 . Since f : U → L(X, Y ), f (x0 ) should be in L(X, L(X, Y )). However, if we identify the space of bounded bilinear mappings with L(X, L(X, Y )), and verify that f (x0 ) as a bilinear mapping is symmetric, see Theorem 1.1.9 below, then we can deﬁne equivalently the second derivative f (x0 ) as follows: For f : U → Y , x0 ∈ U ⊂ X, if there exists a bilinear mapping f (x0 )(·, ·) of X × X → Y satisfying 1 f (x0 +h)−f (x0 )−f (x0 )h− f (x0 )(h, h) = ◦( h 2 ) ∀h ∈ X, as h → 0 , 2 then f (x0 ) is called the second-order derivative of f at x0 . By the same manner, one deﬁnes the mth-order derivatives at x0 successively: f (m) (x0 ) : X × · · · × X → Y is an m-linear mapping satisfying m (j) f (x0 )(h, . . . , h) f (x0 + h) − = ◦( h m ) , j! j=0 as h → 0. Then f is called m diﬀerentiable at x0 . Similar to the ﬁnite-dimensional vector functions, we have: Theorem 1.1.9 Assume that f : U → Y is m diﬀerentiable at x0 ∈ U . Then for any permutation π of (1, . . . , m), we have f (m) (x0 )(h1 , . . . , hm ) = f (m) (x0 )(hπ(1) , . . . , hπ(m) ) . Proof. We only prove this in the case where m = 2, i.e., f (x0 )(ξ, η) = f (x0 )(η, ξ) ∗

∀ξ, η ∈ X .

∗

Indeed ∀y ∈ Y , we consider the function ϕ(t, s) = y ∗ , f (x0 + tξ + sη) . It is twice diﬀerentiable at t = s = 0; so is ∂2 ∂2 ϕ(0, 0) = ϕ(0, 0) . ∂t∂s ∂s∂t Since f (x0 + tξ + sη) is continuous as |t|, |s| small, one has ∂ ϕ(t, ·)|s=0 = y ∗ , f (x0 + tξ)η ; ∂s and then, ∂2 ϕ(t, s)|t=s=0 = y ∗ , f (x0 )(ξ, η) . ∂t∂s Similarly ∂2 ϕ(t, s)|t=s=0 = y ∗ , f (x0 )(η, ξ) . ∂s∂t This proves the conclusion.

10

1 Linearization

Theorem 1.1.10 (Taylor formula) Suppose that f : U → Y is continuously m-diﬀerentiable. Assume the segment {x0 + th | t ∈ [0, 1]} ⊂ U . Then f (x0 + h) =

m 1 (j) f (x0 )(h, . . . , h) j! j=0

1 1 (1 − t)m f (m+1) (x0 + th)(h, . . . , h)dt . + m! 0

Proof. ∀y ∗ ∈ Y ∗ , we consider the function: ϕ(t) = y ∗ , f (x0 + th) . From the Hahn–Banach theorem and the Taylor formula for single-variable functions:

1 m 1 (j) 1 ϕ (0) + (1 − t)m ϕ(m+1) (t)dt , ϕ(1) = j! m! 0 j=0 we obtain the desired Taylor formula for mappings between B-spaces.

Example 1. X = Rn , Y = R1 . If f : X → Y is twice continuously diﬀerentiable, then 2 ∂ f (x) . f (x) = Hf (x) = ∂xi ∂xj i,j=1,...,n Example 2. X = C 1 (Ω, RN ), Y = R1 . Suppose that g ∈ C 2 (Ω × RN , R1 ). Deﬁne

1 |∇u|2 + g(x, u(x)) f (u) = 2 Ω Ω as u ∈ X. By deﬁnition, we have

f (u) · ϕ = [∇u(x)∇ϕ(x) + gu (x, u(x))ϕ(x)]dx , Ω

and f (u)(ϕ, ψ) =

[∇ψ(x)∇ϕ(x) + guu (x, u(x))ϕ(x)ψ(x)]dx .

Ω With some additional growth conditions on guu : 4

(x, u)| ≤ a(1 + |u| n−2 ), a > 0 ∀u ∈ RN , |guu

f is twice diﬀerentiable in H01 (Ω, RN ). As an operator from H01 (Ω, RN ) into itself, f (u) = id + (−)−1 gu (·, u(·)) .

1.1 Diﬀerential Calculus in Banach Spaces

11

is self-adjoint, or equivalently, the operator − + guu (x, u(x))· deﬁned on L2 2 1 N is self-adjoint with domain H ∩ H0 (Ω, R ).

Example 3. Let X = H01 (Ω, R3 ), where Ω is a plane domain. Consider the volume functional

u · (ux ∧ uy ) . Q(u) = Ω

One has Q (u) · ϕ =

ϕ(ux ∧ uy ) + u · [(ϕx ∧ uy ) + (ux ∧ ϕy )] , Ω

and Q (u)(ϕ, ψ) =

ϕ[(ψx ∧ uy ) + (ux ∧ ψy )] + ψ[(ϕx ∧ uy ) + (ux ∧ ϕy )] Ω

+ u[(ϕx ∧ ψy ) + (ψx ∧ ϕy )] , ∀ϕ, ψ ∈ H01 (Ω, R3 ). If further we assume u ∈ C 2 (Ω, R3 ), then from integration by parts and the antisymmetry of the exterior product, we have

u(ϕx ∧ uy ) = − (ϕ ∧ uy ) · ux + (ϕ ∧ uxy ) · u Ω

Ω = (ux ∧ uy )ϕ − (ϕ ∧ uxy ) · u , Ω

and

Ω

u(ux ∧ ϕy ) = − uy (ux ∧ ϕ) + u(uxy ∧ ϕ) Ω Ω

ϕ(ux ∧ uy ) + (ϕ ∧ uxy )u . =

Ω

Therefore, Q (u) = 3

Ω

ϕ · (ux ∧ uy ) . Ω

By the same manner, we obtain

Q (u)(ϕ, ψ) = 3 u[(ϕx ∧ ψy ) + (ϕy ∧ ψx )] . Ω C

2

Geometrically, let u : Ω → R3 be a parametrized surface in R3 ; Q(U ) is the volume of the body enclosed by the surface. As exercises, one computes the ﬁrst- and second-order diﬀerentials of the following functionals:

12

1 Linearization

1. X = W01,p (Ω, R1 ), Y = R1 , 2 < p < ∞,

f (u) = |∇u|p dx . Ω

2. X = C01 (Ω, Rn ), where Ω ⊂ Rn is a domain,

f (u) = det(∇u(x))dx . Ω

3. X = C01 (Ω, R1 ), where Ω ⊂ Rn is a domain,

f (u) = 1 + |∇u|2 dx . Ω

1.2 Implicit Function Theorem and Continuity Method 1.2.1 Inverse Function Theorem It is known that the implicit function theorem for functions of several variables plays important roles in many branches of mathematics (diﬀerential manifold, diﬀerential geometry, diﬀerential topology, etc.). Its extension to inﬁnite-dimensional space is also extremely important in nonlinear analysis, as well as in the study of inﬁnite-dimensional manifolds. Theorem 1.2.1 (Implicit function theorem) Let X, Y , Z be Banach spaces, U ⊂ X × Y be an open set. Suppose that f ∈ C(U, Z) has an F-derivative w.r.t. y, and that fy ∈ C(U, L(Y, Z)). For a point (x0 , y0 ) ∈ U , if we have f (x0 , y0 ) = θ , fy−1 (x0 , y0 ) ∈ L(Z, Y ) ; then ∃r, r1 > 0, ∃|u ∈ C(Br (x0 ), Br1 (y0 )), such that ⎧ ⎨ Br (x0 ) × Br1 (y0 ) ⊂ U , u(x0 ) = y0 , ⎩ f (x, u(x)) = θ ∀x ∈ Br (x0 ) . Furthermore, if f ∈ C 1 (U, Z), then u ∈ C 1 (Br (x0 ), Y ), and u (x) = −fy−1 (x0 , u(x0 )) ◦ fx (x, u(x))

∀x ∈ Br (x0 ) .

(1.1)

1.2 Implicit Function Theorem and Continuity Method

13

Proof. (1) After replacing f by g(x, y) = fy−1 (x0 , y0 ) ◦ f (x + x0 , y + y0 ) , one may assume x0 = y0 = θ, Z = Y and fy (θ, θ) = idY . (2) We shall ﬁnd the solution y = u(x) ∈ Br1 (θ) of the equation f (x, y) = θ

∀x ∈ Br (θ) .

Setting R(x, y) = y − f (x, y) , it is reduced to ﬁnding the ﬁxed point of R(x, ·) ∀x ∈ Br (θ). We shall apply the contraction mapping theorem to the mapping R(x, ·). Firstly, we have a contraction mapping: R(x, y1 ) − R(x, y2 ) = y1 − y2 − [f (x, y1 ) − f (x, y2 )]

1 fy (x, ty1 + (1 − t)y2 )dt · (y1 − y2 ) = y 1 − y2 −

0 1

idY − fy (x, ty1 + (1 − t)y2 ) dt· y1 − y2 . 0

Since fy : U → L(X, Y ) is continuous, ∃r, r1 > 0 such that R(x, y1 ) − R(x, y2 )

0, where f (x, θ)

0 such that Bδ (y0 ) ⊂ V and g : Bδ (y0 ) → g(Bδ (y0 )) is a diﬀermorphism. Furthermore (g −1 ) (x0 ) = g −1 (y0 ), with x0 = g(y0 ) .

(1.5)

1.2 Implicit Function Theorem and Continuity Method

15

Proof. Set f (x, y) = x − g(y),

f ∈ C 1 (X × V, X) .

We use the implicit function theorem (IFT) to f , there exist r > 0 and a unique u ∈ C 1 (Br (x0 ), Br (y0 )) satisfying x = g ◦ u(x) . Since g is continuous, ∃δ ∈ (0, r) such that g(Bδ (y0 )) ⊂ Br (x0 ), therefore g : Bδ (y0 ) → g(Bδ (y0 )) is a diﬀeomorphism. And (1.5) follows from (1.1). In the spirit of the IFT, we have a nonlinear version of the Banach open mapping theorem. Theorem 1.2.4 (Open mapping) Let X, Y be Banach spaces, and let δ > 0 and y0 ∈ Y . Suppose that g ∈ C 1 (Bδ (y0 ), X) and that g (y0 ) : Y → X is an open map, then g is an open map in a neighborhood of y0 . Proof. We want to prove that ∃δ1 ∈ (0, δ) and r > 0, such that Br (g(y0 )) ⊂ g(Bδ1 (y0 )) . With no loss of generality, we may assume y0 = θ and g(y0 ) = θ. Let A = g (θ). Since A is surjective, ∃C > 0 such that inf

z∈ker A

y − z Y C Ay X , ∀ y ∈ Y ,

(1.6)

provided by the Banach inverse theorem. One chooses δ1 ∈ (0, δ) and r > 0, satisfying 1 ∀y ∈ Bδ1 (θ) , g (y) − A 2(C + 1) and r

0, φt ◦ h ◦ ψ−t − id ∈ C 0 (Rn , Rn ) is in a small neighborhood of θ, and the above equation has a unique solution h there; therefore φt ◦ h ◦ ψ−t = h . Then we can extend the procedure step by step to all t, i.e. we have φt ◦ h = h ◦ ψt ∀t ∈ R1 .

20

1 Linearization

Example 2. (Local existence of isothermal coordinates) Given a surface M 2 with a Riemannian metric g, i.e., in local coordinates x = (x1 , x2 ), g = Edx21 + 2F dx1 dx2 + Gdx22 , where E, F and G are functions of local coordinates x = (x1 , x2 ), and ∀(x1 , x2 ), Eξ 2 +2F ξη +Gη 2 is a positive deﬁnite quadratic form. Our problem is to ﬁnd a local coordinate u(x) = (u1 (x1 , x2 ), u2 (x1 , x2 )) in a neighborhood of x0 = (x01 , x02 ) such that there exists a function λ = λ(x1 , x2 ) > 0 satisfying g = λ(x1 , x2 )(du21 + du22 ) . The local coordinate u = (u1 , u2 ) is called an isothermal coordinate. We shall ﬁnd u = (u1 , u2 ) satisfying ⎧ ⎨ λ(x)|∂x1 u(x)|2 = E(x) , λ(x)∂x1 u(x) · ∂x2 u(x) = F (x) , ⎩ λ(x)|∂x2 u(x)|2 = G(x) .

(1.11)

We may assume (x01 , x02 ) = (0, 0) and F (0, 0) = 0 (by translation and rotation of the local coordinates). After eliminating λ, (1.11) is equivalent to: F |∂x1 u|2 = E∂x1 u · ∂x2 u (1.12) F |∂x2 u|2 = G∂x1 u · ∂x2 u . This is a ﬁrst-order nonlinear diﬀerential system. In a neighborhood O of θ = (0, 0), deﬁne F (x) · |∂x1 u(x)|2 − E(x) · ∂x1 u(x) · ∂x2 u(x) ϕ(x, ∇u(x)) = , F (x) · |∂x2 u(x)|2 − G(x) · ∂x1 u(x) · ∂x2 u(x) , ∀x ∈ O, and let f be the map satisfying f (u)(x) = ϕ(x, ∇u(x)) . We want to solve the equation: f (u) = θ .

(1.13)

Let x = εy, where y ∈ D, the unit disk on the coordinate plane, and ε > 0 is a parameter, and let (1.14) εv(y) = u(x) − u(x) . where u = (p1 x1 + p2 x2 , q1 x1 + q2 x2 ), and (p1 , p2 , q1 , q2 ) is chosen such that the following conditions are satisﬁed: ∂(u1 , u2 ) = 0 , ∂(x1 , x2 )

1.2 Implicit Function Theorem and Continuity Method

21

and ϕ(0, ∇u) = θ . In fact, this is an algebraic system with four unknowns, and is trivially solvable. Equation (1.13) is equivalent to ϕ(εy, ∇y v(y) + ∇x u(εy)) = 0 . ¯ denote the space C 1,α (D) ¯ modulo a constant. Set F : R1 × Let C˙ 1,α (D) ˙ 1,α 0 α 2 ∩ C0 (D) C (D)) → C (D) , for some α ∈ (0, 1), where (C 1,α

F (ε, v) = ϕ(εy, ∇y v(y) + ∇x u(εy)) . Thus F (0, θ) = θ . It remains to ﬁnd a pair (ε, v) for small ε > 0, satisfying F (ε, v) = θ . Then by the transform(1.14), the solution u in a small neighborhood of θ εD is obtained. Note that (p1 , q1 , p2 , q2 ) = (u1,x1 , u2,x1 , u1,x2 , u2,x2 ). Let us introduce the notation (E0 , F0 , G0 ) = (E, F, G)|(0,0) ; we have Fv (0, θ)h = −(A1 ∇h1 + A2 ∇h2 ) , where h = (h1 , h2 ) ∈ C 1,α ∩ C00 (D) C˙ 1,α (D), and (E0 p2 − 2F0 p1 ) E0 p1 A1 = , G0 p2 (G0 p1 − 2F0 p2 ) (E0 q2 − 2F0 q1 ) E0 q1 . A2 = G0 q2 (G0 q1 − 2F0 q2 ) This is a constant coeﬃcient ﬁrst-order linear diﬀerential operator with the 2 × 2 matrix symbol: L = (A1 ω, A2 ω) , where ω = (ξ, η). Since det L = −2F0 (E0 ξ 2 − 2F0 ξη + G0 η 2 )

∂(u, v) = 0 ∂(x, y)

∀(ξ, η) ∈ R2 \{θ}, L is elliptic. According to the elliptic theory, Fv (0, ·) has a bounded inverse. The IFT is applied to conclude the existence of ε0 > 0 such that the equation F (ε, v) = θ has a unique solution vε , ∀ε ∈ (−ε0 , ε0 ).

22

1 Linearization

Remark 1.2.7 The boundary condition for the ﬁrst-order system can be deduced from the second-order elliptic theory. In fact, h = (h1 , h2 ) satisﬁes A1 ∇h1 + A2 ∇h2 = f ,

(1.15)

where f is given. Suppose that A2 is invertible (it is available); we have ∇h2 = C∇h1 + A−1 2 f , where

(1.16)

C = −A−1 2 A1 ,

Let E= Then E −1 =

0 1 −1 0

0 1

−1 0

. ,

and E 2 = −I. Since h2,xy = h2,yx ,

(1.17)

we obtain a second-order equation for h1 , in which the principal symbol of the second-order equation reads as 1 E 2 (A2 ω)T E −1 A1 ω det(A2 ) 1 (A2 ω)T E −1 A1 ω = det(A2 ) det(L) . = det(A2 )

−ω T EA−1 2 A1 ω = −

The later is positive (or negative) deﬁnite because the ﬁrst-order system is elliptic. Thus the Dirichlet boundary condition for h1 is well posed, and then h2 follows from (1.16) modulo a constant. Remark 1.2.8 There are many applications of the IFT similar to the above examples, e.g. a necessary and suﬃcient condition for an almost complex structure being a complex structure (Newlander–Nirenberg theorem, see L. Nirenberg [NN]), prescribing Ricci curvature problem (see DeTurck [DT] etc.). For applications to boundary value problems in ordinary and partial diﬀerential equations see S.N. Chow, J. Hale [CH] and J. Mawhin [Maw 3]

1.2 Implicit Function Theorem and Continuity Method

23

1.2.3 Continuity Method We have shown the usefulness of the IFT in the existence of solutions for small perturbations of a given equation which has a known solution. As to large perturbations, the IFT is not enough, we have to add new ingredients. The continuity method is a general principle, which can be applied to prove the existence of solutions for a variety of nonlinear equations. Let X and Y be Banach spaces, and f : X → Y be C 1 . Find the solution of the equation: f (x) = θ . Let us introduce a parameter t ∈ [0, 1] and a map F : [0, 1] × X → Y such that both F and Fx are continuous; in addition, F (1, x) = f (x) . Assume that there exists x0 ∈ X satisfying F (0, x0 ) = θ; we want to extend the solution x0 of the equation F (0, x) = θ to a solution of F (1, x) = θ .

(1.18)

For this purpose, we deﬁne a set S = {t ∈ [0, 1] | such that F (t, x) = θ is solvable} . What we want to do is to prove: (1) S is an open set (relative to [0, 1]). For this purpose, it is suﬃcient to prove that ∀t0 ∈ S, ∃xt0 ∈ X, which solves F (t0 , xt0 ) = θ such that Fx−1 (t0 , xt0 ) ∈ L(Y, X), provided by the IFT. (2) S is a closed set. Usually it depends on the a priori estimates for the solution set {x ∈ X | ∃t ∈ S, such that F (t, x) = θ}. For most PDE problems it requires special knowledge and features of the equations and techniques in hard analysis. We present here two major ideas: (a) If there exist a Banach space X1 , which is compactly embedded in X, and a constant C > 0 such that xt X1 C

∀t ∈ S ,

where xt is a solution of F (t, x) = θ, then S is closed.

24

1 Linearization

In fact, we have {tn }∞ ⊂ S, which implies tn → t∗ , and xtn X1 C . Since the embedding X1 → X is compact, xtn subconverges to some point x∗ ∈ X in X. From the continuity, it follows that F (t∗ , x∗ ) = θ. This proves t∗ ∈ S, i.e., S is closed. (b) If ∀t ∈ S, there exists a unique local solution xt of the equation F (t, ·) = θ, and if there exists C > 0 such that x˙ t X C , where x˙ t is the derivative of xt , then S is a closed set. Proof. Let {tn }∞ 1 be a sequence included in an open interval contained in S, with tn ↑ t∗ . Then

tn xtn − xtm x˙ t dt C(tn − tm ) → 0 , tm

as n m → ∞. Let x∗ be the limit. If the IFT is applicable to (t∗ , x∗ ), then we have t∗ ∈ S, i.e., S is closed. Once (1) and (2) are proved, S is a nonempty (0 ∈ S) open and closed set. Therefore S = [0, 1], and then the equation F (1, ·) = θ is solvable. As an application of the continuity method, we have: Theorem 1.2.9 (Global implicit function theorem) Let X, Y be Banach spaces, and let f ∈ C 1 (X, Y ) with f (x)−1 ∈ L(Y, X) ∀x ∈ X. If ∃ constants A, B > 0 such that f (x)−1 A x +B

∀x ∈ X ,

then f is a diﬀeomorphism. Proof. (1) Surjective. We want to prove that ∀y ∈ Y, ∃x ∈ X satisfying f (x) = y . ∀x0 ∈ X, deﬁne F : [0, 1] × X → Y as follows: F (t, x) = f (x) − [(1 − t)f (x0 ) + ty] . Set S = {t ∈ [0, 1]| F (t, ·) = θ is solvable}. Obviously, 0 ∈ S, and since Fx−1 (t, x) = f (x)−1 ∈ L(Y, X), S is open, from the IFT.

1.2 Implicit Function Theorem and Continuity Method

25

It remains to prove the closeness of S. Indeed, in a component (a, b) of S, there is a branch of solutions xt satisfying F (t, xt ) = θ and then

∀t ∈ (a, b) ,

f (xt )x˙ t = y − f (x0 ) .

Thus, x˙ t f (xt )−1 y − f (x0 ) (A xt +B) y − f (x0 ) . Set c =

a+b 2 ,

(1.19)

so

t

xt xc +

y − f (x0 ) (A xs +B)ds

as t > c .

c

Applying Gronwall’s inequality, ∃ a constant C > 0 such that xt C .

(1.20)

Substituting (1.20) into (1.19), we have another constant C1 > 0 such that x˙ t C1 , ∀t ∈ (a, b) . This proves the closeness of S, therefore f is surjective. (2) Injective. We argue by contradiction. If ∃y ∈ Y and x0 , x1 ∈ X, satisfying f (xi ) = y, i = 0, 1. Let γ : [0, 1] → X be the segment connecting these two points: γ(s) = (1 − s)x0 + sx1 s ∈ [0, 1] . Thus f ◦γ is a loop passing through y. If we could ﬁnd x : [0, 1] → X satisfying x(i) = xi i = 0, 1,

and

f ◦ x(s) = y ∀s ∈ [0, 1] , then this would contradict with the locally homeomorphism of f . Deﬁne I = [0, 1] and T : I × C0 (I, X) → C0 (I, Y ) as follows: (t, u(s)) → f (γ(s) + u(s)) − ty − (1 − t)f (γ(s)) , where C0 (I, X) = {u ∈ C(I, X)| u(0) = u(1) = θ} . We want to solve T (t, u) = θ . Obviously, T (0, θ) = θ; and if we have u ∈ C0 (I, X) satisfying T (1, u(·)) = θ, then x(s) = u(s) + γ(s) is what we need. Now,

26

1 Linearization

f

r

y

x1 x0

r Fig. 1.2.

1.

Tu (t, u) = f (γ(·) + u(·)) ∈ L(C0 (I, X), C0 (I, Y )) , which has a bounded inverse. Therefore S = {t ∈ [0, 1]| T (t, u) = θ is solvable} is open, from the IFT.

2. Let ut (s) be a solution at t ∈ S. Then f (γ(s) + ut (s)) · u˙ t (s) = y − f ◦ γ(s) , where u˙ t denotes the derivative with respect to t. Again we obtain u˙ t C0 (I,X) (A ut C0 (I,X) +B1 ) y − f ◦ γ C0 (I,Y ) , where B1 > 0 is another constant depending on B and x0 , x1 only. As in paragraph (1), ∃C > 0 such that u˙ t C0 (I,X) C

∀t ∈ S .

Again, by the continuity method, 1 ∈ S. This is the contradiction. The injectivity of f is proved. Next we shall give an example showing how a priori estimates give the closeness of S. Theorem 1.2.10 Suppose f ∈ C 1 (Ω × R1 × Rn , R1 ), where Ω ⊂ Rn is a bounded domain of smooth boundary. Assume ∃ constants C > 0, satisfying 1 1 → R+ such that (1) There exists an increasing function c : R+

|f (x, η, ξ)| ≤ c(|η|)(1 + |ξ|2 ), ∀ (x, η, ξ) ∈ Ω × R1 × Rn . (2) ∂f (x, η, ξ) 0 . ∂η

1.2 Implicit Function Theorem and Continuity Method

27

(3) Assume ∃M > 0 such that < 0, if η > M f (x, η, θ) = > 0, if η < −M . Assume φ ∈ C 2,γ , for some γ ∈ (0, 1). Then the equation −u = f (x, u(x), ∇u(x)) x ∈ Ω u|∂Ω = φ

(1.21)

possesses a unique solution in C 2,γ . Lemma 1.2.11 Under the assumption (3), if u ∈ C 2 (Ω) is a solution of (1.21), then u C(Ω) max max |φ(x)|, M . ∂Ω

Proof. Assume that |u(x)| attains its maximum at x0 ∈ Ω. We divide our discussion into two cases. (1) x0 ∈ ∂Ω; the proof is done. ◦

(2) x0 ∈ Ω, then ∇u(x0 ) = 0 and −u(x0 ) = f (x0 , u(x0 ), θ). If u(x0 ) > M , then LHS 0, but RHS < 0. It is impossible. Similarly, if u(x0 ) < −M , then LHS 0, but RHS > 0. Again, it is impossible. We have |u(x0 )| M . Lemma 1.2.12 Assume a ∈ C 0,γ (Ω) and φ ∈ C 2,γ (∂Ω). Then the equation −u + u = a(x)(1 + |∇u|2 ) (1.22) u|∂Ω = φ has a unique solution u ∈ C 2,γ (Ω), and u C 2,γ C( a C 0,γ , φ C 2,γ ) . Proof. We apply the continuity method to study equation (1.22). Deﬁne a map F : X × [0, 1] → Y as follows: (u, τ ) → (−u + u − a(τ + |∇u|2 ), u|∂Ω − τ φ) , where X = C 2,γ (Ω), Y = C 0,γ (Ω) × C 2,γ (∂Ω) and γ ∈ (0, γ). Noticing that Fu (u, τ )v = ((−v + v − a∇u · ∇v), v|∂Ω ) ,

28

1 Linearization

and from the Schauder estimates, ∀(u, τ ) ∈ X × [0, 1], Fu (u, τ ) has a bounded inverse. Deﬁne the set: S = {τ ∈ [0, 1]| ∃uτ solving F (uτ , τ ) = θ} . Equation (1.22) is solvable if 1 ∈ S. Since S is open, and 0 ∈ S, it remains to verify that the set S is closed. To this end, it is suﬃcient to prove that there is a constant C, depending on a and φ such that ∀uτ ∈ S, uτ C 2,γ C .

(1.23)

τ Since uτ satisﬁes F (uτ , τ ) = θ, u˙ τ = du dτ exists, from the IFT, and satisﬁes: −u˙ τ + u˙ τ = a + 2a∇uτ · ∇u˙ τ , (1.24) u˙ τ |∂Ω = φ .

Set g = a + 2a∇uτ · ∇u˙ τ ; we have u˙ τ W 2,p C(p)(1+ g Lp ) C(p, a C )(1+ ∇uτ L2p ∇u˙ τ L2p ) , provided by the Lp estimates. Since u˙ τ satisﬁes (1.24), from Lemma 1.2.11, u˙ τ C(Ω) is bounded by a constant depending on a C(Ω) and φ, and from the Gagliardo–Nirenberg inequality 1

1

∇u˙ τ L2p Cp ∇2 u˙ τ L2 p u˙ τ L2 p +C u˙ τ L∞ , we obtain: u˙ τ W 2,p C(1+ ∇uτ 2L2p ) . Again, by the Gagliardo–Nirenberg inequality, we have 1

1

∇uτ L2p Cp ∇2 uτ L2 p uτ L2 p +C uτ L∞ . Repeating the use of Lemma 1.2.11, uτ C is bounded by a constant depending on φ C , and we obtain u˙ τ W 2,p C( φ C 2,γ , a C , p)(1+ uτ W 2,p ) . From

d uτ W 2,p u˙ τ W 2,p , dτ and the Gronwall inequality, we obtain uτ W 2,p CeCτ , where C depends on p, φC 2,γ and aC .

1.2 Implicit Function Theorem and Continuity Method

As a consequence of the Sobolev embedding theorem, for p > uτ C 1,γ C( φ C 2,γ , a C , γ) .

n 1−γ ,

29

we have (1.25)

Substituting into the equation F (uτ , τ ) = θ, we apply the Schauder estimate: uτ C 2,γ C( φ C 2,γ , a C 0,γ , γ) . The continuity method is applicable; we have a solution u of (1.22). Indeed, the solution is unique. Let u1 , u2 be two solutions and set ω = u1 − u2 , then −ω + ω = a∇(u1 + u2 ) · ∇ω in Ω , ω|∂Ω = 0 . ◦

If max ω > 0, then ∃x0 ∈ Ω such that maxΩ ω = ω(x0 ). Thus, ∇ω(x0 ) = 0, −ω(x0 ) 0 and ω(x0 ) > 0. This is impossible. Similarly, one proves that min ω < 0 is impossible. Therefore ω ≡ 0. Now we come back to the proof of the theorem. Proof. Applying the continuity method, we study the equation: −u = tf (x, u(x), ∇u(x)) t ∈ [0, 1] u|∂Ω = tφ

(1.26)

and turn to considering the operator: F : I × C 2,σ (Ω) → C σ (Ω) × C 2,σ (∂Ω),

I = [0, 1] ,

(t, u) → (−u − tf (x, u(x), ∇u(x)), u|∂Ω − tφ) , where σ ∈ (0, γ). We want to solve F (1, u) = θ. However, ∀u ∈ C 2,σ (Ω), Fu (t, u)v = (−v − tfη (x, u(x), ∇u(x))v − tfξ (x, u(x), ∇u(x))∇v, v|∂Ω ) . By assumption (2) and the maximum principle for linear elliptic equations, ∀g ∈ C σ (Ω) × C 2,σ (∂Ω) Fu (t, u)v = g has a unique solution, i.e., Fu (t, u) : C 2,σ ∩ C0 (Ω) → C σ (Ω) × C 2,σ (∂Ω) has a bounded inverse. Thus the set S = {t ∈ I| F (t, u) = θ is solvable} is open, from the IFT. We prove that S is closed. Noting that if ut satisﬁes F (t, ut ) = θ ,

30

1 Linearization

then we set at (x) =

tf (x, ut (x), ∇ut (x)) + ut (x) . 1 + |∇ut (x)|2

By assumptions (1) and (3) and Lemma 1.2.11, at C ≤ C(ut C ) ≤ C(M, φC ) . Since ut satisﬁes (1.22), in which a is replaced by at , and φ by tφ. According to equation (1.25), ut C 1,γ ≤ C(φC 2,γ , M, γ) , and then at C 0,γ ≤ C(φC 2,γ , M, γ, f ) . Again by Lemma 1.2.12, we have uC 2,γ ≤ C(φC 2,γ , M, γ, f ) . This is the required estimate. Since the embedding C 2,γ (Ω) → C 2,σ (Ω) × C 2,σ (∂Ω) is compact, we conclude that S is closed, as we have seen previously. The existence of the solution for F (1, u) = θ follows from the continuity method. The uniqueness is a consequence of the maximum principle. Once we obtain a solution u in C 2,σ , it follows directly by Schauder esti mates that u ∈ C 2,γ Remark 1.2.13 We shall return to this example by dropping assumption (2) in Chap. 3.

1.3 Lyapunov–Schmidt Reduction and Bifurcation 1.3.1 Bifurcation We often meet an equation with a parameter λ: F (x, λ) = 0 . The following phenomenon has been observed: a branch of solutions x(λ) depending on λ, is either disappeared or split into several branches, as λ attains some critical values. This kind of phenomenon is called bifurcation. For example, a simple algebraic equation: x3 − λx = 0 λ ∈ R1 ,

1.3 Lyapunov–Schmidt Reduction and Bifurcation

31

x

λ

0

Fig. 1.3.

has a solution x = 0 ∀λ ∈ R1 . As λ 0, this is the unique solution; but as λ > 0, we have two more branches of solutions √ x=± λ. See Figure 1.3. Bifurcation phenomena occur extensively in nature. Early in 1744, Euler observed the bending of a rod pressed along the direction of its axis. Let θ be the angle between the real axis and the tangent of the central line of the rod, and let λ be the pressure. The length of the rod is normalized to be π. We obtain the following diﬀerential equation with the two free end point conditions: θ¨ + λ sin θ = 0 , ˙ ˙ θ(0) = θ(π) =0. Obviously, θ ≡ 0 is always a solution of the ODE. Actually the solution is unique, if λ is not large. As λ increasingly passes through a certain value λ0 , it is shown by experiment that there exists a bending solution θ = 0. The same phenomenon occurs in the bending of plates, shells etc. In addition, bifurcation occurs in the study of thermodynamics (B´ernard problem), rotation of ﬂuids, solitary waves, superconductivity and lasers. Mathematically, we describe the bifurcation by the following: Deﬁnition 1.3.1 Let X, Y be Banach spaces, and let ∧ be a topological space. Suppose that F : X × ∧ → Y is a continuous map. ∀λ ∈ ∧, let Sλ = {x ∈ X| F (x, λ) = θ} be the solution set of the equation F (x, λ) = θ, where λ is a parameter (Fig. 1.4). Assume θ ∈ Sλ , ∀λ ∈ ∧. We call (θ, λ0 ) a bifurcation point, if for any neighborhood U of (θ, λ0 ), there exists (x, λ) ∈ U with x ∈ Sλ \{θ}.

32

1 Linearization

θ λ

λ Fig. 1.4.

The following problems are of primary concern: (1) What is the necessary and suﬃcient condition for a bifurcation point (θ, λ0 )? (2) What is the structure of Sλ near λ = λ0 ? (3) How do we compute the solutions near the bifurcation points? (4) How about the global structure of ∪λ∈∧ Sλ ? (5) Let F (x, λ) = θ be the steady equation of the evolution equation: x˙ = F (x, λ) , we study the stability of solutions in Sλ as λ approaches λ0 . In this section, we focus our discussions on problems (1) and (2). (4) will be studied in Chap. 3. For simplicity, we assume that U ⊂ X is an open neighborhood of the origin θ of the Banach space X, and that F : U × ∧ → Y is continuous, and satisﬁes F (θ, λ) = θ ∀λ ∈ ∧ . What is the necessary condition for a bifurcation point λ0 ∈ ∧? (1) Assume that Fx (x, λ) is continuous. If (θ, λ0 ) is a bifurcation point, then Fx (θ, λ0 ) does not have a bounded inverse. Proof. By the IFT directly.

(2) Assume F (x, λ) = Lx − λx + N (x, λ) , where L ∈ L(X, Y ), λ ∈ R1 , and that N : U × R1 → X is continuous with N (x, λ) = ◦( x ) as x → θ uniformly for λ in a neighborhood of λ0 . If (θ, λ0 ) is a bifurcation point, then λ0 ∈ σ(L), i.e., λ0 is a spectrum of L. Proof. If not, λ0 ∈ ρ(L), the resolvent set of L. Since ρ(L) is open, ∃ε > 0 and Cε > 0 such that (L − λI)−1 Cε as |λ − λ0 | < ε . It follows that

1.3 Lyapunov–Schmidt Reduction and Bifurcation

33

x (L − λI)−1 N (x, λ) = ◦( x ) if |λ − λε | < ε, and x → θ, as x ∈ Sλ . Thus ∃δ > 0, such that Bδ × (λ0 − ε, λ0 + ε) ∩ Sλ = {(θ, λ)| |λ − λ0 | < ε} , i.e., (θ, λ0 ) is not a bifurcation point.

(3) The above condition is not suﬃcient. For example, let X = R2 , let u x= , v and let

3 u u −v . F (x, λ) = −λ + u3 v v

Obviously, Fx (θ, λ) = (1 − λ)Id. λ = 1 is in the spectrum, but (θ, λ) is not a bifurcation point, because: F (x, λ) = θ ⇔ u4 + v 4 = 0, i.e., x = θ . In order to study the suﬃcient condition and the local behavior of the solution set near its bifurcation points, we introduce the following: 1.3.2 Lyapunov–Schmidt Reduction Let X, Y be Banach spaces, and let ∧ be a topological space. Assume that F : U × ∧ → Y is continuous, where U ⊂ X is a neighborhood of θ. We assume that Fx (θ, λ0 ) is a Fredholm operator, i.e., (1) Im Fx (θ, λ0 ) is closed in Y , (2) d = dim ker Fx (θ, λ0 ) < ∞, (3) d∗ = codim Im Fx (θ, λ0 ) < ∞. Set X1 = ker Fx (θ, λ0 ), Y1 = Im Fx (θ, λ0 ) . Since both dim X1 , and codim Y1 are ﬁnite, we have the direct sum decompositions: X = X1 ⊕ X2 , Y = Y1 ⊕ Y2 , and the projection operator P : Y → Y1 . ∀x ∈ X, there exists a unique decomposition: x = x1 + x2 , xi ∈ Xi , i = 1, 2 .

Thus F (x, λ) = θ ⇔

P F (x1 + x2 , λ) = θ , (I − P )F (x1 + x2 , λ) = θ .

34

1 Linearization

Now, P Fx (θ, λ0 ) : X2 → Y1 is a surjection as well as an injection. According to the Banach theorem, it has a bounded inverse. If we already have F (θ, λ0 ) = θ, then from the IFT, we have a unique solution u : V1 × V → V2 satisfying P F (x1 + u(x1 , λ), λ) = θ , where Vi is a neighborhood of θ in U ∩ Xi , i = 1, 2, and V is a neighborhood of λ0 . It remains to solve the equation: (I − P )F (x1 + u(x1 , λ), λ) = θ on V1 × V . This is a nonlinear system of d variables and d∗ equations. The above procedure is called the Lyapunov–Schmidt reduction. It reduces an inﬁnite-dimensional problem to a ﬁnite-dimensional system. Many applied mathematicians have been doing the same reduction in their concrete problems in their own language. Before going further, let us study the simple properties of the solution x2 = u(x1 , λ). Lemma 1.3.2 Under the assumptions (1), (2), (3) of the Lyapunov–Schmidt reduction, if F ∈ C p (U × ∧, Y ), p 1, satisﬁes F (θ, λ) = θ, where ∧ is again a Banach space, then we have u(θ, λ) = θ,

u (θ, λ0 ) = θ . If p = 1, then u(x1 , λ) = ◦( x1 +|λ − λ0 |) ; and if p = 2, then u(x1 , λ) = O( x1 2 +|λ − λ0 |2 ) . Proof. According to the IFT, the solution in V1 × V → V2 is unique. Since F (θ, λ) = θ, we have u(θ, λ) = θ. Again, by the IFT, u (θ, λ0 )(x1 , λ) = −(P Fx (θ, λ0 ))−1 (P Fx (θ, λ0 )x1 + P Fλ (θ, λ0 )λ) ∀(x1 , λ) ∈ X1 × ∧. From F (θ, λ) = θ, it follows that Fλ (θ, λ) = 0. Therefore u (θ, λ0 ) = θ , provided x1 ∈ X = ker Fx (θ, λ0 ) . The last two conclusions follow from Taylor’s formula.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

35

Next, we turn to the case d = d∗ = 1. Theorem 1.3.3 (Crandall–Rabinowitz) Suppose that U ⊂ X is an open neighborhood of θ, and that F ∈ C 2 (U × R1 , Y ) satisﬁes F (θ, λ) = θ. If Fx (θ, λ0 ) is a Fredholm operator with d = d∗ = 1, and if Fxλ (θ, λ0 )u0 ∈ / Im Fx (θ, λ0 )

(1.27)

for all u0 ∈ ker Fx (θ, λ0 )\{θ}, then (θ, λ0 ) is a bifurcation point, and there exists a unique C 1 curve (λ, ψ) : (−δ, δ) → R1 × Z satisfying F (su0 + ψ(s), λ(s)) = θ λ(0) = λ0 , ψ(0) = ψ (0) = θ , where δ > 0, and Z is the complement space of span {u0 } in X. Furthermore, there is a neighborhood of (θ, λ0 ), in which F −1 (θ) = {(θ, λ)| λ ∈ R1 } ∪ {(su0 + ψ(s), λ(s))| |s| < δ} . Proof. Decompose the spaces X and Y according to the Lyapunov–Schmidt reduction, and write down the reduction equation. By assumptions, ker Fx (θ, λ0 ) = span{u0 } , and ∃φ∗ ∈ Y ∗ \{θ}, such that ker φ∗ = ImFx (θ, λ0 ), from the Hahn–Banach theorem. The reduction equation reads as: g(s, λ) = φ∗ , F (su0 + u(su0 , λ), λ) = 0 .

(1.28)

We have the trivial solution s = 0, and we look for a nontrivial solution. Noticing gs (0, λ0 ) = φ∗ , Fx (θ, λ0 )(u0 + us (θ, λ0 )) , and u0 ∈ ker Fx (θ, λ0 ), by applying Lemma 1.3.2, we obtain gs (0, λ0 ) = 0. Therefore, it is impossible to get a solution s = s(λ) directly from the IFT. However, we may consider λ as a function of s, in this case, the only diﬃculty is that g(0, λ) = 0 for all λ. Let us introduce a new function: 1 g(s, λ) as s = 0 h(s, λ) = s as s = 0 . gs (0, λ) When s = 0, the solutions of h(s, λ) = 0 are the same as (1.28). Here we deﬁne gs (0, λ) to be the value of h at s = 0, in order to make h ∈ C 1 . We assume this conclusion at this moment, and postpone the veriﬁcation to the end. Since h(0, λ0 ) = gs (0, λ0 ) = 0 , and

36

1 Linearization hλ (0, λ0 ) = gsλ (0, λ0 )

= φ∗ , Fxλ (θ, λ0 )(u0 + us (θ, λ0 )) + Fx (θ, λ0 )usλ (θ, λ0 )

= φ∗ , Fxλ (θ, λ0 )u0

= 0 . Again, the IFT is applied, and we obtain a C 1 -curve λ = λ(s), |s| < δ, satisfying h(s, λ(s)) = 0 , λ(0) = λ0 . Set ψ(s) = u(su0 , λ(s)) ; we have ψ(0) = u(0, λ0 ) = θ , ψ (0) = ∇u(θ, λ0 )(u0 , λ (0)) = θ , and

g(s, λ(s)) = φ∗ , F (su0 + u(su0 , λ(s)), λ(s)) = 0 ;

i.e., F (su0 + ψ(s), λ(s)) = θ . This is what we need. Now we turn to verifying h ∈ C 1 (Bη ) for some η > 0, where Bη = {(s, λ) ∈ 2 R | |s|2 + |λ − λ0 |2 < η 2 }. Indeed, only want to verify that h is C 1 at s = 0. By deﬁnition, 1 lim g(s, λ) = gs (0, λ) , s→0 s which implies the continuity of h. Moreover, 1 hs (0, λ) = lim [h(s, λ) − h(0, λ)] s→0 s 1 = lim 2 [g(s, λ) − g(0, λ) − gs (0, λ)s] s→0 s 1 = gss (0, λ) , 2 therefore hs (s, λ)

−

hs (0, λ)

1 1 2 = 2 gs (s, λ)s − g(s, λ) − gss (0, λ)s s 2 = ◦(1) as |s| → 0 .

Since

hλ (0, λ) = gsλ (0, λ) , 1 (0, λ) → 0 as s → 0 . hλ (s, λ) − hλ (0, λ) = gλ (s, λ) − gsλ s The proof is complete.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

37

Z

( su0 +ψ(s), λ (s) ) λ0

u0

λ

Span{ u0} Fig. 1.5.

We present two simple applications. Example 1. (Euler elastic rod) We study the bending problem raised at the beginning of this section: ϕ¨ + λ sin ϕ = 0 in (0, π) , ϕ(0) ˙ = ϕ(π) ˙ =0. Let ˙ = u(π) ˙ = 0)} , X = {u ∈ C 2 [0, π]| u(0) Y = C[0, π] ; and let F : X × R1 → Y be the map: (u, λ) → u + λ sin u . It is a continuous map satisfying F (θ, λ) = θ. According to the necessary condition, if (θ, λ) is a bifurcation point, then λ is in the spectrum of the d 2 ) + λI, i.e., linearized operator ( dt λ = n2 , for some n = 1, 2, . . . d 2 ) + n2 I) = {s cos nt| s ∈ R1 }, and since the diﬀerential Since ker (( dt d 2 operator ( dt ) under the free end point condition is self-adjoint, we have

coker Fu (θ, n2 ) = {s cos nt| s ∈ R1 } . In addition thus

(θ, n2 ) = cos nu|u=θ = I , Fuλ (θ, n2 ) cos nt = cos nt ∈ / Im Fu (θ, n2 ) . Fuλ

38

1 Linearization

All the assumptions in the Crandall–Rabinowitz theorem are satisﬁed, and we obtain a family of C 1 curves (λn (s), ψn (s)) : (−δ, δ) → R1 × Zn , where Zn is the complement of span{cos nt}, satisfying λn (0) = n2 , d ds ψn (0)

= ψn (0) = θ ,

for n = 1, 2, 3, . . . If we set ϕn (s, t) = s cos nt + (ψn (s))(t)

t ∈ [0, π] ,

then

∂ ∂t

2 ϕn (s, t) + λn (s) sin ϕn (s, t) = 0 0 < t < π, |s| < δ , ∂ ∂ ϕn (s, 0) = ϕn (s, π) = 0 . ∂t ∂t

We obtain the bifurcation diagram of Fig. 1.6.

1

4

9

Fig. 1.6.

Example 2. We consider the following elliptic BVP. Let Ω ⊂ Rn be a bounded open domain with smooth boundary ∂Ω, and p ∈ (1, ∞): −u − λu = |u|p−1 u in Ω , u|∂Ω = 0 . Set X = C 2,γ ∩ C0 (Ω), and Y = C γ (Ω) for some γ ∈ (0, 1) , F : (u, λ) → −u − λu − |u|p−1 u , we have Fu (θ, λ) = − − λI . Thus, for (θ, λ) being a bifurcation point, −λ has to be in the spectrum of the Laplacian under the 0-Dirichlet data.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

39

Let us ﬁrst consider the ﬁrst eigenvalue λ1 , which is simple, i.e., ker − − λI is one dimensional; let ϕ1 be the associate eigenfunction. By the same reasoning, coker Fu (θ, λ1 ) is also one dimensional, and Fuλ (θ, λ1 )ϕ1 = −ϕ1 ∈ / Im Fu (θ, λ1 ) . Again, we apply the Crandall–Rabinowitz theorem, and conclude that (θ, λ1 ) is a bifurcation point, and in its neighborhood, the solution set is the C 1 curve (−δ, δ) → (sϕ1 (x) + ψ(s)(x), λ1 (s)) plus the trivial solution set (θ, λ), where ψ(s) ∈ Z, and Z is the complement of span{ϕ1 } in C 2,γ ∩ C0 (Ω). We have not studied eigenvalues other than λ1 , because we do not know if they are simple, i.e., if the condition d = 1 is satisﬁed. In fact, if λ is a simple eigenvalue, then we have the similar result. For some special cases, in which d = 1, we can extend Theorem 1.3.3 as follows: Theorem 1.3.4 Let ∧, X and Y be Banach spaces. Suppose that F ∈ C 2 (X × ∧, Y ) has the form F (x, λ) = L(λ)x + P (x, λ) , where L(λ) ∈ L(X, Y ) ∀λ ∈ ∧, and P (θ, λ) = θ, Px (θ, λ) = θ, Pxλ (θ, λ0 ) = θ for some λ0 ∈ ∧ . If there exist u0 ∈ ker L(λ0 )\{θ} and a closed linear subspace Z ⊂ X such that (z, λ) → L(λ0 )z + λL (λ0 )u0 : Z × ∧ → Y is a linear homeomorphism, then there exist a neighborhood U of (θ, λ0 ) in (span{u0 } × Z) × ∧, δ > 0, and a C 1 map: (−δ, δ) → Z × ∧, deﬁned by s → (ϕ(s), λ(s)) satisfying F −1 (θ) ∩ U \{(θ, λ)| λ ∈ ∧} = {(s(u0 + ϕ(s)), λ(s))| |s| < δ} , and (ϕ(0), λ(0)) = (θ, λ0 ) .

Proof. Similar to Theorem 1.3.3, we deﬁne 1 F (s(u0 + z), λ) Φ(s, z, λ) = s Fx (θ, λ)(u0 + z)

as s = 0 as s = 0 .

One wants to verify that Φ ∈ C 1 ((R1 × Z) × ∧, Y ). It is suﬃcient to verify the continuous diﬀerentiability at s = 0. Since Φ(s, z, λ) − Φ(0, z, λ) = s−1 [F (s(u0 + z), λ) − F (θ, λ) − Fx (θ, λ)(s(u0 + z))]

1 1 Fxx (rts(u0 + z), λ)tdtdr · (u0 + z)2 , =s 0

0

40

1 Linearization

we have Φs (0, z, λ) =

1 Fxx (θ, λ)(u0 + z)2 . 2

Furthermore as s → 0, Φs (s, z, λ) − Φs (0, z, λ) s2 −2 2 = −s F (s(u0 + z), λ) − sFx (θ, λ)(u0 + z) − Fxx (θ, λ)(u0 + z) 2 +s−1 [Fx (s(u0 + z), λ) − Fx (θ, λ)](u0 + z) = o(1) , and Φλ (s, z, λ) − Φλ (0, z, λ) = s−1 [Fλ (s(u0 + z), λ) − Fλ (θ, λ) − Fxλ (θ, λ)s(u0 , z)] = o(1) . Thus, Φ ∈ C 1 . Now, we have Φ(0, z, λ) = L(λ)(u0 + z) + Px (θ, λ)(u0 + z) , Φ(0, θ, λ0 ) = L(λ0 )u0 + Px (θ, λ0 )u0 = θ , and Φ(z,λ) (0, θ, λ0 )(z, λ) = L(λ0 )z + L (λ0 )u0 · λ + Pλx (θ, λ0 )u0 + Px (θ, λ0 )z , = L(λ0 )z + λL (λ0 )u0 , ∀(z, λ) ∈ Z × ∧. By the assumption, the last linear operator is a homeomorphism; then the IFT is applied. Therefore ∃ a neighborhood U of (θ, λ0 ), and a unique C 1 -curve: s → (ϕ(s), λ(s)) ∈ Z × ∧ ∀ |s| < ε, satisfying (ϕ(0), λ(0)) = (θ, λ0 ) , Φ(s, ϕ(s), λ(s)) = θ, i.e., F (s(u0 + ϕ(s)), λ(s)) = θ .

The proof is complete. The above theorem is applied to Hopf bifurcations in ODE.

Example 3. (Hopf bifurcation) We study the periodic solutions for a linear ordinary diﬀerential system: x˙ = Ax , where x ∈ C 1 ([0, 2π], Rn ) and A ∈ M (n, R), the n × n matrix. Let B 0 A= , 0 C

(1.29)

1.3 Lyapunov–Schmidt Reduction and Bifurcation

where B=

0 1 −1 0

41

, and C ∈ M (n − 2, R) .

We assume that (e2πC − I) is invertible. Obviously the linear system (1.29) admits a family of periodic solutions with two real parameters a and b: ⎞ ⎛ a cos t + b sin t ⎜ −a sin t + b cos t ⎟ ⎟ ⎜ ⎟ ⎜ 0 x(t) = ⎜ ⎟ , ∀a, b ∈ R1 . ⎟ ⎜ .. ⎠ ⎝ . 0 Now, we perturb A by introducing a parameter µ ∈ (−1, 1), and consider the nonlinear diﬀerential system x˙ = A(µ)x + P (x, µ) ,

where A(µ) =

B(µ) 0 0 C(µ)

, B(µ) =

µ −β(µ)

(1.30) β(µ) µ

,

and C(µ) ∈ M (n − 2, R), satisfying e2πC(µ) − I ∈ GL(n − 2, R). We assume that P : Rn × (−1, 1) → Rn satisﬁes P (θ, µ) = Px (θ, 0) = Px,µ (θ, 0) = θ . and that β : (−1, 1) → R1 , satisﬁes β(0) = 1, and β (0) = 0 . Obviously, ∀µ ∈ (−1, 1), x = θ is always a solution. We are interested in nontrivial solutions bifurcating from the branch of trivial solutions. Theorem 1.3.5 Suppose that A(µ) and P (x, µ) are C 2 functions satisfying the above assumptions. Then ∃ positive constants a0 , δ0 and a C 1 map (µ, ω, x) : (−a0 , a0 ) → R1 × R1 × C 1 (R1 , Rn ) , satisfying µ(0) = 0, ω(0) = 1 , where x(a) is a 2πω(a)-periodic function, which satisﬁes (1.30), and is of the form:

42

1 Linearization

⎛

⎞ a sin ω(a)−1 t ⎜ a cos ω(a)−1 t ⎟ ⎜ ⎟ ⎜ ⎟ 0 (x(a))(t) = ⎜ ⎟ + ◦(|a|) . ⎜ ⎟ .. ⎝ ⎠ . 0 Furthermore, every 2πω(a)-periodic solution y(t) of (1.30), satisfying |y(t)| < δ0 ∀t, coincides with x(a)(t) modulating a phase shift when |µ| < α0 and |ω − 1| < δ0 . Proof. Since ω depends on µ, we introduce a new scale τ , and let t = ωτ ; (1.30) is rewritten as dx = ωA(µ)x + ωP (x, µ) . dτ

(1.31)

We ﬁnd the 2π-periodic solution of (1.31). Set ∧ = R2 , λ = (ω, µ) ∈ ∧ , 1 X = C2π (Rn ) = {u ∈ C 1 (R, Rn )| u is 2π periodic} , Y = C2π (Rn ) = {u ∈ C(R, Rn )| u is 2π periodic} ,

and set dx − ωA(µ)x , dτ P(x, λ) = ωP (x, µ) ,

L(λ)x =

λ0 = (1, 0) . Let

⎞ ⎞ ⎛ sin τ cos τ ⎜ cos τ ⎟ ⎜ − sin τ ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 0 ⎟ ⎜ u0 = ⎜ ⎟ , and u1 = ⎜ 0 ⎟ ; ⎜ .. ⎟ ⎜ .. ⎟ ⎝ . ⎠ ⎝ . ⎠ 0 0 ⎛

then we have ker L(λ0 ) = span{u0 , u1 } . Deﬁne Z=

z ∈ X|

2π

z(t)uj (t)dt = 0, j = 0, 1 .

0

It remains to verify the following conditions: ∀y ∈ Y , the linear equation L(λ0 )z + λL (λ0 )u0 dz − A(0)z − ωA(0)u0 − µA (0)u0 = dτ =y has a unique solution (ω, µ, z) ∈ R2 × Z.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

43

X

µ (1,0, θ) ω Fig. 1.7. Nontrivial solutions

According to the Fredholm alternative, this is equivalent to saying that ∀y ∈ Y, ∃|(ω, µ) ∈ R2 such that y + ωA(0)u0 + µA (0)u0 ∈ ker L∗ (λ0 )⊥ . However, ker L∗ (λ0 ) = ker L(λ0 ); this is again equivalent to 2π 2π 0 (A(0)u0 (t))u0 (t)dt 0 (A (0)u0 (t))u0 (t)dt = 0 . det 2π (A(0)u0 (t))u1 (t)dt 2π (A (0)u0 (t))u1 (t)dt 0 0 Computing the determinant directly, it equals −4π 2 β (0). Thus all assumptions in Theorem 1.3.4 are satisﬁed. We obtain our conclusion. This kind of bifurcation phenomenon is shown in Fig. 1.7. 1.3.3 A Perturbation Problem We present here an example of perturbation problems. This is a nonlinear Schr¨ odinger equation (NSE); we look for a nonspreading wave packet solution. Assume that the potential V ∈ C 2 (R1 ) is bounded, and the standing wave ψ = ψ(t, x) satisﬁes the following NSE: ih

h2 ∂ 2 ψ ∂ψ =− + V (x)ψ − ψ 3 . ∂t 2 ∂x2

where h > 0 is a small constant, ψ(t, x) = exp(−iλt/h)ϕ(x) with ϕ(x) → 0 as |x| → ∞, and λ < inf V . It is reduced to the following equation: −

h2 ϕ (x) + V (x)ϕ(x) = λϕ(x) + ϕ(x)3 . 2

If V has a nondegenerate critical point x0 , i.e., V (x0 ) = 0, V (x0 ) = 0, 0 we make a change of variables y = x−x h , and the equation is reduced to

44

1 Linearization

1 − ϕ (y) + (Vh (y) − λ)ϕ(y) = ϕ(y)3 , 2

(1.32)

where Vh (y) = V (x0 + hy) and ϕ(y) → 0, as |y| → ∞. This is a perturbation problem, because h > 0 is small. The special case where h = 0 reads as: 1 (1.33) − ϕ + Eϕ = ϕ3 , 2 where E = V (x0 ) − λ, and ϕ(y) → 0 as |y| → ∞. The equation has a special solution: z(y) = α sech (αy), α = (2E)1/2 . Let τθ ϕ = ϕ(y − θ). Since equation (1.33) is autonomous, it has a onedimensional solution manifold Z = {zθ = τθ z| θ ∈ R1 }. It is easily check: Tz Z = span{e}, where e = z , and then Tzθ Z = span{τθ e}. We write eθ = τθ e. Let us deﬁne X = H 2 (R1 ), Y = L2 (R1 ), and 1 F (h, u) = − u + (Vh − λ)u − u3 . 2 Since V is bounded, F : R1 × X → Y . Thus, F (0, zθ ) = 0, ∀ θ ∈ R1 . Since Lθ := Fu (0, zθ ) = −

1 d2 + (E − 3zθ2 )·, with D(Lθ ) = X , 2 dt2

Lθ is a self-adjoint operator on Y . We verify that Lθ is a Fredholm operator: (1) Im (Lθ ) is closed. (2) Ker (Lθ ) = Coker(Lθ ) = Tzθ Z. In fact, only (1) remains to be veriﬁed; it follows from: Lemma 1.3.6 Assume inf (V ) > λ. Then ∃γ > 0, such that Lθ vY ≥ γvX ∀v ∈ (Tzθ Z)⊥ . Proof. We prove it by contradiction. Suppose that ∃φn ∈ (Tzθ Z)⊥ such that φn X = 1, and Lθ φn Y → 0. Then φn φ in X. We claim that φ = 0. In fact, (1) by weakly convergence, φ ∈ (Tzθ Z)⊥ and (2) since X is dense in Y, ∀ψ ∈ X, Lθ φ, ψ = φ, Lθ ψ = lim φn , Lθ ψ = lim Lθ φn , ψ = 0 . Therefore, Lθ φ = 0. Combining (1) and (2), φ = 0. d2 By the assumption E > 0, let B = − 12 dt 2 + E; we have a constant c > 0 such that

1.3 Lyapunov–Schmidt Reduction and Bifurcation

45

BuY ≥ 2cuX . Noticing that |z(y)| ∼ exp(−α|y|) as |y| → ∞, and that φn 0 in X, in combination with the Sobolev embedding theorem, which implies that φn ∞ is bounded, and φn ⇒ 0 uniformly on any bounded interval, we obtain: Lθ φn Y = Bφn − 3zθ2 φn Y ≥ Bφn Y − 3zθ2 φn Y ≥ 2cφn X − c = c , as n is large. This is a contradiction.

X1 , Y = We have the following Lyapunov–Schmidt reduction: X = Tzθ Z Y1 , and let P : Y → Y1 be the orthogonal projection. The equation Tzθ Z F (h, u) = 0 is equivalent to the system: P F (h, zθ , zθ + ξ) = 0, F (h, zθ + ξ), eθ = 0 . From F (0, zθ ) = 0 and the IFT, for h > 0 small, ∃ a unique ξ = ξ(h, θ) solving the ﬁrst equation, P F (h, zθ + ξ(h, θ)) = 0, and satisfying ξ(h, θ) ≤ CF (h, zθ ), for some constant C > 0. It remains to solve: wh (θ) := F (h, zθ + ξ(h, θ)), eθ = 0 . Once θ = θ(h) is obtained, uh = ξ(h, θ(h)) + zθ(h) solves F (h, u) = 0. We write F (h, uh ) = (F (h, uh ) − F (0, uh )) + F (0, uh ) , and by the Taylor formula, F (0, uh ) = F (0, zθ ) + Lθ ξ(h, θ) − N (h, θ) , where N (h, θ) = 3zθ ξ 2 (h, θ) + ξ 3 (h, θ), and θ = θ(h). Since F (0, zθ ) = Lθ eθ = 0, we have | F (0, uh ), eθ | = | N (h, θ), eθ | ≤ Cξ(h, θ(h))2Y , and

F (h, uh ) − F (0, uh ), eθ | =

(V (x0 + hy) − V (x0 ))(zθ (h) + ξ(h, θ(h))eθ dy = I1 + I2 .

The following estimates hold: I1 = (Vh − V0 )zθ , eθ

= −h zθ , Vh zθ − zθ , (Vh − V0 )eθ

1 = − h zθ , Vh zθ

2

1 = − h Vh (y)|z(y − θ)|2 dy 2

1 = − h V (x0 + h(y + θ))|z(y)|2 dy , 2 I2 = | (Vh − V0 )ξ(h, θ), eθ | ≤ (Vh − V0 )eθ ξ(h, θ) ,

46

1 Linearization

and F (h, zθ )2Y = F (h, zθ ) − F (0, zθ )Y

= |Vh (y + θ) − V0 |2 |z(y)|2 dy

2 2 2 ≤ |Vh (y + θ) − V0 | |z(y)| dy + 4Max|V |

R1 \Bρ 2 −µρ

Bρ

|z(y)|2 dy

≤ Max|y|≤ρ |V (h(y + θ)) − V0 |2 + 4Max|V | e ≤ C1 (h(ρ + |θ|))4 + e−µρ

for every ρ > 0, and for some constant C1 > 0. Let us denote the right-hand side by Mh,θ,ρ . Similarly, we have (Vh − V0 )eθ 2 ≤ Mh,θ,ρ . Since z(y) is even,

1 I1 + 1 V (x0 )z2 hθ = 1 [V (x0 )hθ − V (x0 + h(y + θ))]|z(y)|2 dy h 2 2

1 = [V (x0 )h(y + θ) − V (x0 + h(y + θ))|z(y)|2 dy 2

≤ C2 |h(y + θ)|2 |z(y)|2 dy ≤ C2 (h(|θ| + ρ)|2 + e−µρ ) . 1 2 2 V (x0 )z s,

By rescaling θ = hs , v(s) = h1 wh (θ), v0 (s) = h−ν v(hν s), with ν ∈ (1, 2), we obtain

and vh (s) =

|vh (s) − v0 (s)| ≤ C3 (h−ν ((hν |s| + hρ)2 + e−µρ ) + h−1−ν ((hν |s| + hρ)4 + e−µρ )) . Choosing ρ = h−τ with τ > 0, for |s| ≤ 1, we obtain |vh (s) − v0 (s)| ≤ C3 (h−ν ((hν + h1−τ )2 + h−1−ν ((hν + h1−τ )4 −τ

+ (h−ν + h−1−ν )e−µh

)→0,

uniformly as h → 0. Since v0 (s) changes sign at s = 0, we have a zero of vh in [−1, 1] as h > 0 small. This proves the existence of a zero s0 ∈ [0, 1] of vh and then of wh (θ). Thus we have proved the following theorem: Theorem 1.3.7 Assume that V ∈ C 3 (R1 ) is bounded and that x0 is a nondegenerate critical point of V . If λ < inf V , then ∃h0 > 0 such that ∀h ∈ (0, h0 ) ∃ 0 a nonzero solution φ(x) = z( x−xh0 −s0 ) + ξ(h, s0 (h))( x−x h ) of the equation (1.32), with s0 ∈ [−hν , hν ] ν ∈ (1, 2), and ξ(h, s0 (h)) → 0, as h → 0. 0 This solution φ, which is close to z( x−x h ), becomes more concentrated about x0 as h → 0. It is called a nonspreading wave packet with width α = (2(V (x0 ) − λ))−1/2 .

1.3 Lyapunov–Schmidt Reduction and Bifurcation

47

1.3.4 Gluing Gluing is an important technique in nonlinear analysis. It is a method of joining two heteroclinic trajectories to form a new trajectory such that the end point of the ﬁrst is glued to the starting point of the second, and the new trajectory is closed to the union of the two. For instance, on a Riemannian manifold M , let f ∈ C 1 (M, R1 ). Given (x, y) ∈ K ×K, where K is the critical point set of f , i.e., K = {x ∈ M | f (x) = θ}, let M(x, y) be the space of all trajectories c ∈ H 1 (R1 , M ) satisfying c˙ = −∇f ◦ c,

c(−∞) = x, c(+∞) = y .

The importance of the gluing technique is in the study of the compactness of the totality of the trajectory spaces: {M(x, y) | (x, y) ∈ K × K}. It is proved that the trajectory manifolds M(x, y) are compact up to the ∞ topolexistence of sequences which converge to broken trajectories in the Cloc ogy. And the “gluing” means to map such broken trajectories equipped with an additional suitable parametrization into the appropriate trajectory space: M(x, z) × M(z, y) → M(x, y) . This technique is crucial in Floer homology theory (see Hofer and Zehnder [HZ 2], Floer [Fl 2,3], Taubes [Tau]). In order not to involve over specialized knowledge in that theory, it will suﬃce to introduce the technique by an example of a nonspreading wave packet, which we met in the previous subsection. We have known the existence of a nonspreading wave packet, if V has a single nondegenerate critical point. However, if V has several nondegenerate critical points x1 , x2 , . . . , xn , there are at least n nonspreading wave packets with widthes αj = (2(V (xj ) − λ))−1/2 , j = 1, 2, . . . , n. As h becomes small, these wave packets are separated. Are there multi-peak wave packets? More precisely, are there new solutions of equation (1.32), which are closed to some of these wave packets on their peak intervals simultaneously? For simplicity, we only consider the case n = 2; the result can be extended to any n. Assume that V has onlytwo nondegenerate critical points x± = ±R. Let λ < inf(V ). Deﬁne α± = 2(V (±R) − λ), z± (y) = α± sech (α± y), and ∀(θ+ , θ− ) ∈ R2 , z(θ+ , θ− ) = z+ (θ+ ) + z− (θ− ), where z± (θ± ) = τ±(R+θ± )/h z± . Again let 1 d2 u + (Vh − λ)u − u3 . F (h, u) = − 2 dy 2 We are looking for a solution of the form u = z(θ+ , θ− ) + φ ∈ X for the equation F (h, ·) = 0, in which φ is small as h > 0 is. Deﬁne a two-dimensional manifold: Zh = {(z+ (θ+ ), z− (θ− )) |(θ+ , θ− ) ∈ . R2 }, then Tz(θ+ ,θ− ) Zh = span{τ(R+θ+ )/h e+ , τ−(R+θ− )/h e− }, where e± = z± In contrast with the one-peak solution, now z(θ+ , θ− ) are not zeroes of F (0, u), therefore the zeroes of F (h, u) cannot be obtained directly by the IFT.

48

1 Linearization

Instead, we appeal to the contraction mapping theorem. However, Lyapunov– Schmidt reduction is again useful. Fixing (θ+ .θ− ) ∈ R2 , let u0 = z(θ+ , θ− ), we write down the orthogonal decomposition: X1 , Y = Tu0 Zh Y1 , X = Tu0 Zh where the spaces X and Y were introduced in the previous subsection, and let P : Y → Y1 be the orthogonal projection. Again, P F (h, u0 + ξ) = 0 F (h, u) = 0 ⇔ (I − P )F (h, u0 + ξ) = 0 . In fact, by Taylor’s formula, F (h, u0 + ξ) = F (h, u0 ) + Fu (h, u0 )ξ + N (h, u0 , ξ) , where N (h, u0 , ξ) = 3u0 ξ 2 + ξ 3 . First, ﬁxing (θ+ , θ− ) ∈ R2 , we solve the ﬁrst equation in X1 for h > 0 small. Let L(θ+ , θ− , h) = P Fu (h, u0 ). We have the following lemmas: Lemma 1.3.8 ∃γ > 0, ∃h0 > 0, ∃α0 ∈ (0, 1/2) such that L(θ+ , θ− , h)Y ≥ γuX ∀u ∈ X1 . as h ∈ (0, h0 ), and |α± | < α0 . Lemma 1.3.9 ∀ρ > 0, F (h, u0 )Y ≤ C1 (Σ± Max|y|≤ρh (V (y) − V (±R))2 | ± R ± θ± | + e−2µρ + e−µR/h ), where µ = Min{α+ , α− }. Lemma 1.3.10 N (h, u0 , ξ)Y ≤ C2 ξ2X , N (h, u0 , ξ1 ) − N (h, u0 , ξ2 )Y ≤ C2 Max(ξ1 X , ξ2 X )ξ1 − ξ2 X . Once these lemmas have been proved, according to the contraction mapping theorem, one ﬁnds a ﬁxed point ξ of the operator −L−1 (θ+ , θ− , h)N (h, u0 , ξ) − F (h, u0 ) in a neighborhood of u0 on X1 . Namely, ∃h0 > 0, ∃α0 > 0, ∃C1 > 0 such that ∀h ∈ (0, h0 ), ∀|θ± | < α0 , ∃ a unique ξ = ξ(θ+ , θ− , h) ∈ X1 such that P F (h, u0 + ξ) = 0, and ξ ≤ C1 F (h, u0 )Y . Now the second equation is a two system in two variables (θ+ , θ− ) ∈ [−α0 , α0 ]2 : vh (θ+ , θ− ) := h−1 ( e+ , F (h, z(θ+ , θ− ) + ξ(θ+ , θ− , h) , e− , F (h, z(θ+ , θ− ) + ξ(θ+ , θ− , h) ) = (0, 0) .

1.3 Lyapunov–Schmidt Reduction and Bifurcation

49

Let us deﬁne a two-vector function: v0 (θ+ , θ− ) =

1 (|z+ |2 V (R)θ+ , |z− |2 V (−R)θ− ) . 2

Again we need a Lemma: Lemma 1.3.11 For ν ∈ (1, 2) and 0 < h < Min(h0 , α0 ), we have h−ν vh (hν θ+ , hν θ− ) → v0 (θ+ , θ− ) , uniformly on [−1, 1]2 as h → 0. The diﬀerence between these proofs of the Lemmas 1.3.8 to 1.3.11 and those in the previous subsection lie in the interaction terms (for details refer to [Oh]). From the Brouwer degree theory (cf. Sect. 3.1), deg(h−ν vh (hν θ+ , hν θ− ), [−1, 1]2 , (0, 0)) = deg(v0 (θ+ , θ− ), [−1, 1]2 , (0, 0)) = 0 . This proves the existence of a zero of the above system. Finally we arrive at: Theorem 1.3.12 Suppose that V ∈ C 3 is bounded, and λ < inf (V ). Then for each pair (x1 , x2 ) of nondegenerate critical points of V, ∃h0 > 0, such that for h ∈ (0, h0 ) equation (1.32) has a nonzero solution uh of the form τ(x1 +s+ )/h z+ + τ(x2 +s− )/h z− + ξ(h), where |s± | ≤ h−ν with ν ∈ (1, 2), and ξ → 0 as h → 0. 1.3.5 Transversality Transversality is an important notion in diﬀerential geometry. It is a condition, induced by the IFT, on a map between two manifolds, under which the preimage of a submanifold is again a submanifold. f : X → Y is said to be transversal to the submanifold W ⊂ Y if at every point x ∈ f −1 (W ), Im f (x) + Tf (x) (W ) = Tf (x) (Y ) , and is denoted by f W . Related notions are regular points and regular values deﬁned as follows. Deﬁnition 1.3.13 Suppose that X and Y are C 1 Banach manifolds and f ∈ C 1 (X, Y ). A point x ∈ X is called a regular point of f , if f (x) : Tx (X) → Tf (x) (Y ) is surjective, and is singular (or critical) if it is not regular. The images of the singular points under f are called singular values (or critical values), and their complement, regular values. If y ∈ Y is not in the image of f , i.e., f −1 (y) = ∅, then y is a regular value. The following Sard theorem reveals the smallness of the set of critical values:

50

1 Linearization

z=α sech(h−1α(x−x0))

φ(x)

x0

x

S0

z+

z−

uh x1+S+

x2+S− Fig. 1.8.

Theorem 1.3.14 (Sard) Suppose that X and Y are diﬀerential manifolds with dimensions n and m respectively, and that U ⊂ X is open. If f ∈ C r (X, Y ), where r ≥ 1 and r > max{0, dimX −dimY }, then the set of critical values of f has measure zero in Y . The proof is purely measure theoretic, we shall not give it here. The special case dimX = dimY < ∞ will be given in Sect. 3.1, but for the general case refer to Milnor [Mi 2]. We shall extend the study to maps between inﬁnite-dimensional manifolds. Recall a map f between two Banach spaces X and Y is called Fredholm in U ⊂ X, if f ∈ C 1 (U, Y ) and f (x) is a Fredholm operator ∀x ∈ U . The index of f is deﬁned to be ind f (x) = dim ker f (x) − dim coker f (x) . It is known that if U is connected, then ind f (x) is a constant integer, and then is denoted by ind(f ). Lemma 1.3.15 If f ∈ C 1 (U, Y ) is a Fredholm map, then the set of critical points is closed.

1.3 Lyapunov–Schmidt Reduction and Bifurcation

51

Proof. Since x → dim kerf (x) is u.s.c., and ind(f ) is locally a constant, dim cokerf (x) is also u.s.c. Let S be the critical point set of f , then the set S = {x ∈ U |f (x) is not surjective} = {x ∈ U |dim coker f (x) ≥ 1}

is closed.

Theorem 1.3.16 (Sard–Smale) Suppose that X is a separable Banach space and Y is a Banach space. Let f ∈ C r (U, Y ) be a Fredholm map, where U ⊂ X is open. If r > max (0, ind(f )), then the set of critical values is of ﬁrst category. Proof. Since the ﬁrst category set is a countable union of closed nowhere dense sets, and U is separable, it is suﬃcient to prove that ∀x ∈ U , there is a neighborhood of x, V ⊂ U such that the set of critical values of f |V , S(f, V ) is closed and nowhere dense. By the deﬁnition of Fredholm operators, one has the decomposition X = kerf (x)⊕X1 . Let Q be the projection of Y onto Imf (x); from the IFT, there is a neighborhood U0 × V0 ⊂ kerf (x) × X1 , a neighborhood W ⊂ Imf (x) of f (x), and h ∈ C 1 (U0 × W, V0 ) such that Qf (u + h(u, w)) = w, ∀(u, w) ∈ U0 × W , and ∀u ∈ U0 , h(u, ·) : W → V0 is a diﬀeomorphism. Since U0 is ﬁnite dimensional, it can be chosen compact. We set V = U0 × V0 , 1. First we show that S(f, V ) is closed. Due to Lemma 1.3.15 it is suﬃcient to show that the map f |V is closed. In fact, let xn = un + vn ∈ U0 × V0 be such that yn = f (xn ) → y. From the compactness of U0 , un is subconvergent, and vn = h(un , Qyn ) is convergent, therefore f |V is closed. 2. Next we show that S(f, V ) is a nowhere dense set. Deﬁne H : U0 × W → V by H(u, v) = (u, h(u, w)). Then H is a diﬀeomorphism satisfying Qf ◦ H(u, w) = w. Setting f˜ = f ◦ H, we have S(f, V ) = S(f˜, U0 × W ), and Qf˜ |W = idW , therefore ∀w ∈ W , (x, w) ∈ S(f˜, U0 × W ) ⇐⇒ x ∈ S((I − Q)f˜(·, w), U0 ). From r > ind f (x) = dim kerf (x) − dim cokerf (x) and Sard theorem, S((I − Q)f˜(·, w), U0 ) is a null set. Since S(f, V ) is also closed, it is a nowhere dense set. We conclude that the critical set of f is of the ﬁrst category. Corollary 1.3.17 Let X and Y be Banach spaces, and X be separable. If f ∈ C 1 (X, Y ) is Fredholm with negative index, then f (X) does not contain interior points. Proof. If not, say y0 ∈ f (X) is an interior point, i.e., ∃ a neighborhood V of y0 such that V ⊂ f (X). According to the Sard–Smale lemma, ∃y ∈ V such that f {y}, i.e., ∀x ∈ f −1 (y), f (x) : X → Y is surjective, or codim Im(f (x)) = 0, thus ind(f )(x) = dim kerf (x) ≥ 0. A contradiction. The following transversality theorem will be often used in the sequel.

52

1 Linearization

Theorem 1.3.18 Suppose that X, Z are Banach spaces, where X is separable, and that S is a C r Banach manifold. Assume that F ∈ C r (X × S, Z) satisﬁes (1) F {θ}, and (2) ∀s ∈ S, fs (u) = F (u, s) is a Fredholm map with index satisfying max{0, ind(fs )} < r. Then ∃ a residual set Σ ⊂ S (i.e., the countable intersection of open dense sets) such that ∀s ∈ Σ, fs {θ}. Proof. Deﬁne the injection: i : V −→ X ×S and the projections π : X ×S −→ S, p : X × S → X. Let V = F −1 (θ). In the sequel, both of them are restricted on V . We claim that π ◦ i is a Fredholm map with ind (π ◦ i) = ind (fs ). In fact, ker F (v) = Tv (V ) = Im ((p ◦ i) (v)) ⊕ Im ((π ◦ i) (v)) ∀v = (x, s) ∈ Tv (V ). Since F {θ}, we have a direct sum decomposition: X × Ts (S) = Y ⊕ Tv (V ) such that F (v) : Y → Z is an isomorphism. By the assumption that fs (x) is Fredholm, we have direct sum decompositions: X = Im((p ◦ i) (v)) ⊕ Y1 , Z = Z1 ⊕ Z2 , such that fs (x) : Y1 → Z1 . Thus, Y = Y1 ⊕ Y2 , where Ts (S) = Im(π ◦ i) (v) ⊕ Y2 . From F (v) = fs (x) ⊕ ∂s F (v), we have ∂s F (v) : Y2 → Z2 is an isomorphism. Thus, ker fs (x) = Im(p ◦ i) (v) = ker (π ◦ i) (v), coker fs (x) = Z2 Y2 coker(π ◦ i) (v). Therefore ind (fs ) = ind (π ◦ i). According to the Sard–Smale Theorem, the set of regular values of π◦i Σ is a residual set. We are going to prove that for a regular value s, fs {θ}. From F (x, s) = θ, and F {θ}, we have Im F (x, s) = Z, i.e., ∀a ∈ Z, ∃(α, β) ∈ ∂F X × Ts (S) such that a = ( ∂F ∂x α + ∂s β). We are going to prove Im fs (x) = Z, i.e., ∀ a ∈ Z, ∃y ∈ X such that a = fs (x)y. Thus, if β = 0, we take y = α. Otherwise, since π is a projection, (π ◦ i) (x, s) : X × Ts (S) −→ Ts (S) is a projection. That s is a regular value means that (π ◦ i) (x, s) : T(x,s) (V ) −→ Ts (S) is surjective, i.e., to the given β ∈ Ts (S), ∃w ∈ X such that (w, β) ∈ T(x,s) (V ). Since F −1 (θ) = V , we have F (x, s)(w, β) = θ. Setting y = α − w, we obtain ∂F ∂F α+ β fs (x)y − a = F (x, s)[(α, β) − (w, β)] − a = ∂x ∂s − a − F (x, s)(w, β) = θ . The same conclusion holds if X and Z are Banach manifolds. As an application we study the simplicity of eigenvalues of the Laplacian on bounded domains with Dirichlet data. It is well known that all eigenvalues of second-order ODEs on bounded intervals with Dirichlet data are simple, but it is not true for PDE, e.g., the Laplacian on a ball. However, we shall show that for most domains, this is true. What is the meaning of most domains? Given a bounded open domain Ω0 ⊂ Rn with smooth boundary, we consider the manifold S = Diﬀ 3 (Ω0 ) := {g ∈ C 3 (Ω0 , Rn ) | det (g (x)) =

1.3 Lyapunov–Schmidt Reduction and Bifurcation

53

0 ∀x ∈ Ω0 }. Thus as g ∈ S, the domain Ω = g(Ω0 ) is C 3 . ∀Ω, let X(Ω) = H 2 (Ω) ∩ H01 (Ω) and Y (Ω) = L2 (Ω). ∀g ∈ S, deﬁne g ∗ : X(Ω0 ) → X(Ω) by (g ∗ u)(x) = u(g −1 (x)) ∀x ∈ Ω = g(Ω0 ) . One deﬁnes a map F ∈ C 1 ((X(Ω0 )\{θ}) × S, Y (Ω0 )) by F (u, λ, g) = g ∗−1 (∆ + λI)g ∗ u . We want to show that there exists a residual set Σ of S such that ∀g ∈ Σ, all eigenvalues of the problem: (∆ + λ)u = 0 on H01 (Ω) , are simple, where Ω = g(Ω0 ). Since there are, at most, countable eigenvalues, it is suﬃcient to show that for each eigenvalue λ after suitable perturbation of the domain, i.e., for after a suitable g ∈ S, it is simple. To this end, let us deﬁne fg (u, λ) = F (u, λ, g). We claim that it remains to show fg {θ}, or equivalently, fg (u, λ) : X(Ω) × R1 → Y (Ω) is surjective, i.e., {(∆ + λ)w + µu | (w, µ) ∈ X(Ω) × R1 } = Y (Ω)}, or equivalently, codim Im(∆ + λI) ≤ 1. Since we have assumed that λ is an eigenvalue, i.e., codim (Im(∆ + λI) ≥ 1. Therefore λ is simple. We know that fg (u, λ) is a Fredholm map, from Theorem 1.3.18, it is suﬃcient to verify that F {θ}. To this end, ∀(u0 , λ0 , g0 ) ∈ F −1 (θ), let L = F (u0 , λ0 , g0 ), and Ω = g0 (Ω0 ). With no loss of generality we may assume g0 = Id. It remains to verify that L is surjective. Since L(w, µ, h) = (∆+λ0 )w +µu0 +[h·∇, ∆+λ0 ]u0 = (∆+λ0 )(w −h·∇u0 )+µu0 , L is Fredholm, where [A, B] = AB − BA. Suppose it is not surjective, then codim Im(L) > 0, there exists a nontrivial element φ⊥Im(L). First by taking w = h = θ, we have

φu0 = 0 . Ω

Second, we take h = θ; it follows that

φ(∆ + λ0 I)w = 0 ∀ w ∈ X(Ω) . Ω

¯ α ∈ (0, 1) is a solution of the equation: Then φ ∈ C 2,α (Ω), (∆ + λ0 )φ = 0 , with boundary value zero. Then we have

54

1 Linearization

φ(∆ + λ0 )(h · ∇u0 )

0=

Ω

φ(∆ + λ0 )(h · ∇u0 ) − (h · ∇u0 )(∆ + λ0 I)φ

=

Ω

φ∂n (h · ∇u0 ) − ∂n φ(h · ∇u0 )

=

∂Ω

∂n φ(h · ∇u0 )

=− ∂Ω

¯ Rn ). Thus, for all h ∈ C 3 (Ω, ∂n φ∇u0 = 0 on ∂Ω . According to the uniqueness of the Cauchy problem (see [Hor 1]) for the equa¯ this contions (∆ + λ0 )φ = 0, and (∆ + λ0 )u0 = 0 for φ, u0 ∈ C 2,α ∩ C0 (Ω), tradicts with u0 = θ and φ = θ. We have proved a result due to K. Uhlenbeck [Uh]: Theorem 1.3.19 There exists a residual set Σ ⊂ Diﬀ 3 (Ω0 ) such that ∀g ∈ Σ, all eigenvalues of the equation (∆ + λI)u = 0 on H01 (g(Ω0 )) are simple.

1.4 Hard Implicit Function Theorem The inverse function theorem is established for C 1 mappings between Banach spaces, f : X → Y , where X and Y are Banach spaces, under the assumption f (x0 )−1 ∈ L(Y, X). However, if f (x0 )−1 is not bounded, then the problem will be very complicated. The small divisor problem, arising in the study of the long-time behavior of oscillatory motions in the solar system, is a typical example. The celebrated KAM theory, which states that most (in the sense of measure) quasi-periodic solutions of an integrable smooth Hamiltonian system persist under small perturbations of that Hamiltonian, provided that the Hessian is nondegenerate, is a breakthrough in the progress in this direction. Not only is the result an important conclusion in celestial mechanics, but also the methods developed in this theory have great impact in other nonlinear problems. In fact, Nash–Moser iteration, which was introduced in the study of the isometric embedding problem for Riemann manifolds, see [Na 1,2], and [Mos 1,2], is extensively applied in problems arising in geometry (see for instance Hamilton [Ham]) and in physics (see for instance Hormander [Hor 2]). In recent years, KAM theory has been extended to Hamiltonian systems with inﬁnitely many freedoms (see Poschel [Po], Wayne [Way], Kuksin [Kuk], Bourgain [Bou]etc.)

1.4 Hard Implicit Function Theorem

55

In avoiding too many complicated computations, we take an approach due to Hormander by appealing to the Leray–Schauder degree theory in the proof of the existence of a solution. Readers may read this subsection after Chap. 3. In fact there are other approaches without using topological arguments (see for instance J. Moser [Mos 1,2,3], Hormander [Hor 2] or J. T. Schwartz [Scw]). Actually, the boundedness assumption on f (x0 )−1 guarantees the convergence of the simple iteration method in solving the nonlinear equation: y = f (x) ⇔

(1.34)

x = x − f (x0 )−1 (y − f (x)) .

One produces the iteration process as follows: xn+1 = xn − f (x0 )−1 (y − f (xn )),

n = 1, 2, . . . .

(1.35)

However, if we consider the nonlinear wave equation: utt − F (uxx ) = f u|t=0 = ut |t=0 = 0 , where F is a function with F (0) = 1 and F (0) = 0, then the above simple iteration process fails, because the linearized equation reads as utt − uxx = g u|t=0 = ut |t=0 = 0 . when g ∈ H s , with compact support for some s > 0. We can at most estimate the boundedness of H s+1 norm of u, but not H s+2 , as that in the elliptic case. If we use the simple iteration, then we lose derivatives in each step; after a ﬁnite number of steps, the iteration cannot go on. This phenomenon is called “loss of derivatives”. It occurs in nonelliptic problems. In the analytic category, occurs phenomenon whereby the inverse maps reduce the convergence radii (e.g. the small divisor problem, see below); this is called “loss of convergence radii”. As before, this is also the case that the linearized operator has no bounded inverse. 1.4.1 The Small Divisor Problem For a given f , analytic in a neighborhood of θ, with f (0) = 0, and f (0) = σ, ﬁnd u, analytic in a neighborhood of θ with u(0) = 0, and u (0) = 1, satisfying f (u(z)) = u(σz) . We write it in the IFT form:

(1.36)

56

1 Linearization

Let f (z) = σz + fˆ(z) , u(z) = z + u ˆ(z) . Equation (1.36) is equivalent to Φ(fˆ, u ˆ) = fˆ(z + u ˆ(z)) + σ u ˆ(z) − u ˆ(σz) = θ .

(1.37)

Obviously, Φ(θ, θ) = θ, and Φuˆ (fˆ, u ˆ)w ˆ = fˆ (z + u ˆ(z))w(z) ˆ + σ w(z) ˆ − w(σz) ˆ . Φuˆ (θ, θ) is injective, if σ = e2πiτ , where τ is irrational. In fact, we write vˆ =

∞

vj z j , w ˆ=

j=2

∞

wj z j .

j=2

The equation ˆ = vˆ Φuˆ (θ, θ)w has a unique solution w, ˆ in which wj =

vj j = 2, 3, · · · . σ − σj

Since the denominator tends to zero for a subsequence, one cannot expect a bounded inverse of Φuˆ (θ, θ). A real number τ is of type (b, ν), b > 0, ν > 2, if τ − p ≥ b ∀p, q ∈ Z\{0} . (1.38) q qν Claim. For almost all real numbers τ , there exists (b, ν) depending on τ , such that τ is of type (b, ν). Indeed, for given (b, ν), ﬁxing q, the set of all real numbers τ ∈ [0, 1] such that (1.38) does not hold has a measure less than 2bq −ν+1 . Therefore, the set of all real numbers τ such that (1.38) does not hold has the measure 2b q −ν+1 < ∞. Since b can be arbitrarily small, the conclusion follows. Suppose σ = e2πiτ , where τ is a number of type (b, ν). Then p |σ − σ j | = e2πi(j−1)(τ − j−1 ) − 1 p ≥ sin 2π τ − (j − 1) j−1 p 2 ≥ · 2π(j − 1) τ − π j − 1 b ≥4 , (j − 1)ν−1

1.4 Hard Implicit Function Theorem

therefore

57

j ν−1 1 . |σ − σ j | 4b

In this case, even if vˆ has the convergent radius r, w ˆ can only have a smaller convergent radius. In other words, if we introduce a family of Banach spaces: A(r) = {w(z)| ˆ bounded and analytic in |z| < r, w(0) ˆ =w ˆ (0) = 0} with the norm ˆ , |w| ˆ r = Sup|z| 0 such that ∀(x, y) ∈ Ωrλ , ∀λ ∈ (0, 1] with |Φ(x, y)|λ Cλ

−2(α+τ )

, we have y∗ = u(x) ∈ Y λ satisfying 2

Φ(x, u(x)) = θ . Proof. One chooses sequences λn =

λ (1 + 2−n ), 2

n = 0, 1, 2, . . . ,

and µn+1 =

1 λ (λn + λn+1 ) = (1 + 3 · 2−n−2 ), 2 2

n = 0, 1, 2, . . . .

Then λ0 = λ, λn ↓ λ2 . For simplicity, we write Φ(yn ) = Φ(x, yn ), T (yn ) = T (x, yn ). The Newton iteration sequence reads as yn+1 = yn − T (yn )Φ(yn ) n = 0, 1, 2, . . . (1.48) y0 = y . We want to prove: (1) yn ∈ Brλn (y), ∞ nτ (2) let cn = |Φ(yn )|λn , n=1 2 cn < ∞, (n+3)τ −τ (3) yn+1 − yn µn+1 M cn (2 λ ).

60

1 Linearization

If they are proved, then ∞

∞

−τ 3τ

yn+1 − yn λ M λ 2

n=1

2

2nτ cn < ∞ .

n=1

Therefore ∃y∗ = u(x) such that yn → y∗ in Y λ , and 2

|Φ(x, y∗ )| λ lim |Φ(yn )|λn = lim cn = 0 , 2

n→∞

n→∞

i.e., Φ(x, y∗ ) = θ. First, we prove (3) . In fact, From (1.48) and assumption (2), yn+1 − yn µn+1 T (yn )Φ(yn )µn+1 M (2(n+3)τ λ

−τ

(n+3)τ

= M cn (2

)|Φ(yn )|λn

−τ

λ

).

Next, we turn to proving (2) . By (1.48), we have Φ(yn+1 ) = [Φ(yn+1 ) − Φ(yn ) − Φy (x, yn )(yn+1 − yn )] − [Φy (x, yn )T (yn ) − I]Φ(yn ) .

(1.49)

Combining (3) , with assumptions (1) and (3), for n large, cn+1 = |Φ(yn+1 )|λn+1 M 22α(n+3) λ + M c2n (2n+3 λ where a = λ Set

−1 2(α+τ )

)

−2α

yn+1 − yn 2µn+1

≤ aq n c2n ,

−2(α+τ ) 3

q (M + M 3 ) and q = 4α+τ . αn = aq n cn ;

we obtain αn+1 = aq n+1 cn+1 a2 q 2n+1 c2n = qαn2

∀n .

One chooses k ∈ (1, 2), 0 ∈ (0, 1) satisfying 1

q k−1 0 < 1 , and n+1 = qkn ∀n. Thus n+1 = q 1+k+k

2

+···+kn kn+1 0

1

1

= q − k−1 (q k−1 0 )k

n+1

0, such that yn+1 − y0 µn+1 ac0 M1 . If c0 is so small that yn+1 − y0 µn+1 < r − y − yλ

∀n ,

then yn − yλn yn − yµn < r − y − yλn . Therefore yn − yλn < r .

The proof is complete.

As an application, now we return to the small divisor problem. We have the following theorem. Theorem 1.4.2 (Siegel) Suppose that τ is of type (b, ν), b > 0, ν > 2. Let σ = e2πiτ . If f (z) = σz + fˆ(z) is an analytic function on |z| < 1, with fˆ(0) = fˆ (0) = 0, then there exist γ, r0 ∈ (0, 1) and an analytic function u 3r0 such that on |z| < 4(1+γ) sup |fˆ(z)| γ |z| 0, A(r) = {w(z)|analytic and bounded in |z| < r, with w(0) = w (0) = 0} , with norm |w|r = sup |w(z)| , |z| 0 small. Since all assumptions (1)–(3) of Theorem 1.4.1 are satisﬁed (cf. (1.46), (1.44), and (1.43) respectively), the conclusion follows from Theorem 1.4.1. 1.4.2 Nash–Moser Iteration Let us turn to the “loss of derivative” problem. Besides the simple iteration method, there are other ways to solve equation (1.34). Given c > 0 and a curve x(t) satisfying f (x(t)) = (1 − e−ct )y , x(t) satisﬁes the ODE: x(t) ˙ = (f (x(t)))−1 (y − f (x(t))) . If, for some initial data x(0) = x0 , the solution globally exists, and has a limit x(t) → x∞ , then f (x∞ ) = y . Discretizing the equation, we return to Newton’s approximation scheme: xn+1 − xn = cf (xn )−1 (y − f (xn )) . This is (1.41) for c = 1. There is enough room to generalize the above method. Let f (x0 ) = y0 be a special solution. In order to overcome the problem of “loss of derivatives”, we introduce a family of smoothing operators St , t ∈ [1, ∞), which regularizes the function x, and satisﬁes St → Id, as t → ∞. Find a suitable function g(t), which will be described later so that the curve x(t) tends to a solution x∞ of the equation f (x) = y. We intend to solve the ODE: ⎧ ˙ = f (v(t))−1 g(t) ⎨ x(t) v(t) = St x(t) ⎩ x(0) = x0 .

1.4 Hard Implicit Function Theorem

63

Since d f (x(t)) = f (x(t))x(t) ˙ dt ˙ + g(t) , = (f (x(t)) − f (v(t)))x(t) we should have

f (x∞ ) − f (x0 ) =

∞

e(t)dt + 0

∞

g(t)dt , 0

where e(t) = (f (x(t)) − f (v(t)))x(t). ˙ Design the iteration scheme: ⎧ ⎨ δn := xn+1 − xn = n f (vn )−1 gn , vn = Sθn xn , ⎩ n = θn+1 − θn ,

(1.51)

where θ1 < θ2 < . . . < θn → ∞. Then f (xn+1 ) − f (xn ) = n (en + gn ) , where n en = f (xn + δn ) − f (xn ) − f (xn )δn + (f (xn ) − f (vn ))δn . It follows that f (xn+1 ) − f (x0 ) =

n

j (gj + ej ) .

j=0

Since we are only interested in the limiting result, sometimes we modify it to be: n j gj + Sθn En = Sθn (y − f (x0 )) , j=0

where En =

n−1 j=0

j ej , so that gn can be determined step by step.

gn = −1 n [(Sθn − Sθn−1 )(y − f (x0 ) − En−1 ) − n−1 Sθn en−1 ], n = 1, 2, . . . , (1.52) and g0 = −1 0 Sθ0 (y − f (x0 )) . We start with an abstract framework. {Ea }a≥0 is called a family of Banach spaces with smoothing operators, if Eb → Ea for b ≥ a is an injection, and ∃C = C(a, b) such that ua Cub . Let E∞ = ∩a≥0 Ea be endowed with the weakest topology, such that E∞ → Ea is continuous. Moreover, we assume that ∃ a family of linear operators Sθ : E0 → E∞ , depending on a parameter θ ≥ 1, such that

64

1 Linearization

(1) Sθ ub Cua b a, d Sθ ub Cθb−a−1 ua , (2) dθ where C is a constant depending on a and b. The following inequalities hold: (1) Sθ ub Cθb−a ua , ∀ a < b. (2) (I − Sθ )ub Cθb−a ua ∀ b < a. ∀λ ∈ (0, 1) ∀ a, b. (3) (Interpolation inequality) uλa+(1−λ)b Cuλa u1−λ b Both (1) and (2) follow from the second inequality of the deﬁnition. In fact, if a > b, then ∞ d S u u − Sθ ub = t dt θ b

∞ b−a−1 C t ua dt = Cθb−a ua if b < a . θ

and if a < b, then

θ d Sθ ub = St u + S1 ub 1 dt b

θ C tb−a−1 dtua + Cua Cθb−a ua 1

where C denotes various constants. (3) is a consequence of (1) and (2). In fact, may assume a < b. For c = λa + (1 − λ)b, uc Sθ uc + (I − Sθ )uc C(θc−a ua + θc−b ub ) 1

b b−a , we obtain the desired inequality. By choosing θ = ( C u u a )

Example. H¨ older Spaces with Smoothing Operators Let Ω be a bounded domain in Rn we write H α (Ω) as the H¨older space, deﬁned as follows: H 0 (Ω) = C(Ω). If k ≥ 0 is an integer, k < a k + 1, C α (Ω) if a < k + 1 α H (Ω) = C α−0 (Ω) if a = k + 1 . The seminorm is deﬁned to be |u|a =

|α|=k

|∂ α u|a−k ,

1.4 Hard Implicit Function Theorem

and for 0 < a 1 ,

65

|u(x) − u(y)| , |x − y|α x,y∈Ω

|u|a = sup and the norm is deﬁned as:

ua = uC + |u|a . We now come to deﬁne the smoothing operators Sθ , θ ≥ 1. For the sake of simplicity, we assume Ω = Rn , the bounded domain case can be modiﬁed by standard argument. For a compact set K in Rn , one chooses X ∈ C0∞ (Rn ) to be 1 in a neighborhood of K, a function ψ ∈ C0∞ (Rn ) to be 1 in a neighborhood of 0, and let ϕ be the Fourier transform of ψ, ϕ = Fψ, i.e.,

exp (2πixξ)ψ(ξ)dξ . ϕ(x) = Rn

Let ϕθ (x) = θn ϕ(θx) θ ≥ 1 . If u has support in K, we deﬁne Sθ u = X (ϕθ ∗ u) ∈ C0∞ (Rn ) . Since ϕθ (x) = Fψ(ξ/θ), and ψ( θξ ) → 1 (in the distribution sense), we have ϕθ → δ, so Sθ u → u, as θ → +∞. The operators Sθ , which approximate the identity and map to smooth functions, are call smoothing operators. Theorem 1.4.3 The smoothing operators Sθ have the following properties, for θ > 1 and u ∈ H α : (1) Sθ ub Cua b a, d Sθ ub Cθb−a−1 ua . (2) dθ Proof. (1) following directly from the translation invariance and the convexity of the norms: ϕθ ∗ ua ϕL1 ua . We verify (2). Noticing d Sθ u = X · F −1 (θ−1 ψ1 dθ

ξ · (Fu)(ξ)) θ

where ψ1 (ξ) = −ξ · ∇ψ(ξ) , −1

is the inverse Fourier transform. Again ψ1 ∈ C0∞ , and vanishes in and F the neighborhood of the origin. According to (1), (2) holds for b = a. We shall only need to verify (2) for b = 0 and b = a + k, k ∈ N, because the remaining cases follow from the interpolation inequality.

66

1 Linearization

For b = a + k, d d Sθ u ∂ α S = u θ dθ dθ a+k a |α|=k α −1 k−1 ξ ξ = θ ψ1 · (Fu)(ξ) F θk θ a |α|=k

Cθk−1 ua because ξ α ψ1 (ξ) is again in L1 . For b = 0, we only need to prove for a = k ∈ N. Let us write ψ1 (ξ) =

ξα

ξ α ψα (ξ) where ψα (ξ) =

|α|=k

|α|=k

|ξ α |2

ψ1 (ξ) ,

then ψα ∈ C0∞ and vanishes in the neighborhood of the origin: d −1 −k−1 ξ α F Sθ u θ u)(ξ) ψ (∂ α dθ θ 0 0 |α|=k

[∂ α u(x − y) − ∂ α u(x)](ψα )θ (y)dy θ−k−1 |α|=k

=θ

−k−1

·

|y||(ψα )θ (y)|dyuk+1

= Cθ−k−2 uk+1 .

The proof is complete.

Now we are going to introduce a family of Banach spaces associated with Ea as follows: Let θj = 2j , j = 0, 1, 2, . . . , j = θj+1 − θj , j ≥ 1, 0 = 1: −1 R0 = −1 0 Sθ1 , Rj = j (Sθj+1 − Sθj ), j ≥ 1

According to (2), Rj ub

−1 j

b−a θj+1 d − θjb−a θj+1 St u C ua . θj dt (b − a)j b

We obtain: (4) Rj ub Cab θjb−a−1 ua , and (5)

∞ j=0

j Rj u = u.

1.4 Hard Implicit Function Theorem

67

The series is convergent in Eb if u ∈ Ea , a > b. This is called the Paley–Littlewood decomposition. Conversely, suppose we have a sequence {uj } ⊂ Ea , a ∈ [a1 , a2 ], satisfying uj ai Cθjai −a−1 ,

i = 1, 2 ∀j ;

then, by interpolation inequality, (6) uj b Cθjb−a−1 ∀b ∈ [a1 , a2 ], ∀j, and then the series

j

j uj converges in Eb if b < a.

Deﬁnition 1.4.4 For a ∈ [a1 , a2 ], we deﬁne Ea = {u = Cθjai −a−1 i = 1, 2, ∀j} with norm: ua =

inf u=

j

∞ j=0

j uj | uj ai

sup θj−ai +a+1 uj ai .

j uj i,j

Then Ea is a family of Banach spaces. The following properties hold: (7) ub Cua C1 ua if b < a. To prove the ﬁrst inequality. ∀ > 0 ∀ u ∈ Ea ∃{uj } ⊂ Eai , i = 1, 2 such that u = j j uj , and uj ai Cθjai −a−1 (ua + ) . From (6), uj b Cθjb−a−1 . It follows that ub C(ua + ). The second inequality follows from (4) and (5). (8) The space Ea does not depend on a1 , a2 . It follows from the interpolation inequality directly. for b < a. Indeed, ∀u = Ea one chooses (9) (I − Sθ )ub Cθb−a ua a1 < a2 such that b < a1 < a < a2 . ∀ > 0, ∃{uj } ⊂ Eai , i = 1, 2, such that u = Σj uj and uj ai ≤ Cθjai −a−1 (ua + ). Then j uj − Sθ uj b , (I − Sθ )ub and (I − Sθ )uj b Cθb−ai uj ai Cθb−ai θjai −a−1 (ua + ), i = 1, 2 , therefore (I − Sθ )ub C(ua + ) ⎞ ⎛ j θb−a1 θja1 −a−1 + j θb−a2 θja2 −θ−1 ⎠ . × ⎝ θj >θ

Thus,

θj a. Let α, β, r > 0,and Brα be a ball with radius r centered at the origin in Eα . Assume that f : Brα → Fβ is C 2 with f (θ) = θ, and satisﬁes (1) f (v)−1 exists for v ∈ Brα ∩ E∞ , and ∀g ∈ F∞ , the map (v, g) → f (v)−1 g: (E∞ ∩ Brα ) × F∞ → Ea2 is continuous for some a2 > α > a1 0, and satisﬁes f (v)−1 ga ≤ C(gβ+a−α + g0 vβ+α ) (2) f (u)(v, w)β+δ C

∀a ∈ [a1 , a2 ] ,

max(l−α,0)+max(m,a1 )+n 0. Then ∀y ∈ Fβ with yβ small, ∃u ∈ Eα satisfying f (u) = y . Proof. One uses the iteration scheme (1.51), so one should determine g ﬁrst. As we have seen from above, after decomposition, it satisﬁes a series of recursive equations. ∀g ∈ Fβ we have the decomposition: g=

j gj ,

j

with Deﬁne

gj b Cb θjb−β−1 gβ .

(1.53)

⎧ ⎨ uj+1 = uj + j f (vj )−1 gj , u0 = θ , vj = S θ j u j , ⎩ δj = j f (vj )−1 gj .

(1.54)

We are going to prove f (vj )−1 gj a C1 gβ θja−α−1 vj a C2 gβ θja−α uj − vj a C3 gβ θja−α

a ∈ [a1 , a2 ] ,

(1.55)

a ∈ (α, a2 ] ,

(1.56)

a a2 ,

(1.57)

inductively. Suppose (1.56) and (1.57) are true for j k and (1.55) holds for j < k, we prove (1.55) for j = k. By the assumption (1) and (1.53): f (vk )−1 gk a C(gk β+a−α + gk 0 vk β+a ) # " C θka−α−1 gβ + θk−β−1 gβ · θkβ+a−α gβ Cθka−α−1 gβ ,

1.4 Hard Implicit Function Theorem

69

if gβ is small. Now we prove (1.57) for j = k + 1, from (9), in the case a < α, uk+1 − vk+1 a = uk+1 − Sθk+1 uk+1 a a−α Cθk+1 uk+1 α .

Since uk+1 =

k

j f (vj )−1 gj ,

j=0

and by the deﬁnition of Fα norm, we have uk+1 α Cgβ , ∀α ∈ [a1 , a2 ] . Thus

a−α gβ . uk+1 − vk+1 a Cθk+1

In the case a = a2 , uk+1 − vk+1 a2 Cuk+1 a2 C

k

j f (vj )−1 gj a2

j=0 k

j θja2 −α−1

Cgβ

a2 −α Cθk+1 gβ

j=0

.

The other cases, α a a2 , are veriﬁed by the interpolation property. Finally, we prove (1.56) for j = k + 1: vk+1 a uk+1 a + vk+1 − uk+1 a a−α 2Cgβ θk+1 .

Thus the construction of the sequence uk is possible. And uk , vk are all in Brα if gβ is small. Now f (uj+1 ) − f (uj ) = (f (uj + δj ) − f (uj ) − f (uj )δj ) + (f (uj ) − f (vj ))δj + j gj = j (ej + ej + gj ) where and

ej = −1 j (f (uj+1 ) − f (uj ) − f (uj )(uj+1 − uj )) ,

(1.58)

70

1 Linearization

ej =

1

f (vj + t(uj − vj ))(−1 j δj , uj − vj )dt .

0

We obtain from the assumption (2), for n < α < l, (1 + tuj + (1 − t)vj l )f (vj )−1 gj n uj − vj n (1.59) ej β+δ C Cθj−1− gβ

2

(1.60)

where = 3α − l − 2n. Similarly by Taylor’s formula, ej β+δ θj−1− gβ

2

(1.61)

Let T (g) = j j (ej + ej ). Then T (g)β+δ Cgβ . According to (7), and the assumption on the compactness of embedding, Fb → Fa as b > a, we conclude that T : Fβ → Fβ is compact. According to the recursive formula (1.52), g is uniquely determined by y, therefore I + T is locally injective. Now one can apply the Leray–Schauder invariance of domain theorem (see Chap. 3, Corollary 3.4.12) to conclude that ∀y ∈ Fβ with small yβ , there exists g ∈ Fβ such that (I + T )(g) = y. Substituting this g into the iteration scheme (1.54), un is convergent to some u in Ea , provided by (1.55), and (7). Again, by (1.58), f (u) = (I + T )(g) = y. 2

2 Fixed-Point Theorems

It is well known that the contraction mapping theorem is one of the most important ﬁxed-point theorems in analysis. It is based on the metric of the underlying space. It is simple and is strongly dependent on the chosen metric, but useful. In particular, the unique solution can be computed by iteration. In fact, the implicit function theorem and then the ﬁrst three Sects. of Chap. 1 are based on it. The Brouwer ﬁxed theorem (1911), which says that every continuous selfmapping on a closed ball B n has a ﬁxed-point, is a fundamental ﬁxed-point theorem in topology. It is based on the notion of retraction. Because of its importance, there are a lot of proofs, roughly speaking, divided into three classes according to methods: (1) Combinatorics (Sperner lemma) by which computing methods are developed. (2) Algebraic topology; the topological degree and other algebraic topological invariants are introduced. (3) Diﬀerential topology; the proofs are simple and beautiful (see Dunford and Schwartz [DS], Milnor [Mi 2] etc.). In the study of analysis, we need inﬁnite-dimensional versions of this theorem. A new ingredient – the compactness – is added, while the ball is replaced by its topological equivalent – the convex set. The Schauder ﬁxedpoint theorem and its extensions are all based on convexity and compactness. They are widely used in combining with a priori estimates for solutions in diﬀerential equations. In an ordered space, the Bourbaki–Kneser principle is another basic ﬁxedpoint theorem with applications in analysis, in case compactness is unavailable. The chapter is organized as follows: The order method is studied in Sect. 2.1. Several ﬁxed-point theorems based on the Bourbaki–Kneser principle are derived, by which, the sub- and super-solutions method in ordered Banach space are developed with applications in PDEs. The convexitycompactness method is developed in detail in Sect. 2.3. We start with the KKM map and the Ky Fan inequality. All other ﬁxed-point theorems, including the Schauder ﬁxed-point theorem and its generalizations, the Nash equilibrium, and the Von Neumann–Sion saddle point theorem, are derived

72

2 Fixed-Point Theorems

as consequences. Various applications are studied, in particular, the Ky Fan ﬁxed-point theorem for set valued mappings is applied to free boundary problems. Section 2.4 is devoted to the existence and iteration method for ﬁxed points of nonexpansive maps, which are on the borderline of the contraction mappings. The prototype of the monotone mapping is the subdiﬀerential of a convex function. Due to the special feature of monotonicity, the compactness requirement can be reduced considerably. Since monotone mappings map a Banach space into its dual, one studies the surjectivity of monotone mappings. The main results in this aspect are due to Minty [Min] and Browder [Bd 3,4]. Applications to variational inequalities and quasi-linear elliptic problems are studied as well. We shall show how the monotonicity is applied instead of the compactness in the existence theory. These are the contents of Sects. 2.5 and 2.6. Convex sets and convex functions will be used form time to time; we collect their most important properties in Sect. 2.2.

2.1 Order Method A kind of nonlinear operator deﬁned on ordered spaces is order preserving; this special feature makes the ﬁxed-point problem easy to handle. A set E is said to be ordered if a partial order is deﬁned by the following axioms: ∀x, y, z ∈ E (i) x x, (ii) x y, and y x imply x = y, (iii) x y, and y z imply x z. A chain (or totally ordered set) E is an ordered set, on which ∀x, y ∈ E either x y or y x. The following terminologies are introduced: The smallest (greatest) element x (x resp.): x x (x x), ∀x ∈ E; the minimum (maximum) element x∗ (x∗ resp.): if x x∗ (x∗ x), x ∈ E then x∗ = x (x∗ = x); the lower (upper) bound of a subset F ⊂ E: a ∈ E (a ∈ E) and a x (x a) ∀x ∈ F ; the inﬁmum (supremum) of F inf F (sup F respectively) is the greatest (smallest) element of the subset, which consists of all lower (upper) bounds of F . ∀a ∈ E, we denote S+ (a) = {x ∈ E| a x} , and call it the right section of a. Similarly we deﬁne the left section S− (a). Set [a, b] = S+ (a) ∩ S− (b) for a b, we call it an order interval. Zorn’s lemma, which is equivalent to the Zermelo selection axiom, is our starting point: In a nonempty ordered set (E, ≤), if every chain has an upper bound in E, then the set has a maximum element.

2.1 Order Method

73

Let (E, ) and (F, ) be two ordered sets. A map f : E → F is called (strict) order preserving, if x ( 0 ∃yλ ∈ E satisfying ϕ(yλ ) ϕ(x) ρ(x, yλ ) λ1

(2.2) (2.3)

ϕ(y) > ϕ(yλ ) − ελρ(y, yλ ) ∀y ∈ E\{yλ } .

(2.4)

Proof. Deﬁne a subset of E: E1 = {y ∈ E| ϕ(y) + λερ(x, y) ϕ(x)} , E1 is closed (ϕ is l.s.c.) and nonempty (x ∈ E1 ), so is complete. We claim that ∃yλ ∈ E1 such that ϕ(y) > ϕ(yλ ) − λερ(y, yλ )

∀y ∈ E1 \{yλ } .

Indeed, if not, ∃λ > 0, ∀yλ ∈ E1 , ∃y ∈ E1 \{yλ } such that ϕ(y) ϕ(yλ ) − λερ(y, yλ ) . Deﬁne f (yλ ) = y = yλ , then f : E1 → E1 satisfying λερ(yλ , f (yλ )) ϕ(yλ ) − ϕ(f (yλ )). According to the Caristi theorem, f has a ﬁxed point; this is a contradiction. Since (2.2) and (2.3) are trivially true, it remains to verify (2.4) for y ∈ E\E1 . In fact, if ϕ(y) ϕ(yλ ) − λερ(y, yλ ) and y ∈ E\E1 , then ϕ(y) ϕ(x) − λερ(x, yλ ) − λερ(y, yλ ) ϕ(x) − λερ(x, y) . It follows that y ∈ E1 . This is a contradiction.

2.1 Order Method

75

Theorem 2.1.4 (Amann) Assume that every chain of (E, ) has a supremum, and that f : E → E is order-preserving mapping. If f has a subsolution a f (a) then f has a least ﬁxed point in S+ (a). Proof. (1) We verify the existence of a ﬁxed point. Deﬁne E+ = {x ∈ E| x f (x)} ∩ S+ (a). According to the Bourbaki–Kneser principle, we only want to show: (i) E+ = Ø, (ii) f (E+ ) ⊂ E+ , (iii) every chain of E+ has an upper bound in E+ . Indeed, a ∈ E+ ; (i) follows. Since f is order preserving, f (E+ ) ⊂ E+ . Let X be a chain of E+ . By assumption, it has a supremum b ∈ E. From x f (x) f (b) ∀x ∈ X, we have b f (b), and then b ∈ E+ , i.e., b is an upper bound of X in E+ . (2) Let FixS+ (a) (f ) be the ﬁxed point set of f in S+ (a). Deﬁne G+ = {y ∈ E+ | y z ∀z ∈ FixS+ (a) (f )} . We claim that FixG+ (f ) = Ø, then ∀y0 ∈ FixG+ (f ), y0 is the least ﬁxed point in S+ (a). To this end, from the Bourbaki–Kneser principle, it is suﬃcient to verify: (i) G+ = Ø, (ii) f (G+ ) ⊂ G+ , (iii) every chain of G+ has an upper bound in G+ . In fact, a ∈ G+ ; (i) follows. ∀y ∈ G+ , ∀z ∈ FixS+ (a) (f ), f (y) f (z) = z , which implies f (y) ∈ G+ , and then (ii) follows. Let X be a chain of G+ , by assumption, it has a supremum b ∈ E, so is b ∈ E+ as in (1) and b z ∀z ∈ FixS+ (a) (f ). i.e., b ∈ G+ . This is an upper bound of X in G+ . An ordered set is called chain complete if every chain has an inﬁmum and a supremum. Corollary 2.1.5 Let E be a chain complete ordered set, and let f : E → E be order preserving. Suppose that ∃ a pair of sub- and super- solutions a a. Then f has at least a least and a greatest ﬁxed point in [a, a]. An ordered set E is called a lattice, if for every pair x, y ∈ E, x ∨ y = sup{x, y} and x ∧ y = inf{x, y} are in E. A lattice is called complete if every nonempty subset possesses an inﬁmum and a supremum. Corollary 2.1.6 (Birkhoﬀ–Tarski) Let E be a complete lattice and let f be an order-preserving self-map on E, then there exist a smallest and a greatest ﬁxed point of f . Proof. Let a = inf E and a = sup E, then a and a are sub- and super- solutions of f respectively, satisfying a a. We study chain complete ordered sets. Let X be a Banach space, and let P ⊂ X be a nonempty closed convex positive cone with P ∩ (−P) = {θ}. It reduces a partial order structure on X:

76

2 Fixed-Point Theorems

x y if and only if y − x ∈ P . The order deﬁned above matches the linear structure and the topology on X, but not necessarily the magnitude of the norm. xi yi , i = 1, 2 =⇒ x1 + x2 y1 + y2 , x y, λ 0 =⇒ λx λy , and xn yn , xn → x, yn → y =⇒ x y . A Banach space X with a closed convex positive cone P, induces an ordered Banach space (OBS) (X, P). In particular ∀a ∈ X, S± (a) is closed. Lemma 2.1.7 Every compact subset E of an OBS is chain complete. Proof. For every chain X ⊂ E, we want to show that it has a supremum. In fact ∀x ∈ X, the family of sets: {S+ (x) | x ∈ X} is a family of closed n

subsets with ﬁnite intersection property, i.e., ∀x1 , x2 , . . . , xn ∈ X, ∩ S+ (xi ) = i=1

S+ (x∗ ), where x∗ = sup{x1 , . . . , xn } ∈ X, provided that X is a chain. Since X is compact, ∩ S+ (x) ∩ X = Ø. Let x be an element in the intersection. x∈X

Then (1) x x ∀x ∈ X, i.e., x is an upper bound of X, and (2) ∀ upper ¯ d. Thus x ¯ is a supremum. bound d of X, from x ≤ d and x ¯ ∈ X, it follows x Similarly, one proves that it has an inﬁmum. A positive cone P is called normal if every ordered interval [a, b] in X is bounded in the norm. Lemma 2.1.8 P is normal if and only if ∃ a constant C > 0 such that 0 x y =⇒ x C y . Proof. “=⇒” If not, ∃θ xn yn such that xn n3 yn n = 1, 2, . . . . Let zn = n2x ynn , then zn n. But θ zn n2y ynn . Deﬁne ∞ y = n=1 n12 yynn ; we have θ zn y, i.e., zn ∈ [θ, y]. But zn is unbounded. A contradiction. “⇐=” ∀x ∈ [a, b], we have θ x − a b − a. Since x − a C b − a , it follows that x x − a + a a +C b − a . Corollary 2.1.9 Suppose that (X, P) is a normal OBS, and that D = [a, b] is an order interval. If f is an order-preserving compact self-mapping on D, then it has a smallest and a greatest ﬁxed point.

2.1 Order Method

77

Proof. By normality, D is bounded in the norm. Let E = f (D). E is a compact subset of an OBS, provided by the compactness of f . Since f is a self-mapping on D, a ≤ f (a), f (b) ≤ b, so a < b is a pair of sub- and super-solutions of f . The conclusion follows from Corollary 2.1.5 and Lemma 2.1.7. Noticing that a bounded closed convex set of a reﬂexive Banach space is weakly compact, when we make use of the weak topology, Corollary 2.1.9 can be modiﬁed as follows: Corollary 2.1.10 Suppose that (X, P) is a reﬂexive normal OBS, and that D = [a, b] is an order interval. If f is an order-preserving self-mapping on D, then f has a smallest and a greatest ﬁxed point. Example 1. Let M be a compact space, and let X = C(M ) with norm x = max |x(ξ)| . ξ∈M

Deﬁne P = {x ∈ X| x(ξ) 0}. Then P is normal and (X, P ) is an OBS. Example 2. Let (Ω, B, µ) be a measure space and let Lp (Ω, B, µ), 1 p ∞ be the Lp space over (Ω, B, µ). Deﬁne P = {x ∈ Lp (Ω, B, µ)| x(ξ) 0 a.e. ξ ∈ Ω}. Then (X, P ) is an OBS, and P is normal. Example 3. Let X = C 1 ([0, 1]) with norm x = max |x(t)| + max |x (t)| , t∈[0,1]

t∈[0,1]

and P = {x ∈ X| x(t) 0 ∀t ∈ [0, 1]}. Then (X, P ) is an OBS, but P is not normal. Corollaries 2.1.9 and 2.1.10 are the foundation of the sub- and supersolutions method in the theory of diﬀerential equations. Example 4. Consider the following semi-linear elliptic BVP: −u(x) = ϕ(x, u(x)) in Ω , u|∂Ω = 0

(2.5)

where Ω is a bounded domain with smooth boundary in Rn , and ϕ : Ω × R1 → R1 is continuous. Assume that ∀x ∈ Ω, the function t → ϕ(x, t) is nondecreasing. Deﬁne K = (−)−1 with 0-Dirichlet condition; it is known that K is a positive linear operator from the maximum principle. Let F be the mapping u(x) → ϕ(x, u(x)). Since ϕ is nondecreasing in u, the map F is order preserving. Setting X = C0 (Ω), we consider the following mappings: F

i

K

◦

j

C0 (Ω) → C0 (Ω) → Lp (Ω) → Wp2 ∩ Wp1 (Ω) → C0 (Ω) ,

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2 Fixed-Point Theorems

where i and j are injections, and p > n2 . The later is compact, provided by the Sobolev embedding theorem. Deﬁne f = j ◦ K ◦ i ◦ F : C0 (Ω) → C0 (Ω); the problem (2.5) is equivalent to ﬁnding the ﬁxed point of f . Since f is compact and order preserving, it has a ﬁxed point if one can ﬁnd a sub-solution u and ◦

super-solution u of (2.5) in Wp2 ∩ Wp1 (Ω). i.e., u u satisfying −u ϕ(x, u(x)) and − u ϕ(x, u(x)) a.e. in Ω . Namely, we have proved: Statement 2.1.11 Suppose that ϕ : Ω × R1 → R1 is continuous and ∀x ∈ Ω, t → ϕ(x, t) is nondecreasing. If there is a pair of sub- and super-solutions ◦

u u in Wp2 ∩ Wp1 (Ω),

n 2

< p < ∞, then equation (2.5) has the smallest and ◦

the greatest solutions in the order interval [u, u] ⊂ Wp2 ∩ Wp1 (Ω). Remark 2.1.12 The nondecreasing condition on ϕ can be weakened as follows: ∃ω 0, such that ∀x ∈ Ω, the function: ϕω (x, t) = ϕ(x, t) + ωt is nondecreasing in t. Indeed, instead of K and F , we introduce the linear positive operator Kω = (− + ωI)−1 and the order-preserving mapping Fω : u(x) → ϕω (x, u(x)). Then we ﬁnd the ﬁxed point of fω = j ◦ Kω ◦ i ◦ Fω . Moreover, the continuity condition on ϕ can be replaced by the Caratheodory condition. In this case, we consider the map f : Lp (Ω) → Lp (Ω), with f = i ◦ j ◦ ◦K ◦ F instead, where F is the Nemytski operator. Remark 2.1.13 The sub- and super-solutions method has an advantage in numerical analysis, because one can get the smallest and the greatest solutions in the order interval [u, u] by iteration. Setting ui+1 = f (ui ), ui+1 = f (ui ) u0 = u, u0 = u .

i = 0, 1, 2, . . . ,

Then one has u0 u1 · · · un un · · · u1 u0 . The sequences {un } and un converge to the least solution u∗ and the greatest solution u∗ respectively. If u∗ = u∗ , then this is the unique solution in [u, u]; and the two sequences provide lower and upper bounds estimates for u∗ = u∗ . Remark 2.1.14 In contrast to Corollaries 2.1.9 and 2.1.10, we consider an order-reversing compact mapping f on a normal OBS (X, P). Let a ∈ X be a

2.1 Order Method

79

sub- (or super-) solution of both f and f 2 . Setting xi+1 = f (xi ), i = 0, 1, 2, . . ., and x0 = a, we have either x0 x2 x4 · · ·

· · · x3 x1 if a f 2 (a) f (a) ,

or x1 x3 x5 · · ·

· · · x2 x0 if f (a) f 2 (a) a, respectively .

Thus x2n # x and x2n+1 $ x if a f (a) , (x2n+1 # x and x2n $ x if f (a) a respectively.) If the ﬁxed-point set of f , Fix(f ) ∩ S+ (a) (or Fix(f ) ∩ S− (a)) is not empty, then it is contained in [x, x]. Example 5. Let (M, g) be a compact Riemannian manifold without boundary, and let 2 1 ∂i (g ij det(g)∂j ) ∆M = det(g) i,j=1 be the Laplace–Beltrami operator with respect to g. Assume the Gaussian curvature k(x) < 0 ∀x ∈ M . Given a function K(x) < 0 on M , we consider the following equation: M u(x) − k(x) + K(x)e2u(x) = 0 x ∈ M .

(2.6)

This equation arises from geometry. The solution u deﬁnes a conformal metric g! = e2u g. Under the new metric g!, K(x) is the Gaussian curvature of M. We construct a pair of sub- and super-solutions: Choose C > 0 suﬃciently large such that −k(x) + K(x)e2C 0 −k(x) + K(x)e−2C

∀x ∈ M .

Then (−C, C) is a pair of sub- and super-solutions. Statement 2.1.15 Suppose that K, k ∈ C(M ) satisfy K(x) < 0, k(x) < 0 ∀x ∈ M , then equation (2.6) has a unique solution. Proof. Since ϕ(x, u) = −k(x) + K(x)e2u is decreasing in u, one cannot apply Statement 2.1.11 directly. Assume K = max(−K(x)) and set ω = 2Ke2C , x∈M

then the function t → ωt + K(x)e2t is nondecreasing. After Remark 2.1.13, Statement 2.1.11 is applied, and we conclude that there is a smallest and a greatest solution of (2.6) in the order interval [−C, C] of C(M ) for any suﬃciently large constant C > 0. However, since K(x) < 0 and e2u is increasing in u, from the maximum principle, the solution is unique. References can be found in Kazdan [Ka], Kazdan and Warner [KW 1][KW 2] and Aubin [Au 1].

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2 Fixed-Point Theorems

Example 6. Consider the following nonlinear heat equation: ⎧ ∂u k 1 ⎪ ⎨ ∂t − u = λu − au in Ω × R+ u=0 on ∂Ω × R1+ ⎪ ⎩ u(·, 0) = ϕ on Ω × {0}

(2.7)

where λ > 0, a > 1, k > 1 and ϕ ∈ Wp2 (Ω) is nonnegative, for some p > n2 . Notice that one can ﬁnd a large constant M > 0 such that λM − aM k < 0 and max ϕ M . Since ∃ω > 0 such that (λ + ω)u − auk is increasing in the interval [0, M ], one can use the sub- and super-solution method to prove the existence of a positive solution of (2.7). In fact, ∀T > 0 deﬁne a mapping A : g → v = Ag, Lp (Ω × [0, T ]) → 2,1 Wp (Ω × [0, T ]) → C(Ω × [0, T ]) as follows: ⎧ ∂v ⎪ ⎨ ∂t − v + ωv = g v=0 ⎪ ⎩ v(·, 0) = ϕ

in Ω × [0, T ] on ∂Ω × [0, T ] on Ω × {0} .

From the maximum principle for heat equations, A is order preserving. Deﬁne F (u) = (λ + ω)u − auk ; again F is order-preserving in the order interval D = [θ, M ] of the OBS C(Ω × [0, T ]). Setting f = A ◦ F ◦ i, where i is the injection C(Ω × [0, T ]) → Lp (Ω × [0, T ]), f is a compact order-preserving map on D. Now θ f (θ), and f (M ) M . From the parabolic maximum principle, we have a ﬁxed point u = f (u) in [θ, M ]. This is a solution of (2.7). The sub- and super-solutions method is extensively used in the study of nonlinear diﬀerential equations. The analytic basis is the maximum principle. As to systems, the method is also applicable if the associated maximum principle is extended to the linearized operator. However, the construction of a pair of sub- and super-solutions is technical, but crucial. It depends on the special feature of the nonlinear term.

2.2 Convex Function and Its Subdiﬀerentials 2.2.1 Convex Functions Let X be a vector space, a function f : X → R ∪ {+∞} is convex, if ∀x, y ∈ X, ∀λ ∈ [0, 1], we have f (λx + (1 − λ)y) λf (x) + (1 − λ)f (y) . It is called concave, if −f is convex. f is called aﬃne if f is both convex and concave. The domain, the epigraph and the level set of f are deﬁned as follows:

2.2 Convex Function and Its Subdiﬀerentials

81

dom(f ) = {x ∈ X| f (x) < +∞} , epi(f ) = {(x, λ) ∈ X × R1 | f (x) λ} , fa = {x ∈ X| f (x) a} ∀a ∈ R1 . f is said to be proper if f ≡ +∞. By deﬁnition, f is convex ⇐⇒ ∀x1 , . . . , xn ∈ X, ∀λ1 , . . . , λn 0, f

n i=1

λi = 1 ,

i=1

λi xi

n

n

λi f (xi ) ⇐⇒ epi(f ) is convex .

i=1

Also f is convex =⇒ ∀a ∈ R1 , fa is convex, but the converse is not true. A function f is called quasi convex (quasi concave) if fa is convex (concave resp.) ∀a ∈ R1 . For a subset C of X, the indicator function is deﬁned to be 0 if x ∈ C χC (x) = +∞ if x ∈ C . Thus C is convex ⇐⇒ χC is a convex function. The following simple propositions hold: (1) If f and g are convex, then ∀α, β 0 αf + βg is convex. (2) If A is a linear mapping: X → Y and if f : Y → R ∪ {+∞} is convex, then f ◦ A : X → R ∪ {+∞} is convex. (3) If {fι | ι ∈ ∧} is a set of convex functions, then sup{fι | ι ∈ ∧} is convex. We study the continuity of convex functions on Banach spaces. Theorem 2.2.1 Suppose that f : X → R ∪ {+∞} is a convex function on a Banach space X. If f is bounded from above in a neighborhood of a point x, then f is continuous at x. Moreover, f is locally Lipschitzian in int(dom(f )). Proof. (1) We may assume x = θ and f (θ) = 0. From the assumption, ∃η > 0, ∃M > 0 such that (2.8) f (y) M ∀y ∈ Bη (θ) . On one hand, ∀z ∈ Bη (θ)\{θ}, we have ηz z ηz z M f z . f (z) = f η z η z η One the other hand, ∀z ∈ X

(2.9)

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2 Fixed-Point Theorems

η z z+ 0 = f (θ) = f η+ z η+ z η M z f (z) + , η+ z η+ z

η z − z

It follows that

M z . η Combining (2.9) with (2.10), we obtain f (z) −

|f (z)|

M z η

(2.10)

∀z ∈ Bη (θ) .

(2) To prove the second assertion, we ﬁrst show that ∀y ∈ int(dom(f )), f is bounded above in a neighborhood of y, i.e., ∃η1 > 0, ∃M1 > 0 such that f (z) M1

∀z ∈ Bη1 (y) .

(2.11)

To this end one chooses w = (1+t)y such that the segment yw ⊂ int(dom(f )). t η and M1 = max {M, f (w)}, we have f (w) M1 , and Setting η1 = 1+t 1 + t t (z − y) η , if z ∈ Bη1 (y). Since t z= 1+t

w 1+t (z − y) + t 1+t

it follows from the convexity that 1+t 1 t f (z − y) + f (w) M1 . f (z) 1+t t 1+t Thus (2.11) holds. (3) We turn to proving that f is locally Lipschitzian in int(dom(f )). In fact, after (2), we only want to show that |f (z) − f (z )|

M z − z η−δ

∀z, z ∈ Bδ (θ), 0 < δ < η .

Dividing the segment zz into n equal length subsegments yi yi+1 , i = 1, . . . , n with y1 = z, yn+1 = z , and n > z−z η−δ , we have yi+1 ∈ Bη−δ (yi ) ⊂ Bη (θ). Applying (1), it follows that |f (yi+1 ) − f (yi )|

M yi+1 − yi η−δ

Then |f (z) − f (z )|

i = 1, 2, . . . , n .

M z − z . η−δ

2.2 Convex Function and Its Subdiﬀerentials

83

We also need to study lower semi-continuity of functions deﬁned on a topological space X. Deﬁnition 2.2.2 A function f : X → R1 ∪ {+∞} is said to be lower semicontinuous (l.s.c.), if ∀λ ∈ R1 , the level set fλ = {x ∈ X| f (x) λ} is closed. It is called sequentially lower semi-continuous (s.l.s.c.), if for any sequence {xn }, with xn → x ∈ X, we have limf (xn ) f (x) . From the deﬁnition, it follows directly that (1) f is l.s.c. ⇐⇒ epi(f ) is closed in X × (R1 ∪ {+∞}). (2) If {fα |α ∈ ∧} is a family of l.s.c. functions. Then f (x) = sup {fα (x)|α ∈ ∧} is l.s.c. (3) If f, g are l.s.c. and λ, µ 0, then λf + µg is l.s.c. Theorem 2.2.3 Let X be a Banach space and let f : X → R1 ∪ {+∞} be quasi convex. Then (1) f is l.s.c. ⇐⇒ f is weakly lower semi-continuous (i.e., l.s.c. in weak topology). (2) If X ∗ is separable, f is weakly l.s.c. ⇐⇒ f is sequentially weakly l.s.c. (3) If X = Y ∗ , where Y is a separable B-space, then f is w∗ l.s.c. ⇐⇒ f is sequentially w∗ l.s.c. Proof. (1) “⇐=” is trivial; we prove “=⇒”. Since ∀λ ∈ R1 , fλ is convex and closed. By the Hahn–Banach theorem, it is also weakly closed, i.e., f is weakly l.s.c. (2) In this case, the weak topolopy restricted to a normed bounded set is metrizable. (3) In this case, the w∗ -topology restricted to a normed bounded set is metrizable. We collect the following theorems on weak compactness, w∗ -compactness, weak sequential compactness and w∗ -sequential compactness for further reference. Theorem 2.2.4 (Banach–Alaoglu) Let X be a Banach space, and let E ⊂ X ∗ . Then E is w∗ -compact if and only if E is normed bounded and w∗ -closed. In particular, if further, X is reﬂexive, then for any E ⊂ X, E is weakly compact if and only if E is normed bounded and weakly closed. See for instance Larsen [La], pp. 254–257. Theorem 2.2.5 Let E be a weakly closed set in a Banach space, then it is weakly compact if and only if it is weakly sequentially compact. See for instance Larsen [La], pp. 303–309.

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2 Fixed-Point Theorems

2.2.2 Subdiﬀerentials Convex functions on Banach spaces are not diﬀerentiable in general. However, the notion of subdiﬀerentials is introduced as a replacement of derivatives of diﬀerentiable functions. Deﬁnition 2.2.6 (Subdiﬀerential) Let f : X → R ∪ {+∞} be a convex function on a vector space X. ∀x0 ∈ dom(f ), x∗ ∈ X ∗ is called a subgradient of f at x0 if x∗ , x − x0 + f (x0 ) f (x) ∀x ∈ X . The set of all subgradients at x0 is called the subdiﬀerential of f at x0 , and is denoted by ∂f (x0 ). Geometrically, x∗ ∈ ∂f (x0 ) if and only if the hyperplane y = x∗ , x − x0 + f (x0 ) lies below the epigraph of f , i.e., it is a support of epi(f ). y

x∗2

f (x)

f (x0) x∗1 x

x0

x

Fig. 2.1.

Obviously, ∂f (x0 ) may contain more than one point. The following propositions hold, if X is a Banach space. (1) ∂f (x0 ) is a w∗ -closed convex set. (2) If x0 ∈ int(dom(f )), then ∂f (x0 ) = Ø. Proof. We apply the Hahn–Banach separation theorem to the convex set epi(f ): ∃(x∗ , λ) ∈ X ∗ × R1 \{(θ, 0)} such that

2.2 Convex Function and Its Subdiﬀerentials

x∗ , x0 + λf (x0 ) x∗ , x + λt

85

∀(x, t) ∈ epi(f ) .

Since (x0 , f (x0 ) + 1) ∈ epi(f ), it follows that λ 0. However, λ = 0. Otherwise, we would have x∗ , x − x0 0 ∀x ∈ dom(f ). From x0 ∈ int(dom(f )), we conclude that x∗ = θ. It contradicts with (x∗ , λ) = (θ, 0). 1 ∗ x , we obtain x∗0 ∈ ∂f (x0 ). Setting x∗0 = −λ (3) ∀λ 0 ∂(λf )(x0 ) = λ∂f (x0 ). (4) If f, g : X → R1 ∪ {+∞} are convex, then ∀x0 ∈ int(dom(f )∩ dom(g)) ∂(f + g)(x0 ) = ∂f (x0 ) + ∂g(x0 ) . Proof. “⊃” is trivial; we are going to prove “⊂”. We may assume x0 = θ, f (θ) = g(θ) = 0, and θ ∈ ∂(f + g)(θ). We want to show that ∃x∗0 ∈ ∂f (θ) such that −x∗0 ∈ ∂g(θ). Since the set C = {(x, t) ∈ dom(g) × R1 | t −g(x)} is convex, and C ∩ int(epi(f )) = (θ, 0), from the fact that f (x) + g(x) 0. According to the Hahn–Banach separation theorem, ∃(x∗ , λ) ∈ X ∗ × R1 such that x∗ , x + λf (x) 0 x∗ , x + λt

∀(x, t) ∈ C .

In the same manner as the proof of (2), we verify λ > 0. Setting x∗0 = we have x∗0 , x f (x), − x∗0 , x g(x) ∀x ∈ X . i.e., x∗0 ∈ ∂f (θ), and −x∗0 ∈ ∂g(θ).

−x∗ λ ,

(5) If f : X → R ∪ {+∞} is convex, and is G-diﬀerentiable at a point x0 ∈ int(dom(f )), then ∂f (x0 ) is a single point x∗0 satisfying x∗0 , h = df (x0 , h) ∀h ∈ X. 1

Proof. We may assume x0 = θ and f (θ) = 0. Deﬁne the functional on X: L(h) = df (θ, h) . It is homogeneous. From the convexity of f , we have L(h1 + h2 ) L(h1 ) + L(h2 ), ∀ h1 , h2 ∈ X . Then L is linear. Again by the convexity: −f (−h) L(h) f (h)

∀h ∈ X .

Combining with Theorem 2.2.1, L is continuous. Therefore ∃x∗0 ∈ X ∗ such that L(h) = x∗0 , h . It follows that x∗0 ∈ ∂f (θ). Now, suppose x∗ ∈ ∂f (θ), i.e., x∗ , h f (h) ∀h ∈ X, and then x∗ , h 1 ∗ t f (th) ∀h ∈ X, ∀t > 0. By taking limit t → +0, it follows that x , h ∗ ∗ ∗ x0 , h ∀h ∈ X. Then x = x0 .

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2 Fixed-Point Theorems

(6) If f : X → R1 ∪ {+∞} is convex and attains its minimum at x0 , then θ ∈ ∂f (x0 ). Let us present a few examples in computing the subdiﬀerentials of convex functions. Example 1. (Normalized duality map) Let X be a real Banach space, and let f (x) = 12 x 2 . Then ∂f (x) = F (x) := {x∗ ∈ X ∗ | x∗ = x , x∗ , x = x 2 }. It is called the normalized duality map. In the particular case where X is a real Hilbert space, it is known that f (x) = x, and indeed, F (x) = x. However, for Banach spaces, f may not be diﬀerentiable. We verify ∂f (x) = F (x) as follows: “⊂” ∀x∗ ∈ ∂f (x), we have x∗ , y − x

1 ( y 2 − x 2 ) , 2

and then ∀h ∈ X, x∗ , h

1 t ( x + th 2 − x 2 ) x h + h 2 , 2t 2

as t > 0. It follows that x∗ , h x h , and then x∗ x . On the other hand, setting y = λx, we have (λ−1) x∗ , x 12 (λ2 −1) x 2 . Dividing by (λ − 1) as λ ∈ (0, 1), and then letting λ = 1 we obtain x∗ , x ≥ x 2 ; it follows that x x∗ . Thus x = x∗ , and x∗ , x = x2 . “⊃” ∀x∗ ∈ F (x), one has x∗ , y − x = x∗ , y − x 2 x∗ y − x 2 1 ( y 2 − x 2 ) 2 = f (y) − f (x) ∀y ∈ X, i.e., x∗ ∈ ∂f (x). Example 2. Let C be a convex subset of X. The support function of C is deﬁned to be SC (x∗ ) = sup x∗ , x . Geometrically, SC (x∗ ) = x∗ , x0 if and x∈C

only if {x ∈ X| x∗ , x = SC (x∗ )} is a support hyperplane of C at x0 . We consider the subdiﬀerential of the indicator function of C: ⎧ ◦ ⎪ if x0 ∈ C ⎨{θ} ∂χC (x0 ) = {x∗ ∈ X ∗ | SC (x∗ ) = x∗ , x0 } if x0 ∈ ∂C ⎪ ⎩ Ø if x0 ∈ C .

2.3 Convexity and Compactness

87

Example 3. Let β : R1 → R1 be a monotone nondecreasing function. Then

t ϕ(t) = β(s)ds 0

is a continuous convex function, and ∂ϕ(t0 ) = [β(t0 − 0), β(t0 + 0)] ∀t0 ∈ R1 . If further, ∃p > 1, C1 , C2 > 0 such that |β(t)| C1 + C2 |t|p−1 , then the functional

ϕ(u(x))dx J(u) = Ω

is a convex functional on Lp (Ω), where Ω is a measurable set in Rn with bound measure. We conclude that ∂J(u0 ) = [β(u0 (x) − 0), β(u0 (x) + 0)]

:= {v ∈ Lp (Ω)| β(u0 (x) − 0) v(x) β(u0 (x) + 0) ∀ a.e. Ω} where

1 p

+

1 p

= 1. In fact,

v ∈ ∂J(u0 ) ⇐⇒

v(x)(u(x) − u0 (x))

Ω

[ϕ(u(x)) − ϕ(u0 (x))] ∀u ∈ Lp (Ω) Ω

⇐⇒ v(x)(u(x) − u0 (x)) ϕ(u(x)) − ϕ(u0 (x)) a.e. ⇐⇒ v(x) ∈ ∂ϕ(u0 (x)) a.e.

2.3 Convexity and Compactness The Schauder ﬁxed-point theorem is one of the most important ﬁxed-point theorems in nonlinear analysis. In this section we shall study a series of ﬁxedpoint theorems on compact convex set, including the Schauder ﬁxed-point theorem and its extensions. All these theorems are set up by considering the Ky Fan inequality and KKM mapping from the outset. Deﬁnition 2.3.1 Suppose that X is a vector space, E ⊂ X is a subset of X. A set-valued mapping G : E → 2X is called a KKM mapping, if ∀x1 , . . . , xn ∈ n E, conv{x1 , . . . , xn } ⊂ ∪ G(xi ). i=1

Knaster, Kuratowski, and Mazurkiweicz (1929) discovered the following (KKM) theorem: Let [p0 , . . . , pn ] be an n-simplex generated by n + 1 points p0 , p1 , . . . , pn in a vector topological space, and let M0 , . . . , Mn be (n + 1) k

closed sets, satisfying [pi0 , . . . , pik ] ⊂ ∪ Mij ∀ index subset {i0 , . . . , ik } ⊂ j=0

n

{0, 1, . . . , n}. Then ∩ Mi = Ø. i=0

The following version of the KKM theorem is due to Ky Fan.

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2 Fixed-Point Theorems

G( x1 ) x1 G ( x3 )

x3

x2 G ( x2 ) Fig. 2.2.

Theorem 2.3.2 (FKKM) Suppose that X is a locally convex Hausdorﬀ space (LCS), E ⊂ X, and that G : E → 2$X is a closed-valued KKM map. If ∃x0 ∈ E such that G(x0 ) is compact, then x∈E G(x) = Ø. Proof. Deﬁne F (x) = G(x) ∩ G(x0 ). Then {F (x)| x ∈ E} is a family of closed subsets of G(x0 ). If one can show the ﬁnite intersection property of {F (x)| x ∈ E}, then our conclusion is proved. To this end, we only want to verify the ﬁnite intersection property for n {G(x)| x ∈ E}. If not, ∃{x1 , . . . , xn } ⊂ E, such that ∩ G(xi ) = Ø. Let i=1

us introduce the Euclidean metric d on L = span{x1 , . . . , xn }. Since L is ﬁnite dimensional, the metric derives the same topology on L, from the following facts: Each ﬁnite-dimensional LCS is normable (a consequence of the Kolmogorov normable theorem), and all norms on a ﬁnite-dimensional space are equivalent. Then the function λ(x) =

n

d(x, L ∩ G(xi )) > 0

∀x ∈ L .

i=1

Deﬁne βi (x) =

1 d(x, L ∩ G(xi )) i = 1, . . . , n . λ(x)

∀x ∈ L, we have βi (x) > 0 ⇐⇒ x ∈ G(xi ) i = 1, · · · , n . Again, deﬁne C = conv{x1 , . . . , xn } and a map ϕ : C → C to be ϕ : x →

n

βi (x)xi .

i=1

According to the Brouwer ﬁxed-point theorem, ∃x0 ∈ C such that x0 = ϕ(x0 ). Letting I(x0 ) = {i| βi (x0 ) > 0}, we have

2.3 Convexity and Compactness

89

ϕ(x0 ) ∈ conv{xi | i ∈ I(x0 )} . But x0 ∈ G(xi ) ∀i ∈ I(x0 ) implies x0 ∈

∪

i∈I(x0 )

G(xi ). This contradicts with

the KKM mapping.

Ky Fan’s inequality is the foundation of this section, by which all ﬁxedpoint theorems are derived. Theorem 2.3.3 (Ky Fan’s inequality) Suppose that X is an LCS and that E ⊂ X is a nonempty convex set. Assume φ : E × E → R1 satisfying (1) ∀y ∈ E x → φ(x, y) is l.s.c., (2) ∀x ∈ E y → φ(x, y) is quasi-concave, (3) ∃y0 ∈ E such that {x ∈ E| φ(x, y0 ) sup φ(x, x)} is compact . x∈E

Then ∃x0 ∈ E such that sup φ(x0 , y) sup φ(x, x) .

y∈E

x∈E

Before going to the proof of Theorem 2.3.3, we ﬁrst introduce a geometric version of the statement. Given a set E, let A ⊂ E × E be a subset. Deﬁne the sections: ∀x ∈ E, A1 (x) = {y ∈ E| (x, y) ∈ A} , and ∀y ∈ E, A2 (y) = {x ∈ E| (x, y) ∈ A} . Denote the diagonal {(x, x)| x ∈ E} by . A1 (x)

y

x

A2 (y) Fig. 2.3.

In this manner, we deﬁne two set-valued mappings: E → 2E , A1 : x → A1 (x) and A2 : y → A2 (y) .

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2 Fixed-Point Theorems

Theorem 2.3.4 Suppose that X is an LCS and that E ⊂ X is a nonempty convex set. Assume that Γ ⊂ E × E is a subset satisfying (1 ) (2 ) (3 ) (4 )

∀y ∈ E, Γ2 (y) is open, ∀x ∈ E, Γ1 (x) is convex, ∩ Γ = Ø, ∃y0 ∈ E such that E\Γ2 (y0 ) is compact.

Then ∃x0 ∈ E such that Γ1 (x0 ) = Ø. We are going to show that Theorem 2.3.3 ⇐⇒ Theorem 2.3.4. Proof. “ =⇒” Deﬁne a function on E × E: 1 (x, y) ∈ Γ φ(x, y) = 0 (x, y) ∈ Γ ∀y ∈ E, we see ⎧ ⎪ ⎨E {x ∈ E| φ(x, y) > λ} = Γ2 (y) ⎪ ⎩ Ø

if λ < 0 if 0 λ < 1 if 1 λ ,

then x → φ(x, y) is l.s.c., from (1 ). Similarly, ∀x ∈ E ⎧ ⎪ if λ < 0 ⎨E {y ∈ E| φ(x, y) > λ} = Γ1 (x) if 0 λ < 1 ⎪ ⎩ Ø if 1 λ , then y → φ(x, y) is quasi-concave, from (2 ). Since ∩ Γ = Ø, φ(x, x) = 0 ∀x ∈ E. The set {x ∈ E| φ(x, y0 ) 0} = E\Γ2 (y0 ) is compact, from (4 ). Theorem 2.3.3 is applied to conclude that ∃x0 ∈ E such that φ(x0 , y) 0 ∀y ∈ E, i.e., Γ1 (x0 ) = Ø. “ ⇐=” Deﬁne Γ = {(x, y) ∈ E × E| φ(x, y) > µ} , where µ = supx∈E φ(x, x). Then (1) and (2) imply (1 ) and (2 ) respectively. By deﬁnition Γ ∩ = Ø. Also E\Γ2 (y0 ) = {x ∈ E| φ(x, y0 ) µ} is compact. We then apply Theorem 2.3.4, and conclude that ∃x0 ∈ E such that Γ1 (x0 ) = Ø, i.e., φ(x0 , y) µ ∀y ∈ E.

2.3 Convexity and Compactness

91

Now we turn to giving a proof of Theorem 2.3.4. Proof. Deﬁne a set-valued mapping G : y → E\Γ2 (y) . From (1 ), it is a closed set-valued mapping. (4 ) implies that G(y0 ) is compact. If we can show that G is a KKM map, then the FKKM theorem is applied to conclude that ∩ G(y) = Ø, i.e., y∈E

∃x0 ∈ ∩ G(y) ⇐⇒ (x0 , y) ∈ (E × E)\Γ ∀y ∈ E ⇐⇒ Γ1 (x0 ) = Ø . y∈E

Veriﬁcation of the KKM map. If not, ∃{y1 , . . . , yn } ⊂ E and ∃λ1 ≥ 0, . . . , λn ≥ 0, with Σn1 λi ≤ 1, such that w=

n

n

n

i=1

i=1

λi yi ∈ ∪ G(yi ) ⇐⇒ w ∈ ∩ Γ2 (yi ) ,

i=1

⇐⇒ yi ∈ Γ1 (w) ∀i , by (2 )

=⇒ w ∈ Γ1 (w) , =⇒ ∩ Γ = Ø .

This contradicts with (3 ).

First we mention that this theorem is very useful in game theory. In particular, the Nash equilibrium theorem and Von Neumann–Sion saddle point theorem can be derived from it. In an n-person game, there are n players {1, 2, . . . , n}. The ith person has a strategy set Ei and a payoﬀ function fi , i = 1, 2, . . . , n. The payoﬀ functions depend on the strategies of all players, i.e., fi , i = 1, 2, . . . , n are functions of x = (x1 , . . . , xn ), where xj ∈ Ej , j = 1, . . . , n. Nash’s solution of the game is such a strategy (x∗1 , . . . , x∗n ) ∈ E1 × · · · × En : no player has any incentive to change his strategy as long as his enemies don’t change theirs. Mathematically, let X1 , . . . , Xn be n-LCS, let Ei ⊂ Xi be nonempty compact convex sets, ∀i = 1, . . . , n, and let f1 , . . . , fn be n continuous functions on E = E1 × · · · ×En . A point x∗ = (x∗1 , . . . , x∗n ) ∈ E is called a Nash equilibrium if fi (x∗ ) fi (x∗1 , . . . , yi , . . . , x∗n ) ∀i = 1, 2, . . . , n ∀y = (y1 , . . . , yn ) ∈ E . Corollary 2.3.5 (Nash) If the continuous functions xi → fi (x1 , . . . , xi , . . . , xn )

∀i = 1, . . . , n

are quasi-concave ∀(x1 , . . . , x ˆi , . . . , xn ) ∈ Ej ∀i. Then the Nash equilibrium j=i

exists.

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2 Fixed-Point Theorems

In fact, let us deﬁne a function on E × E: φ(x, y) =

n

(fi (x1 , . . . , yi , . . . , xn ) − fi (x1 , . . . , xn ))

i=1

It is easy to verify that all the assumptions of the Ky Fan inequality are satisﬁed (in particular, y → φ(x, y) is quasi-concave), and φ(x, x) = 0 ∀x ∈ E. Then ∃x∗ ∈ E satisfying φ(x∗ , y) 0 ∀y ∈ E. This is the Nash equilibrium. Corollary 2.3.6 (Von Neumann–Sion) Suppose that X, Y are reﬂexive Banach spaces, E ⊂ X and F ⊂ Y are nonempty closed convex sets. Assume that f : E × F → R1 satisﬁes (1) ∀y ∈ F , x → f (x, y) is l.s.c. and quasi-convex. (2) ∀x ∈ E, y → f (x, y) is u.s.c. and quasi-concave. (3) ∃(x0 , y0 ) ∈ E ×F such that f (x0 , y) → −∞ as y → +∞ and f (x, y0 ) → +∞ as x → +∞. Then there exist (x∗ , y ∗ ) ∈ E × F such that f (x∗ , y) f (x∗ , y ∗ ) f (x, y ∗ )

∀(x, y) ∈ E × F .

(2.12)

Proof. Deﬁne φ(u, v) = f (x, w) − f (z, y) , where u = (x, y), v = (z, w) ∈ E × F . Then we have the following: From Theorem 2.2.3, ∀v ∈ E × F, u → φ(u, v) is w.l.s.c. It is easy to verify directly that∀u ∈ E × F, v → φ(u, v) is quasi-concave, φ(u, v0 ) = f (x, y0 ) − f (x0 , y) → +∞ as x + y → +∞, where v0 = (x0 , y0 ) . The set {u ∈ E × F | φ(u, v0 ) 0} is weakly compact. We apply Ky Fan’s inequality to the space E × F with weak topology. There exists u∗ = (x∗ , y ∗ ) such that f (x∗ , w) − f (z, y ∗ ) 0 ∀(z, w) ∈ E × F , which implies f (x∗ , y) f (x∗ , y ∗ ) f (x, y ∗ ) ∀(x, y) ∈ E × F . The solution (x∗ , y ∗ ) is called the saddle point of f . As a special case where dim Y = 0, we have:

2.3 Convexity and Compactness

93

Corollary 2.3.7 Suppose that X is a reﬂexive Banach space. If f : X → R1 ∪ {+∞} is an l.s.c. convex function, satisfying f (x) → +∞ as x → ∞, then there exists x0 ∈ X such that f (x0 ) = min {f (x)| x ∈ X}. In the case where E and F are compact, the Von Neumann–Sion theorem is also rewritten as follows: min max f (x, y) = max min f (x, y) . x∈E y∈F

y∈F x∈E

(2.13)

Indeed, (2.12) ⇐⇒ (2.13). Proof. “=⇒” We always have β := max min f (x, y) min max f (x, y) =: α . y∈F x∈E

x∈E y∈F

However, (2.12) implies α max f (x∗ , y) f (x∗ , y ∗ ) min f (x, y ∗ ) β . y∈F

x∈E

Therefore α = β, and then (2.13) holds. “⇐=” Since both E and F are compact and the functions x → f (x, y) is l.s.c., y → f (x, y) is u.s.c., the functions x → maxy∈F f (x, y) is l.c.s, and y → minx∈E f (x, y) is u.s.c., we have (x∗ , y ∗ ) ∈ E × F such that min f (x, y ∗ ) = max min f (x, y) x∈E

y∈F x∈E

= min max f (x, y) x∈E y∈F

= max f (x∗ , y) . y∈F

Since

min f (x, y ∗ ) ≤ f (x∗ , y ∗ ) ≤ max f (x∗ , y) , x∈E

these three are equal. It (2.12) follows.

y∈F

In this sense, the Von Neumann–Sion theorem is called the mini-max theorem, which is the fundamental theorem in two-person game theory, and has many applications in other ﬁelds of mathematics (linear programming, convex programming, potential theory, the dual variational theory in mechanics etc.) Next, we turn to ﬁxed-point theorems. For a set-valued mapping Γ : E → 2F , where E, F are Hausdorﬀ topological vector spaces, we say Γ is upper (or lower) semi-continuous (u.s.c. or l.s.c., respectively for short), if for any closed (open resp.) set W ⊂ F , the pre-image Γ−1 (W ) = {x ∈ E| Γ(x) ∩ W = Ø} is closed (open resp.). For single-valued mappings, both u.s.c. and l.s.c. are all reduced to continuity. An alternative description of the upper semi-continuity is that for any open set W ⊂ F , the set {x ∈ E| Γ(x) ⊂ W } is open. A point x0 ∈ E is called a ﬁxed point of Γ if x0 ∈ Γ(x0 ).

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2 Fixed-Point Theorems

Theorem 2.3.8 (Ky Fan–Glicksberg) Suppose that X is an LCS, and that E ⊂ X is a nonempty compact convex set. If Γ : E → 2E is u.s.c., and ∀x ∈ E, Γ(x) is a nonempty closed convex subset of E, then ∃x0 ∈ E such that x0 ∈ Γ(x0 ). Proof. We prove by contradiction. If ∀x ∈ E, x ∈ Γ(x), then from the Hahn– Banach theorem ∃x∗ ∈ X ∗ ∃t ∈ R1 such that x∗ , y < t < x∗ , x

∀y ∈ Γ(x) .

Since Γ is u.s.c. there exists a neighborhood U (x) of x such that x∗ , y < t < x∗ , z

∀y ∈ Γ(z) ∀z ∈ U (x) . n

Since E is compact, we have x1 , . . . , xn ∈ E such that E ⊂ ∪ U (xi ). Let {x∗1 , . . . , x∗n } ⊂ X ∗ be the associated elements; one has x∗i , y < x∗i , x

i=1

∀y ∈ Γ(x) ∀x ∈ U (xi ) .

We construct a partition of unity of E: {βi : E → R1 , i = 1, . . . , n| βi 0,

n

βi = 1 on E and βi (x) > 0 iﬀ x ∈ U (xi )} .

i=1

Deﬁne a continuous map f : E → X ∗ by f (x) =

n

βi (x)x∗i ,

i=1

one has f (x), y =

n

βi (x) x∗i , y

0, substituting y 0 = f (x0 ) − ε, z 0 = f (x), y i = −ε, z i = gi (x) i = 1, . . . , n into (2.17) ∀x ∈ X, and letting ε → 0, we have

ω0 f (x0 )

n

ωi gi (x) + ω0 f (x)

∀x ∈ X .

(2.18)

i=1 n

Claim ω0 > 0. Otherwise, ω0 = 0, thus i=1 ωi gi (x) 0. Since we have assumed gi (x∗ ) < 0, combining with the fact that ωi 0, i = 0, 1, . . . , n, it follows that ωi = 0, i = 1, . . . , n. This is impossible. Dividing (2.18) by ω0 , we obtain f (x0 )

n

ξj gj (x) + f (x)

∀x ∈ X ,

j=1

where ξj =

ωj ω0 .

Then n

ξj gj (x0 ) = 0 ,

j=1

it follows that ξj gj (x0 ) = 0 j = 1, . . . , n. Thus f (x0 ) +

n

yj gj (x0 ) f (x0 ) +

j=1

n

ξj gj (x0 ) f (x) +

j=1

n

ξj gj (x)

j=1

∀(x, y) ∈ X × Rn+ , i.e., (x0 , ξ) is a saddle point of L. Statement 2.3.13 In the convex programming problem, assume (1) f (x) → +∞ as x → ∞, (2) ∃x0 ∈ X such that gi (x0 ) < 0, i = 1, 2, . . . , n. The problem (2.16) has a solution. Proof. Consider the Lagrangian on X × Rn+ , L(x, y) = f (x) +

n

yi gi (x) .

i=1

Obviously, ∀y ∈ Rn+ , x → L(x, y) is l.s.c. and convex, ∀x ∈ X, y → L(x, y) is linear , and by (2) and (1),

98

2 Fixed-Point Theorems

L(x0 , y) → −∞ as y → ∞ , L(x, θ) = f (x) → +∞ as x → ∞ . Applying Corollary 2.3.6, there is a saddle point of L, which is the solution of the problem (2.15), according to Statement 2.3.12. 2. Periodic solutions of ODE We study the 2π-periodic solutions of the following ODE: x ¨ + g(x) = f (t)

(2.19)

where g ∈ C(R1 ) and f ∈ L2 ([0, 2π]). Assume that ∃n ∈ N and ε > 0 such that g(x) (n + 1)2 − ε for |x| large . (2.20) n2 + ε x Statement 2.3.14 The problem (2.19) has a 2π-periodic solution under the assumption (2.20). Proof. Let γ = n2 + n + 1/2. Then ∃R > 0 such that 1 |g(x) − γx| n + − |x| as |x| R . 2 Rewrite (2.19) as x ¨ + γx + g(x) − γx − f (t) = 0

(2.21)

and deﬁne the linear operator L : x → −(¨ x + γx) with domain D(L) = {x ∈ H 2 (0, 2π)| x(0) = x(2π), x(0) ˙ = x(2π)}. ˙ Then L has a bounded inverse: L−1 : L2 (0, 2π) → D(L). Since the injection D(L) → L2 (0, 2π) is compact, (2.21) is equivalent to x = L−1 · G(x) , where G : x → g(x) − γx − f is a bounded continuous mapping on L2 (0, 2π). Then F = L−1 ◦ G is a compact map on L2 (0, 2π). 1 , Gx (n + 12 − ε) x + C, where Moreover L−1 n+1/2 √ C = 2π max |g(x) − γx|+ f 2 . Setting R1 Cε , we have F : B R1 (θ) → |x|R

B R1 (θ). The Schauder ﬁxed-point theorem is applied to ensure the existence of x0 ∈ BR1 (θ) satisfying F (x0 ) = x0 , thus x0 ∈ D(L) and satisﬁes (2.21). 3. Obstacle problem Find the equilibrium position u of a membrane with a ﬁxed boundary, acted upon by an external force f , and obstructed by an ﬁxed obstacle ψ.

2.3 Convexity and Compactness

99

The problem is posed as a minimizing problem on a closed convex set. Let Ω ⊂ Rn be a bounded domain, f ∈ L2 (Ω), ψ ∈ H 1 (Ω). Find u ∈ H01 (Ω) such that u is the minimizer of the problem: min {J(v)| v ∈ E}, where

1 |∇v|2 − f · v , J(v) = 2 Ω and

& % E = v ∈ H01 (Ω)| v(x) ψ(x) a.e. .

From J(u) J(u + t(v − u)) ∀v ∈ E and t ∈ (0, 1), and letting t → 0, we obtain the following variational inequality:

∇u · ∇(v − u) f (v − u) ∀v ∈ E . (2.22) Ω

Ω

The existence of a weak solution for (2.22) can be studied by variational method, see Chap. 4. However, we present here a ﬁxed-point approach. First we reduce the problem to a system of inequalities. One has u ψ a.e. ,

(2.23)

Denote the positive cone of H01 (Ω) by C = {w ∈ H01 (Ω)| w(x) 0 a.e.}. Let v∈ H01 (Ω), and v ≤ u, then v ∈ E, and let w = u − v, then (2.22) implies that Ω ∇u∇w Ω f w ∀w ∈ C i.e., −u − f 0 a.e. ,

(2.24)

Substituting v = ψ and 2u − ψ, respectively into (2.22), we obtain

(−u − f ) · (ψ − u) ≥ 0 , Ω

and

(−u − f ) · (u − ψ) ≥ 0 , Ω

respectively, therefore Ω (−u − f )(u − ψ) = 0. If u ∈ Wp2 (Ω) then, (−u − f )(u − ψ) = 0 a.e.

(2.25)

Conversely, from (2.24), (2.25) and v ≤ ψ, we obtain Ω (−∆u−f )(v−u) ≥ 0, i.e., (2.22) Thus, for u ∈ Wp2 ∩ H01 (Ω), it is a solution of the variational inequality (2.22) if and only if it satisﬁes the system of inequalities (2.23), (2.24), (2.25). Second, the system of inequalities is equivalent to a PDE with discontinuous nonlinearity, i.e., for ψ ∈ Wp2 (Ω) the system of inequalities (2.23), (2.24), (2.25) is again equivalent to

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2 Fixed-Point Theorems

min {f (x), −ψ(x)} −u = f (x)

if u(x) ψ(x) if u(x) < ψ(x) .

(2.26)

Veriﬁcation: “=⇒” On the set {x ∈ Ω| u(x) < ψ(x)} the equality follows from (2.25). On the complement set {x ∈ Ω| u(x) = ψ(x)} we have −ψ(x) = −∆u(x) f (x) a.e., and then −∆u(x) = −∆ψ(x) = min {f (x), −ψ(x)}. “⇐=” (2.24) is trivial. If the set U = {x ∈ Ω| u(x) > ψ} = ∅, (2.26) implies that −(u − ψ) 0 on U . This contradicts the maximum principle; then (2.23) holds, and consequently (2.25). Now, (2.26) is a PDE with discontinuous nonlinearity: −u = φ(x, u) where

min {f (x), −ψ(x)} φ(x, u) = f (x)

(2.27) if u ψ(x) if u < ψ(x) .

We shall solve the equation by the Ky Fan ﬁxed-point theorem. First, let us deﬁne a set-valued mapping p

F : C(Ω) → 2L in Lp (Ω) ,

(Ω)

: u → F u = [φ(x, u(x)), φ(x, u(x))], the order interval

where φ(x, t) = lim φ(x, t ), and φ(x, t) = lim φ(x, t ). t →t

t →t

We consider the following maps: F

p

C(Ω) −→ 2L

(Ω)

K

2

1

j

−→ 2Wp ∩H0 (Ω) −→ 2C(Ω) ,

where K = (−)−1 and j is the embedding map. It is not diﬃcult to verify that Γ := j ◦K◦F is u.s.c., and ∀u ∈ C(Ω), Γ(u) is a nonempty closed convex set of C(Ω). Second, we consider the set E = K(B R (θ)) of C(Ω), where R = max { f p , ψ p }. It is a nonempty compact convex set in C(Ω), and Γ : E → 2E . Ky Fan’s ﬁxed-point theorem is applied to ensure the existence of u0 ∈ E such that u0 ∈ Γ(u0 ), i.e., u0 ∈ Wp2 ∩ H01 (Ω) satisfying ⎧ ⎪ ⎨min {f (x), −ψ(x)} −u0 (x) ∈ [min {f (x), −ψ(x)}, f (x)] ⎪ ⎩ f (x)

if u0 (x) > ψ(x) if u0 (x) = ψ(x) if u0 (x) < ψ(x) .

Third, however, on the set U = {x ∈ Ω| u0 (x) = ψ(x)}, −u0 (x) = −ψ(x) a.e. This implies that −ψ(x) f (x) a.e., i.e.,

2.3 Convexity and Compactness

min {f (x), −ψ(x)} −u0 (x) = f (x)

101

if u0 (x) ψ(x) if u0 (x) < ψ(x) .

This is exactly the solution of the PDE with discontinuous nonlinearity (2.26). Statement 2.3.15 If ψ ∈ Wp2 (Ω), f ∈ Lp (Ω), p > ◦

n 2,

then there exists

u ∈ Wp2 ∩ Wp1 (Ω) satisfying (2.26). The obstacle problem is a free boundary problem for PDEs. The boundary where the membrane detaches from the obstacle is unknown, and is to be determined simultaneously with the position of the membrane. A kind of free boundary problem, having the feature that the boundary values and the normal derivatives of the unknown function coincide on both sides of the unknown free boundary, can be studied via the above approach, i.e., pose the free boundary problem as a PDE with discontinuous nonlinearity, in which the unknown boundary is automatically absorbed into the unknown function. Fixed-point theorems for set-valued mappings can be used to study the existence of the solutions. Examples of the obstacle problem are the seepage surface in dams, the free boundary of conﬁned plasma, water cones in oil reservoirs etc. (see K.C. Chang [Ch 1,2]). 4. Stefan problem Consider a container Ω with boundaries Γ and Γ , ﬁlled with ice. Cooling on Γ and heating on Γ (see Fig. 2.4), we study the melting process.

Γ ’’ C(t) Γ’ Ω

Fig. 2.4.

At time t, the domain occupied by water is denoted by C(t), and let θ(x, t) 0 be the temperature at the space–time (x, t) as x ∈ C(t). Both θ and C are unknown, but g(x, t) = θ(x, t)|Γ ×[0,T ] is given ∀T > 0. Let S(t) = ∂C(t) be the unknown boundary. Let us suppose that S(t) is given by an equation σ(x) = t. The temperature distribution is governed by the following equations:

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2 Fixed-Point Theorems

⎧ ∂ ⎪ ⎪ ∂t θ − θ = 0 ⎪ ⎪ ⎪ ⎪ ⎨θ(x, t)|Γ ×[0,T ] = g(x, t), θ(x, t)|S(t) = 0 ⎪ ⎪ ⎪ ∇θ · ∇σ|S(t) = −L, ⎪ ⎪ ⎪ ⎩θ(x, 0) = 0

∀t ∈ [0, T ], ∀x ∈ C(t), ∀t ∈ [0, T ],

(2.28)

∀x ∈ Ω,

where L > 0 is a constant. The boundary conditions on S(t) are derived from the continuity of the temperature and the conservation of the energy, respectively. We set θ(x, t) if x ∈ C(t) ! θ(x, t) = 0 if x ∈ C(t) , and

t

u(x, t) =

! s)ds . θ(x,

0

The latter is called the modiﬁed Baiocchi transformation of θ. Let

t ψ(x, t) = g(x, s)ds . 0

The problem (2.28) is reduced to a parabolic equation with discontinuous nonlinearity: ⎧ ∂u ⎪ ∂t − u = −LH(u) in QT := Ω × (0, T ) , ⎪ ⎪ ⎨u| Γ ×[0,T ] = ψ , (2.29) ⎪ u|Γ ×[0,T ] = 0 , ⎪ ⎪ ⎩ u(x, 0) = 0, on Ω . Again, the advantage of the formulation (2.29) is that the unknown domain C(t) is implicitly involved. The price we paid is the nonlinear term being a discontinuous nonlinear function: the Heaviside function H(u) = 1 as u > 0, and = 0 as u 0. The problem can be solved by the Ky Fan ﬁxed-point theorem as before. Let us introduce the anisotropic Sobolev space Wp1,2 (QT ) = {u ∈ Lp (QT )| ∂t u, ∇x u, ∇2x u ∈ Lp (QT )} , with norm: u Wp1,2 = u p + ∂t u p + ∇2x u p . The boundary values of functions in Wp1,2 (QT ) span a fractional Sobolev 1 1 1− 2p ,2− p

space Wp

(Γ × (0, T )), where Γ = Γ ∪ Γ . In particular, C 1,2 (Γ × 1 1 1− 2p ,2− p

[0, T ]) = {u ∈ C(Γ×[0, T ])| ∂t u, ∇x u ∇2x u ∈ C(Γ×[0, T ])} ⊂ Wp (0, T )).

(Γ×

2.3 Convexity and Compactness 1−

1

103

,2− 1

p Statement 2.3.16 Assume ψ ∈ Wp 2p (Γ × (0, T )), p > n, and ψ 0; the equation (2.29) has a unique solution u ∈ Wp1,2 (QT ).

Proof. Deﬁne a set-valued mapping ⎧ ⎪ ⎨1 u → F (u) = [0, 1] ⎪ ⎩ 0

u(x, t) > 0 , u(x, t) = 0 , u(x, t) < 0 ,

p

then F : C(QT ) → 2L (QT ) is u.s.c. Let Kψ : Lp (QT ) → Wp1,2 (QT ) be the aﬃne operator f → u, where u satisﬁes the linear parabolic equation: ⎧ ∂u − u = f in QT , ⎪ ⎪ ⎪ ∂t ⎨ u|Γ ×(0,T ) = ψ , ⎪ u|Γ ×(0,T ) = 0, ⎪ ⎪ ⎩ u(x, 0) = 0, on Ω . Then the set-valued mapping Γ := Kψ ◦ (L ◦ F ) : C(Ω) → 2C(Ω) is a u.s.c. closed convex set-valued mapping. Deﬁne a compact convex set: 1

E = {Kψ u| u ∈ Lp (QT ), u p Lm(Ω) p } . Obviously Γ : E → 2E . According to Ky Fan’s ﬁxed-point theorem, one ﬁnds u ∈ Wp1,2 (QT ) satisfying:

− u ∈ −LF (u) u|Γ ×(0,T ) = ψ, u|Γ ×(0,T ) = 0, u|Ω×{0} = 0 . ∂u ∂t

On the set u−1 (0) = {(x, t) ∈ QT | u(x, t) = 0}, we have ∂u ∂t − u = 0. This means u automatically satisﬁes (2.29). Now, we prove the uniqueness of the solution u of equation (2.29). In fact, let u1 and u2 be two solutions of (2.29), and u = u1 − u2 , then ∂u ∂t − u = −L(H(u1 ) − H(u2 )) in QT , u|∂QT \(Ω×{T }) = 0 . By the maximum principle for parabolic equations, both sets {(x, t) ∈ QT | ± u(x, t) > 0} are null sets. Thus u1 = u2 a.e. Returning to the original problem (2.28), we deﬁne C(t) = {x ∈ Ω| u(x, t) > ! C(t) ∀t ∈ (0, T ). ! t) = ∂ u(x, t), and θ(·, t) = θ| 0}, θ(x, ∂t The veriﬁcation of (2.28) is omitted.

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2 Fixed-Point Theorems

2.4 Nonexpansive Maps Let X be a Banach space, and E be a closed subset of X. A map T : E → X is called a nonexpansive map, if T (x) − T (y) x − y

∀x, y ∈ E .

In contrast with the fact that a contraction mapping T : E → E, (i.e., ∃α ∈ (0, 1) such that T (x) − T (y) α x − y ) always has a ﬁxed point, the nonexpansive map does not in general. This can be seen by the following example: Example 1. X = c0 , the space consisting of all sequences x = {ξi }∞ 1 converging to zero, with the norm x = max |ξi | . i1

Let T (x) = (1− x , ξ1 , ξ2 , . . .) , and E be the closed unit ball. Then T : E → E is non-expansive: T (x) − T (y) = max | x − y |, max |ξi − ηi | x − y . i1

However, T does not have a ﬁxed point. In fact, if x = T (x), then ξ1 = ξ2 = . . . = 1− x . This is impossible. We are going to study the ﬁxed points for a nonexpansive map on a closed convex subset of a Banach space X with a normal structure, which means that for every closed convex subset E of X containing at least two points, there exists a point x0 ∈ E such that sup x − x0 < diam (E) .

(2.30)

x∈E

It is easy to see that Hilbert spaces are Banach spaces with normal structure: Assume a, b ∈ E, then ∀x ∈ E, by parallelogram identity, 2 2 x − a + b + a − b = 1 ( x − a 2 + x − b 2 ) diam (E) . 2 2 2 Taking x0 = 12 (a + b), (2.30), is satisﬁed. Now, for any closed bounded convex subset E, one deﬁnes a number rE = inf sup x − y . x∈E y∈E

By deﬁnition, rE diam (E). If further, X is a Banach space with a normal structure, then rE < diam (E).

2.4 Nonexpansive Maps

105

Lemma 2.4.1 If X is a reﬂexive Banach space and if E is a bounded closed convex set, then ∃x0 ∈ E such that sup x0 − y = rE .

y∈E

Proof. The function f (x) = sup x − y is convex, l.s.c., and coercive. We y∈E

now apply Corollary 2.3.7; there exists x0 ∈ E such that f (x0 ) = minx∈E f (x). Setting G(y) = {x ∈ E| y − x rE } ∀y ∈ E, we deﬁne ct (E) = ∩ G(y) . y∈E

Provided by Lemma 2.4.1, x0 ∈ ct (E). Therefore ct (E) is a nonempty closed convex subset of E. It is called the center of E. By deﬁnition, we have sup

x − y ≤ rE .

(2.31)

x,y∈ct(E)

Theorem 2.4.2 Suppose that E is a bounded closed convex set of a reﬂexive Banach space with a normal structure, and that T : E → E is nonexpansive, then the ﬁxed-point set of T Fix (T ) is closed and nonempty. Proof. (1) Let X be the set of all nonempty closed convex subsets of E. Deﬁne the partial order on X by inclusion: C1 C2 if C2 ⊂ C1

∀C1 , C2 ∈ X .

Then (X, ) is an ordered set, in which every chain X1 has a supremum: ∩ C. The intersection is not empty, because E is weakly compact and C is C∈X1

weakly closed. (2) Deﬁne f (C) = ct (conv(T (C))) , then f : X → X is order preserving and E ≤ f (E). Applying the Amann theorem, f has a ﬁxed point C0 ∈ X, i.e., C0 ⊂ E is a nonempty closed convex T -invariant set satisfying: C0 = ct(conv(T (C0 ))) . (3) It remains to verify that C0 consists of a single point. If not, from the fact that rC < diam (C) for all bounded closed convex sets C containing more than one point, we obtain: diam(C0 ) = diam(ct(conv(T (C0 ))))

106

2 Fixed-Point Theorems

=

sup

x−y

x,y∈ct(conv(T (C0 )))

≤ rconv(T (C0 )) < diam (conv(T (C0 ))) = diam (T (C0 )) diam (C0 ) . This is a contradiction. Thus, C0 is a single-point set in E, say C0 = {x0 }. It follows that x0 = f ({x0 }) = ct(conv (T x0 )) = T x0 . Obviously Fix (T ) is closed.

Remark 2.4.3 A Banach space X is called uniformly convex, if ∀ε > 0 ∃δ > 0 such that ∀x, y ∈ X x , y 1, x − y ε implies that x+y 2 1 − δ. Obviously, a Hilbert space is a uniformly convex Banach space. Moreover, Lp (Ω, B, µ), 1 < p < +∞ are also, but C(Ω) is not. It is known [DS] that a uniformly convex Banach space is a reﬂexive Banach space with normal structure. One has the following: Theorem 2.4.4 (Browder–G¨ ohde) Suppose that X is a uniformly convex Banach space, and that E is a nonempty bounded closed convex set of X. If T : E → E is nonexpansive, then T has a ﬁxed point. A natural question: can we ﬁnd out a ﬁxed point of a nonexpansive map T by iteration as we did for contraction mappings? Starting from any point x0 ∈ E, set xi = T i x0 , ∀i, from xi+1 − xi xi − xi−1 . . . x1 − x0 , we know that { xi+1 − xi }∞ i=0 is a nonincreasing sequence. It is obviously true that even if {xi } has a convergent subsequence xni → x∗ , we still cannot conclude that x∗ is a ﬁxed point of T , e.g., T x = −x, and x0 = θ. To this end, Mann [Man] introduced an iteration method for the following perturbed mappings: ∀α ∈ (0, 1), set Tα x = αx + (1 − α)T x . One easily veriﬁes that if E is convex and T : E → E is nonexpansive then: (1) (2) (3) (4)

Tα : E → E. Tα is nonexpansive. Tα shares the same ﬁxed point set with T . Let X be a uniformly convex Banach space, and E be a bounded closed convex subset of X. Let p be the ﬁxed point of T . If further, I − T is proper (a map f is called proper if the inverse image of any compact set C, f −1 (C) is compact), then ∀α ∈ (0, 1), Tαi x−p → 0 as i → ∞ ∀x ∈ E.

2.4 Nonexpansive Maps

107

The proofs of (1)–(3) are trivial. In order to give a proof of (4). Let zi = Tαi x . Since the sequence zi+1 − zi is nonincreasing, either (1) ∃ > 0 such that zi+1 − zi ≥ , or (2) zi+1 − zi → 0. But case (1) is impossible. In fact, from the deﬁnition of the uniform convexity ∃δ0 = δ(α, ) > 0, such that zi+1 − p = α(zi − p) + (1 − α)(T zi − T p) (1 − δ0 ) max { zi − p , T zi − T p } = β zi − p ... βi x − p , where β = 1 − δ0 . This proves that zi → p, a contradiction. In case (2), by the assumption that I − T is proper, there exists a convergent subsequence zni → p. It follows that p = T p, i.e., p ∈ Fix (T ) = Fix (Tα ). Thus zn − p ≤ zni − p → 0 as n ≥ ni → ∞. We have thus proved: Theorem 2.4.5 Assume that X is a uniformly convex Banach space, and that E is a nonempty bounded closed convex set of X. If T : E → E is nonexpansive and I − T is proper, ∀α ∈ (0, 1), let Tα = αI + (1 − α)T , then ∀x ∈ E, the sequence Tαi x converges to a ﬁxed point of T. Applications (1) The iteration method is an eﬀective computing method in the convex feasibility problem: Given a family of closed convex sets {Ei | i = 1, . . . , n} in a Hilbert space n

H, ﬁnd a point x ∈ ∩ Ei , if the intersection is nonempty. i=1

In fact, given a closed convex set E in H, ∀x ∈ H, there is a projection onto E: PE x ∈ E, which is deﬁned by the minimizer of the problem: c = min x − y . y∈E

The existence of PE x follows from the parallelogram identity: 2 yn + ym , x − yn − ym 2 = 2( x − yn 2 + x − ym 2 ) − 4 2 where {yn } is a minimizing sequence: x − yn → c. Again by the parallelogram identity, the minimizer is unique. We deﬁne the minimizer y ∗ to be the projection PE x. The projection can be characterized by the following variational inequality: (x − PE x, PE y − PE x) 0 ∀x, y ∈ H .

(2.32)

108

2 Fixed-Point Theorems

x

E PE x

Fig. 2.5.

In fact, ∀t ∈ (0, 1), tPE y + (1 − t)PE x ∈ E. One has x − tPE y − (1 − t)PE x 2 x − PE x 2 . i.e., −2t(x − PE x, PE y − PE x) + t2 PE y − PE x 2 0 . Letting t → 0, we obtain the desired result (2.32). Statement 2.4.6 PE is a nonexpansive map on H. Proof. By deﬁnition, ∀x, y ∈ H (x − PE x, PE y − PE x) 0 , (y − PE y, PE x − PE y) 0 . Summing the two inequalities: (x − y + (PE y − PE x), PE y − PE x) 0 , it follows that PE y − PE x 2 (y − x, PE y − PE x) x − y PE y − PE x . Therefore PE is nonexpansive.

One may use Mann iteration to compute a feasible solution. Various algorithms have been introduced to improve the convergent rate and the computation. (2) Periodic solution for nonlinear evolution equations: Let X be a real Hilbert space. Given ϕ ∈ X, and f : R1+ × X → X. We consider the following nonlinear evolution equation: Find x ∈ C 1 (R1+ , X) satisfying x(t) ˙ = f (t, x(t)) t ∈ R1+ , x(0) = ϕ .

(2.33) (2.34)

2.5 Monotone Mappings

109

Statement 2.4.7 Assume ω > 0 and that (1) (2) (3) (4)

f is ω-periodic in t, i.e., f (t, x) = f (t + ω, x), (f (t, x) − f (t, y), x − y) 0 ∀t ∈ R1+ , ∀x, y ∈ X, ∃R > 0 such that (f (t, x), x) < 0, ∀t ∈ [0, ω], ∀x ∈ ∂BR (θ), ∀ϕ ∈ B R (θ), the initial-value problem (2.33) (2.34) has a solution.

Then (2.33) has a unique ω-periodic solution, i.e., x(t + ω) = x(t). Proof. From (4), we consider the Poincar´e map on X: T : x(0) → x(ω) , and want to show that T has a unique ﬁxed point. First we show that T : B R (θ) → B R (θ) is a well-deﬁned nonexpansive map: 1. We show that the solution for the initial-value problem is unique, i.e., if x(t), y(t) are solutions with the same initial data ϕ, then x(t) = y(t). Indeed, from d x(t) − y(t) 2 = 2(x (t) − y (t), x(t) − y(t)) dt = 2(f (t, x(t)) − f (t, y(t)), x(t) − y(t)) 0, it follows that x(t) − y(t) x(0) − y(0) .

(2.35)

Thus x(t) = y(t), and then T is well deﬁned. 2. Moreover, (2.35) implies that T is nonexpansive. 3. We turn to showing that T maps B R (θ) into itself. Because of assumption (3) d x 2 = 2(x (t), x(t)) = 2(f (t, x(t)), x(t)) < 0 , dt whenever x(t) ∈ ∂BR (θ). If the conclusion is not true, then ∃t > 0 such that x(t) ∈ / BR (θ) and then ∃t0 ∈ (0, t), such that x(t0 ) ∈ ∂BR (θ) and x(t) ∈ / BR (θ) as t0 < t < t0 + δ for some δ > 0. But this contradicts the diﬀerential inequality. Now, we apply Theorem 2.4.2; there is a ﬁxed point of T in B R (θ). Again applying step 1, the ω-periodic solution is also unique.

2.5 Monotone Mappings Let us consider the subdiﬀerential of a convex function f on a vector space X: ∀x, y ∈ dom (f ), ∀x∗ ∈ ∂f (x), ∀y ∗ ∈ ∂f (y ∗ ), we have

110

2 Fixed-Point Theorems

x∗ , y − x + f (x) f (y) , y ∗ , x − y + f (y) f (x) . By addition,

y ∗ − x∗ , y − x 0

(2.36)

This is what we call the monotonicity of ∂f . Deﬁnition 2.5.1 Suppose that X is a real Banach space and that E ⊂ X is ∗ a nonempty subset. A set-valued mapping A : E → 2X is called monotone if ∗ ∗ ∀x, y ∈ E, ∀x ∈ A(x), ∀y ∈ A(y), one has y ∗ − x∗ , y − x 0 . The set {x ∈ E| A(x) = Ø} is called the domain of A, denoted by D(A), and the set ΓA = {(x, x∗ ) ∈ E × X ∗ | x ∈ D(A), x∗ ∈ A(x)} is called the graph of A. A single-valued monotone mapping is called a monotone operator. Example 1. Suppose that X is a real Hilbert space, and that A is a linear positive operator, i.e., ∀x ∈ D(A), (Ax, x) 0. Then A is a monotone operator. Example 2. Suppose that X is a real Hilbert space, and that T is a nonexpansive map, then A = id − T is a monotone operator: (Ay − Ax, y − x) = y − x 2 −(T y − T x, y − x) 0 . Example 3. Let X be a real Banach space, and let f : X → R1 ∪ {+∞} be convex, then ∂f is a monotone mapping. Example 4. (p-Laplacian) Let Ω be a bounded domain of Rn . For 1 < p < ∞, the operator ⎛⎡ ⎞ ⎤ p−2 2 n n 2 ∂ ⎜ ∂u " # ∂u ⎟ ⎦ Au = −div |∇u|p−2 ∇u = − ⎝⎣ ⎠ ∂xi ∂xj ∂xi i=1 j=1 ◦

deﬁnes a map from Wp1 (Ω) to Wp−1 (Ω),

Au, v =

1 p

+

1 p

= 1, as follows:

⎡ ⎤ p−2 2 2 n ∂u ∂u ∂v ⎣ ⎦ dx ∂xj ∂xi ∂xj Ω j=1

n

i=1

◦

∀v ∈ Wp1 (Ω) .

In fact, by the H¨ older inequality:

|∇u|

p−1

Ω

|∇v|

|∇u|

p

Ω

1

p

Ω

|∇v|

p

p1 ,

2.5 Monotone Mappings

111

◦

Au ∈ (Wp1 (Ω))∗ = Wp−1 (Ω). We verify that A is monotone: According to the elementary inequalities: |b − a|p if p 2 , p−2 p−2 (|b| b − |a| a)(b − a) cp p−2 2 (1 + |b| + |a|) |b − a| if 1 < p < 2 , older inequality, we obtain where cp > 0 is a constant, and a, b ∈ Rn , and the H¨ Au − Av, u − v

|∇u − ∇v|p Ω cp 2 2 ( Ω (1 + |∇u| + |∇v|)p )1− p ( Ω |∇u − ∇v|p ) p

if p 2 , if 1 < p < 2 .

The requirement of the continuity for monotone operators is very weak. Deﬁnition 2.5.2 Let X be a real Banach space and let E ⊂ X be a nonempty subset. A map A : E → X ∗ is called hemi-continuous at x0 ∈ E, if ∀y ∈ X, ∗ ∀tn ↓ 0 with x0 + tn y ∈ E, imply that A(x0 + tn y) A(x0 ). It is called demi∗ continuous at x0 ∈ E, if ∀{xn } ⊂ E, xn → x0 implies that A(xn ) A(x0 ), ∗ where ∗ is the w -convergence. Obviously, “continuous” =⇒ “demi-continuous” =⇒ “hemi-continuous” . ◦

The monotone operator in Example 4 is hemi-continuous. In fact, ∀u, v, w ∈

Wp1 (Ω), we consider the function t → A((1 − t)x + ty), w , and verify the continuity. Now ⎡ ⎤ p−2 2 2 n ∂v ∂u ⎣ ⎦ +t (1 − t) A((1 − t)u + tv), w = ∂xj ∂xj Ω j=1

×

n i=1

∂v ∂u +t (1 − t) ∂xi ∂xi

∂w . ∂xi

Since the integrand on the RHS is dominated by an integrable function (|∇u|+ |∇v|)p−1 |∇w|, the Lebesgue dominance theorem is applied. An important property for monotone operators reads as: Lemma 2.5.3 Let E be a convex subset of a real Banach space X. If A : E → X ∗ is hemi-continuous and monotone, then for any sequence {xj } ⊂ E with xj x ∈ E and lim A(xj ), xj − x 0, we have lim A(xj ), xj − y A(x), x − y

∀y ∈ E .

112

2 Fixed-Point Theorems

Proof. First we claim that lim A(xj ), xj − x = 0 .

(2.37)

Indeed, since A is monotone, it follows that 0 = lim A(x), xj − x lim A(xj ), xj − x lim A(xj ), xj − x 0 . Again by the monotonicity and (2.37), ∀z ∈ E lim A(xj ), x − z = lim A(xj ), xj − z lim A(z), xj − z = A(z), x − z . (2.38) Now, ∀y ∈ E, ∀tn ↓ 0, substituting z = zn := (1 − tn )x + tn y into (2.38), we obtain (2.39) lim A(xj ), x − y A(zn ), x − y . By using the hemi-continuity, the RHS of (2.39) tends to A(x), x − y . Combining (2.37) (2.39) and the last fact, it follows that lim A(xj ), xj − y A(x), x − y .

(2.40)

Remark 2.5.4 The deduction argument from (2.37) to (2.40) is called Minty’s trick, in which a combination of the monotonicity and the hemicontinuity is applied. This lemma is substantial in the study of monotone operators. The following notion on pseudo-monotonicity is abstracted from it. Deﬁnition 2.5.5 Let X be a reﬂexive Banach space and let E ⊂ X be a nonempty closed convex subset. An operator A : E → X ∗ is called pseudo monotone, if (1) ∀ ﬁnite-dimensional linear subspace L ⊂ X, A|L∩E : L ∩ E → X ∗ is demi-continuous. (2) ∀ sequence {xj } ⊂ E with xj x ∈ E, the condition lim A(xj ), xj −x 0 implies that lim A(xj ), xj − y A(x), x − y , ∀y ∈ E. Thus, a hemi-continuous monotone operator is pseudo monotone. Moreover, a completely continuous mapping A : X → X ∗ (i.e., for any xj x in X, we have Axj → Ax in X ∗ ) is pseudo monotone. In contrast with the ﬁxed-point problem for compact maps and nonexpansive mappings, we shall study the surjection of pseudo monotone operators, because the latter maps a subset of a Banach space into its dual space. The following generalized Ky Fan’s inequality is the basis of this section. Theorem 2.5.6 (Brezis, Nirenberg, Stampacchia) Assume that E is a nonempty convex set of an LCS X. Assume Φ : E × E → R1 satisfying

2.5 Monotone Mappings

113

(1) ∀ ﬁnite-dimensional linear subspace L of X, ∀y ∈ E ∩ L, x → Φ(x, y)|L∩E is l.s.c. (2) ∀x ∈ E, y → Φ(x, y) is quasi concave. (3) ∃ a compact set K ⊂ X, ∃y0 ∈ E such that {x ∈ E| Φ(x, y0 ) 0} ⊂ K. (4) sup Φ(x, x) = 0. x∈E

(5) ∀x, y ∈ E and for any net xα → x, ∀t ∈ [0, 1], Φ(xα , (1 − t)x + ty) 0 implies that Φ(x, y) 0. Then ∃x0 ∈ E such that sup Φ(x0 , y) 0. y∈E

Proof. Again we transform the problem into its geometric version by setting Γ = {(x, y) ∈ E × E| Φ(x, y) > 0}. Then (1)–(5) are transformed into: (1 ) ∀L, letting EL = E ∩ L and ΓL = Γ ∩ (L × L), ∀y ∈ EL ΓL 2 (y) is open in E. (2 ) ∀x ∈ E, Γ1 (x) is convex. (3 ) ∃y0 ∈ E such that E\Γ2 (y0 ) ⊂ K. (4 ) ∩ Γ = Ø, where is the diagonal of E × E. (5 ) Let xy be the segment connecting x and y, then Γ1 (xα ) ∩ xy = Ø implies y ∈ Γ1 (x) ∀y ∈ E for any net xα → x. Now we set X = {L| ﬁnite-dimensional linear subspace, with y0 ∈ L and EL = Ø}. ∀L ∈ X, applying Ky Fan’s inequality, ∃xL ∈ EL such that ΓL 1 (xL ) = Ø. From (3 ), xL ∈ K as L ∈ X. ∀L ∈ X, setting L L N = {x ∈ K| Γ1 (x) = Ø}, from the above discussion, it is nonempty. L We claim that the family {N | L ∈ X} has the ﬁnite intersection property: Indeed, ∀ L1 , . . . , Ln ∈ X, setting L = span {L1 , . . . , Ln } we have n

n

i=1

i=1

Li L Li L ∈ X. Then ΓL 1 (xL ) = ∪ Γ1 (xL ) and N = ∩ N . Since K is compact, L

Z := ∩ N = Ø. L∈X

We shall prove that ∀x ∈ Z, Γ1 (x) = Ø. To this end, ∀y ∈ E, we choose L L L ∈ X such that the segment xy ⊂ L. Since x ∈ N , ∃ a net xL α ∈ N , such L L that Γ1 (xα ) ∩ xy = Ø, and xα → x. From assumption (5 ), y ∈ Γ1 (x). Since y ∈ E is arbitrary, we obtain Γ1 (x) = Ø, Therefore there exists x0 ∈ Z such that sup Φ(x0 , y) 0. y∈E

In many of the applications, the LCS is taken to be a reﬂexive Banach space with its weak topology. Since then for weakly closed set it is weakly compact if and only if it is sequentially weakly compact (cf. Theorem 2.2.5), the net convergence in condition (5) of the above theorem can be replaced by the sequential convergence. The following theorem is the foundation of the theory of variational inequalities, which arise in free boundary problems for partial diﬀerential equations.

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2 Fixed-Point Theorems

Theorem 2.5.7 (Hartman–Stampacchia) Suppose that X is a real reﬂexive Banach space, and that E ⊂ X is a nonempty closed convex set. Assume (1) A : E → X ∗ is pseudo monotone, (2) ϕ : E → R1 is convex and l.s.c., (3) ∃y0 ∈ E ∃R0 > 0 such that Ax, x − y0 + ϕ(x) − ϕ(y0 ) > 0

as x R0 and x ∈ E .

Then ∃x0 ∈ X such that Ax0 , x0 − y + ϕ(x0 ) − ϕ(y) 0,

∀y ∈ E .

Proof. (1) Assign the weak topology on X, and deﬁne Φ(x, y) = Ax, x − y + ϕ(x) − ϕ(y) . It is easy to see that conditions (1)–(4) of Theorem 2.5.6 are satisﬁed. Let us verify (5). ∀x, y ∈ E ∀t ∈ [0, 1], assume xj x and Φ(xj , (1 − t)x + ty) 0. Setting t = 0 and 1 separately, we have lim A(xj ), xj − x lim [ϕ(x) − ϕ(xj )] 0

(2.41)

and lim A(xj ), xj − y lim [ϕ(y) − ϕ(xj )] ϕ(y) − ϕ(x) .

(2.42)

From the pseudo monotonicity, it follows that lim A(xj ), xj − y A(x), x − y .

(2.43)

Combining (2.42) and (2.43), we obtain Φ(x, y) 0. (2) Now we apply Theorem 2.5.6 and conclude the existence of x0 ∈ E satisfying Φ(x0 , y) 0. This is the desired conclusion. Theorem 2.5.8 (F. Browder) Suppose that X is a real reﬂexive Banach space, and that A : X → X ∗ is pseudo monotone and coercive, i.e., x −1 A(x), x → +∞ as x → +∞. Then A is surjective. Proof. We shall prove that ∀z ∈ X ∗ , ∃x0 ∈ X satisfying A(x0 ) = z. Deﬁne T : x → A(x) − z. Then T is pseudo monotone and satisﬁes T x, x > 0 as x > R0 , for some R0 > 0, provided by the coerciveness of A. We apply the Hartman– Stampacchia theorem to conclude the existence of x0 ∈ X satisfying T x0 , x0 − y 0 ∀y ∈ X. Since y is arbitrary in the linear space X, it follows that Ax0 = z.

2.5 Monotone Mappings

115

As a consequence, we return to Example 4, the p− Laplacian −∆p : W01,p (Ω) → W −1,p (Ω), 1 < p < ∞, is a homeomorphism. The surjection follows from Browder’s theorem, and the injection as well as the continuity of the inverse mapping follow from the inequality in the veriﬁcation of the monotonicity. Corollary 2.5.9 (Hartman–Stampacchia) Suppose that A is a hemicontinuous monotone operator deﬁned on the unit ball B centered at θ of a Hilbert space H. If Ax = λx, ∀x ∈ ∂B, ∀λ < 0 then ∃x0 ∈ B such that Ax0 = θ. Proof. Since A is pseudo monotone, we apply Theorem 2.5.7 to conclude the existence of x0 ∈ B satisfying (Ax0 , x0 − y) 0 ∀y ∈ B . ◦

If x0 ∈ B, then we have Ax0 = θ. Otherwise x0 ∈ ∂BR ∀y ∈ BR , we decompose y = (1 − t)x0 + y ⊥ , and Ax0 = λx0 + y0⊥ , where t > 0, y ⊥ ⊥x0 , y0⊥ ⊥x0 , and λ ∈ R1 . Then (2.44) −λt x0 2 +(y0⊥ , y ⊥ ) 0 . Since y ⊥ is arbitrary, ﬁrst letting t → +0, we obtain y0⊥ = θ, i.e., Ax0 = λx0 . By the assumption, λ 0. Again by (2.44) λt x0 2 0 ,

which implies that λ = 0. Therefore Ax0 = θ.

y

θ

x0 y⊥ (1 − t)x0

Fig. 2.6.

Corollary 2.5.10 (Minty) Suppose that H is a real Hilbert space, and that A is a continuous strongly monotone operator, i.e., ∃c > 0 such that (Ax − Ay, x − y) c x − y 2 Then A is a homeomorphism.

∀x, y ∈ H .

(2.45)

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2 Fixed-Point Theorems

Proof. Obviously, A is pseudo monotone and coercive. As a consequence of the Browder theorem, A is surjective. The injectivity of A as well as the continuity of A−1 follows from the inequality (2.45). Applications 1. Elliptic variatiational inequality Let A be the p-Laplacian deﬁned in Example 4 and let Ω ⊂ Rn be a ◦

bounded domain. We consider the closed convex subset of X = Wp1 (Ω) with 1 < p < ∞: ◦

E = {u ∈ Wp1 (Ω)| |∇u(x)| 1 a.e.} . Given f ∈ Wp1 (Ω), where p =

p p−1 ,

ﬁnd u ∈ E such that

Au, v − u f, v − u ∀v ∈ E . The existence of a solution follows from the Hartman–Stampacchia theorem directly. This is a free boundary problem, in fact the solution u, if it is regular, satisﬁes the equation: ⎧ p−2 ⎪ in Ω1 , ⎨−div (|∇u| ∇u) = f |∇u| = 1 in Ω2 , ⎪ ⎩ u and ∇u coincide on the interface of Ω1 and Ω2 , where Ω1 = {x ∈ Ω| |∇u(x)| < 1} , and Ω2 = Ω\Ω1 . The same method can be applied to the obstacle problem and the Stefan problem studied in Sect. 2.3. But in these problems, the working spaces should be taken to be the following Hilbert spaces: H01 (Ω) and H 1 ([0, T ] × Ω) respectively. 2. Weak solutions for quasi-linear elliptic equations Let Ω be a bounded domain of Rn . We study the weak solution of the quasi-linear elliptic equation: n ∂ i=1 ∂xi Ai (x, u(x), ∇u(x)) + B(x, u(x), ∇u(x)) = f (x) in Ω , u|∂Ω = 0 ,

where we make use of the following structural condition on A = {Ai }n1 and B:

(1) B ∈ C(Ω × R1 × Rn , R1 ), A ∈ C(Ω × R1 × Rn , Rn ),

2.5 Monotone Mappings

(2) ∃p ∈ (1, ∞), ∃g ∈ Lp (Ω), p =

p p−1 ,

117

∃C > 0 such that

A(x, u, ξ) g(x) + C(|u| + |ξ|)p−1 , |B(x, u, ξ)| g(x) + C(|u| + |ξ|)p−1 , (3)

[A(x, u, ξ)− A(x, u, ξ )]·(ξ−ξ ) > 0, ∀(x, u) ∈ Ω×R1 , ∀ξ, ξ ∈ Rn , ξ = ξ , and (4) 1 A(x, u, ξ) · ξ → +∞ as |ξ| → ∞, (x, u, ξ) ∈ Ω × R1 × Rn . p−1 1 + |ξ| + |ξ|

Remark 2.5.11 Condition (3) (i.e., the monotone condition) implies the el

lipticity of the diﬀerential operator: u(x) → div A(x, u(x), ∇u(x)). ◦

Let X = Wp1 (Ω); we deﬁne a form on X × X:

a(u, v) = [A(x, u(x), ∇u(x)) · ∇v(x) + B(x, u(x), ∇u(x))v(x)]dx Ω

× ∀(u, v) ∈ X × X . From (1) and (2), |a(u, v)| C( u p−1 1,p + g p ) v 1,p . We deﬁne T : X → X ∗ by T u, v = a(u, v), where , is the duality between X ∗ and X. In order to verify that T is pseudo monotone, we deﬁne A : X × X → X ∗ as follow:

A(x, u, ∇w) · ∇v + B(x, u, ∇u)v ∀u, v, w ∈ X . A(u, w), v = Ω

Ω

We conclude: (a) A(u, u) − A(u, w), u − w 0, ∀u, w ∈ X. (b) ∀u ∈ X, w → A(u, w) is bounded and demi-continuous. (c) If uj u in X and if A(uj , uj ) − A(uj , u), uj − u → 0, then A(uj , w) A(u, w) in X ∗ ∀w ∈ X. (a) and (b) are obviously true; we prove (c). Proof. From uj u in X, we have uj → u(in Lq (Ω)) and uj → u a.e. after a subsequence, and then ∀w ∈ X, → − − → A (x, uj (x), ∇w(x)) → A (x, u(x), ∇w(x))

a.e.

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2 Fixed-Point Theorems

Combining with the growth condition (2), → − → − A (·, uj , ∇w) → A (·, u, ∇w) in Lp (Ω, Rn ) . Thus

− → A (x, uj , ∇w)∇uj →

Ω

and

Ω

− → A (x, u, ∇w)∇u ,

(2.46)

Ω

− → A (x, uj , ∇w)∇v →

A(x, u, ∇w)∇v .

(2.47)

Ω

It remains to verify that gj := B(·, uj , ∇uj ) g := B(·, u, ∇u) (Lp (Ω)). The diﬃculty is that we only have ∇uj ∇u in Lp (Ω, Rn ), and that B is nonlinear with respect to ∇u. How can we pass the limit? If one can show that ∇uj → ∇u a.e., then gj → g a.e., and then we claim that gj g in Lp (Ω). In fact, let Ek = {x ∈ Ω||gj (x)−g(x)| ≤ 1, j ≥ k}; we have E1 ⊂ E2 ⊂ . . . , and mes(Ek ) → mes(Ω). Since ∀χ ∈ Lp (Ω) with support in Ek , we have (g − g)χ → 0, from the Lebesgue dominance theorem, and those χ span a Ω j dense set in Lp (Ω); in combination with the fact that gj p is bounded, the conclusion follows. We turn to verifying that ∇uj → ∇u a.e., and deﬁne the function: → − → − Fj (x) = [ A (x, uj (x), ∇uj (x)) − A (x, uj (x), ∇u(x))] · ∇(uj (x) − u(x)) . By (3), Fj (x) 0. As a consequence of our assumption, Ω Fj (x)dx → 0. From Fatou’s lemma, it follows that Fj (x) → 0 a.e. Therefore there exists a / Z. null set Z such that uj (x) → u(x), and Fj (x) → 0 ∀x ∈ Firstly, we claim that ∃ a measurable function M (x) < +∞, such that |∇uj (x)| M (x) ∀x ∈ Z .

(2.48)

From (2), → − Fj (x) ≥ A (x, uj (x), ∇uj (x))∇uj (x) − C1 (x)(1 + |∇uj (x)| + |∇uj (x)|p−1 ) where C1 (x) depends on u(x) and ∇u(x). If (2.48) were false, then there would be a positive measure set E, on which |∇uj (x)| → +∞. It follows that Fj (x) → +∞, from (4). But, this is a contradiction. Secondly, we claim that ∀x ∈ Z, ∇uj (x) → ∇u(x). In fact, let ξ(x) be a limit point of ∇uj (x). From Fj (x) → 0 it follows that: → − → − [ A (x, u(x), ξ(x)) − A (x, u(x), ∇u(x))] · (ξ(x) − ∇u(x))dx = 0 . Applying the monotone relation (3), we have ξ(x) = ∇u(x). Since all limit points of ∇uj (x) equal ∇u(x), we obtain ∇uj (x) → ∇u(x) a.e. Combining (2.47) with this fact, we obtain A(uj , w) A(u, w) in X ∗ ∀w ∈ X.

2.5 Monotone Mappings

119

(d) If uj u in X and A(uj , v) ψ in X ∗ , then A(uj , v), uj → ψ, u . → − − → In fact, we already have uj → u (Lp (Ω, R1 )) and A (·, uj , ∇v) → A (·, u, ∇v) (Lp (Ω, Rn )). Thus

→ − A (x, uj , ∇v)∇(uj − u) + B(x, uj , ∇uj )(uj − u) A(uj , v), uj = Ω

+ A(uj , v), u → ψ, u . After these preparations, we turn to verifying that the operator T , deﬁned by the form a, is pseudo monotone. Since u → T (u) is obviously demicontinuous, it is suﬃcient to prove that uj u and lim T (uj ), uj − u 0 implies that lim T (uj ), uj − v T (u), u − v ∀v ∈ X. First we show that T (uj ), uj − u → 0, and deﬁne the sequence: bj = A(uj , uj ) − A(uj , u), uj − u . By (a), bj 0. Since A(uj , u) is bounded in X ∗ , there exists a subsequence, for which we don’t change the subscript, such that A(uj , u) ψ in X ∗ . By (d), A(uj , u), uj → ψ, u . Thus, A(uj , u), uj − u → 0. But by assumption, lim A(uj , uj ), uj − u = lim T (uj ), uj − u 0 , therefore bj → 0. It follows T (uj ), uj − u → 0. It remains to show: lim T (uj ), u − v ≥ T (u), u − v , from (a), T (uj ) − A(uj , w), uj − w 0, ∀w ∈ X . Letting w = (1 − θ)u + θv, θ ∈ (0, 1), ∀v ∈ X, we have θ T (uj ), u − v θ A(uj , w), u − v + A(uj , w), uj − u − T (uj ), uj − u . By taking limit, lim T (uj ), u − v A(u, w), u − v

or lim T (uj ), u − v A(u, (1 − θ)u + θv), u − v ∀θ ∈ (0, 1) letting θ → 0, by (b), the RHS = T (u), u − v , this is the desired inequality. Statement 2.5.12 Under the above structural conditions, we assume further: (5) ∃C1 > 0 ∃h ∈ L1 (Ω) such that − → A (x, u, ξ) · ξ > C1 |ξ|p − h(x)

∀(x, u, ξ) ∈ Ω × R1 × Rn . ◦

Then ∀f ∈ Lp (Ω), there exists a weak solution u ∈ Wp1 (Ω) of the quasilinear elliptic equation:

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2 Fixed-Point Theorems

− → A (x, u(x), ∇u(x)) · ∇v(x) + B(x, u(x), ∇u(x))v(x) =

Ω

f (x)v(x) Ω ◦

× ∀v ∈ Wp1 (Ω) . Proof. It is reduced to ﬁnd the ﬁxed point of the pseudo monotone operator T . Since (5) implies the coerciveness: 1 1 T u, u = a(u, u) → +∞ . u 1,p u 1,p The existence of u follows from Browder’s theorem.

It is worth noting that in the above statement, no a priori estimate has been used. In the veriﬁcation of the pseudo monotonicity, a crucial point is the ellipticity condition (or the monotone condition (3)). The statement can be extended to high-order quasi-elliptic equations. This is an approach where no compactness is concerned.

2.6 Maximal Monotone Mapping To a set-valued mapping A, ∀x ∈ D(A), one may take any nonempty subset D(A1 ) of D(A) and a nonempty subset A1 (x) of A(x), and deﬁne a new set-valued mapping A1 : x → A1 (x) ∀x ∈ D(A1 ). If A is monotone, so is A1 . In order to avoid the indetermination of domain and the image of setvalued monotone mappings in the study of their surjectivity, one introduces the notion of maximal monotone mappings. First, we deﬁne the graph ΓA := {(x, y) ∈ X × Y | x ∈ D(A), y ∈ A(x)} of a set-valued mapping A : X → 2Y . Deﬁnition 2.6.1 Assume that A1 and A2 are two monotone set-valued map∗ pings from X to 2X . If ΓA1 ⊂ ΓA2 , then we say that A2 is a monotone extension of A1 . A monotone mapping, which does not have any proper monotone extension, is called a maximal monotone mapping. ∗ In other words, A : X → 2X is maximal monotone iﬀ for (x, x∗ ) ∈ X ×X ∗ , y ∗ − x∗ , y − x ≥ 0 ∀ (y, y ∗ ) ∈ ΓA ⇒ (x, x∗ ) ∈ ΓA . Example 1. Assume that φ : R1 → R1 is a nondecreasing function. The map A : u → [φ(u−0), φ(u+0)] is maximal monotone, but the map B : u → φ(u−0) is not, if φ(u − 0) = φ(u + 0). Example 2. A hemi-continuous monotone operator A is maximal monotone. In fact, for (x, x∗ ) ∈ X × X ∗ , if Ay − x∗ , y − x ≥ 0 ∀ y ∈ X ,

2.6 Maximal Monotone Mapping

121

then ∀z ∈ X, ∀t ∈ [0, 1], we have A((1 − t)x + tz) − x∗ , z − x ≥ 0 . Letting t → 0, it follows that Ax−x∗ , z −x ≥ 0. Since z ∈ X is arbitrary, we obtain Ax = x∗ . Moreover, we shall prove later that the subdiﬀerential of a proper l.s.c. convex function is maximal monotone. Since maximal monotone mappings occur in convex analysis, free boundary boundary problems in mathematical physics, and nonlinear semigroups. we shall extend most of the results of monotone operators to their set-valued counterpart-maximal monotone mappings. The central problem is on the surjectivity. Now we are going to extend the notions of the demi-continuity and the hemi-continuity for single-valued mappings to set-valued mappings. Deﬁnition 2.6.2 Suppose that E is a nonempty subset of a real Banach ∗ space X. A mapping A : E → 2X is called weakly ∗ upper hemi-continuous ∗ (w u.h.c.,) if ∀x, y ∈ E ∀z ∈ X, the function t → A((1 − t)x + ty), z is a u.s.c. set-valued function at t = 0. It is called upper demi-continuous (u.d.c.) 1 if ∀z ∈ X, x → A(x), z is u.s.c. from E to 2R . Obviously, u.d.c. implies w∗ u.h.c., and for single-valued mappings, w∗ u.h.c. is hemi-continuous, and u.d.c. is demi-continuous. ∗ Note that for any set-valued mapping A : D(A) → 2X , the inverse mapping A−1 : X ∗ → 2X , x∗ → {x ∈ D(A)| x∗ ∈ A(x)} is well deﬁned (in case x∗ ∈ rang (A), one deﬁnes A−1 (x∗ ) = Ø). In particular, if X is reﬂexive then A is (maximal) monotone ⇐⇒ A−1 is (maximal) monotone. Lemma 2.6.3 Assume that A : X → 2X ∃ε > 0 such that A is bounded on Bε (x0 ).

∗

is monotone. Then ∀x0 ∈ X,

Proof. We may assume x0 = θ. We prove it by contradiction. Assume that A is not bounded in any neighborhood of θ, i.e., ∃xn → θ ∃x∗n ∈ A(xn ) such that x∗n → ∞. We claim that ∀ε > 0 ∃z ∈ Bε (θ) and a subsequence {n } such that (2.49) x∗n , xn − z → −∞ . Then from the monotonicity, ∀z ∗ ∈ A(z), we have x∗n − z ∗ , xn − z 0 . (2.49) implies that z ∗ , z = +∞. Obviously, this is impossible. Let us return to proving the existence of z satisfying (2.49). If it is not true, then ∃ε > 0 ∀z ∈ B ε (θ) ∃Cz ∈ R1 such that x∗n , xn − z Cz

∀n .

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2 Fixed-Point Theorems

Now, ∀k ∈ N, let Ek = {u ∈ B ε (θ)| x∗n , xn − u −k, ∀n} . ∞

Obviously, Ek are closed, and ∪ Ek = B ε (θ). From the Baire category ark=1

gument, ∃r > 0, ∃y0 ∈ B (θ) and ∃k0 ∈ N such that Br (y0 ) ⊂ Ek0 . Setting C = C−y0 , we have x∗n , xn + y0 C ∀n , and

x∗n , xn − u ≥ −k0 , ∀u ∈ B r (y0 ) .

By addition, x∗n , 2xn + y0 − u C − k0

∀u ∈ B r (y0 ) .

Setting v = 2xn + y0 − u, for suﬃciently large n, such that xn < 4r , we obtain x∗n , v C − k0 ∀v ∈ B r2 (θ) . This means x∗n proved.

2|C−k0 | , r

which contradicts x∗n → ∞; (2.49) is ∗

Proposition 2.6.4 Assume that A : X → 2X is a maximal monotone mapping with D(A) = X on a reﬂexive real Banach space X. Then: (1) ∀x ∈ X A(x) is a nonempty closed convex set. (2) A is u.d.c. (3) If there is a sequence {xj }, and a sequence x∗j ∈ A(xj ) satisfying xj x and lim x∗j , xj − x 0, then ∃x∗ ∈ A(x) such that lim x∗j , xj − y x∗ , x − y

∀y ∈ X .

Proof. (1) If x∗i ∈ A(x) i = 1, 2, then ∀y ∈ X, ∀y ∗ ∈ A(y) we have y ∗ − x∗i , y − x 0,

i = 1, 2 .

Therefore y ∗ − (λx∗1 + (1 − λ)x∗2 ), y − x 0 ∀λ ∈ [0, 1] . By the maximality, λx∗1 + (1 − λ)x∗2 ∈ A(x). Then A(x) is convex. Similarly, A(x) is closed. (2) In order to show that A is u.d.c., i.e., ∀z ∈ X, x → A(x), z is a u.s.c. 1 set-valued mapping: X → 2R . We prove it by contradiction. If ∃z0 ∈ X ∃ε0 > ∗ 0 ∃xn → x0 and ∃xn ∈ A(xn ) such that x∗n , z0 ∈ A(x0 ), z0 +(−ε0 , ε0 ). From Lemma 2.6.3, {x∗n } is bounded, and then it possesses a weakly convergent subsequence: x∗n x∗0 in X ∗ . Since A is maximal, x∗0 ∈ A(x0 ). This is a contradiction. (3) The proof is the same as that for single-valued monotone operators; cf. Lemma 2.5.3.

2.6 Maximal Monotone Mapping

123

Conversely, we have: Proposition 2.6.5 Assume that X is a reﬂexive real Banach space and that ∗ A : X → 2X is a w∗ .u.h.c. monotone mapping. If ∀x ∈ X, A(x) is a nonempty closed convex set, then A is maximal monotone. Proof. We shall verify that for (x, x∗ ) ∈ X × X ∗ , if y ∗ − x∗ , y − x 0 ∀(y, y ∗ ) ∈ ΓA ,

(2.50)

then (x, x∗ ) ∈ ΓA . For otherwise, according to the Ascoli separation theorem, ∃z ∈ X such that x∗ , z > sup { w∗ , z | w∗ ∈ A(x)} . Since A is w∗ .u.h.c. ∃δ > 0 such that x∗ , z > yt∗ , z

∀yt∗ ∈ A(yt ) ∀t ∈ (0, δ) ,

where yt = x + tz. But from the monotonicity, we have t yt∗ − x∗ , z = yt∗ − x∗ , yt − x 0 . This is a contradiction.

The following theorem is a characterization of maximal monotone mappings on Hilbert spaces. Theorem 2.6.6 (Minty) Suppose that H is a Hilbert space, and that A : H → 2H is a set-valued mapping. The following statements are equivalent: (1) A is maximal monotone. (2) A is monotone and I + A is surjective. (3) ∀λ > 0 (I + λA)−1 is nonexpansive. Before going on to prove Theorem 2.6.6, we now introduce the following notion. Deﬁnition 2.6.7 Suppose that E is a nonempty subset of a real Hilbert space H. A set-valued mapping T : E → 2H is called expansive if ∀x, y ∈ E, ∀x∗ ∈ T (x), ∀y ∗ ∈ T (y), x∗ − y ∗ x − y . Obviously, T : H → 2H is surjective and expansive iﬀ T −1 is nonexpansive. In this case, T −1 is single valued and injective. Lemma 2.6.8 A : E → 2H is monotone if and only if ∀λ > 0, Tλ = I + λA is expansive.

124

2 Fixed-Point Theorems

Proof. Since ∀x, y ∈ E, ∀x∗ ∈ A(x), ∀y ∗ ∈ A(y), we have y−x+λ(y ∗ −x∗ ) 2 = y−x 2 +2λ(y ∗ −x∗ , y−x)+λ2 y ∗ −x∗ 2 . (2.51) “=⇒” If A is monotone, then the RHS of (2.51) y − x 2 , i.e., Tλ is expansive. “⇐=” If Tλ is expansive, then the LHS of (2.51) y − x 2 . Letting λ → 0, A is monotone. Lemma 2.6.9 (Deburnner–Flor) Suppose that E is a nonempty closed convex subset of a real Hilbert space H. If A : E → 2H is monotone, then ∀y ∈ H, ∃x ∈ E such that (η + x, ξ − x) (y, ξ − x)

∀(ξ, η) ∈ ΓA .

Proof. ∀(ξ, η) ∈ ΓA , let us deﬁne C(ξ, η) = {x ∈ E| (η + x − y, ξ − x) 0} . $ Ø. Since C(ξ, η) Our conclusion is equivalent to {C(ξ, η)| (ξ, η) ∈ ΓA } = is a bounded closed convex set, it is weakly closed and E is weakly compact. It is suﬃcient to verify that {C(ξ, η)| (ξ, η) ∈ ΓA } has the ﬁnite intersection property. Indeed, ∀ξ1 , . . . , ξn ∈ E ∀ηi ∈ A(ξi ), i = 1, . . . , n, we consider the simplex λi = 1 , n = λ = (λ1 , . . . , λn )| λi 0 ∀i, and the function Φ(λ, µ) =

n

µi (x(λ) + ηi − y, x(λ) − ξi )

∀(λ, µ) ∈ n × n ,

i=1

where x(λ) =

n i=1

λi ξi . It is easy to see that ∀µ ∈ n , λ → Φ(λ, µ) is continuous, ∀λ ∈ n , µ → Φ(λ, µ) is linear ,

and Φ(λ, λ) = =

=

n

[λi (x(λ) − y, x(λ) − ξi ) + λi (ηi , x(λ) − ξi )]

i=1 n

λi λj (ηi , ξj − ξi )

i,j=1 n

1 λi λj (ηi − ηj , ξj − ξi ) ≤ 0 , 2 i,j=1

2.6 Maximal Monotone Mapping

125

because A is monotone. According to Ky Fan’s inequality, ∃λ0 ∈ n such that Φ(λ0 , µ) 0 ∀µ ∈ n , i.e., (x(λ0 ) + ηi − y, x(λ0 ) − ξi ) 0 ∀i . Therefore x(λ0 ) ∈

n $

C(ξi , ηi ). This proves the ﬁnite intersection property for

i=1

{C(ξ, η) | (ξ, η) ∈ ΓA }, and thus the lemma.

Proof of Minty’s theorem. Proof. (1)=⇒(2) We only want to show that ∀y ∈ H, ∃x ∈ H such that y − x ∈ A(x). In fact, according to the Deburnner–Flor lemma, ∃x ∈ H satisfying (η − (y − x), ξ − x) 0, ∀(ξ, η) ∈ ΓA . By maximality, y − x ∈ A(x). (1)⇐=(2) Assume that B is a monotone mapping with ΓA ⊂ ΓB ; we want to show that B = A. Since I + A is surjective, ∀(x, y) ∈ ΓB , ∃x ∈ D(A) such that x + y ∈ (I + A)(x ), so is x + y ∈ x + B(x ), i.e, y = y + x − x ∈ B(x ). Also from (x, y) ∈ ΓB , x + y ∈ x + B(x). Provided by the monotonicity of B, we have x − x 2 = (y − y, x − x ) 0 , i.e., x = x . Therefore B = A and then A is maximal. Since A is maximal monotone if and only if λA is ∀λ > 0, we may assume λ = 1 in proving the equivalence of (2) and (3). Also, by Lemma 2.6.8, (I + A) is expansive ⇐⇒ A is monotone. (2) ⇐⇒ (3) follows from the fact that (I + A) is expansive and surjective ⇐⇒ (I + A)−1 is nonexpansive. In the literature, for a maximal monotone mapping A, ∀λ > 0, setting Jλ = (I + λA)−1 we call the operator Aλ (x) = λ1 (I − Jλ )(x) the Yosida regularization of A. By deﬁnition, Aλ (x) ∈ A(Jλ x), but in general, Aλ (x) = A(Jλ x). Theorem 2.6.10 If f : H → R1 ∪ {+∞} is a proper, l.s.c., convex function, then ∂f is maximal monotone. Proof. The monotonicity has been known since the beginning of the last section. Due to Minty’s theorem, it remains to verify the surjection of I + ∂f , i.e., ∀y0 ∈ H, ∃x0 ∈ H such that y0 − x0 ∈ ∂f (x0 ). To this end, we deﬁne a function: ϕ(x) =

1 x 2 +f (x) − y0 , x . 2

It is proper, l.s.c., and convex, and ϕ(x) → +∞ as x → ∞. There exists R1 > 0 s.t. ϕ(x) > ϕ(θ) as x R1 . Applying the Hartman–Stampacchia theorem, ∃x0 ∈ D(ϕ) = D(f ) such that

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2 Fixed-Point Theorems

ϕ(x0 ) ϕ(x)

∀x ∈ H .

Therefore, θ ∈ ∂ϕ(x0 ), i.e., y0 − x0 ∈ ∂f (x0 ).

In fact, the above conclusion can be extended to general real Banach ∗ spaces. In this extension, the normalized duality map J : X → 2X plays ∗ ∗ ∗ 2 ∗ 2 an important role, J(x) = {x ∈ X | < x , x >= x = x }, (cf. Example 1 in Sect. 2.2). It is easy to verify that J is an odd, positively homogeneous, bounded set-valued mapping, and it is known that J(x) = ∂ϕ(x), where ϕ(x) = 12 x2 , so it is monotone. The Browder theorem (Theorem 2.5.8) is also extended to maximal monotone mappings: Let X be a uniformly convex Banach space, if D(A) ⊂ X, and A : D(A) → X ∗ is maximal monotone and coercive: y,x x → +∞ ∀(x, y) ∈ ΓA , as x → ∞, then A is surjective. For a proper l.s.c. convex function on a real Banach space, f : X → R1 ∪ {+∞}, the conjugate function of f, f ∗ : X ∗ → R1 ∪ {+∞} will be deﬁned in Sect. 4.1.3 to be f ∗ (x∗ ) = sup { x∗ , x − f (x)} . x∈X

It will be proved therein that f ∗ is again a proper l.s.c. convex function, satisfying f (x) + f ∗ (x∗ ) = x∗ , x ⇔ x∗ ∈ ∂f (x) ⇔ x ∈ ∂f ∗ (x∗ ) . Thus we have the following: Corollary 2.6.11 If f : X → R1 is proper, l.s.c., and convex, then ∂f ∗ = (∂f )−1 is maximal monotone. Proof. According to the above discussion, ∂f ∗ = (∂f )−1 . The maximality of ∂f ∗ follows from Theorem 2.6.10.

3 Degree Theory and Applications

The Leray–Schauder degree is an important topological tool introduced by Leray and Schauder in the study of nonlinear partial diﬀerential equations in the early 1930s. The nontriviality of the degree ensures the existence of a ﬁxed point of the compact mapping in the domain. It enjoys the properties of homotopy invariance and additivity, which make the topological tool more convenient in application, and provides more information on ﬁxed points. In Sects. 3.5 and 3.6 we shall see how better results could be obtained by using this tool than by using any of the other methods that we discussed previously. The Leray–Schauder degree is an extension of the Brouwer degree from ﬁnite–dimensional spaces to inﬁnite–dimensional Banach spaces, while the Brouwer degree is a powerful tool in algebraic topology. We introduce the notion of the Brouwer degree in Sect. 3.1, and investigate its fundamental properties in Sect. 3.2. On the one hand, we introduce the deﬁnition of the Brouwer degree from the point of view of diﬀerential topology so that it closely ties with the counting of zeroes of mappings; on the other hand, we build up the relationship between this deﬁnition and that used in algebraic topology. The applications to the Brouwer ﬁxed-point theorem and Borsuk–Ulam theorem, as well as the intersection number etc. are studied in Sect. 3.3. The notion of the Leray–Schauder degree is deﬁned by approximation. All its fundamental properties are transferred from those of the Brouwer degree directly. These are the contents of Sect. 3.4. We emphasize the computation of the degree, because the more precisely we know the degree the sharper we can estimate the number of ﬁxed points. This opens a door to the study of multiple solutions in nonlinear analysis. Rabinowitz’s global bifurcation theorem, based on the computation of the Leray–Schauder degree, is an important part of nonlinear functional analysis and probably the only global result in bifurcation theory. It provides global information on the branch of solutions emanating from a bifurcation point. The applications to nonlinear Sturm–Liouville problems as well as nonlinear elliptic problems are presented. They are studied in Sect. 3.5.

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3 Degree Theory and Applications

In Sect. 3.6, we introduce various applications of the Leray–Schauder degree. Schaefer’s ﬁxed-point theorem is more convenient in application to partial diﬀerential equations. Relying on the degree arguments, multiple solutions for semilinear elliptic equations are studied. In particular, some interesting multiple results are obtained in combination with the super- and sub-solutions method. When we are concerned with positive solutions of diﬀerential equations, one modiﬁes the degree theory to that for compact mappings deﬁned on closed convex sets. Moreover, we present some other applications: The Krein–Rutmann theory for positive linear operators, the existence of a positive solution for the superlinear elliptic problem, and bridging solutions of disjoint domains etc. in order to expose the extensive applicability of the degree theory. In Sect. 3.7, various extensions of the Leray–Schauder degree are discussed, including the degrees for set contraction mappings, condensing mappings, setvalued mappings, Fredholm mappings, etc.

3.1 The Notion of Topological Degree The topological degree was originally introduced by H. Poincar´e in the qualitative study of ODEs. Firstly, he deﬁned the index of a singularity P0 as follows: Let P0 = (x0 , y0 ) be a singularity of the plane vector ﬁeld (F(x,y), G(x,y)), i.e., F (x0 , y0 ) = G(x0 , y0 ) = 0. Let C be a closed curve surrounding P0 in the phase plane. We count the variation of the angles of the tangents of the ﬂow: x˙ = F (x, y) y˙ = G(x, y) generated by (F, G) along C counterclockwise. This amount should be a multiple of 2π, and is denoted by ind(P0 ). One may measure this amount by observing the compass moving along the curve C in a magnetic ﬁeld. Afterwards, he deﬁned the winding number of a closed curve C, on which there are no singularities of the vector ﬁeld. Diﬀerent from the index, at this time, C is not necessarily just enclosing one single singularity. He noticed: (1) The winding number is invariant under deformation of curves C, if there is no singularity on these curves C. (2) The winding number for a given curve C is invariant under deformation of vector ﬁelds, if there is no singularity on C. A direct consequence follows: (3) Let D be the domain enclosed by C. If the winding number of C is nonzero, then there must be a singularity of the vector ﬁeld in D. For a given vector ﬁeld f = (F, G), a closed curve C (or equivalently, the enclosed domain D), we denote the winding number by deg (f, D), More generally, instead of the zeroes of f , sometimes we study the solutions of

3.1 The Notion of Topological Degree

129

f (x, y) = P0 for a given P0 ∈ R2 ; we denote the winding number of f −P0 by deg (f, D, P0 ). In order to extend the concept of winding numbers to higher-dimensional vector ﬁelds, we would rather make some stronger restrictions on f at ﬁrst. It makes the geometric characterization of winding numbers easy to understand. Let us ﬁrst assume that f is an analytic function: f (z) = F (z) + iG(z), where z = x + iy and w0 = p0 + iq 0 are complex numbers. Let Ω be a bounded open domain in C with boundary ∂Ω. Suppose f (z) = w0 ∀z ∈ ∂Ω, then we have

1 d arg(f (z) − w0 ) deg(f, Ω, w0 ) = 2π ∂Ω

1 = d log(f (z) − w0 ) 2πi ∂Ω

f (z) 1 = dz 2πi ∂Ω f (z) − w0 = σj , (3.1) zj ∈f −1 (w0 )∩Ω

where σj is the multiplicity of zj , i.e., f (z) = w0 + cj (z − zj )σj + ◦(|z − zj |σj ),

(3.2)

as |z − zj | → 0, and cj = 0. If f is not analytic but diﬀerentiable, let f = (F, G), z = (x, y) and w0 = (p0 , q0 ). Assume f (z) = w0 , ∀z ∈ ∂Ω and det f (z) = det

∂(F, G) (z) = 0 ∀z ∈ f −1 (w0 ) , ∂(x, y)

then the point set Ω ∩ f −1 (w0 ) is ﬁnite, from the IFT; and

1 d arg(f (z) − w0 ) deg(f, Ω, w0 ) = 2π ∂Ω

G − q0 1 = d arctan 2π ∂Ω F − p0

(F − p0 )dG − (G − q 0 )dF 1 = . 2π ∂Ω (F − p0 )2 + (G − q 0 )2 Applying Green’s formula, it equals

(F − p0 )dG − (G − q 0 )dF = (F − p0 )2 + (G − q 0 )2 ∂Bε (zj ) −1 zj ∈f

(w0 )∩Ω

sgn det f (zj ),

zj ∈f −1 (w0 )∩Ω

(3.3)

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3 Degree Theory and Applications

−1 where ε > 0 is small enough that + Bε (zi )∩Bε (zj ) = Ø ∀zi = zj in f (w0 )∩Ω, provided det f (z) = 0 ∀z ∈ zi ∈f −1 (w0 )∩Ω Bε (zi ). The last formula provides a clue to deﬁning the topological degree for higher-dimensional mappings. For each root of f (z) = w0 , we assign a signed number, which is the signature of the determinant at this point, and then the topological degree is the summation of all these signed numbers. Before going to the deﬁnition, we need a special case of the Sard theorem. Since we have not proved the Sard theorem in Chap. 1, now we shall present a proof for the special case, due to its simplicity. However, we refer the reader to Milnor [Mi 2] for a proof of the general theorem, which can also be obtained by combining our proof for the special case with an inductive argument.

Theorem 3.1.1 (Sard) Let Ω ⊂ Rn be an open set, and let f : Ω → Rn be a C 1 map. Write Z = {x ∈ Ω| det f (x) = 0} . Then f (Z) is a zero measure set. Proof. Consider a closed cube C ⊂ Ω, each side with length a. We divide C (1) (1) (1) into N n conﬁguration cubes Ci , i = 1, . . . , N N , with int(Ci ) ∩ int(Cj ) = a ∅, i = j, each side with length N . Thus on one hand, √ na , f (x) − f (x0 ) KC x − x0 KC N ∀x, x0 ∈ Ci for some i, where KC = max { f (x) | x ∈ C}. On the other hand, ∀ε > 0 ∃ an integer N such that √ na . f (x) − f (x0 ) − f (x0 )(x − x0 ) < ε x − x0 ε N (1)

Let x0 ∈ Z, f (x0 ) is not invertible. Thus f (x0 )Rn cannot span the whole space Rn , but is included in a (n−1)-dimensional linear subspace H. It follows that √ ε na (1) ∀x ∈ Ci , dist(f (x), f (x0 ) + H) N (1)

(1)

in which Ci is the small cube including x0 . √We conclude that f (C ) is √ √i included in a cube, whose sides have length 2ε Nna , 2KCN na , . . . 2KCN na , respectively. Therefore (1) m∗ (f (Ci ∩ Z)) m∗ (f (C ∩ Z)) (1)

Ci

(1)

Ci

∩Z=Ø

m∗ (f (Ci )) (1)

∩Z=Ø n

n−1 2 n 2 KC n a ε, n

3.1 The Notion of Topological Degree

131

where m∗ is the Lebesgue outer measure. Since ε > 0 and the cube C are arbitrary, we obtain m∗ (f (Z)) = 0. Suppose that Y is an n-manifold, a subset W ⊂ Y is called a null set, if for each chart (ϕ, U ) of Y , the set ϕ(U ∩ W ) ⊂ Rn is a null set. Let us recall the deﬁnitions of regular/critical values/points of a map between two Banach manifolds. Sard’s theorem asserts that if X and Y are n-manifolds, then for any f ∈ C 1 (X, Y ), the set of critical values of f is a null set. Let X0 , Y be two oriented smooth n-dimensional manifolds, and X ⊂ X0 be an open subset satisfying the condition that X = X ∪ ∂X is compact. If f ∈ C(X, Y ) ∩ C 1 (X, Y ), and if y0 is a regular value of f then f −1 (y0 ) = {x ∈ X| f (x) = y0 } must be a ﬁnite set, from the IFT. Deﬁne f −1 (y0 ) = {x1 , . . . , xk } . Assuming that y0 ∈ / f (∂X) is a regular value, we deﬁne deg(f, X, y0 ) =

k

sgn det f (xj ) .

(3.4)

j=1

Again we let Z denote the set of critical points of f . We shall extend the deﬁnition to continuous mappings in three steps: I. The special case f ∈ C(X, Y ) ∩ C 2 (X, Y ), y0 ∈ f (Z) ∪ f (∂X). In order to remove the assumptions on the regularity and the condition f ∈ C 1 (X, Y ), we would rather express (3.4) in an integration form. Let Uj be a neighborhood of xj , such that f : Uj → f (Uj ) is a diﬀeomorphism, and that Ui ∩ Uj = Ø, ∀i = j, where i, j = 1, 2, . . . , k. Set k , V = f (Uj ) . j=1

This is a neighborhood of y0 . Let us choose a C ∞ n-form on Y : µ = ψ(y)dy1 ∧ . . . ∧ dyn , (in the following, we write simply dy = dy1 ∧ . . . ∧ dyn ) such that supp ψ ⊂ V ∩ (Y \f (∂X)) ,

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3 Degree Theory and Applications

and

µ=1. Y

Since detf (x) has the same sign in each Uj , we have

µ◦f = X

k

j=1

=

k

j=1

=

k

µ◦f

Uj

ψ(f (x)) det f (x)dx

Uj

sgn det f (xj )

Uj

j=1

=

k

sgn det f (xj )

k

ψ(y)dy f (Uj )

j=1

=

ψ ◦ f (x)| det f (x)|dx

sgn det(f (xj ))

j=1

= deg(x, X, y0 ) .

(3.5)

In the integration form, µ is arbitrarily chosen. We shall prove that the integral X µ ◦ f does not depend on the special choice of µ. We call the chart (g, Ω) a good chart at y0 ∈ Y , if (1) g(y0 ) = θ, (2) g(Ω) = In = (−1, +1)n in Rn . A C ∞ − n-form µ on Y is called admissible if µ = ψ(y)dy satisﬁes (1) supp ψ ⊂ Ω ∩ (Y \f (∂X)) for a good chart (g, Ω) at y0 . (2) Y µ = 1. Lemma 3.1.2 Suppose that µ = ψ(y)dy is a C ∞ − n-form on Y satisfying (1) supp ψ ⊂ Ω, and (g, Ω) is a good chart at y0 , (2) Y µ = 0. Then there exists an (n − 1) form ω such that supp ω ⊂ Ω and dω = µ. Proof. In the coordinates, g

Ω −→ In ∩ ∩ Y Rn one may assume suppψ ⊂ In and ψ(y)dy = 0. What we want to prove is the existence of a function ν ∈ C 1 (Rn , Rn ) such that

3.1 The Notion of Topological Degree

133

supp ν ⊂ In and div ν = ψ . Indeed, let ν = (ν1 , ν2 , . . . , νn ), then ω = ν1 dy2 ∧ . . . ∧ dyn + . . . + (−1)n νn dy1 ∧ . . . ∧ dyn−1 satisﬁes dω = µ . We prove the conclusion by mathematical induction. For n = 1, one takes

y ψ(t)dt , ν(y) = −∞

then ν is as required. Assume the lemma is true for n = k. Let

∞ φ(ˆ y) = ψ(ˆ y , yk+1 )dyk+1 , −∞

where y = (ˆ y , yk+1 ), yˆ = (y1 , . . . , yk ). Then supp φ ⊂ Ik (−1, 1)k ,

and

φ(ˆ y )dˆ y=

ψ(y)dy = 0 .

By the hypothesis of induction, there exists ω ∈ C 1 (Rk , Rk ) satisfying supp ω ⊂ Ik , div ω = φ .

Choose τ ∈ C 1 (R1 ), supp τ ⊂ I1 , then

∞

−∞

τ (t)dt = 1 , −∞

[ψ(ˆ y , yk+1 ) − φ(ˆ y )τ (yk+1 )]dyk+1 = 0 .

Let y , yk+1 ) = νk+1 (ˆ then

∞

yk+1

−∞

[ψ(ˆ y , t) − φ(ˆ y )τ (t)]dt ,

∂ νk+1 (ˆ y , yk+1 ) = ψ(ˆ y , yk+1 ) − φ(ˆ y )τ (yk+1 ) . ∂yk+1

This means that ψ = div ν, where ν = (ω1 (ˆ y )τ (yk+1 ), . . . , ωk (ˆ y )τ (yk+1 ), νk+1 (ˆ y , yk+1 )) , and (ω1 , . . . , ωk ) = ω .

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3 Degree Theory and Applications

In order to prove that the integral X µ ◦ f does not depend on µ, we assume that ν, µ are two admissible n forms with respect to a good chart (g, Ω) at y0 . According to Lemma 3.1.2, there exists an (n − 1) form ω such that dω = ν − µ, and then

ν◦f − µ◦f = dω ◦ f . X

If we can prove

X

X

(dω) ◦ f =

d(ω ◦ f ),

X

(3.6)

X

then by the Stokes theorem, we conclude:

ν◦f = µ◦f . X

X

We are going to prove (3.6). First, we need Lemma 3.1.3 If f ∈ C(X, Y ) ∩ C 2 (X, Y ), then for any (n − 1) form ω, we have d(ω ◦ f ) = dω ◦ f . Proof. In local coordinates, ω=

n

νj (y)dy1 ∧ . . . ∧ dˆ yj ∧ . . . ∧ dyn ,

j=1

where yj ∧ . . . ∧ dyn = dy1 ∧ . . . ∧ dyj−1 ∧ dyj+1 ∧ . . . ∧ dyn . dy1 ∧ . . . ∧ dˆ Thus ω◦f =

n n j=1

νk ◦

f (x)Jfkj (x)

dx1 ∧ . . . ∧ dˆ xj ∧ . . . ∧ dxn ,

k=1

where Jfki is the (k, i) cofactor of the Jacobian determinant det f (x) = Jf (x). Hence d(ω ◦ f ) n

n ∂νk ◦ f (x) ∂Jfki (x) ∂fl ki Jf (x) · + νk ◦ f (x) dx1 ∧ . . . ∧ dxn = ∂yl ∂xi ∂xi i,k=1 l=1 ⎤ ⎡ n n n ∂Jfki (x) ∂ν ◦ f (x) ∂f k l ⎦ =⎣ Jfki (x) + νk ◦ f (x) ∂yl ∂x ∂x i i i=1 i k,l=1

× dx1 ∧ . . . ∧ dxn

k=1

3.1 The Notion of Topological Degree

135

in which f = (f1 , . . . , fn ) . Let g = (−1)k−1 (f1 , . . . , fˆk , . . . , fn ) ,

then Jfki

i−1

= (−1)

det

ˆ ∂g ∂g ∂g ,..., ,..., ∂x1 ∂xi ∂xn

.

Hence n n 2 ˆ ∂ ki g ∂g ∂ ∂g ∂g Jf = (−1)i−1 det ,..., ,..., ,..., ∂x ∂x1 ∂xi ∂xl ∂xi ∂xn i i=1 i=1 l=i n 2 ˆ g ∂ ∂g ∂g ∂g = (−1)i−1+l−2 det , ,..., ,..., ∂xi ∂xl ∂x1 ∂xi ∂xn i=1 l>i - ˆ ∂2g ∂g ∂g ∂g i−1+l−1 (−1) det , ,..., ,..., =0. + ∂xl ∂xi ∂x1 ∂xi ∂xn i>l

From this, together with n ∂fl ki Jf (x) = δkl Jf (x) ∂x i i=1

follows d(ω ◦ f ) =

n ∂νk ◦ f (x) k=1

∂yk

Jf (x)dx1 ∧ . . . ∧ dxn = (dω) ◦ f .

Now we arrive at: / f (Z) ∩ f (∂X) and Theorem 3.1.4 Assume f ∈ C(X, Y ) ∩ C 2 (X, Y ). If y0 ∈ respect to a good chart (g, Ω) at y0 , with µ is an admissible C ∞ n form with supp(µ) ⊂ Ω ∩ (Y \f (∂X)), then X µ ◦ f is a constant independent of µ. II. Removal of the restriction y0 ∈ f (Z) Combining Theorem 3.1.4 and (3.5), we know that

deg(f, X, y0 ) = µ◦f , X

if f ∈ C(X, Y )∩C 2 (X, Y ) and y0 ∈ f (Z)∪f (∂X), where µ is an arbitrary admissible C ∞ n form with respect to a good chart at y0 . Note that the integral on the right-hand side of the above formula does not contain y0 explicitly.

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3 Degree Theory and Applications

Further recall that the set of regular valuesis dense in f (X) according to the Sard theorem, thus if we take the integral X µ ◦ f to be the deﬁnition of deg(f, X, y0 ), the assumption y0 ∈ f (Z) is not necessary. In fact, this integral is deﬁned for all y0 ∈ f (∂X), and is integer valued (the left-hand side of the formula (3.5)) if y0 belongs to the dense (regular value) set. Hence X µ ◦ f is integer valued. Therefore the restriction y0 ∈ f (Z) can be removed when we use X µ ◦ f as the deﬁnition of the deg (f, X, y0 ). III. Extension to continuous mappings We ﬁrst prove: C

Lemma 3.1.5 Let φ : X × [0, 1] −→ Y satisfy φ(·, t) ∈ C 2 (X, Y ) ∀t ∈ [0, 1]. If y0 ∈ φ(∂X × [0, 1]), then deg(φ(·, t), X, y0 ) is a constant independent of t. Proof. Since φ(∂X × [0, 1]) is closed, there is a good neighborhood Ω at y0 such that Ω ∩ φ(∂X × [0, 1]) = Ø. If we choose ψ ∈ C ∞ (Y, R1 ) such that suppψ ⊂ Ω, Y ψdy = 1, then µ = ψ(y)dy is admissible, and

deg(φ(·, t), X, y0 ) = µ ◦ φ(·, t) . (3.7) X

The right-hand side is integer valued and is continuous in t, hence it must be a constant. Now we may extend the deﬁnition of degree to mappings in C(X, Y ) by approximation. By the use of the Whitney embedding theorem, we embed Y into RN as a regular submanifold for large N . Thus there exist a tubular neighborhood W of Y and a C ∞ retract r : W → Y . Since we assumed y0 ∈ f (∂X), one may choose 0 < ε < dRN (r−1 (y0 ), f (∂X)) such that the ε neighborhood of Y is in W , where dRN (·, ·) is the distance in RN . According to the Weierstrass approximation theorem and the partition of unity, there exists g ∈ C 2 (X, RN ) satisfying (3.8) f − g C(X,RN ) < ε . Thus for such g, deg (r ◦ g, X, y0 ) is well deﬁned. Moreover, we shall prove that deg(r ◦ g1 , X, y0 ) = deg(r ◦ g2 , X, y0 ) for g1 , g2 ∈ C 2 (X, RN ) both satisfying (3.8). Indeed, deﬁne h : X × [0, 1] → RN by h(x, t) = (1 − t)g1 (x) + tg2 (x) , and φ : X × [0, 1] → Y by φ = r ◦ h. Then distRN (φ(x, t), y0 ) = 0 ⇔ distRN (h(x, t), r−1 (y0 )) = 0 .

3.2 Fundamental Properties and Calculations of Brouwer Degrees

137

But, distRN (h(∂X, t), r−1 (y0 )) distRN (f (∂X), r−1 (y0 )) − ε > 0 , this proves that y0 ∈ φ(∂X × [0, 1]). Applying Lemma 3.1.5, we obtain deg(r ◦ g1 , X, y0 ) = deg(r ◦ g2 , X, y0 ) . We are now in a position to deﬁne the Brouwer degree for continuous mappings. Deﬁnition 3.1.6 (Brouwer degree) Let X0 , Y be two oriented smooth n− manifolds, and let X ⊂ X0 be an open subset with compact closure X. For / f (∂X), to the triple (f, X, y0 ) we deﬁne f ∈ C(X, Y ), y0 ∈ deg(f, X, y0 ) = deg(r ◦ g, X, y0 ) , where r is a C ∞ retract of a tubular neighborhood W of Y to Y , and g is a map deﬁned in (3.8). This is called the Brouwer degree of f . The Brouwer degree is integer valued. It is not diﬃcult to verify that Brouwer degree does not depend on the special choice of W and r. By deﬁnition, if X = Ø, then deg (f, X, y0 ) = 0. In dealing with mappings between complex manifolds, we identify C n with R2n by the canonical isomorphism z → (x, y), where z = x + iy for z ∈ C n , x, y ∈ Rn . Similarly for the map f → (u, v), where f : Ω → C n , and f = u + iv. As an exercise, readers can verify that if f is analytic then det(f (z)) > 0.

3.2 Fundamental Properties and Calculations of Brouwer Degrees The Brouwer degree has the following fundamental properties. (1) (Homotopy invariance). If φ : X × [0, 1] → Y is continuous and y0 ∈ φ(∂X × [0, 1]), then deg(φ(·, t), X, y0 ) = constant . Proof. Using the above notations, we choose 0 < < dRN (r−1 (y0 ), φ(∂X × [0, 1])) , where dRN (·, ·) is the distance in RN . By the above approximation, there is φˆ ∈ C 2 (X × [0, 1], Y ) such that ˆ 0) . One may choose g0 ∈ C 2 (X, RN ) such that dC(X,RN ) (g, f ) < r ◦ g0 , and choose y1 ∈ g(Z) satisfying dRN (y0 , y1 ) < 2ε , then

ε 2

for g =

dist(y1 , g(X\(X1 ∪ X2 ))) > 0 . On the one hand, by (3.5) we have deg(g, X, y1 ) = sgn det g (xi ) xi ∈g −1 (y1 )

=

+

xi ∈g −1 (y1 )∩X1

sgn det g (xi )

xi ∈g −1 (y1 )∩X2

= deg(g, X1 , y1 ) + deg (g, X2 , y1 ) . On the other hand, by property (2), deg(f, X, y0 ) = deg(g, X, y1 ), deg(f, Xi , y0 ) = deg(g, Xi , y1 ), i = 1, 2 . This is due to the fact that θ = tf (x) + (1 − t)g(x) − ty0 − (1 − t)y1 , as x ∈ ∂X ∪ ∂X1 ∪ ∂X2 ⊂ X\(X1 ∪ X2 ), t ∈ [0, 1] . Corollary 3.2.3 (Excision) If K ⊂ X is compact and y0 ∈ f (K) ∪ f (∂X), then deg(f, X, y0 ) = deg(f, X\K, y0 ) . Proof. Let X1 = X\K, X2 = Ø, then X\(X1 ∪ X2 ) = ∂X ∪ K . Corollary 3.2.4 (Kronecker existence) If y0 ∈ f (∂X) and deg(f, X, y0 ) = 0, then f −1 (y0 ) = Ø. Proof. Assume the contrary: f −1 (y0 ) = Ø, i.e., y0 ∈ f (X). By Corollary 3.2.3, deg(f, X, y0 ) = deg(f, X\X, y0 ) = 0 .

3.2 Fundamental Properties and Calculations of Brouwer Degrees

141

Kronecker existence theorem is often used in the study of the solvability of the equation f (x) = y0 . (5) (Normality). If X ⊂ Y , and y0 ∈ ∂X, then ◦ deg(id, X, y0 ) = 1, y0 ∈ X 0, y0 ∈ /X. More generally, we have: (6) If L is a n × n nondegenerate real matrix, and if X is a bounded open subset and θ ∈ X ⊂ Rn , then deg(L, X, θ) = (−1)β , where β = λj 0 such that deg (f, Bε (x0 ), y0 ) = deg (f (x0 ), Bε (θ), θ) from the homotopy invariance and translation invariance. The conclusion then follows from (6).

3.2 Fundamental Properties and Calculations of Brouwer Degrees

145

(10) If f ∈ C(X, Y ) and y0 ∈ f (∂X) with f −1 (y0 ) = {x1 , . . . , xp }, then deg (f, X, y0 ) =

p

i(f, xj , y0 ) .

i=1

Proof. It follows from the excision property plus the additivity property. At the end of this section we establish the connection between the definition of the Brouwer degree given above with that in algebraic topology. Following Corollary 3.2.2, for f ∈ C 2 (X, Y ), we have µ◦f deg (f, X, y0 ) = X . µ Y Let f ∗ : Ωn (Y ) → Ωn (X), µ → µ ◦ f be the pullback of f , where Ωn (X) is the space of n− forms over X. This means that deg (f, X, y0 ) is the ratio of the integrations of f ∗ µ and µ, ∀µ ∈ Ωn (Y ). In algebraic topology, the Brouwer degree for maps f from the sphere into itself is deﬁned to be the multiplier of the homomorphism f ∗ : H n (S n ) → H n (S n ), in which H n (S n ) stands for the cohomology group of the n− sphere, and the map f maps the generator [ω] into λ[ω] for some integer λ. According to de Rham theory, for a compact, oriented, connected n manifold without boundary X, the cohomology group H k (X) is deﬁned to be ker dk /Im dk−1 , where dk : Ωk (X) → Ωk+1 (X) is the exterior diﬀerentiation, k = 0, 1, . . . , n. We shall prove that Ωn (S n ) and then H n (S n ) is one dimensional. Namely: Lemma 3.2.7 For a compact, oriented, connected n-manifold X without boundary, dim H n (X) = 1. Proof. We want to show that H n (X) is generated by one generator. Let us choose an atlas {(Ui , ψi )| i = 1, . . . , p} such that Ui ∩ Uj = Ø ⇒ Ui ∪ Uj is contained in a good chart . ∈ Ωn (X) with suppµ0 ⊂ U1 , and It is suﬃcient to prove that1 for ∀µ, µ0 n−1 µ = 1, there exist λ ∈ R and ω ∈ Ω (X) such that X 0 µ − λµ0 = dω . Since X is connected, for ∀i, ∃ a curve C starting from U1 ending at Ui . Let p {Ui0 , Ui1 , . . . , Uil } be a chain of neighborhoods in {Ui }1 , such that Ui0 = U1 , Uil = Ui and Uik ∩ Uik−1 = Ø k = 1, 2, . . . , l .

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3 Degree Theory and Applications

We choose αk ∈ Ωn (X) satisfying

supp αk ⊂ Uik , αk = 1, k = 1, 2, . . . , l , X

and α0 = µ0 . Since supp(αk − αk−1 ) ⊂ Uik ∪ Uik−1 is contained in a good neighborhood, and

(αk − αk−1 ) = 0 , X

we have ωki ∈ Ωn−1 (X) satisfying αk − αk−1 = dωki , k = 1, 2, . . . , l , provided by Lemma 3.1.2. Hence αl = µ0 + d

l

ωki .

(3.15)

k=1

Since Uil = Ui , αl depends on the index i, we write it as βi , i = 1, . . . , p. p p Now, let {χi }1 be a partition of unity with respect to {Ui }1 , i.e., supp χi ⊂ Ui χi 0,

p

χi ≡ 1 .

i=1

Set µi = χi µ , then Denote ci =

supp µi ⊂ Ui .

X

µi . There exists ω !i ∈ Ωn−1 (X) such that µi − ci βi = d! ωi ,

from Lemma 3.1.2, i = 1, 2, . . . , p. Combining (3.15) and (3.16), we have µ=

p

χi µ =

i=1

=

p i=1 p i=1

µi ci βi + d

p i=1

ω !i

(3.16)

3.2 Fundamental Properties and Calculations of Brouwer Degrees

=

p

ci µ0 + d

i=1

p

ω !i + ci

i=1

l

147

ωki

k=1

= λµ0 + dω , p p l where λ = µ , and ω = ωi + ci k=1 ωki ). Thus i=1 ci = i=1 (! X i n we have proved that all elements in H (X) are multipliers of [µ0 ], i.e., dim H n (X) = 1. Combining Corollary 3.2.2 and Lemma 3.2.7, we obtain Theorem 3.2.8 Let X and Y be compact, oriented, connected n-manifolds without boundaries. Then for any f ∈ C 2 (X, Y ), the following diagram commutes: f∗ H n (Y ) −−−−→ H n (X) ⏐ ⏐ ⏐ ⏐ Y0 X0 R1

−−−−→ deg f

R1

Finally, we turn to studying the relationship between degrees on balls and those on spheres. Let f : B n → Rn be a continuous map satisfying θ ∈ f (∂B n ), so deg (f, B n , θ) is well deﬁned. Let us deﬁne Φ(x) =

f (x) f (x)

∀x ∈ ∂B n .

Then Φ : S n−1 → S n−1 deﬁnes a Brouwer degree: deg (Φ, S n−1 ). What is the relationship between deg(f, B n , θ) and deg(Φ, S n−1 )? Deﬁne a homotopy θ if x = θ φ(x, t) = x 2 x f ( ) if x = θ . x f ( ) t x x

Since f |∂B n = φ(·, 0)|∂B n , from Corollary 3.2.1 and the homotopy invariance, deg(f, B n , θ) = deg(φ(·, 0), B n , θ) = deg(φ(·, 1), B n , θ) . In the case where f ∈ C 1 , and y0 ∈ S n−1 is a regular value of Φ, for ∀ε > 0 small, ε2 y0 is a regular value of φ(·, 1), and Φ−1 (y0 ) = {x1 , . . . , xk } if and only if φ(·, 1)−1 (ε2 y0 ) = {εx1 , . . . , εxk }. Since sgn JΦ (xj ) = sgn Jφ(·,1) (εxj ), j = 1, 2, . . . , k , we obtain deg(φ(·, 1), B n , θ) = deg(Φ, S n−1 ). The assumptions on f and on y0 can easily be dropped. Namely, we have proved: Corollary 3.2.9 deg(f, B n , θ) = deg(Φ, S n−1 ) = deg (Φ). Thus the notation deg (Φ, S n−1 ) is in coincidence with the notation deg (fˆ, ∂B n , θ) introduced after Corollary 3.2.1.

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3.3 Applications of Brouwer Degree 3.3.1 Brouwer Fixed-Point Theorem The Brouwer degree is a fundamental tool in algebraic topology. It is widely used in topological arguments. We are satisﬁed having a glimpse of these applications. Theorem 3.3.1 (Brouwer ﬁxed-point theorem) A continuous map f n n from B into itself has a ﬁxed point x0 ∈ B , i.e., f (x0 ) = x0 . Proof. Let g = id − f . With no loss of generality, we may assume θ ∈ g(∂B n ). Since ∀x ∈ ∂B n , ∀λ 1, f (x) ∈ λx, we have θ ∈ φ(∂B n ) × [0, 1], where φ(x, t) = (1 − t)g(x) + tx . Thus by the homotopy invariance and the normality, ◦

◦

deg (g, B n , θ) = deg (id, B n , θ) = 1 . Our conclusion follows from the Kronecker existence theorem.

The following topological fact follows from a simple degree argument: Theorem 3.3.2 There is no continuous map f : B n f |∂B n = id|∂B n , i.e., ∂B n is not a retraction of B .

n

→ ∂B n such that

n

Proof. If not, there is a continuous map f : B → ∂B n . f |∂B n = id|∂B n . On one hand, deg (f, B n , θ) = 0 due to the Kronecker existence theorem. On the other hand, by Corollary 3.2.1 and the normality deg (f, B n , θ) = deg (id, B n , θ) = 1 .

This is a contradiction. 3.3.2 The Borsuk-Ulam Theorem and Its Consequences

Let Ω ⊂ Rn be a bounded open set, which is symmetric with respect to the origin θ, i.e., −Ω = Ω. A map f : Ω → Rn is called odd, if f (−x) = −f (x) ∀x ∈ Ω . We are going to study the Brouwer degree of odd mappings. Theorem 3.3.3 (Borsuk) Suppose that Ω ⊂ Rn is a bounded open set containing θ and is symmetric with respect to θ. If f : Ω → Rn is an odd continuous map, then deg (f, Ω, θ) = odd , whenever θ ∈ f (∂Ω).

3.3 Applications of Brouwer Degree

149

Proof. One chooses ε > 0 such that Bε = Bε (θ) ⊂ Ω. According to Tietze’s theorem, there exists a continuous map fε : Ω → Rn satisfying x x ∈ Bε fε (x) = f (x) x ∈ ∂Ω . If we use 12 (fε (x) − fε (−x)) to replace fε (x), then one may assume that fε is odd. Since fε |∂Ω = f |∂Ω , according to Corollary 3.2.1, the additivity and the normality, deg (f, Ω, θ) = deg (fε , Ω, θ) = deg (fε , Bε , θ) + deg (fε , Ω\B ε , θ) = 1 + deg (fε , Ω\B ε , θ) . In the following, we want to prove that deg (fε , Ω\B ε , θ) is even. Noticing that for a C 1 odd map g with regular value θ, elements in g −1 (θ) ∩ (Ω\B ε ) occur in pairs. Our strategy is to approximate fε by a C 1 odd map g with regular value θ. Once it is constructed, we have deg(f , Ω\B , θ) = deg(g, Ω\B , θ) ≡ 0 (mod 2) . By the Weierstrass approximation theorem, we may assume that fε ∈ C ∞ . It remains to approximate it such that θ is a regular value. To this end we ¯ (θ))×Mn×n −→ Rn deﬁned by F (x, A) = f (x)+ consider the map F : (Ω\B Ax. If we can show that F {θ}, then according to the transversality theorem (Theorem 1.3.14), gA {θ}, for almost every A ∈ Mn×n , where gA (x) = F (x, A). In fact, F (x, A)(y, B) = f (x)y + Ay + Bx, ∀(y, B) ∈ Rn × Mn×n . Since x = θ, ∀z ∈ Rn , we take B = x 2 z, and y = θ, we have f (x, A)(y, B) = z, i.e., F (x, A) is surjective, or F {θ}. Since A can be chosen arbitrarily small, the proof is complete.

Theorem 3.3.4 (Borsuk–Ulam) Suppose that Ω ⊂ Rn is a symmetric bounded open set including θ, and that g : ∂Ω → Rm , m < n, is odd and continuous. Then there exists x0 ∈ ∂Ω such that g(x0 ) = 0. Proof. We prove by contradiction. Suppose θ ∈ g(∂Ω). We deﬁne a continuous extension g! : Ω → Rm by the Tietze theorem. With no loss of generality, one may assume g! is odd. Now we apply Theorem 3.3.3, deg (! g , Ω, θ) = 0. Choosing y0 ∈ Rn \Rm , with small norm, it follows that deg (! g , Ω, y0 ) = 0 Thus, g!−1 (y0 ) = Ø, but this is impossible.

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3 Degree Theory and Applications

Corollary 3.3.5 Suppose that Ω ⊂ Rn is a symmetric bounded open set including θ, and that g : ∂Ω → Rm , with m < n, is continuous. Then there exists x0 ∈ ∂Ω such that g(x0 ) = g(−x0 ) Proof. Set g1 (x) = g(x) − g(−x), then apply Theorem 3.3.4 directly.

Remark 3.3.6 We consider a continuous vector ﬁeld on S n , i.e., a continuous map g : S n → Rn . Corollary 3.3.5 means that there exists a pair of antipodal points ±x0 , which have the same vector g(x0 ) = g(−x0 ). Theorem 3.3.7 (Ljusternik–Schnirelmann–Borsuk) Suppose that Ω ⊂ Rn is a symmetric bounded open set including θ, and that {A1 , . . . , Ap } is a closed covering of ∂Ω, satisfying Ai ∩ (−Ai ) = Ø ∀i. Then p n + 1. Proof. We may assume that ∩pi=1 Ai = Ø. For otherwise, if ∃x0 ∈ ∩pi=1 Ai , then by covering ∂Ω = ∪pi=1 (−Ai ), there must be a j such that x0 ∈ Aj ∩ (−Aj ). But this is impossible. Now we set di (x) = dist (x, Ai ), i = 1, 2, . . . , p , and f : ∂Ω → Rp−1 ⊂ Rn−1 as follows: f (x) = (d1 (x), d2 (x), . . . , dp−1 (x)) . Supposing p n, we apply Corollary 3.3.5, ∃x0 ∈ ∂Ω satisfying f (x0 ) = f (−x0 ). Since {A1 , A2 , . . . , Ap } is a covering, there exists j p such that x0 ∈ Aj . There are two possibilities: 1. j p − 1. It follows that dj (−x0 ) = dj (x0 ) = 0, thus −x0 ∈ Aj or x0 ∈ Aj ∩ (−Aj ); this is a contradiction. +p−1 / j=1 Aj , thus j = p. In this case, dj (−x0 ) = dj (x0 ) > 0, ∀j = 2. x0 ∈ +p−1 / j=1 (−Aj ). Again, by covering, x0 ∈ Ap ∩ (−Ap ), 1, 2, . . . , p − 1, so x0 ∈ a contradiction. The following theorem is an extension of the open mapping theorem to continuous mappings in ﬁnite-dimensional spaces. Theorem 3.3.8 (Invariance of domains) If Ω ⊂ Rn is a nonempty open set, and if f : Ω → Rn is continuous and locally injective, then f is an open map. Proof. We want to show that ∀x0 ∈ Ω, ∀ε > 0 with Bε (x0 ) ⊂ Ω, ∃δ > 0 such that f (Bε (x0 )) ⊃ Bδ (f (x0 )). One may assume x0 = θ = f (θ), and that f : B (θ) → f (B (θ)) is 1–1. Since θ ∈ f (∂B (θ)), ∃δ > 0 such that B δ (θ) ⊂ Rn \f (∂B (θ)). Our problem is reduced to proving that ∀y0 ∈ Bδ (θ), ∃x ∈ B (θ) such that f (x) = y0 . Due to the Kronecker existence theorem, it is suﬃcient to prove that deg (f, B (θ), y0 ) = 0 .

3.3 Applications of Brouwer Degree

151

Since δ > 0 can be chosen arbitrarily small, according to property (2) of the Brouwer degree, deg (f, B (θ), y0 ) = deg (f, B (θ), θ) .

Let H(x, t) = f

x 1+t

−f

−tx 1+t

∀(x, t) ∈ B (θ) × [0, 1], .

It is continuous on B (θ) × [0, 1], and satisﬁes H(x, 0) = f (x), H(x, 1) = f ( 12 x) − f (− 12 x). We claim that θ ∈ H(∂Bε × [0, 1]). For otherwise, there are (x0 , t) ∈ ∂B (θ) × [0, 1] satisfying x0 −tx0 f =f . 1+t 1+t Since f is 1 − 1 on B (θ), we would have x0 −tx0 = , 1+t 1+t i.e., x0 = θ. But, this is impossible. Therefore by the homotopy invariance and Borsuk–Ulam theorem deg (f, Bε (θ), θ) = deg (H(·, 1), Bε , θ) = 0 . and then deg (f, Bε (θ), y0 ) = 0. This proves our conclusion.

Comparing with the global implicit function theorem, the following theorem is an extension to continuous mappings in ﬁnite-dimensional spaces: Corollary 3.3.9 If f : Rn → Rn is continuous and locally injective, and if f (x) → +∞ as x → +∞, then f is surjective. Proof. From Theorem 3.3.8, f (Rn ) is open. We shall prove that f (Rn ) is also closed. Suppose yn ∈ f (Rn ) with yn → y0 . We shall prove that ∃x0 ∈ Rn satisfying f (x0 ) = y0 . Indeed we have xn ∈ Rn satisfying f (xn ) = yn ∀n. According to the assumption, {xn } is bounded, and then there exists a convergent subsequence xn → x0 . Thus y0 = f (x0 ). f (Rn ) is both open and closed. Therefore f (Rn ) = Rn . 3.3.3 Degrees for S 1 Equivariant Mappings We now turn to a computation of the degree of an S 1 group action equivariant map. Let us consider an S 1 representation on C n : Tφ = ei diag {λ1 φ,...,λn φ} , where φ ∈ [0, 2π], and λ1 , . . . , λn ∈ Z.

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3 Degree Theory and Applications

Theorem 3.3.10 Suppose that Ω ⊂ C n is a T (S 1 ) invariant open domain containing θ, and that f ∈ C(∂Ω, C n \{θ}) satisﬁes f (Tφ z) = eikφ f (z), ∀z ∈ ∂Ω, ∀φ ∈ [0, 2π] . Then deg (f, Ω, θ) =

kn . λ1 . . . λn

Proof. 1. We may assume that λ1 = · · · = λn = 1. In fact, if it is proved for the special case, then for general λ1 , . . . , λn ∈ Z+ , we deﬁne the transform ˜ = g −1 (Ω), then the map f ◦ g : Ω ˜ → C n satisﬁes g(z) = (z1λ1 , . . . , znλn ), let Ω Tφ ◦ g(z) = g ◦ T˜φ z, where T˜φ = eiφ In . Therefore f ◦ g(T˜φ z) = f (Tφ ◦ g(z)) = eikφ f ◦ g(z) . Provided by the product formula, we have ˜ θ) = deg (f, Ω, θ) · deg (g, Ω, ˜ θ) . deg (f ◦ g, Ω, We assume the conclusion holds for λ1 = · · · = λn = 1, therefore ˜ θ) = k n . deg (f ◦ g, Ω, But, ˜ θ) = λ1 · · · λn , deg (g, Ω, we obtain our conclusion. If λ1 , . . . , λn are not all positive, then we introduce a transform g : (z1 , . . . , zj , . . . , zn ) → (z1 , . . . , z¯j , . . . , zn ) for all negative λj s. Again by the product formula, we have deg (f ◦ g, g −1 (Ω), θ) = (−1)sgn λ1 ···λn deg (f, Ω, θ) . Thus it is suﬃcient to prove the theorem for λ1 = · · · = λn = 1. 2. We may assume that f is smooth. This can be done by a molliﬁer; the argument is standard, so we omit it. The only thing that has to be veriﬁed is the equivariance of the molliﬁed map. 1 : η(t) = 1, as t ≤ , and 0, as t > 2, 3. Taking a smooth function on R+ 1 where < 2 dist (∂Ω, θ), and deﬁne fˆ = η(|z|)(z1k , . . . , znk ) + (1 − η(|z|))f (z) . Then fˆ has the same degree as f on Ω. In summary, we may assume that f is smooth and S 1 -equivariant, with λ1 = . . . = λn = 1, and has the form (z1k , . . . , znk ) in B (θ). ¯ε (θ). Deﬁne a map F : 4. Now, we consider the degree of f on Ω\B ¯ε (θ)) × Mn×n → C n by (Ω\B

3.3 Applications of Brouwer Degree

153

F (z, A) = f (z) + Az k , where z k = (z1k , . . . , znk ). We show that F {θ}. In fact, n n k ¯ B) = Fz · w + Fz¯ · w ¯+ zi Bij . F (z, A)(w, w, 1

1

¯ (θ), ∀ξ ∈ C n , the system Since z ∈ /B n

Bij zjk = ξi

1

has a solution Bij . By taking w = w ¯ = 0 the map F (z, A) is surjective. Applying the transversality theorem, fA := F (·, A) {θ} for almost all A ∈ Mn×n . Since A can be chosen small, the critical set S(fA , Ω\B (θ)) of fA on Ω\B (θ) consists of isolated points, but ∀z ∗ ∈ S(fA , Ω\B (θ)), Tφ z ∗ ∈ S(fA , Ω\B (θ)). This is impossible if it is nonempty. This proves that ¯ε (θ), θ) = 0. Therefore deg(fA , Ω\B deg (fA , Ω, θ) = deg (fA , B (θ), θ) . But on Bε (θ), fA (z) = z k + Az k , for suﬃciently small A, we obtain: deg (fA , B (θ), θ) = deg (f, B (θ), θ) = k n . Combining all together, we have proved the conclusion. 3.3.4 Intersection The Brouwer degree is useful in the intersection theory. We brieﬂy introduce it here. Let Ω1 , Ω2 ⊂ Rn be two compact manifolds with ∂Ω1 = ∅ and ∂Ω2 = ∅, where dim Ω1 = k + 1, dim Ω2 = n − k − 1. Let φ1 : ∂Ω1 → Rn , φ2 : Ω2 → Rn be two continuous mappings with φ1 (∂Ω1 ) ∩ φ2 (Ω2 ) = ∅. We say φ1 and φ2 link, if for any continuous extension of φ1 , φ!1 : Ω1 → Rn , we have φ!1 (Ω1 ) ∩ φ2 (Ω2 ) = ∅ . In other words, 11 (x) = φ2 (y) . ∃(x, y) ∈ Ω1 × Ω2 , such that φ Deﬁne a mapping F : Ω1 × Ω2 → Rn by F (x, y) = φ!1 (x) − φ2 (y); it is / F (∂Ω1 × Ω2 ) = equivalent to saying that F has a zero in Ω1 × Ω2 . Since θ ∈ F (∂(Ω1 ×Ω2 )), deg (F, Ω1 × Ω2 , θ) is well deﬁned and depends on (φ1 , φ2 ) only,

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3 Degree Theory and Applications

on account of Corollary 3.2.1. It is suﬃcient to verify deg (F, Ω1 × Ω2 , θ) = 0. Thus the Brouwer degree is a tool in the study of linking. Example 1. Let r1 > 1, r2 > 0. Let Ω1 , Ω2 ⊂ Rn be deﬁned as follows: Ω1 = Brk2 (θ) × [0, r1 ] & % := (x1 , x2 , . . . , xk+1 , 0, . . . , 0) x21 + . . . x2k = r22 , 0 ≤ xk+1 ≤ r1 , % & Ω2 = S n−k−1 := (0, 0, . . . , xk+1 , . . . , xn ) x2k+1 + . . . + x2n = 1 , and let φ1 = id|∂Ω1 , φ2 = idΩ2 . Then φ1 and φ2 link. 2 In fact, let F (x1 , x2 , . . . , xn ) = (x1 , . . . , xk , xk+1 − (1 − x2k+2 − . . . − x2n ), xk+2 , . . . , xn ), it has a unique zero in Ω1 × Ω2 : x1 = · · · = xk = xk+2 = · · · = xn = 0, xk+1 = 1. Therefore, deg(F, Ω1 × Ω2 , θ) = 1 . x3

Ω2 r1 θ

x1

Ω1 r2 x2 Fig. 3.1.

The Borsuk–Ulam theorem is applied to the study of the intersection of symmetric sets. Assume that X is a Banach space. To a closed symmetric set A ⊂ X\{θ}, we deﬁne the genus γ(A) of A to be inf{k ∈ N | ∃ an odd φ ∈ C(A, Rk \{θ})}. Example 2. Assume that Ω ⊂ Rn is a symmetric bounded open set containing θ, and that there is an odd homeomorphism h : A → ∂Ω. Then γ(A) = n. In fact, by deﬁnition, γ(A) ≤ n. Suppose γ(A) < n, then there exist k < n and an odd φ such that φ ∈ C(A, Rk \{θ}). Let g = φ ◦ h−1 , then g : ∂Ω → Rk \{θ} is continuous and odd. According to the Borsuk–Ulam theorem, there exists x0 ∈ A such that g(x0 ) = θ. This is a contradiction.

3.4 Leray–Schauder Degrees

155

3.4 Leray–Schauder Degrees In analysis we study continuous mappings in inﬁnite-dimensional spaces, so we should be concerned with extending the Brouwer degree from ﬁnitedimensional spaces to inﬁnite-dimensional Banach spaces. However, the generalization cannot suit arbitrary continuous mappings. This can be seen from the fact that if this sort of degree theory, which possesses the fundamental properties homotopy invariance, additivity and normality established in Sect. 3.2, had been built up, then the Brouwer ﬁxed-point theorem would be extended directly to Banach spaces as follows. Let B be the unit open ball at the origin in an inﬁnite-dimensional real Banach space X, φ ∈ C(B, B). Let ft = id − tφ, t ∈ [0, 1] and θ ∈ ft (∂B). If deg (ft , B, θ) had been extended such that all the above properties of the Brouwer degree hold, then there would be a zero of f1 = id − φ, i.e., a ﬁxed point x ∈ B of φ: φ(x) = x . The “proof” is the same as in the previous section; we repeat it as follows: From the hypothesis θ ∈ ft (∂B), according to the homotopy invariance and the normality, deg (f1 , B, θ) = deg (id, B, θ) = 1 . Hence by the Kronecker existence, f1−1 (θ) ∩ B = Ø, i.e., ∃x ∈ B such that φ(x) = x. However, for an inﬁnite-dimensional Banach space X, the above conclusion cannot be true for any continuous mapping φ. Example. Let X=l = 2

x = (x1 , x2 , . . . , xn , . . .)| x = 2

∞

x2n

0, there is a continuous operator Kε taking values in a ﬁnite-dimensional linear subspace Emε such that K(x) − Kε (x) ε

∀x ∈ Ω .

Proof. Let Bε (yj ) j = 1, . . . , mε , be ﬁnitely many ε− balls covering K(Ω). Let ψj (x) = (ε− x − yj )+ λ, λ 0 , λ+ = 0, λ < 0 .

where

Let ϕi (x) =

ψi (x) , mε j=1 ψj (x)

and Kε (x) =

mε

i = 1, 2, . . . , mε ,

ϕi (K(x))yi ,

i=1

then Kε (x) ∈ span {y1 , . . . , ymε } and K(x) − Kε (x)

m

ϕi (K(x)) K(x) − yi

i=1 mε

ε

ϕi (K(x)) = ε .

i=1

The idea in extending the degree to mapping f = id − K on bounded open set Ω ⊂ X, where K is a compact map, is by approximation: ∀ε > 0 arbitrarily small, we already have the Brouwer degree for fε = id − Kε , then we want to deﬁne the degree for f by Brouwer degrees for these fε . In doing so, we should verify: (1) If y0 ∈ X satisﬁes y0 ∈ f (∂Ω), then for small ε > 0, y0 ∈ fε (∂Ω). (2) For any two such fε1 and fε2 , the Brouwer degrees for fε1 , fε2 are equal. In proving (1), we need: Lemma 3.4.3 Let K : Ω → X be compact, f = id − K and S ⊂ Ω be closed. Then f (S) is closed. Proof. Let {xn } ⊂ S with f (xn ) → z ∗ . We want to show that z ∗ ∈ f (S). Indeed, there are a subsequence {ni } and y ∗ ∈ X such that Kxni → y ∗ , and then xni → z ∗ − y ∗ , which we write as x∗ . Then x∗ ∈ S. By the continuity of K, Kx∗ = y ∗ , and hence f (x∗ ) = z ∗ , i.e., z ∗ ∈ f (S).

3.4 Leray–Schauder Degrees

157

Thus, y0 ∈ f (∂Ω) implies dist (y0 , f (∂Ω)) > 0. Choosing 0 < ε < dist (y0 , f (∂Ω)), we have dist (y0 , fε (∂Ω)) > 0 . Now we turn to (2). Consider the mapping fε : Ω ∩ Rmε → Rmε where mε = dim span{ Kε (Ω)}. The Brouwer degree deg (f , Ω ∩ Rm , y0 ) is well deﬁned. Since mε depends on ε, we should compare all the degrees for these fε . Namely, we have: Lemma 3.4.4 Let Ω ⊂ Rn be a bounded open set, Rm ⊂ Rn , and i : Rm → Rn be the canonical immersion: ˆ = (x1 , . . . , xm , 0, . . . , 0) . x = (x1 , . . . , xm ) → x Let K : Ω → Rm be continuous, f = id − K and p ∈ Rm satisfying pˆ ∈ f (∂Ω). Then deg (f, Ω, pˆ) = deg (f |Rm ∩Ω , Rm ∩ Ω, p) .

ˆ ∈ C 1 (Ω, Rm ) satisfying Proof. Let ε = dist (ˆ p, f (∂Ω)) > 0. Choose K ε ˆ ˆ K − K C(Ω) < 2 . Let g = id − K. Then idm − Jg (y) = det 0

ˆi ∂K ∂xj

ˆ

Ki − ∂∂x k idn−m

i, j = 1, . . . , m, ¯. , ∀y ∈ Ω k = m + 1, . . . , n,

By the Sard theorem the critical value set of g|Rm ∩Ω : Rm ∩ Ω → Rm is an m-dimensional set of measure zero. It has a regular value q ∈ Rm such that p − q < ε/2, and qˆ ∈ g(∂Ω). Thus deg (f, Ω, pˆ) = deg (g, Ω, qˆ) (homotopy invariance) sgn Jg (yi ) = yi ∈g −1 (ˆ q)

=

yi ∈g|−1 m R

∩Ω

sgn Jg |Rm ∩Ω (yi ) (g −1 (ˆ q ) ⊂ Rm ∩ Ω) (ˆ q)

= deg (g|Rm ∩Ω , Rm ∩ Ω, q) = deg (f |Rm ∩Ω , Rm ∩ Ω, p) (homotopy invariance)

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3 Degree Theory and Applications

Now we are ready to deﬁne the topological degree for f = id − K when K is compact. Deﬁnition 3.4.5 (Leray–Schauder degree) Let X be a real Banach space, and Ω ⊂ X be bounded and open. Let K : Ω → X be compact, f = id − K and p ∈ X\f (∂Ω). Deﬁne deg (f, Ω, p) = deg (fε , Ω ∩ Eε , p) , where ε ∈ (0, dist(p, f (∂Ω))), fε = id − Kε , and Kε is a continuous operator assuming values in a ﬁnite-dimensional space Eε with p ∈ Eε and satisﬁes Kx − Kε x ε

∀x ∈ Ω .

Let us explain the legitimacy of the deﬁnition. 1. By Lemma 3.4.3, dist(p, f (∂Ω)) > 0, according to Theorem 3.4.2, there exist such Kε and Eε . 2. We verify that deg (fε , Ω ∩ Eε , p) is well deﬁned for all such fε and Eε and takes the same value. Since dist(p, fε (∂Ω ∩ Eε )) dist(p, f (∂Ω)) − ε > 0 , deg (fε , Ω ∩ Eε , p) is well deﬁned. Let (fε0 , Eε0 ) and (fε1 , Eε1 ) be two arbitrary pairs of mappings and ﬁnite-dimensional linear subspaces satisfying the hypotheses of the deﬁnition, and let ˆε , ˆ = span{Eε , Eε }, fˆε = id − K E 0 1 i i ˆ ε (x) = (Kε (x), 0) and 0 is the zero element in E ˆ & Eε , i = 0, 1. where K i i i By Lemma 3.4.4, we have ˆ ∩ Ω, p), i = 0, 1 . deg (fεi , Eεi ∩ Ω, p) = deg (fˆεi , E ˆ × [0, 1] → E ˆ by Let us deﬁne φ : (Ω ∩ E) φ(x, t) = tfˆε0 (x) + (1 − t)fˆε1 (x),

∀t ∈ [0, 1] .

Applying the homotopy invariance of the Brouwer degree we obtain ˆ ∩ Ω, p) = deg (fˆε , E ˆ ∩ Ω, p) . deg (fˆε0 , E 1 This proves deg (fε0 , Eε0 ∩ Ω, p) = deg (fε1 , Eε1 ∩ Ω, p) .

3.4 Leray–Schauder Degrees

159

Remark 3.4.6 A set Ω ⊂ X is called ﬁnitely bounded, if for all linear ﬁnitedimensional subspaces E ⊂ X, E ∩ Ω is bounded. In Deﬁnition 3.4.5, one may assume that Ω is ﬁnitely bounded. More generally, for a given {e1 , . . . , ek } ⊂ E, if for all linear ﬁnitedimensional subspaces E ⊂ X with {e1 , . . . , ek } ⊂ E, E ∩ Ω is bounded, then Ω is called {e1 , . . . , ek }- ﬁnitely bounded. Again, the Leray–Schauder degree is well deﬁned on open sets, which are {e1 , . . . , ek }-ﬁnitely bounded. Similarly to the Brouwer degree, the Leray–Schauder degree enjoys the following fundamental properties: (1) (Homotopy invariance) Let K : Ω × [0, 1] → X be compact and p ∈ (id − K)(∂Ω × [0, 1]), then deg (id − K(·, t), Ω, p) = constant . (2) (Translation invariance) deg (id − K, Ω, p) = deg (id − K − p, Ω, θ) . (3) (Additivity) Let Ω1 , Ω2 ⊂ Ω, Ω1 ∩ Ω2 = Ø and p ∈ (id − K)(Ω\(Ω1 ∪ Ω2 )), then deg (id − K, Ω, p) = deg (id − K, Ω1 , p) + deg (id − K, Ω2 , p) . (4) (Normality)

1, deg (id, Ω, p) = 0,

p∈Ω, p ∈ Ω .

Similarly, all the other properties are consequences of these properties. Since the proofs are standard (reducing to ﬁnite-dimensional spaces by ε-approximations, and applying the corresponding properties of the Brouwer degree), we omit them. In particular, we have: (5) (Kronecker existence) Let Ω ⊂ X be a bounded open set and let K : Ω → X be compact. If y0 ∈ (id − K)(∂Ω) and deg (id − K, Ω, y0 ) = 0, then there exists x0 ∈ Ω satisfying x0 = Kx0 + y0 . (6) Let K be a compact linear operator, 1 ∈ σ(K) (the spectrum of K) and θ ∈ Ω, then deg (id − K, Ω, θ) = (−1)β , where β=

λj >1,λj ∈σ(K)

k βj , βj = dim ∪∞ k=1 ker (λj I − K) .

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3 Degree Theory and Applications

Proof. According to the Riesz–Schauder theory, σ(K) has only a point spectrum except 0. Let E1 be the ﬁnite-dimensional subspace spanned by all the generalized eigenvectors corresponding to eigenvalues > 1, then we have the direct sum decomposition X = E1 ⊕ E2 . Both E1 and E2 are invariant subspaces of K. Deﬁne KEi = Ki , then, for ∀t ∈ [0, 1], for ∀x ∈ E2 \{θ}, tK2 x = x. The homotopy invariance and the excision ensure that ∃ε > 0 such that deg (id − K, Ω, θ) = deg (id − K, Bε , θ) = deg (id − (K1 ⊕ tK2 ), Bε , θ) = deg (id − K1 , Bε , θ) = (−1)β . Now, let us return to the discussion in the beginning of this section. We have the special form of the Schauder ﬁxed-point theorem: if K : B → B is compact, then K has a ﬁxed point in B. However, the unit ball B may be replaced by any bounded closed convex set. Because it is known from Dugundji’s theorem (see Sect. 3.6) that for any closed convex subset C of X, there is a retract r : X → C, i.e., r is continuous, and satisﬁes r ◦ i = id|C , and i ◦ r ∼ id|X , where i is the injection C → X. The following Schauder ﬁxed-point theorem, which we have met in Sect. 2.2 is a consequence of the Kronecker existence property of the Leray–Schauder degree. Theorem 3.4.7 (Schauder ﬁxed point) Let C be a bounded closed convex set in X. If K : C → C is compact. Then K has a ﬁxed point. Proof. We choose R > 0 large enough such that C ⊂ BR (θ). Let r : B R (θ) → C be a retract, and i : C → B R (θ) be the injection. We have a compact mapping f = i ◦ K ◦ r : B R (θ) → B R (θ): B R⏐(θ) ⏐ r0 C

B R3(θ) ⏐ i⏐ K

−→

C

Following the steps in the proof of the Brouwer ﬁxed-point theorem, there is a ﬁxed point x0 ∈ B R of f , i.e., x0 = i ◦ K ◦ r(x0 ). Thus x0 ∈ C and then r(x0 ) = x0 . It follows that x0 = Kx0 . Another ﬁxed-point theorem, which is also frequently used in the theory of diﬀerential equations in studying the existence of solutions by a priori estimates, reads as: Theorem 3.4.8 (Schaefer) Suppose that K : X × [0, 1] → X is a compact mapping, satisfying K(x, 0) = θ. If the set S = {x ∈ X| ∃t ∈ [0, 1] such that x = K(x, t)} is bounded. Then K = K(·, 1) has a ﬁxed point.

3.4 Leray–Schauder Degrees

161

◦

Proof. Taking r > 0 large such that S ⊂ B r , we deﬁne f (x, t) = x − K(x, t)

∀(x, t) ∈ B r × [0, 1]

then θ ∈ f (∂Br × [0, 1]). According to the homotopy invariance deg (id − K, Br , θ) = deg (id, Br , θ) = 1 . Corollary 3.4.9 Suppose that K : X → X is compact and that the set S = {x ∈ X| ∃t ∈ [0, 1] such that x = tK(x)} is bounded. Then K has a ﬁxed point. Proof. Set K(x, t) = tK(x) As an application of Schaefer’s ﬁxed-point theorem, we study the following semi-linear elliptic BVP, which we have met in Sect. 1.2. Let Ω ⊂ Rn be a bounded open domain with smooth boundary. Theorem 3.4.10 Suppose that f ∈ C γ (Ω×R1 ×Rn , R1 ), for some γ ∈ (0, 1), satisﬁes (1) ∃ an increasing function c : R1+ → R1+ such that |f (x, η, ξ)| c(|η|)(1 + |ξ|2 )

∀(x, η, ξ) ∈ Ω × R1 × Rn .

(2) ∃ a constant M > 0 such that f (x, η, θ)

0

as η > M as η < −M.

Assume φ ∈ C 2,γ (∂Ω). Then the equation −u = f (x, u(x), ∇u(x)) u|∂Ω = φ

(3.17)

has a solution u ∈ C 2,γ (Ω). Proof. Fixing γ ∈ (0, γ), we deﬁne a map T from C 2,γ (Ω) = X into itself by u → v, where v is the solution of the following BVP: −v = f (x, u(x), ∇u(x)) v|∂Ω = φ .

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3 Degree Theory and Applications

By the Schauder estimate, we obtain v C 2,γ (Ω) C( u C 2,γ (Ω) , φ C 2,γ (∂Ω) ) . Thus, T is compact. Let us introduce a parameter t, and deﬁne v = T (u, t) to be the solution of the equation: −v = tf (x, u(x), ∇u(x)) v|∂Ω = φ . We intend to apply Schaefer’s ﬁxed-point theorem; it is suﬃcient to verify the boundedness of the set S = {u ∈ X| ∃t ∈ [0, 1] such that u = T (u, t)} , i.e., ∃ a constant C > 0 such that solutions ut of the equations −ut = tf (x, ut (x), ∇ut (x)) ut |∂Ω = φ

(3.18)

satisfy the following a priori estimate of the solution of (3.17): ut C 2,γ C

(3.19)

However, this has been done in Sect. 1.2.

Comparing the continuity method with the ﬁxed-point method based on Schaefer’s theorem, the latter does not require the invertible of the linearized equation. Moreover, less regularity on the nonlinear term is assumed. Of course, there is only the existence of a solution but not uniqueness. The Borsuk theorem is also extended. Theorem 3.4.11 Let Ω ⊂ X be a symmetric bounded open set including θ. If K : Ω → X is an odd compact map with θ ∈ f (∂Ω), where f = id − K, then deg (f, Ω, θ) is odd. Proof. We approximate f by ﬁnite-dimensional odd maps. Indeed, in Theorem 3.4.2, let K be approximated by Kε , with Im Kε ⊂ span{y1 , . . . , ymε }, and let ˆ ε (x) = 1 [Kε (x) − Kε (−x)] . K 2 Then ˆ ε (x) K(x) − K

1 1 K(x) − Kε (x) + K(−x) − Kε (−x) < ε , 2 2

and ˆ ε ⊂ span{y1 , . . . , ym } . Im K ε The conclusion follows from Theorem 3.3.3 directly.

3.4 Leray–Schauder Degrees

163

Corollary 3.4.12 Let Ω ⊂ X be a symmetric bounded open set including θ. If K : Ω → X is compact, f = id − K, and if f (x) = tf (−x)

∀(x, t) ∈ ∂Ω × [0, 1] ,

then deg (f, Ω, θ) is odd. Proof. Deﬁne

1 t K(x) − K(−x) 1+t 1+t 1 t = f (x) − f (−x). 1+t 1+t

φ(x, t) = id −

Thus θ ∈ (∂Ω × [0, 1]), φ(·, 0) = f and φ(·, 1) = 12 (f (x) − f (−x)) is odd. We obtain deg (f, Ω, θ) = deg (φ(·, 1), Ω, θ) = odd number . By the same proof of the invariance of domains theorem for the Brouwer degree, we have its inﬁnite-dimensional version: Corollary 3.4.13 (Invariance of domains) Assume that Ω ⊂ X is a nonempty open set, and that K : Ω → X is compact. If f = id − K is locally injective, then f is an open map. The notion of the index of an isolated solution is also extended. For f = id − K, where K is a compact map, we deﬁne the index of f at an isolated ﬁxed point x0 as follows: i(f, x0 , θ) = deg (f, Bε (x0 ), θ) for suﬃciently small ε > 0. (7) In particular, if K is diﬀerentiable at x0 , where f (x0 ) = θ, and if f (x0 ) is invertible, then we have i(f, x0 , θ) = (−1)β , where β =

{λj >1 | λj ∈σ(K (x0 ))}

k βj and βj = dim ∪∞ k=1 ker (λj I − K (x0 )) .

Proof. By homotopy invariance, ∃ε > 0 such that i(f, x0 , θ) = deg (f, Bε (x0 ), θ) = deg (id − K (x0 ), Bε (x0 ), θ) . It is suﬃcient to verify that K (x0 ) is a compact linear operator, because if it is so, then the conclusion follows directly from (6). The veriﬁcation is as follows:

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3 Degree Theory and Applications

If T = K (x0 ) were not compact, then there would be {xj } ⊂ B1 (θ) and ε > 0 such that T xi − T xj ε as i = j . Choose δ > 0 small such that ∀k = 1, 2, . . . , K(x0 + δxk ) − K(x0 ) − δT xk

εδ , 4

thus εδ K(x0 + δxi ) − K(x0 + δxj ) − δT (xi − xj ) 2 δ T (xi − xj ) − K(x0 + δxi ) − K(x0 + δxj ) δε− K(x0 + δxi ) − K(x0 + δxj ) ,

that is K(x0 + δxi ) − K(x0 + δxj )

1 εδ, i = j . 2

It contradicts with the compactness of K.

The Leray-Schauder degree theory is also applied to the study of the intersections of inﬁnite-dimensional manifolds, in which the continuous mappings should be replaced by compact vector ﬁelds.

3.5 The Global Bifurcation We have studied in Sect. 1.3, the local bifurcation phenomenon, which describes a branch of solutions splitting into several branches. Having the topological tools at hand, we are able to provide more precise local information and to investigate the global behavior of bifurcating branches. Consider F : X × R1 → X in the following form: F (x, λ) = Lx − λx + N (x, λ) ,

(3.20)

where X is a real Banach space, L ∈ L(X, X), λ ∈ R1 , and N (x, λ) = o ( x ) as x → θ uniformly in any ﬁnite interval of λ. As we know, bifurcation points (θ, λ0 ) only occur at λ0 ∈ σ(L), i.e., λ0 is a spectrum of L. In particular, if L is compact, and λ0 = 0, then λ0 is an eigenvalue of L. Conversely, we have: Theorem 3.5.1 (Knasnoselski) Suppose that L is a linear compact operator on X,+and that λ0 = 0 is an eigenvalue of L with odd multiplicity, ∞ i.e., β = dim k=1 ker(L − λ0 I)k is odd. If ∀λ, N ( · , λ) is compact, and N is continuous in x and λ, and satisﬁes N (x, λ) = o ( x )

3.5 The Global Bifurcation

165

uniformly in any ﬁnite interval of λ, then (θ, λ0 ) is a bifurcation point of the equation F (x, λ) = θ. Proof. Set S = {(x, λ) ∈ X × R1 | F (x, λ) = θ} , and S+ = S \ ({θ} × R1 ) . If (θ, λ0 ) is not a bifurcation point, then there is a closed interval [λ− , λ+ ] not including 0 and satisfying: 1. σ(L) ∩ [λ− , λ+ ] = {λ0 }, 2. ∃r > 0 such that (Br (θ) × [λ− , λ+ ]) ∩ S+ = ∅. In the sequel we write Br = Br (θ) brieﬂy. Deﬁne a deformation: Φ(x, t) = x −

1 (Lx + N (x, λ(t))) λ(t)

t ∈ [0, 1] ,

where λ(t) = tλ− + (1 − t)λ+ . Hence θ ∈ / Φ(∂Br × [0, 1]). It follows that 1 1 (L + N (· , λ+ )), Br , θ = deg id − (L + N (· , λ− )), Br , θ . deg id − λ+ λ− For suﬃciently small r > 0, again by the homotopy invariance, we obtain 1 1 L, Br , θ = deg id − L, Br , θ , deg id − λ+ λ− i.e., βj

(−1)λj >λ+

βj

= (−1)λj >λ−

,

where λj ∈ σ(L), and βj is the multiplicity of λj . This is (−1)β = 1 . By assumption β is odd. This is impossible. The oddness of the algebraic multiplicity of λ0 is a suﬃcient condition for bifurcation points, but not necessary (see Chap. 5, Theorem 5.1.37.) We turn to studying some global results. Lemma 3.5.2 Let K be a compact metric space, K1 , K2 be disjoint closed subsets. Then the following alternatives hold: either 1. ∃ a component of K intersecting K1 and K2 , or 4 2 such that 41, K 2. ∃ compact subsets K 4 i , i = 1, 2, Ki ⊂ K 41 ∪ K 42, K=K and

41 ∩ K 42 = ∅ . K

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3 Degree Theory and Applications

Proof. If (1) is not true, then ∃ ε0 > 0 such that each ε0 − chain cannot intersect with K1 and K2 simultaneously. For otherwise, ∀ ε > 0 ∃ aεi ∈ Ki 1

and ε-chain Cε connecting aεi , i = 1, 2. Since Ki , i = 1, 2, are compact, {ain } has a limiting point ai ∈ Ki , i = 1, 2. Set Ca1 = {x ∈ K| ∀ ε > 0 a1 can be connected with x by an ε-chain} . Since K is compact, Ca1 is a closed connected set. Therefore, a2 ∈ Ca1 . This means that (1) holds. We arrive at a contradiction. 4 1 = {y ∈ K| ∃ x ∈ K1 and ∃ε0 chain connecting x and y}. To the ε0 , let K 41. 4 1 ∩ K2 = ∅ and K1 ⊂ K Obviously, K 4 1 is both open and closed. We shall prove that K 4 1 , Bε (x) ∩ K 4 1 = ∅, which implies x ∈ K 41, On the one hand, ∀ x ∈ K 0 4 1 , Bε (x) ⊂ K 4 1 , so it is also 4 1 is closed. On the other hand, ∀x ∈ K therefore K 0 4 4 4 4 open. We set K2 = K\K1 . Then K1 and K2 meet all conditions we need. Theorem 3.5.3 (Leray-Schauder) Let X be a real Banach space, T : X × R1 → X be a compact map satisfying T (x, 0) = θ, and f (x, λ) = x − T (x, λ). Let S = {(x, λ) ∈ X × R1 | f (x, λ) = θ} ,

$

ζε S

X

O ζ+

U

λ λ∗

(θ, 0)

Fig. 3.2.

3.5 The Global Bifurcation

167

and let ζ be the component of S passing through (θ, 0). If ζ ± = ζ ∩ (X × R1± ) , then both ζ + and ζ − are unbounded. Proof. Since T is compact, if ζ + (or ζ − ) is bounded, then it is compact. We consider a ε− neighborhood ζε of ζ + . Let K = ζε ∩ S. This is a compact metric space. 4 i , i = 1, 2, such Set K1 = ζ + , K2 = ∂ζε ∩ S. According to Lemma 3.5.2, ∃K that 4i i = 1, 2, Ki ⊂ K 41 ∪ K 42, K=K and Choosing

41 ∩ K 42 = ∅ . K

4 2 ), dist(K 4 1 , ∂ζε )} , 41, K 0 < δ < min{dist(K

4 1 , we have and deﬁning a δ/2− neighborhood O of K ∂O ∩ S = ∅, ζ+ ⊂ O . Setting

2 U = O ∪ {(x, λ) ∈ X × R1− x + λ2 < δ 2 } ,

and λ∗ larger than the projection of O onto R1+ , we consider the map F : (X × R1 ) × [0, 1] → X × R1 as follows: (x, λ, t) → (f (x, λ), λ − tλ∗ ) .

Then ⇐⇒

F (x, λ, t) = (θ, 0)

f (x, λ) = θ . λ = tλ∗

Now, ∀(x, λ) ∈ ∂O, by construction, f (x, λ) = θ; ∀(x, 0) ∈ U, if f (x, 0) = θ, then x = θ; ∀(x, λ) ∈ X × R1− with x + λ2 = δ 2 , we have λ = tλ∗ . 2

Since ∂U ⊂ ∂O ∪

δ 2 (x, 0) ≤ x ≤ δ ∪ {(x, λ) ∈ X × R1− x + λ2 = δ 2 } , 2

we have (θ, 0) ∈ / F (∂U × [0, 1]). Thus deg( F ( · , · , 0), U, (θ, 0) ) = deg( F ( · , · , 1), U, (θ, 0) ) .

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3 Degree Theory and Applications

On the one hand, since λ = λ∗ on U , from Knonecker existence theorem, we have deg( F ( · , · , 1), U, (θ, 0) ) = 0 . On the other hand, ∀t, x − T (x, tλ) = θ, λ = 0,

⇐⇒

x = θ, λ = 0.

This implies (x − T (x, tλ), λ) = (θ, 0) on ∂U . Then, by the homotopy invariance and the excision property, deg( F ( · , · , 0), U, (θ, 0) ) = deg( idX×R1 , B δ (θ, 0), (θ, 0) ) = 1 . 2

This is the contradiction.

Although the Leray–Schauder theorem is not directly related to bifurcation problems (if T (θ, λ) = θ, then the conclusion is trivial: ζ = {θ} × R1 ), but in applications, it can be used in the spirit of the following global bifurcation theorem due to P. Rabinowitz and improved by Ize [Iz]. Theorem 3.5.4 (Rabinowitz) Let X be a real Banach space and F (x, λ) = x − λLx − N (x, λ), where L ∈ L(X, X) and N : X × R1 → X are compact. Let S be the solution set of F (x, λ) = θ, S+ = S \ ({θ} × R1 ), and let ζ be the component of S+ , containing (θ, λ1 ). Assume that N (x, λ) = o( x) uniformly on any ﬁnite interval in λ and that λ−1 1 ∈ σ(L) is an eigenvalue of odd multiplicity. Then the following alternatives hold: Either 1. ζ is unbounded; or 2. there are only ﬁnite number of points {(θ, λi ) i = 1, , . . . , l} lying on ζ, where λ−1 ∈ σ(L), i = 1, 2, . . . , l. Furthermore, if βi is the algebraic muli l tiplicity of λ−1 i , then i=1 βi is even. Proof. If ζ is bounded, then ζ is compact, because both L and N are compact. Since L is a compact linear operator, there are at most ﬁnitely many points {(θ, λj ) λj ∈ σ(L−1 ), j = 1, . . . , l } lying on ζ. Let 0 < ε < dist(ζ, {λ | λ−1 ∈ σ(L), λ = λ1 , λ2 , . . . , λl }), and let ζε be the ε− neighborhood of ζ. Set K = ζε ∩ S+ . Then K is a compact metric space, and ζ ∩ ∂ζε = ∅. Set K1 = ζ, 4 i , i = 1, 2, such K2 = S+ ∩ ∂ζε . According to Lemma 3.5.2, ∃ compact sets K 4 i , i = 1, 2, K = K 41 ∪ K 4 2 and K 41 ∩ K 4 2 = ∅. Set 0 < δ < that Ki ⊂ K 4 2 ), dist(K 4 1 , ∂ζε )}, and let O be the 1 δ neighborhood of K 41. 41, K min{dist(K 2 Then we have ζ ⊂ O ⊂ O ⊂ ζε , . S+ ∩ ∂O = ∅ If λ−1 ∈ σ(L), and λ = λ1 , λ2 , . . . , λl , then (θ, λ) ∈ / O. We consider the following map:

3.5 The Global Bifurcation

169

X

λ1

λ2

λ3

λl+1

λl

λ

ζ

O

Fig. 3.3.

Φ : O × R1 → X × R1 , Φ(x, λ; t) = (F (x, λ), x2 − t2 ) . We claim that (θ, 0) ∈ / Φ(∂O × {t}) as t = 0. Indeed, if Φ(x, λ; t) = (θ, 0), i.e., F (x, λ) = θ ; x2 = t2 = 0 then (x, λ) ∈ S+ , but S+ ∩ ∂O = ∅. By the homotopy invariance, for 0 < r < R, we have deg(Φ( · , R), O, (θ, 0)) = deg(Φ( · , r), O, (θ, 0)) . We choose R large enough such that O ⊂ BR (θ, 0), the ball in X × R1 . From the Knonecker existence theorem, deg(Φ( · , R), O, (θ, 0)) = 0 . Then we choose ε > 0 such that the ε-balls Bε ((θ, λj )) ⊂ O, j = 1, 2, . . . , l. For small r > 0, one wishes to prove:

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3 Degree Theory and Applications

deg(Φ( · , r), O, (θ, 0)) =

l

deg(Φ( · , r), B√r2 +ε2 ((θ, λj )), (θ, 0)) .

(3.21)

i=1

Let K > 0 be an upper bound: (id − λL)−1 K , l +

for λ in the projection of O onto R1 subtracting the set

(λj − ε, λj + ε).

j=1

We choose r > 0 such that N (x, λ)

0. Let us extend this conclusion to nonlinear problems (3.22). Set Sj = {v ∈ C01 ( [0, π] ) v has exactly (j − 1) simple roots in (0, π), and v (0), v (π) = 0} j = 1, 2, . . .. Thus Sj is an open set in C01 ( [0, π] ), and Si ∩ Sj = ∅

as

i = j ,

i, j = 1, 2, . . .. We shall prove that ζi ∩ ζj = ∅ as i = j. It is suﬃcient to prove that ζj ⊂ (Sj × R1 ) ∪ {(θ, λj )},

j = 1, 2, . . .

3.5 The Global Bifurcation

173

Lemma 3.5.5 There exists a neighborhood Oj of (θ, λj ) such that if (u, λ) ∈ Oj \({θ} × R1 ) is a solution of (3.22) then u ∈ Sj , ∀j. Proof. We prove by contradiction. If ∃ solutions (un , αn ) of (3.23), with un = / Sj . Let vn = uunn , then θ, satisfying (un , αn ) → (θ, λj ), but un ∈ vn = αn Lvn + un −1 N (un , αn ) . Since vn = 1, αn → λj , un −1 N (un , αn ) = o(1), and L is compact, we have vn → v, v = 1, satisfying v = λj Lv . According to the Sturm-Liouville theorem, v ∈ Sj . This contradicts the open ness of Sj . Now, we are going to prove: Theorem 3.5.6 Suppose that h satisﬁes (3.24). Then the branch of nontrivial solutions ζj passing through (θ, λj ) is included in (Sj × R1 ) ∪ {(θ, λj )}. In particular, ζj is unbounded. Proof. From Lemma 3.5.5, ζj ∩ Oj ⊂ (Sj × R1 ) ∪ {(θ, λj )}. We shall prove that ζj is entirely contained in the above set. If not, ∃(u∗ , λ∗ ) ∈ ζj ∩ ∂(Sj × R1 )\(θ, λj ). Since ∂(Sj × R1 ) = ∂Sj × R1 , and u∗ ∈ ∂Sj implies that u∗ has at least a double zero, i.e., ∃ξ ∈ [0, π] such that u∗ (ξ) = u∗ (ξ) = 0, it follows from ∈ σ(L), i = j, the uniqueness of ODE, u∗ = θ. Thus (u∗ , λ∗ ) = (θ, λi ), λ−1 i according to the necessary condition of bifurcation. Again, by Lemma 3.5.5, i = j. This is a contradiction. Our conclusion follows from Theorem 3.5.4 directly. Remark 3.5.7 One may consider other boundary conditions than the Dirichlet condition. Let us return to the example in Sect. 1.3: x ∈ (0, π) , −u = λ sin u (3.25) u(0) = u(π) = 0 . Applying Theorem 3.5.6, we conclude that ∀ n = 1, 2, . . . , there is an unbounded branch of solutions ζn passing through the bifurcation point (θ, n2 ). Again, one may prove that if (u, λ) ∈ ζn , then λ n2 . Indeed, if λ < n2 , then sin u < n2 . u Comparing (3.25) with the equation λ

(3.26)

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3 Degree Theory and Applications

−v = n2 v x ∈ (0, π) v(0) = v(π) = 0,

(3.27)

we would have v ∈ Sn+1 , on account of Sturm’s theorem. But v ∈ Sn . This is a contradiction. Moreover, ∀(u, λ) ∈ ζn ,

ξ u (x) = λ sin u(t)dt , x

where ξ ∈ (0, π) is a root of u (x) = 0. Thus u C λπ , and uC 1 λπ(1 + π) .

(3.28)

Combining (3.26) with (3.28), we have ζn ⊂ {(u, λ) ∈ C01 ([0, π]) × R1 λ n2 , uC 1 λπ(1 + π)} . Hence, the projection of ζn onto R1 is the interval (n2 , ∞). For any λ0 ∈ (n , ∞), there exist at least n branches ζ1 , ζ2 , . . . , ζn that meet λ = λ0 . On (j ) each ζj there exists at least a pair of nontrivial solutions ±uλ0 , j = 1, 2, . . . , n. Namely: 2

C01([0, π])

1

4

9

Fig. 3.4.

λ

3.6 Applications

175

Theorem 3.5.8 ∀λ ∈ (n2 , (n + 1)2 ] equation (3.26) has at least n distinct pairs of nontrivial solutions. Remark 3.5.9 It is natural to ask if we can extend the above method to study elliptic nonlinear eigenvalue problems as that for ODE. The main obstruction is that there is no counterpart of the Sturm–Liouville theory for PDEs. There are only a few partial results in this direction. Remark 3.5.10 (Bifurcation at inﬁnity) We study the bifurcation at inﬁnity by transforming the problem into the bifurcation at zero. We consider the equation: x = λLu + M (x, λ) ,

(3.29)

where L is a linear compact operator on X, and M : X × R1 → X satisﬁes M (x, λ) = o(x) uniformly in any ﬁnite interval as x → ∞. (λ0 , ∞) is called a bifurcation point at inﬁnity, if there exists a sequence of solutions (λn , xn ) of (3.29), such that xn → ∞, and λn → λ0 . Now we deﬁne y = x−2 x, then y = x−1 . Equation (3.29) is transformed into: y = λLy + N (y, λ) , (3.30) where N (y, λ) = y2 M (y−2 y, λ). Since N (y, λ) = o(y) as y → θ , (3.30) is exactly the equation we studied previously.

3.6 Applications 3.6.1 Degree Theory on Closed Convex Sets In applications of the Leray–Schauder degree theory, boundary conditions should be carefully investigated, i.e., on the boundary there is no ﬁxed point of the compact vector ﬁeld. However, for mappings with images in a closed convex set, the notion of the boundary can be reduced. Let us present an extension version of the Tietze theorem, Dugundji’s theorem, concerning the extension of continuous mappings on closed convex sets, which we have met in Sect. 3.4. Theorem 3.6.1 (Dugundji) Suppose that A is a closed subset of a metric space X, and that C is a convex subset of a Banach space Y . Then for any continuous map f : A → C, there exists a continuous extension f! : X → C. Proof. Since X \ A is a metric subspace, it is paracompact. ∀x ∈ X\A, let 0 < r(x) < 12 dist(x, A), and let Br(x) (x) be the ball with radius r(x) and

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3 Degree Theory and Applications

center x . All these balls form a covering of X\A. The covering has a locally ﬁnite reﬁnement {Uα | α ∈ ∧}. On the set X\A, we deﬁne dist(x, X\Uα ) (β∈Λ) dist(x, X\Uβ )

λα (x) =

∀α ∈ Λ .

They satisfy:

supp λα ⊂ Uα , 0 λα (x) 1, and

λα (x) = 1 .

α∈Λ

∀α ∈ Λ we choose aα ∈ A, such that dist(aα , Uα ) < 2dist(A, Uα ) ,

and let f!(x) =

λα (x)f (aα ) f (x)

x ∈ X\A x∈A.

Since {Uα | α ∈ Λ} is a locally ﬁnite covering, the sum is ﬁnite. We claim that f! is continuous. It is suﬃcient to prove that f! is continuous on the boundary ∂A, i.e., for ∀x0 ∈ ∂A. ∀ε > 0, ∃δ > 0 such that f!(x) − f (x0 ) < ε , whenever dist(x, x0 ) < δ. In fact, ∃δ > 0, such that if dist(x, x0 ) < 6δ and x ∈ A, then f (x) − f (x0 ) < ε . Now, ∀x ∈ X, if dist(x, aα ) < 3δ, from dist(x0 , aα ) ≤ dist(x0 , x) + dist(x, aα ) , it follows that dist(x0 , aα ) < 4δ, then f (aα ) − f (x0 ) < .

(3.31)

Otherwise, dist(x, aα ) ≥ 3δ, then dist(x, aα ) ≥ 3dist(x, x0 ) ≥ 3dist(x, A) . In this case we claim that x ∈ Uα . If not, i.e., x ∈ Uα , then we would have dist(x, aα ) diam(Uα ) + dist(Uα , aα ) 2r(x) + 2dist(A, Uα ) dist(A, x) + 2dist(A, x) = 3dist(A, x)

3.6 Applications

177

This is a contradiction, therefore, x ∈ Uα and then λα (x) = 0 .

(3.32)

Combining equations (3.31) and (3.32), we have λα (x) f (aα ) − f (x0 ) < ε . f!(x) − f (x0 ) dist(x,aα ) 0 is a large number such that U ⊂ BR (θ). Firstly, we point out that the deﬁnition makes sense, i.e, θ ∈ (id − K ◦ r)(∂(BR (θ) ∩ r−1 (U ))). In fact, if it is not true, then ∃p ∈ ∂(BR (θ) ∩ r−1 (U )) such that p = K ◦ r(p). Thus, p ∈ C and then p = Kp, which implies p ∈ C ∩ ∂(BR (θ) ∩ r−1 (U )), i.e., p ∈ ∂U . This is a contradiction. Secondly, one veriﬁes that the degree so deﬁned does not depend on the retraction r, and the radius R. In fact, let R1 < R2 such that U ⊂ int(BR1 (θ)). All ﬁxed points of id − K in U must be in BR1 (θ). By excision, deg (id − K ◦ r, r−1 (U ) ∩ BR2 (θ), θ) = deg (id − K ◦ r, r−1 (U ) ∩ BR1 (θ), θ) . Again let r1 , r2 be two retractions: X → C, by excision, deg (id − K ◦ ri , ri−1 (U ) ∩ BR (θ), θ) = deg (id − K ◦ ri , r1−1 (U ) ∩ r2−1 (U ) ∩ BR (θ), θ) . for i = 1, 2. We deﬁne F (x, t) = id − (tK ◦ r1 + (1 − t)K ◦ r2 ) ∀t ∈ [0, 1] . From the homotopy invariance, degC (id − K, U, θ) is independent of ri , i = 1, 2. In particular, if X is a real OBS, i.e., X is a real Banach space with a closed positive cone P (see Sect. 2.1), then P is a closed convex subset, and then degP (id − K, U, θ) makes sense. Now, we present a few degree computations for special boundary conditions in an OBS.

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3 Degree Theory and Applications

Lemma 3.6.2 Suppose that (X, P ) is an OBS, and that U ⊂ P is bounded and open. Assume that K : U → P is compact, satisfying ∃y ∈ P \{θ}, such that x − Kx = ty ∀t ≥ 0, ∀x ∈ ∂U ; then degP (id − K, U, θ) = 0 . Proof. From the homotopy invariance, deg (id − K − ty, U, θ) = constant, ∀t ≥ 0 . Since K is compact, ∃C > 0 such that x − Kx ≤ C ∀x ∈ U . One chooses C t > y , then x − Kx = ty ∀x ∈ U . It follows that degP (id − K, U, θ) = 0. Theorem 3.6.3 Suppose that (X, P ) is an OBS, U ⊂ P is bounded open and contains θ. Assume that ∃ρ > 0 such that Bρ (θ) ∩ P ⊂ U and that K : U → P is compact and satisﬁes: (1) ∀x ∈ P, with x = ρ, ∀λ ∈ [0, 1), x = λK(x), (2) ∃y ∈ P \{θ}, such that x − K(x) = ty, ∀x ∈ ∂U, ∀t ≥ 0. Then K possesses a ﬁxed point on Uρ , where Uρ = U \Bρ (θ). Proof. We may assume that K has no ﬁxed point on ∂Bρ (θ) then degP (id − K, U, θ) = degP (id − K, Uρ , θ) + degP (id − K, Bρ (θ) ∩ P, θ) . According to Lemma 3.6.2 and assumption (2), LHS = 0, and from assumption (1) and the homotopy invariance, the second term on the RHS = 1. Therefore, degP (id − K, Uρ , θ) = −1. Theorem 3.6.4 Suppose that (X, P ) is an OBS, U ⊂ P is bounded and open, and contains θ. Assume that ∃ρ > 0 such that U ⊂ Bρ (θ) ∩ P , and that Uρ := (Bρ (θ) ∩ P )\U possesses an interior point. If K : B ρ (θ) ∩ P → P is compact and satisﬁes: (1) ∀x ∈ P, x = ρ, ∀λ ∈ [0, 1), x = λK(x). (2) ∃y ∈ P \{θ} such that x − K(x) = ty, ∀t ≥ 0, ∀x ∈ ∂U. Then K has a ﬁxed point in U ρ . Proof. The proof is similar to the previous theorem: deg (id − K, Uρ , θ) = 1. We omit it. Remark 3.6.5 Let (X, P ) be an OBS, and let 0 < ρ1 < ρ2 , and Pρi = P ∩Bρi (θ). We write ∂Pρi = P ∩∂Bρi (θ), i = 1, 2. A map f : Pρ2 \Pρ1 → P is called a cone compression, if x f (x), ∀x ∈ ∂Pρ2 , and f (x) x, ∀x ∈ ∂Pρ1 . It is called a cone expansion if f (x) x, ∀x ∈ ∂Pρ2 , and x f (x) ∀x ∈ ∂Pρ1 . Noticing that

3.6 Applications

179

f (x) x implies ∀y ∈ P \{θ}, ∀t ≥ 0, x − f (x) = ty , and x f (x) implies ∀λ ∈ [0, 1], x = λf (x) , Theorems 3.6.3 and 3.6.4 extend the earlier results of Krasnoselski on cone compression and cone expansion mappings. Let (X, P ) be an OBS satisfying X = P − P . A map T : P → P is called positive. We shall study the bifurcation problem for positive mappings on OBS. Let R+ = [0, +∞) and f : P × R+ → P be a continuous mapping satisfying f (θ, λ) = θ ∀λ ∈ R+ . We are looking for λ0 ∈ R+ , such that ∀ > 0, ∃(x, λ) ∈ (B+ \{θ}) × (λ0 − , λ0 + ) satisfying x = f (x, λ) , where B+ = B (θ) ∩ P . At this time, (θ, λ0 ) is a bifurcation point of positive solutions. Lemma 3.6.6 Let (X, P ) be deﬁned above. Let f : P × R+ → P be of the form: f (x, λ) = λT x + g(x, λ) , where T ∈ L(X, X) is a positive and compact operator, and g : P × R+ → P is compact, satisfying g(x, λ) = o(x) as x → 0 and g(x, 0) = θ. If (θ, λ0 ) is an eigenvalue of is a bifurcation point with λ0 ≥ 0, then λ0 > 0, and λ−1 0 T with positive eigenvector. Proof. By deﬁnition, ∃(xn , λn ) ∈ (P \{θ}) × R+ , such that (xn , λn ) → (θ, λ0 ), with xn = f (xn , λn ). Setting yn = xxnn , we have yn − λ0 T yn = (λn − λ0 )T yn + g(xn , λn )xn −1 . Deﬁne S+ = ∂B1 (θ) ∩ P , then yn ∈ S+ , and then, after a subsequence we have T yn → z0 . Let y0 = λ0 z0 , then y0 ∈ S+ . Obviously, y0 = λ0 T y0 , and λ0 > 0. One has the following variant of the global bifurcation theorem. Theorem 3.6.7 (Dancer) Let (X, B) and f, T, g be as in Lemma 3.6.6. Assume that T has ﬁnitely many positive eigenvalues with positive eigenvectors. Let Σ = {(x, λ) ∈ P ×R+ | x = f (x, λ)} and Σ+ = Σ ∩ ((P \{θ}) × (R+ \{0})). Then Σ+ contains an unbounded connected branch emanating from the reciprocal of one of these eigenvalues. be the smallest positive eigenvalue of T , and let ζ be the Proof. Let λ−1 0 connected component of Σ+ ∪ ({θ} × [0, λ0 ]), containing {θ} × [0, λ0 ]. If ζ is bounded, then ∃µ > λ0 such that ζ ⊂ Qµ := Bµ+ × [0, µ], and ζ ∩ ∂Qµ = Ø, where Bµ+ = Bµ (θ) ∩ P .

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3 Degree Theory and Applications

Since µ−1 is not an eigenvalue with positive eigenvector of T , ∃α > 0 such that x − µT x ≥ αµx, ∀x ∈ P . One chooses ρ ∈ (0, min{ αµ 2 , µ}), such that g(x, µ) ≤

αµ + 2 x as x ∈ Bρ . ((Bµ+ \B+ × {0}) ∪

Choose ∈ (0, min{µ, ρ}). Let ζ1 = ζ ∪ ({θ} × [0, µ]), D = ((∂Bµ (θ) ∩ P ) × [0, µ]) ∪ ((Bµ+ \B+ ) × {µ}). Then D ∩ ζ1 = Ø. Similar to the proof in the global bifurcation theorem, ∃U ⊂ P × [0, µ] open, such that Σ ∩ ∂U = Ø, ζ1 ⊂ U , and U ∩ D = Ø. Thus, 1 = degP (id, B+ , θ) = degP (id − f (·, 0), U0 , θ) = degP (id − f (·, µ), Uµ , θ) = degP (id − f (·, µ), B+ , θ) , where Uλ := {x ∈ P |(x, λ) ∈ U }. with norm 1. Let φ ∈ S + be a positive eigenvector associated with λ−1 0 We claim that the equation x − µT x = βφ ∀β > 0 has no positive solution. For otherwise, if ∃β0 > 0, x0 ∈ P, satisfying x0 − µT x0 = β0 φ . Then

x0 ≥ (µτ0 λ−1 0 + β0 )φ > (τ0 + β0 )φ ,

where τ0 = sup {τ ∈ R+ |x0 ≥ τ φ}. This contradicts the deﬁnition of τ0 . Setting β ∈ (0, ρ 2 ), we have x − µT x + (1 − λ)βφ + λg(x, µ) ≥ αµx −

ρ αµ − x > 0, as x = ε . 2 2

Then, by the homotopy invariance, degP (id − f (·, µ), B+ , θ) = degP (id − µT − βφ, B+ , θ) = 0 . The contradiction shows that ζ is unbounded.

3.6.2 Positive Solutions and the Scaling Method The above degree computation is applied to the study of the positive solution of certain superlinear elliptic equations. Let Ω ⊂ Rn be an open bounded domain with smooth boundary. Assume that aij , bj ∈ C(Ω), i, j = 1, . . . , n, φ ∈ C 1 (∂Ω) is nonnegative, and f ∈ C(Ω × R1 ). Let

3.6 Applications

Lu(x) =

n n ∂ ∂u ∂u bj (x) . ai,j (x) + ∂x ∂x ∂x i j j i,j=1 j=1

We study positive solutions of the equation: Lu + f (x, u) = 0, in Ω , u = φ,

on ∂Ω ,

181

(3.33)

(3.34)

in which f is superlinear in u. In order to apply degree theory, we should estimate a priori bounds for positive solutions u. We introduce here a useful method – the scaling method – based on the following global results of Liouville-type theorems, cf. Gidas and Spruck [GS], Chen and Li [ChL 1] and Y. Li [Li 1]. n+2 ) for n ≥ 3, and p > 1 for n = 1, 2. If u ∈ Theorem. Assume p ∈ (1, n−2 2 n C (R ) is a nonnegative solution of the equation:

∆u + up = 0 in Rn ,

(3.35)

then u ≡ 0. n , Proof. (We only prove the theorem under a stronger condition: 1 < p < n−2 for general case see the above-mentioned references.) Let ϕ1 > 0 be the ﬁrst eigenfunction of the Laplacian on B1 (θ) ⊂ Rn , i.e., −∆ϕ1 = λ1 ϕ1 , in B1 (θ) , (3.36) ϕ1 = 0 on ∂B1 (θ) . x ), we have For ∀R > 0, let ϕR (x) = ϕ1 ( R λ1 in BR (θ) , −∆ϕR = R 2 ϕR ,

ϕR = 0

on ∂BR (θ) .

Integrating by parts,

up ϕpR = − ∆uϕpR BR BR

∇u∇ϕpR = BR

∂ϕR p−1 ϕ =− u∆ϕpR + p u ∂n R BR ∂BR

u∆ϕpR . =−

(3.37)

(3.38) (3.39) (3.40) (3.41)

BR

(3.42)

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3 Degree Theory and Applications

Since

BR

u∆ϕpR

2 uϕp−2 R |∇ϕR |

= p(p − 1) BR

it follows that

λ1 p (uϕR ) ≤ 2 R BR

BR

uϕpR ,

(3.43)

p

λ1 p ≤ 2 R

pλ1 − 2 R

BR

uϕpR

(3.44)

p

p1

(uϕR ) BR

BR

ϕpR

p−1 p .

(3.45)

ϕp1 .

(3.46)

Therefore,

(uϕR )p ≤ BR

We obtain limR→0

λ1 p R2

BR

p p−1

BR

p

2p

ϕpR = (λ1 p) p−1 Rn− p−1

B1

(uϕR )p = 0. Thus u ≡ 0.

Similarly, one has the Liouville theorem for half space: n+2 ) for n ≥ 3, and p > 1 for n = 1, 2. If u ∈ Theorem. Assume p ∈ (1, n−2 n C 2 (Rn ) ∩ C(R+ ) is a nonnegative solution of the equation:

n

∆u + up = 0, in R+ , u=0 on xn = 0 ,

(3.47)

then u ≡ 0. n+2 ) for n ≥ 3, and p > 1 for n = 1, 2. Theorem 3.6.8 Assume p ∈ (1, n−2 1 ¯ Suppose that f ∈ C(Ω × R ) satisﬁes

lim

t→+∞

f (x, t) = h(x) uniformly in x ∈ Ω, tp

(3.48)

where h is a positive function. If u ≥ 0 is a solution of equation (3.34), then there exists a constant C depending on p, Ω, φ and f only such that u ≤ C. Proof. One proves the theorem by contradiction. If the conclusion is not true, then there exist a sequence of solutions uk of equation (3.34) and points Pk ∈ Ω satisfying Mk := sup uk (x) = uk (Pk ) → +∞ as k → ∞ . x∈Ω

We may assume Pk → P ∈ Ω as k → ∞. There are two possibilities: either (1) P ∈ Ω◦ , or (2) P ∈ ∂Ω. 2

In case (1): Let d = 12 dist(P, ∂Ω), λkp−1 Mk = 1, y =

x−Pk λk ,

and set

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183

2

vk (y) = λkp−1 uk (x) . It is deﬁned on B

d λk

(θ) and satisﬁes vk (y) = vk (θ) = 1 .

sup y∈B

d λk

(θ)

and n ∂ ∂vk (y) k aij (y) ∂yj ∂yi i,j=1 + λk

n j=1

bkj (y)

2p ∂vk (y) + λkp−1 f ∂yj

−2 λk y + Pk , λkp−1 vk (y) = 0 ,

where akij (y) = aij (λk y + Pk ) → aij (P ), λk bkj (y) = λk bj (λk y + Pk ) → 0, and the last term is asymptotic to h(λk y + Pk )vkp . ∀R > 0, ∃k such that BR (θ) ⊂ B d (θ). Since vk is uniformly bounded λk

on BR (θ), from the Lq estimates, 1 < q < ∞, we have vk W 2,q (BR (θ)) ≤ C, a uniform constant. According to the Sobolev embedding theorem and the diagonal principle, one has a subsequence vkj → v in C 1,β ∩ W 2,q (BR (θ)) for some β ∈ (0, 1), ∀R > 0, and v(θ) = 1. Thus as a weak solution v satisﬁes n ∂ 2 v(y) p n i,j=1 aij (P ) ∂yi ∂yj + h(P )v (y) = 0, in R . v(θ) = 1. After scaling and rotation, it is reduced to equation (3.35). Again by bootstrap iteration, v ∈ C 2 (Rn ), it follows from the Liouville-type theorem that v ≡ 0. This contradicts v(θ) = 1. In case (2), P ∈ ∂Ω. Without loss of generality, we may assume ∃δ > 0 such that Bδ (P ) ∩ ∂Ω is contained in the hyperplane xn = 0. One deﬁnes vk , λk as before. Let dk = d(Pk , ∂Ω), then dk = Pk · en , where en is the unit normal vector of ∂Ω near P . Therefore vk is well deﬁned in B δ (θ) ∩ {yn > − λdkk }. λk

Again we have sup vk (y) = vk (θ) = 1. We claim that there is a constant C1 > 0 such that λdkk ≥ C1 . In fact, according to elliptic regularity up to the boundary, |∇vk | is uniformly bounded, so is vk (θ) − vk 0, . . . , 0, − dk ≤ C −1 dk , 1 λk λk i.e., 2

1 − λkp−1 sup φ(x) ≤ C1−1 x∈∂Ω

Since λk → 0, our claim is proved.

dk . λk

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3 Degree Theory and Applications dkj λkj

If there is a subsequence

→ ∞, then the discussion is reduced to case

(1). Therefore we may assume that there is another constant C2 such that C1 ≤ λdkk ≤ C2 , after a subsequence, λdkk → s. By taking the limit, we obtain:

n i,j=1

2

v aij (P ) ∂y∂i ∂y + h(P )v p (y) = 0 in yn > −s , j v(y) = 0 on yn = −s .

After scaling and rotation, it is reduced to equation (3.47). By the same reasoning, we have v ≡ 0, which contradicts v(θ) = 1. Combining these two cases, we have proved the theorem. Lemma 3.6.9 Under the assumptions of Theorem 3.6.8, if further f (x, t) ≥ 0, and f (x, t) > λ1 , uniformly in x ∈ Ω , (3.49) lim inf t→+∞ t where λ1 is the ﬁrst eigenvalue of the linear elliptic operator −L (see equation (3.33)) with eigenfunction ϕ1 . Then there exists a constant C > 0 such that for all solutions u of the equation: Lu + f (x, u) + tϕ1 = 0, in Ω, ∀ t ≥ 0 , (3.50) u=0 on ∂Ω ,

we have

f (x, u(x))ϕ1 (x)dx ≤ C , Ω

and t≤C. Proof. We may assume ϕ1 L2 = 1. By the condition (3.49), there are constants M > 0, k ∈ (λ1 , lim inf t→+∞ f (x,t) t ) such that f (x, t) ≥ kt − M, ∀ t > 0 . For any solution u of equation (3.50), by integration, we have

ϕ1 (x)Lu(x)dx = ϕ1 (x)f (x, u(x))dx + t , − Ω

Ω

thus

ϕ1 (x)u(x)dx ≥ k

λ1 Ω

ϕ1 (x)u(x)dx − M

Ω

i.e.,

ϕ1 (x) + t , Ω

ϕ1 (x)u(x)dx ≤ M

t + (k − λ1 ) Ω

ϕ1 (x)dx . Ω

Since f (x, t) ≥ 0, from the maximum principle, u ≥ 0. The conclusions follow.

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185

Theorem 3.6.10 Suppose that f ∈ C(Ω × R1 ) is positive, which satisﬁes the conditions (3.48) (3.49) and lim sup t→0

f (x, t) < λ1 uniformly in t

x∈Ω.

(3.51)

Then equation (3.34) has a positive solution in which φ = 0. Proof. Deﬁne F : u(x) → f (x, u(x)) from C(Ω) into itself, and K = (−L)−1 being the linear compact operator on C(Ω). Then T = K ◦ F is a compact mapping from C(Ω) into its positive cone. Applying Lemma 3.6.9 and Theorem 3.6.8 we take R > 0 large enough such that equation (3.50) has no positive solution u with C− norm u = R, and it follows from Lemma 3.6.2 that degP (id − T, BR (θ) ∩ P, θ) = 0. According to condition (3.51) and the homotopy invariance, we have ρ > 0 small such that degP (id − T, Bρ (θ) ∩ P, θ) = 1. Therefore there is a nontrivial positive solution. 3.6.3 Krein–Rutman Theory for Positive Linear Operators Let X be a real Banach space, and P ⊂ X be a positive closed cone with ◦

nonempty interior P . A linear continuous operator L ∈ L(X, X) is called ◦

positive if L(P ) ⊂ P ; it is called strictly positive if L(P \{θ}) ⊂ P . Example. Let M be a compact topological space with a Radon measure µ. Let X = C(M ), P = {u ∈ C(M )|u ≥ 0}, then (X, P ) is an OBS. Given a nonnegative continuous function K : M × M → R1 , we deﬁne:

(Lu)(x) = K(x, y)u(y)dµ . M

Then L is a positive operator. Example. We consider the second-order elliptic operator: 2 Lu = −Σni,j=1 aij ∂ij u + Σni=1 bi ∂i u + cu ,

(3.52)

where aij , bi , c ∈ C(Ω), i, j = 1, 2, . . . , n, Ω ⊂ Rn is an open domain with smooth boundary (cf. Protter and Weinberger [PW]). Assume the ellipticity condition, i.e., ∃α > 0 such that Σni,j=1 aij (x)ξi ξj ≥ α|ξ|2 , ∀ξ = (ξ1 , . . . , ξn ) ∈ Rn , ∀x ∈ Ω , and c≥0. It is known that ∀f ∈ Lp , 1 < p < ∞, the equation:

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3 Degree Theory and Applications

Lu = f , u|∂Ω = 0 , has a unique solution u ∈ W 2,p (Ω); the inverse operator K = L−1 : f → u is positive and bounded on Lp (Ω)(and also on C0 (Ω) the subspace of C(Ω) with boundary value zero), but not strictly positive. However, we have: Lemma 3.6.11 Let X = C01 (Ω), and P = {u ∈ X|u ≥ 0}. Then K = L−1 is a strictly positive compact operator. Proof. We only want to prove the strict positivity, i.e., ∀f ∈ C01 (Ω) if f ≥ 0, ◦

but f = θ, then u = Kf ∈ P . In fact, by the strong maximum principle, u(x) > 0 ∀x ∈ Ω. Again by the Hopf maximum principle, ∂n u|∂Ω < 0, where ∂n is the outer normal derivative. Since ∂Ω is compact, ∃δ > 0 such that supx∈∂Ω ∂n u(x) < −δ < 0. Now, u ∈ C 1 (Ω), there is a neighborhood N of ∂Ω on which ∂ν u ≤ −δ/2, where ν is the direction connecting x ∈ N to the closest point on ∂Ω. Setting α = inf {u(x)|x ∈ Ω\N } and β = Min{α, δ/2}, then the open ball, centered ◦

at u with radius β, is contained in P .

The ﬁrst eigenvalue λ1 and the ﬁrst eigenfunction φ play important role in the study of second-order elliptic equations. It is known that λ1 > 0 is algebraically simple, and that the normalized φ > 0 is unique. In fact, the conclusion is a special case of the following Krein–Rutman theorem: Theorem 3.6.12 (Krein–Rutman) Suppose that (X, P ) is an OBS with non◦

empty interior P , and that T is a compact strictly positive operator. Then T ◦

possesses a unique positive eigenvector φ ∈ P with φ = 1, associated with a 1 algebraically simple eigenvalue λ11 = limn→∞ T n n > 0, satisfying: λ1 ≤ |λ|

∀λ−1 ∈ σ(T ) .

(3.53)

Proof. (1). (The existence of a positive eigenvector.) ∀x ∈ P \{θ}, ∃M > 0, P, ∀n ∈ N large, such that M T x ≥ x. For otherwise, it must be T x − n1 x ∈ ◦

and then T x ∈ P ; this contradicts the strict positiveness of T . According to Dugundji’s theorem, there exists a retraction r : X → P . We deﬁne T : X × R1 → P as follows: T (y, λ) = λr ◦ T (y + x) . Thus, ∀λ, T (·, λ) is compact, and satisﬁes T (y, 0) = θ, ∀y ∈ X. We notice that for λ ≥ 0, y = T (y, λ) is equivalent to y ∈ P , and y = λT (y + x). From the Leray–Schauder Theorem, there is an unbounded connected branch of solutions ζ + ⊂ P × R1+ passing through (θ, 0).

3.6 Applications

187

We claim that if (y, λ) ∈ ζ + , then λ ∈ [0, M ]. In fact, since M T x ≥ x, y = λ x, we obtain λT (y + x) ≥ M y ≥ λT y ≥

λ M

n x, ∀n ∈ N .

If λ > M , then x = θ. This is a contradiction. Therefore, ∃(yn , λn ) ∈ ζ+ such that λn → λ∗ ∈ [0, M ], yn > n. Setting zn = yynn , we have zn = λ n T z n +

λn Tx . yn

Since T is compact, we have zn → z ∗ = θ satisfying z ∗ = λ∗ T z ∗ . This implies ◦

that z ∗ ∈ P , λ∗ > 0, z ∗ = 1. Let φ = z ∗ , and λ1 = λ∗ ; these are what we need. ◦ (2). (The uniqueness of the positive eigenvector.) ∀x ∈ P, ∀y ∈ P , let us deﬁne δy (x) = sup {λ ≥ 0|y + λx ∈ P }. It is easily seen that δy (x) > 0 is a continuous function of x ∈ X\P . By deﬁnition, ⎧ ⎨λ ∈ [0, δy (x)] implies y + λx ∈ P , ⎩λ > δ (x) implies y + λx ∈ P . y ◦

Now, suppose that φi ∈ P satisfy φi = 1, φi = λi T φi , i = 1, 2. Set γ1 = δφ1 (−φ2 ), γ2 = δφ2 (−φ1 ) . Then

⎧ ⎨T (φ1 − γ1 φ2 ) =

1 λ1 (φ1

− γ1 λλ12 φ2 ) ,

⎩T (φ − γ φ ) = 2 2 1

1 λ2 (φ2

− γ2 λλ21 φ1 ) . ◦

If φ2 − γ2 φ1 = θ, then T (φ2 − γ2 φ1 ) ∈ P . This implies λ2 < λ1 . But T (φ1 −γ1 φ2 ) ∈ P implies λ1 ≤ λ2 . This is a contradiction. Therefore, φ1 = φ2 . Again, by the normality, λ1 = λ2 . (3). (The inequality (3.52)) Assume that ψ ∈ X\(P ∪ (−P )) satisfying λT ψ = ψ, and ψ = 1. In the case where λ is real: From φ ± δφ (±ψ)ψ = θ, it follows that ◦ 1 λ1 φ ± δφ (±ψ)ψ = T (φ ± δφ (±ψ)ψ) ∈ P . λ1 λ Therefore λ1 < |λ|. In the case where λ is not real: We write λ = |λ|eiθ , then ∃x1 , x2 ∈ X, such that x1 + x2 = λ(T x1 + iT x2 ) .

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3 Degree Theory and Applications

1 Let H = span{x1 , x2 }, then T |H = |λ| Rθ , where Rθ is the following matrix: cos θ − sin θ sin θ cos θ

We claim that P ∩ H = {θ}. For otherwise, since both P and H are invariant with respect to T , T |P ∩H is positive, according to the conclusion of the ﬁrst paragraph, it should have a positive eigenvalue. But this contradicts the representation of Rθ . Now, ∀z ∈ H\{θ}, ◦ λ1 1 δφ (z)Rθ z = T (φ + δφ (z)z) ∈ P . φ+ λ1 |λ| λ1 Thus |λ| δφ (z) < δφ (Rθ z). We choose z0 ∈ H\{θ} such that δφ (z0 ) = sup {δφ (z)|z ∈ C}, where C = {cos θx1 + sin θx2 |θ ∈ [0, 2π]}, it follows that λ1 < |λ|. (4). (The algebraic simplicity of λ1 ). From (1), we have shown that ker(id− λ1 T ) = span{φ}. According to the Riesz–Schauder theory, it is suﬃcient to show that ker(id−λ1 T )2 = ker(id−λ1 T ), i.e., if x ∈ X satisﬁes x−λ1 T x = −φ, then x = θ. To this end, it is suﬃcient to show that x ∈ P ∩ (−P ). In fact, if x ∈ P , let γ = δφ (−x), then

T (φ − γx) =

1 γ φ − (x + φ) . λ1 λ1

It follows that

◦ γ 1 (φ − γx) = T (φ − γx) + φ ∈ P . λ1 λ1 It contradicts the deﬁnition of γ, therefore x ∈ P . Similarly, x ∈ −P . Therefore x ∈ P ∩ (−P ) = θ.

Corollary 3.6.13 Let L be the linear second-order elliptic operator (3.52) with the Dirichlet boundary condition, deﬁned on a bounded open domain Ω in Rn , the boundary ∂Ω being smooth. Then the eigenvalue problem: Lu = λu in Ω u = 0 on ∂Ω . has a positive eigenfuction φ with positive, algebraically and geometrically simple eigenvalue λ1 , satisfying λ1 ≤ |λ| ∀λ ∈ σ(L) . Corollary 3.6.14 ([Tu]). Let (X, P ) be an OBS, and let f, T, g, Σ+ be as in Theorem 3.6.7, in which T is a strictly positive compact operator. Then (θ, λ1 ) is a bifurcation point and Σ+ contains an unbounded connected component in P emanating from (θ, λ1 ). Also, we have the following result on the bifurcation at inﬁnity:

3.6 Applications

189

Corollary 3.6.15 Let (X, P ) be an OBS. Let T ∈ L(X, X) be strictly positive and compact, and g : P ×R+ → P be compact, satisfying g(x, 0) = θ, g(x, λ) = o(x), as x → ∞ uniformly on each ﬁnite interval. Deﬁne f (x, λ) = λT x + g(x, λ) ∀(x, λ) ∈ P × R+ . Let Σ = {(x, λ) ∈ P × R+ |f (x, λ) = x} and Σ+ = Σ ∩ (P \{θ} × R+ \{0}). Then there are only two possibilities for the connected component ζ of Σ+ , passing through the point (∞, λ1 ), where λ1 is the eigenvalue of T with positive eigenvector: 1 (1) The projection of ζ onto R+ is unbounded. 1 (2) ζ ∩ ({θ} × R+ ) = Ø.

3.6.4 Multiple Solutions In this subsection, we present a few examples showing that degree theory is applicable to the study of multiple solutions for certain nonlinear elliptic equations. Example 1. Let L be the second-order elliptic operator deﬁned in (3.52), and let g ∈ C(R1 ) satisfy g(0) = 0, |g(t)| ≤ C, and g(t) − λt = o(t), as |t| → 0, where C is a constant, and λ ∈ R1 is between the ﬁrst and the second eigenvalues of L. We consider the equation: Lu = g(u) in Ω , (3.54) u = 0. on ∂Ω . Obviously, u = 0 is a trivial solution. We shall show that there exists a nontrivial solution. This can be seen by the following degree computation. Let X = C01 (Ω), since λ is between λ1 and λ2 , by homotopy invariance of the index, i(id − L−1 g, θ) = i(id − λL−1 , θ) = −1 . Since all solutions are bounded in X, again by the homotopy invariance of the degree, deg (id − L−1 g, BR (θ), θ) = deg (id, BR (θ), θ) = 1 . If there were no nontrivial solution, the above identities would contradict the additivity of the degree. If one slightly changes the above assumptions then there exists a positive solution. Example 2. Let L be as in (3.52), and let g ∈ C 1 (R1 ) satisfy g(0) = 0, 0 ≤ g ≤ C, and g(t) − λt = o(t), as t → 0 with λ > λ1 . Then there exists a positive solution of (3.54). This can be shown by the sub- and super-solutions method. Obviously, u = L−1 C is an super-solution. Let u = φ1 , where φ1 is the positive eigenfunction

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3 Degree Theory and Applications

of L, and > 0 small. By the condition of g near zero, it is a sub-solution of L. From the Hopf maximum principle, we have u ≤ u. Thus we obtain a positive solution. We are inspired by Example 2 to compute the degree of an order-preserving compact map on an order interval. ◦

Theorem 3.6.16 Let (X, P ) be an OBS with P = ∅, and let [u, u] ⊂ X be a {e1 , . . . , ek } ﬁnitely bounded order interval for some {e1 , . . . , ek } ⊂ X. If f : [u, u] → X is order preserving, compact and satisfying: ◦

◦

f (u) ∈ u + P , f (u) ∈ u − P ; Then U := [u, u]◦ = ∅ and deg (id − f, U, θ) = 1 .

Proof. Since f is order preserving, ∀u ∈ [u, u], f (u) ≤ f (u) ≤ f (u). Moreover, we have f ([u, u]) ⊂ U . In fact, ∃ > 0, such that f (u) + B (θ) ⊂ u + P, f (u) + B (θ) ⊂ u − P . Therefore f (u) + B (θ) ∈ (u + P ) ∩ (u − P ) , i.e., f (u) ∈ U . This implies that U is a nonempty open convex set. Choosing arbitrarily x ∈ U, ∀t ∈ [0, 1], deﬁne φ(u, t) = t(u − f (u)) + (1 − t)(u − x) . Since f (∂U ) ⊂ U, tf (u) + (1 − t)x ∈ U ∀(u, t) ∈ ∂U × [0, 1]. Thus θ ∈ φ(∂U × [0, 1]). From the homotopy invariance of the degree, deg (id − f, U, θ) = deg (φ(·, 1), U, θ) = deg (φ(·, 0), U, θ) = deg (id, U, x) = 1 . The above theorem is due to Amann [Am 3]. ◦

Corollary 3.6.17 (Amann) Let (X, P ) be an OBS with P = ∅, and let ui , ui , i = 1, 2, satisfy u1 ≤ u1 ≤ u¯2 , u1 ≤ u2 ≤ u2 ; and [u1 , u1 ] ∩ [u2 , u2 ] = Ø. If [u1 , u¯2 ] is {e1 , . . . , ek }- ﬁnitely bounded for some {e1 , . . . , ek } ⊂ X, and if f : [u1 , u2 ] → X is an order-preserving compact map satisfying: ◦

◦

f (ui ) ∈ ui + P , f (ui ) ∈ ui − P , i = 1, 2 . Then there are at least three distinct ﬁxed points of f on the order interval [u1 , u2 ].

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191

Proof. Set Ui = [ui , ui ])◦ , i = 1, 2, and U = [u1 , u2 ]◦ . From the additivity, deg (id − f, U, θ) = deg (id − f, U1 , θ) + deg (id − f, U2 , θ) + deg (id − f, U \(U1 ∪ U2 ), θ) . Applying the above theorem, we have deg (id − f, U \(U1 ∪ U2 ), θ) = −1 . According to the Kronecker existence property, ∃ ﬁxed points ui , i = 1, 2, 3, such that ui ∈ Ui , i = 1, 2. and u3 ∈ U \(U1 ∪ U2 ). In the applications to PDEs, let Ω ⊂ Rn be a bounded domain with smooth boundary. We take X = C01 (Ω), and e ∈ X is a positive function in Ω and satisﬁes ∂n e(x) < 0, ∀ x ∈ ∂Ω, where ∂n is the outer normal derivative. Then it is easy to verify that any order interval [u, v] is e− ﬁnitely bounded. Example 3. Let L be the operator deﬁned in (3.52), and let g ∈ C 1 (R, R1+ ) be nondecreasing and satisfying g(0) = 0, g (0) = 1, t < g(t) ∀t ∈ (0, t0 ] for some small t0 > 0, and g(t) ≤ C as t > 0, where C > 0 is a constant. Then ∃λ∗ ∈ (0, λ1 ) such that the equation: Lu = λg(u), in Ω (3.55) u = 0, on ∂Ω , possesses at least one positive solution ∀λ ≥ λ∗ and at least two distinct positive solutions ∀λ ∈ (λ∗ , λ1 ). Indeed, we extend g to R1 oddly; the mapping f (u) = L−1 g(u) is order preserving. ∀λ > 0, it has a pair of strict sub- and super-solutions {−λCe, +λCe}, where e = L−1 1. Thus, vλ := limn→∞ (λf )n (λC) exists and is the maximal solution of (3.55). Obviously, vλ is upper semi-continuous and monotone non1 |vλ > 0}. We claim that λ∗ < λ1 , decreasing in λ. Deﬁne λ∗ = inf {λ ∈ R+ where λ1 is the ﬁrst eigenvalue of L. In fact, near the bifurcation point (θ, λ1 ) we ﬁnd uλ > 0, a solution of (3.55) with small uλ C < t0 . One has

Luλ φ1 dx = λ g(uλ )φ1 dx , Ω

Ω

where φ1 > 0 is the ﬁrst normalized eigenfunction of L∗ . Then

λ1 uλ φ1 dx > λ uλ φ1 dx . Ω

It follows that λ < λ1 , Since vλ ≥ uλ > 0, then λ∗ < λ1 . For ∀λ ∈ (λ∗ , λ1 ) ∃δ > 0, such that λg(t) < λ1 t, as |t| < δ. Let φ1 be the normalized ﬁrst eigenfunction; as ∈ (0, δ), ±φ1 is a pair of sub∗ and super-solutions of (3.55). However, as λ = λ 2+λ , vλ is a sub-solution of (3.55). Set > 0 small enough such that φ1 < vλ . Then we ﬁnd two pairs of sub- and super solutions: [−φ1 , +φ1 ], [vλ , λCe]. According to the Amann theorem, there must be two distinct nontrivial solutions of (3.55): u1λ ∈ [−φ1 , λCe]\([−φ1 , φ1 ] ∪ [vλ , λCe]), and u2λ ∈ [vλ , λCe]. Since > 0 can be arbitrarily small, both of u1λ , u2λ are positive. This is our conclusion.

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3 Degree Theory and Applications

3.6.5 A Free Boundary Problem We turn to an application of the bifurcation at inﬁnity to a free boundary problem for the ﬂux equation in the conﬁned plasma. Let Ω ⊂ R2 be the section of the container, a bounded domain with smooth boundary rotated along and oﬀ the z axis in R3 , which contains plasma conﬁned by an external magnetic ﬁeld. Let u be the ﬂux function of the magnetic ﬁeld. The current I is a given positive constant. The following equation is derived from the Maxwell equation: ⎧ ∂2 ∂ 1 ∂ ⎪ Lu := −[ ∂r ( r ∂r ) + 1r ∂z ]u = λu+ in Ω, ⎪ ⎪ ⎨ (3.56) u = C, on ∂Ω, ⎪ ⎪ ⎪ ⎩ ∂u − ∂Ω 1r ∂n = I, where C is a constant to be determined, and u+ = max {u, 0}. The domain Ωp := {x ∈ Ω | u(x) > 0}, which is the space occupied by the plasma, is unknown. It will be solved simultaneously with the ﬂux function u. Physically, we are interested in the case where Ωp ⊂ Ωp ⊂ Ω, and Ωp = ∅. From Green’s formula,

1 ∂u =I. u+ = Lu = − λ Ω Ω ∂Ω r ∂n We see that the problem is solvable only if λ > 0. Again, from Green’s formula,

1 ∂φ1 , u + φ1 = λ 1 uφ1 + C λ Ω Ω ∂Ω r ∂n where φ1 is the ﬁrst eigenfunction of L with respect to Dirichlet boundary condition. Combining with the maximum principle it is seen that C > 0 ⇒ λ < λ1 , and C = 0 ⇒ λ = λ1 . In all these cases, the solution u(x) > 0 ∀ x ∈ Ω, from the maximum principle. i.e., Ωp = Ω. They are not of interest in physics. We reduce the problem (3.56) to the following: Given a constant C1 < 0, ﬁnd (v, λ) satisfying ⎧ ⎨ Lv = λv+ in Ω , v = C1 on ∂Ω , (3.57) ⎩ λ>0. Indeed, if (v, λ) is a solution of (3.57), let v = αu, C1 = αC, where α > 0 is an adjustment constant satisfying

1 ∂u = −I . ∂Ω r ∂n Then (u, λ) is the solution of equation (3.56). We have the following conclusion:

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193

Statement 3.6.18 ∀I > 0, ∃u ∈ C 2,γ (Ω) γ ∈ (0, 1) with C < 0, which solves equation (3.56) if and only if λ > λ1 . Proof. After the above discussion, we just take C1 = −1. Let w = v + 1, then equation (3.57) is reduced to ⎧ ⎨ Lw = λ(w − 1)+ in Ω , w = 0 on ∂Ω , (3.58) ⎩ λ>0. Let us take X = C(Ω), F : w → (w − 1)+ , K = L−1 , our problem is reduced to ﬁnding the solution of w = λK ◦ F (w). It is easily seen that there is no 1 . According to Corollary 3.6.15, the bifurcation point located at {θ} × R+ projection of the connected component ζ of the positive solution set passing 1 is unbounded. It through the bifurcation point at inﬁnity (∞, λ1 ) onto R+ must include the interval (λ1 , +∞). Therefore, ∀λ ∈ (λ1 , +∞), there is a solution wλ on ζ. On the other hand, if there is a nontrivial solution wλ of equation (3.58), then wλ ≥ 0 and F (wλ ) < wλ . Thus

1 wλ φ1 dx = wλ Lφ1 dx λ1 Ω Ω

1 Lwλ φ1 dx = λ1 Ω

λ = F (wλ )φ1 dx λ1 Ω

λ < wλ φ1 dx . λ1 Ω This proves λ > λ1 . The proof of the regularity is standard.

3.6.6 Bridging We present an example to express the fact that solutions of a diﬀerential equation on disjoint domains can be glued up as a new solution in a larger domain. A typical example we consider here is a dumbbell D ⊂ R2 , i.e., two disks B± = B1/2 ((±1, 0)) connected by a segment E = {(x, y) ∈ R2 | y = 0, x ∈ [−1/2, 1/2]}. Let B = B+ ∪B− , and D = B ∪E. Assuming that f ∈ C 1 is of power growth with f (0) = 0, and that u± ∈ H01 (B± ) are solutions of the equation: − u = f (u) . (3.59) We consider a sequence of domains Dn squeezing to D, and want to ﬁnd a solution un ∈ H01 (Dn ) of (3.59) such that un is closed to u± in certain sense.

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Ω Dn -1

0

1

Fig. 3.5.

Let Ω be a bounded domain containing Dn , and D ⊂ Dn . Assuming that U ⊆ Ω is a bounded domain, the composition operator u → f (u) is C 1 from H 1 (U ) to Lq (U ), ∀q ∈ (1, ∞). Let r : H 1 (Ω) → H 1 (U ) be the restriction u → u|U , and e : H01 (U ) → H01 (Ω) be the extension (eu)(x) = u(x) if x ∈ U and 0 elsewhere. Let K = (−∆)−1 : Lq (U ) → H01 (U ), 1 ≤ q < ∞, then K is linear and compact. We deﬁne T = e · K · f · r : H01 (Ω) → H01 (Ω). The following statements are easy to verify: (1) ∀u ∈ H01 (U ), we have (r · e)u = u. (2) ∀v ∈ H01 (Ω), if rv ∈ H01 (U ), then (e · r)v = v. (3) For u0 ∈ H01 (U ), u0 is a solution of (3.59) if and only if e · u0 is a ﬁxed point of T . (4) If u0 is nondegenerate (i.e., ker−∆ − f (u0 )I) = {θ}), then I − T (u0 ) is invertible on H01 (Ω). Let us choose X = H01 (Ω). Let Tn , TB and en , eB be the operators and the extensions on Dn ∀n ∈ N and B = B+ ∪ B− , respectively. Statement 3.6.19 For any nondegenerate ﬁxed point eB u0 of TB , u0 ∈ H01 (B), there exists a ﬁxed point en un of Tn for n large, such that en un − e0 u0 X → 0 as n → ∞. Proof. By the nondegeneracy assumption, eB u0 is isolated and then i(I − T, eB u0 ) = ±1. In order to show the existence of a ﬁxed point en un of Tn , un ∈ H01 (Dn ), it is suﬃcient to verify that for all small δ > 0, I − Tn is homotopic to I − TB on ∂V , where V = Bδ (eB u0 ) is a ball in H01 (Ω). If it is so, then deg(I − Tn , V, θ) = deg(I − TB , V, θ) = 0 . Our conclusion follows. Now, we verify it by contradiction. Suppose that ∃tn ∈ [0, 1], ∃un ∈ H01 (Dn ), en un ∈ ∂V , such that en un = tn TB en un + (1 − tn )Tn en un . Since

3.7 Extensions

195

en un is bounded, and Tn , TB are compact, there exists a subsequence unj such that enj unj converges to some v and then v ∈ ∂V . We verify that v is a solution of (3.59), and that v(x) = 0, a.e., on Ω\B. Since V can be chosen arbitrarily close to eB u0 , it contradicts the nondegeneracy of u0 . (1) ∀φ ∈ H01 (B), from

∇un ∇φ = f (un )φ , B

it follows that

B

∇v∇φ = B

f (v)φ ; B

i.e., v is a solution (3.59) on B. (2) From en un = 0 a.e., on Ω\Dn , we have v = 0 a.e., on Ω\D. Since |E| = 0, v = 0, a.e., on Ω\B. Therefore, v ∈ H01 (Ω) is in fact a solution of (3.59). In other words, u+ and u− are nondegenerate solutions of (3.59) on H01 (B+ ) and H01 (B− ), respectively. We deﬁne u0 (x) = u± (x) as x ∈ B± , then eB u0 is a degenerate ﬁxed point of TB . By Statement 3.6.19 there are a sequence of domains Dn squeezing to D and a sequence of solutions un ∈ H01 (Dn ) of the equation (3.59) such that un is close to u0 .

3.7 Extensions We brieﬂy introduce a few directions in generalizing the Leray-Schauder degree. 3.7.1 Set-Valued Mappings We have used the ﬁxed-point theorem for set-valued mappings in Chap. 2. It is natural to ask if the degree theory for set-valued mappings can be set up. Let X be a Banach space, let Γ(X) be the set of all nonempty closed convex subsets of X. Deﬁnition 3.7.1 (Compact convex set-valued mapping) Let Ω ⊂ X be a bounded subset, and let φ : Ω → Γ(X). It is called a compact convex setvalued mapping, if (1) it is upper semi-continuous, and (2) φ(Ω) is compact. Obviously, any compact single-valued mapping is a compact convex setvalued mapping. The main idea in extending the Leray–Schauder degree to compact convex set-valued mappings is to approximate the set-valued mapping by singlevalued mappings. The following notion is useful:

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Deﬁnition 3.7.2 (Single-valued approximation) Let (M, d) be a metric space, and let Y be a Banach space. Let T : M → 2Y and T˜ : M → Y. We say that T˜ is a − single-valued approximation of T for > 0, if 1. T˜(M ) ⊂ convT ¯ (M ), 2. ∀x ∈ M, ∃y ∈ M and z ∈ T (y) satisfying d(x, y) < , and T˜(x) − z < . The following picture shows the − approximation of a set-valued mapping.

T

T˜

Fig. 3.6.

Theorem 3.7.3 Let (M, d) be a metric space, X be a Banach space, and let φ : M → Γ(X) be upper semi-continuous. Then ∀ > 0 there exists an − approximation φ of φ. Proof. By the upper semi-continuity, ∀x ∈ M, ∃δx > 0, such that φ(Bδx (x)) ⊂ φ(x) + B (θ). One may choose δx < 2 . Then A = {Bδx (x) | x ∈ M } is an open covering of M . It has an open locally ﬁnite star-reﬁnement B = {Vβ | β ∈ Λ} (see Dugundji [Du 2]). The so-called star-reﬁnement means that + ∀β ∈ Λ, ∃xβ ∈ M such that St(Vβ , B) ⊂ Bδxβ (xβ ), where St(Vβ , B) = {Vα ∈ $ B | Vα Vβ = ∅}. Deﬁne a partition of unity with respect to B : {λβ | β ∈ Λ}, and φ (x) = Σβ∈Λ λβ (x)zβ , where zβ ∈ φ(Vβ ). We are going to verify that φ is an − approximation of φ. Indeed, ∀x0 ∈ M , if x0 ∈ Vβ , then there must be y ∈ M and δy < 2 such that Vβ ⊂ Bδy (y). Thus, d(x0 , y) < 2δy < . However, zβ ∈ φ(Vβ ) ⊂ φ(Bδy (y)) ⊂ φ(y) + B (θ), we have φ (x0 ) = Σni=1 λβi (x0 )zβi ∈ φ(y) + B (θ), provided by the convexity of φ(y), and the fact {Vβi }ni=1 = {Vβ | x0 ∈ Vβ , β ∈ Λ}. We may ¯ ). choose z ∈ φ(y) such that φ (x0 ) − z < . Obviously, φ (M ) ⊂ convφ(M What is the homotopy equivalence for compact convex set-valued mappings? Deﬁnition 3.7.4 Let Ω be a bounded open set in a Banach space X, and let ¯ → Γ(X) be two compact convex set-valued mappings. We say that φ1 , φ2 : Ω φ1 is homotopically equivalent to φ2 , denoted by φ1 φ2 , if there exists a ¯ → Γ(X), such family of compact convex set-valued mappings Φ : [0, 1] × Ω that θ ∈ / F ([0, 1] × ∂Ω), and Φ(i, ·) = φi (·), where F (t, x) = x − Φ(t, x).

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197

¯ → Γ(X) is a compact convex set-valued Lemma 3.7.5 Suppose that φ : Ω mapping and f = id − φ. If θ ∈ / f (∂Ω), then ∃0 > 0 such that ∀ ∈ (0, 0 ), for any − single-valued approximation φ˜ of φ, we have θ ∈ / F ([0, 1] × ∂Ω), ˜ where F (t, x) = x − (1 − t)φ(x) − tφ(x). Proof. We prove the lemma by contradiction. If there exist single-valued continuous mappings φ˜n , and tn ∈ [0, 1], xn ∈ ∂Ω, yn ∈ Ω, zn ∈ φ(yn ) such that: xn ∈ (1 − tn )φ˜n (xn ) + tn φ(xn ) , 1 xn − yn < , n 1 ˜ . φn (xn ) − zn < 2n Since φ(Ω) is compact, and φ˜n (Ω) ⊂ convφ(Ω), after a subsequence, we have xn → x∗ ∈ ∂Ω, tn → t∗ , and then yn → x∗ . Since φ is u.s.c., for n large, we ˜n (xn ) ∈ 1 (θ). It follows, φ have φ(xn ) ⊂ φ(x∗ ) + B n1 (θ), φ(yn ) ⊂ φ(x∗ ) + B 2n ∗ φ(x ) + B n1 (θ), then (1 − tn )φ˜n (xn ) + tn φ(xn ) ⊂ φ(x∗ ) + B n1 (θ) , and then xn ∈ φ(x∗ ) + B n1 (θ) as n large. Thus x∗ ∈ φ(x∗ ). This is a contradiction. Lemma 3.7.6 If φ1 , φ2 are two homotopically equivalent compact convex setvalued mappings, then there exists > 0 such that for any − approximation φ˜1 , φ˜2 of φ1 and φ2 , respectively, we have φ˜1 φ˜2 . Proof. According to Lemma 3.7.5, we have φ˜0 φ0 φ1 φ˜1 in the sense of set-valued mappings. We shall prove that φ˜0 φ˜1 is also true in single-valued sense. ¯→ In fact there exists a compact convex set-valued mapping h : [0, 1] × Ω Γ(X) satisfying h(i, x) = φ˜i (x), i = 0, 1 and := dist(H([0, 1] × ∂Ω), θ) > 0, where H = id − h. By Lemma 3.7.3, and Lemma 3.7.5, there exists a 2 − ˜ : [0, 1] × Ω ˜ ¯ → X of h such that h(µ, ·) h(µ, approximation h ·). More precisely, let ˜ p(λ, µ; x) = (1 − λ)h(µ, x) + λh(µ, x) , we have θ ∈ / P ([0, 1] × [0, 1] × ∂Ω), where P (λ, µ; x) = x − p(λ, µ; x). Deﬁne a family of single-valued mappings: ⎧ 1 ⎪ ⎨p(1 − 3t, 0; x) if 0 ≤ t ≤ 3 g(t, x) = p(0, 3t − 1; x) if 13 ≤ t ≤ 23 ⎪ ⎩ p(3t − 2, 1; x) if 23 ≤ t ≤ 1 . It is easy to verify that this is the required homotopy.

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Now, we are ready to deﬁne the degree of id − φ, where φ is a compact convex set-valued mapping. Deﬁnition 3.7.7 Let Ω be an open bounded set in a Banach space X, let ¯ → Γ(X) be a compact convex-valued mapping, and f = id − φ. If φ : Ω θ∈ / f (∂Ω), deﬁne deg (f, Ω, θ) = lim deg (f˜ , Ω, θ) , →0

where f˜ = id − φ˜ and φ˜ is an − approximation of φ. According to Lemmas 3.7.5 and 3.7.6, the degree is well deﬁned. Readers can easily verify that the degree enjoys the same basic properties as the Leray–Schauder degree: Homotopy invariance, additivity, translation invariance and normality. So do the Kronecker existence and the excision. The results of Examples 1–3 in Sect. 3.6 can be extended to case where g has jump discontinuity by the use of the degree of set-valued mappings, see K. C. Chang [Ch 1], [Ch 2]. 3.7.2 Strict Set Contraction Mappings and Condensing Mappings The motivation of introducing the strict set contraction mapping is to extend the notion of compactness and then to extend the L-S degree. Let X be a Banach space, and let A ⊂ X be a bounded subset. ∀ > 0, we consider a − net N of A. Let α(A) = inf{ > 0 | ∃a ﬁnite − net N of A} . By deﬁnition, for α(A) = 0 if and only if A¯ is compact. It is easy to verify the following simple properties of α: If A ⊂ B, then α(A) ≤ α(B). α(A ∪ B) ≤ α(A) + α(B). ¯ = α(A). α(A) α(A + B) ≤ α(A) + α(B), α(λA) = |λ|α(A), ∀λ ∈ C. For a sequence of bounded closed nonempty subsets · · · ⊂ An ⊂ · · · ⊂ A2 ⊂ A1 satisfying α(An ) → 0, we have A = ∩∞ n=1 An = ∅, and α(A) = 0. 6. (Mazur) α(conv(A)) ¯ = α(A).

1. 2. 3. 4. 5.

Deﬁnition 3.7.8 (k-set contraction map) Let X be a Banach space, Ω ⊂ X be a bounded subset. A continuous map φ : Ω → X is called a k− set contraction mapping, k ≥ 0, if α(φ(A)) ≤ kα(A) ∀ bounded A ⊂ Ω . Remark 3.7.9 We assign ∞ · 0 = 0, then continuous mappings can be considered as ∞ – set contraction mappings. Example 1. If φ : Ω → X is compact, the φ is a 0− set contraction. Example 2. If φ : Ω → X is a Lipschitz mapping with Lipschitz constant L, then φ is an L− set contraction.

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199

Deﬁnition 3.7.10 (Strict set contraction mapping and condensing mapping) A k− set contraction mapping φ is called strict if k ∈ [0, 1). It is called a condensing mapping if α(φ(A)) < α(A) ∀ bounded A . According to (4): If φ, ψ : Ω → X are k1 − and k2 − set contraction mappings respectively, then φ + ψ is a k1 + k2 − set contraction. In particular, if φ is a contraction mapping (i.e., a Lipschitz mapping with Lipschitz constant L < 1), and if ψ is compact, then φ + ψ is a strict set contraction mapping. By the deﬁnition, if φ : Ω → X is a k1 − set contraction, and ψ : φ(Ω) → X is a k2 − set contraction. (In case, k1 = ∞, we assume further φ maps bounded set to bounded set); then ψ ◦ φ is a k1 k2 − set contraction. First we are going to deﬁne the degree for strict set contraction mappings. We intend to reduce it to compact mappings. Let f = id − φ, where φ is a k− contraction mapping, where 0 ≤ k < 1. ¯ then we deﬁne deg (f, Ω, θ) = 0. If θ ∈ / f (Ω), ¯ Otherwise the ﬁxed-point set of φ is not empty. Deﬁne A1 = convφ(Ω), ¯ ∩ Ak ), k = 1, 2, . . .. Since the ﬁxed point set of φ is not Ω and Ak+1 = convφ( $∞ empty, A := n=1 An = ∅. However, we have ¯ α(An+1 ) = α(convφ(An ∩ Ω)) ≤ kα(An ) ≤ ... ≤ k n α(A1 ) → 0 . ¯ ∩ A → A. Therefore α(A) = 0, A is a compact convex set, and then φ : Ω According to the Dugundji extension theorem, there exists a continuous ˜ Ω∩A ¯ → A such that φ| = φ|Ω∩A . Then φ˜ is compact and shares the same φ˜ : Ω ¯ ¯ ﬁxed point set with φ. ˜ We claim that θ ∈ / (id − φ)(∂Ω) ⇒ θ ∈ / (id − φ)(∂Ω). In fact, if x ∈ ∂Ω ˜ ˜ satisﬁes x = φ(x), then x ∈ ∂Ω ∩ A, so is φ(x) = φ(x) and then θ ∈ (id − φ)(∂Ω). With the above notations, we deﬁne the degree for strict set contraction mappings. Deﬁnition 3.7.11 Let φ be a strict set contraction mapping; we deﬁne ˜ Ω, θ) . deg (id − φ, Ω, θ) = deg (id − φ, To verify that the degree is well deﬁned, we shall prove that the def˜ i.e., if φ˜1 , φ˜2 are two inition does not depend on the special choice of φ, ˜ ˜ ¯ = φ|Ω∩A , i = 1, 2, then we deﬁne such extensions: φi : Ω → A, φi |Ω∩A ¯ ¯ / (id − φ)(∂Ω), it follows that F (t, x) = x − [(1 − t)φ˜1 (x) + tφ˜2 (x)]. From θ ∈

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θ∈ / (id−F )([0, 1]×∂Ω), i.e., φ˜1 φ˜2 . According to the homotopy invariance of the Leray–Schauder degree, we have deg (id − φ˜1 , Ω, θ) = deg (id − φ˜2 , Ω, θ). It is easy to verify that the degree enjoys the homotopy invariance, additivity, translation invariance and normality. They are left to readers as exercises. Accordingly, this enables us to apply the degree theory to a map, which can be decomposed into a summation of a contraction mapping and a compact map. Finally, we extend the degree to condensing mappings. If φ is a condensing ¯ is bounded, say it is included in the ball centered at θ with mapping, φ(Ω) radius R > 0. For any > 0, setting λ ∈ (1 − R , 1), and φλ = λφ, we have φ(x) − φλ (x) ≤ , and α(φλ (A)) ≤ λα(φ(A)) ≤ λα(A) ∀ bounded A , i.e., φλ is a strict set contraction mapping which is closed to φ. We deﬁne deg (id − φ, Ω, θ) = deg (id − φλ , Ω, θ) . Again, it is easy to verify that the degree is also well deﬁned and enjoys all basic properties of the Leray–Schauder degree. Again, this are left to readers. 3.7.3 Fredholm Mappings We know that the Leray–Schauder degree can be applied to quasilinear elliptic ¯ γ ∈ (0, 1) satisfying equations: Find u ∈ C 2,γ ∩ C0 (Ω), n

aij (x, u, ∇u)

i,j=1

∂2u + f (x, u, ∇u) = 0, in Ω . ∂xi ∂xj

(3.60)

¯ → C 1 (Ω) ¯ such that It is executed as follows. Deﬁne a mapping K : C01 (Ω) 0 1 ¯ 2,γ ¯ ∀u ∈ C0 (Ω), v = Ku ∈ C ∩ C0 (Ω) satisﬁes the following linear equation: n

aij (x, u, ∇u)

i,j

∂2v + f (x, u, ∇u) = 0 in Ω . ∂xi ∂xj

Thus the ﬁxed points of K are the solutions of equation (3.60). However, if we consider a general elliptic equation: A(u) := A(x, u, ∇u, ∇2 u) = g(x) in Ω ,

(3.61)

where g is a given function and the quadratic form is positive deﬁnite: n

∂A ξi ξj ≥ α|ξ|2 , α > 0 , ∂u x x i j i,j=1 then it seems that the Leray–Schauder degree argument is not applicable, because we do not know how to recast it as a ﬁxed-point problem for compact

3.7 Extensions

201

mappings. However, the linearization of A(u) is a linear second-order elliptic operator, so is a Fredholm operator with index zero. People have made great eﬀorts in deﬁning an integer-valued degree theory for C 1 Fredholm mappings of index 0 between Banach manifolds. Among them we should mention Caccioppoli [Cac 2], Smale [Sm 3](for Z2 valued), Elworthy and Tromba [ElT 1],[ElT 2], Borisovich, Zvyagin, and Sapronov [BZS] (for C 2 mappings and Z valued), and Fitzpatrick, Pejsachowicz, and Rabier [FP],[FPR] and [PR 1] [PR 2] etc. We are satisﬁed to introduce the idea of the deﬁnition; details are to be found in the above-mentioned references. The main diﬃculty in extending the Leray–Schauder degree theory to Fredholm mappings lies in the fact discovered by Kuiper [Kui] that the general linear group of inﬁnite-dimensional separable Hilbert space is connected and even contractible. A new ingredient has to be introduced to deﬁne the orientation of Fredholm mappings. Following [FPR], one deﬁnes the parity of curves of Fredholm operators. Let us denote by K(X) the space of all compact linear operators, and by Φ0 (X, Y ), (GL(X, Y )) the space of all Fredholm operators of index 0 (and isomorphisms resp.) between Banach spaces X and Y . For a curve of linear compact operators K ∈ C([0, 1], K(X)), if I−K(i), i = 0, 1 are invertible, then we deﬁne the parity by σ(I − K) = iLS (I − K(0), θ)iLS (I − K(1), θ) , where iLS (I − K(i), θ), i = 0, 1, are the Leray–Schauder indices. The notion of parity is extended to curves of Fredholm mappings with index 0: ∀A ∈ C([0, 1], Φ0 (X, Y )), if A(i) ∈ GL(X, Y ), then there exists N ∈ C([0, 1], GL(Y, X)) such that N (t)A(t) = I − K(t), ∀t ∈ [0, 1], where K ∈ C([0, 1], K(X)). If A(i) ∈ GL(X, Y ), i = 0, 1, then we deﬁne σ(A) = σ(I − K) . One can show that σ(A) does not depend on the special choice of N . In particular, if A ∈ C([0, 1], GL(X, Y )), then we take N (t) = A−1 (t), and then σ(A) = 1. If X = Y is a ﬁnite-dimensional Banach space, by deﬁnition, σ(A) = ±1 if and only if A(0) and A(1) lie in the same/diﬀerent connected component(s) in GL(X). But this is not true for inﬁnite-dimensional space due to the above-mentioned Kuiper’s theorem. A geometric interpretation is given in [FP]: Let Sj = {L ∈ Φ(X, Y +)∞| dimkerL = j}, j = 1, 2, . . ., and S = Φ0 (X, Y )\GL(X, Y ), then S = j=1 Sj , and σ(A) is the number of points of transversal intersection of a generic path A with S1 . The parity of the curve A ∈ C([0, 1], Φ0 ) enjoys the following homotopy invariance and the invariance under reparametrizations: (Homotopy invariance) Let H ∈ C([0, 1]2 , Φ0 (X, Y )) and suppose that H(t, i) ∈ GL(X, Y ), i = 0, 1, ∀t ∈ [0, 1]. Then σ(H(t, ·)) is a constant.

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(Invariance under reparametrizations) Let A ∈ C([0, 1], Φ0 (X, Y )) with A(i) ∈ GL(X, Y ), i = 0, 1, and let γ ∈ C([0, 1], [0, 1]) satisfy γ(i) = i, i = 0, 1. Then σ(A ◦ γ) = σ(A). Let Ω ⊂ U ⊂ X be open subsets, and U be connected and simply con¯ and the boundary ∂Ω are understood nected. In the following the closure Ω relative to U . Deﬁnition 3.7.12 A Fredholm mapping F ∈ C 1 (U, Y ) of index 0 is called Ω-admissible if F |Ω¯ is proper. A Fredholm homotopy H ∈ C 1 ([0, 1] × U, Y ) of index 1 is called Ω-admissible if H|[0,1]×Ω¯ is proper. Let p ∈ U be a regular point of a C 1 Ω-admissible Fredholm mapping F ; we take it as a base point. Let y ∈ Y \F (∂Ω) be a regular value of F |Ω , then F −1 (y) ∩ Ω is a ﬁnite set (may be empty): {x1 , x2 , . . . , xk }. If it is not empty, let γi ∈ C([0, 1], U ), with γi (0) = p, γi (1) = xi , be curves connecting p and xi , i = 1, 2, . . . , k. Then the parities σi = σ(F ◦ γi ), i = 1, 2, . . . , k, are all well deﬁned, and are independent of γi . Thus for a regular value y, we deﬁne the degree of F with the base point p by degp (F, Ω, y) =

k

σi .

i=1

If F −1 (y) ∩ Ω = ∅, then we deﬁne degp (F, Ω, y) = 0. Quinn and Sard [QS] avoided the requirement of the separability of the spaces X and Y , and obtained an improved version of Sard–Smale Theorem, by which one shows that for any y ∈ Y \F (∂Ω) there exists > 0 such that B (y) ⊂ Y \F (∂Ω) contains a regular value z of F . In combining with an approximation theorem for C 1 Fredholm mappings of any index in [PR 2], we can deﬁne Deﬁnition 3.7.13 Assume that F ∈ C 1 (U, Y ) is Ω-admissible, p ∈ U is a base point of F , and y ∈ Y \F (∂Ω). We deﬁne the degree degp (F, Ω, y) = degp (F, Ω, z), where z ∈ B (y) ⊂ Y \F (∂Ω) is a regular value of F |Ω . Again, the degree is independent of the choice of z. For diﬀerent base points p0 , p1 , one has degp0 (F, Ω, y) = σ(F ◦ γ) degp1 (F, Ω, y) , where γ ∈ C([0, 1], U ), γ(i) = i, i = 0, 1. The following fundamental properties hold: (Homotopy invariance) Let H be an Ω-admissible homotopy. Suppose that y ∈ Y \([0, 1] × ∂Ω) and that p is a base point of H(t, ·) ∀ t ∈ [0, 1]. Then degp (H(1, ·), Ω, y) = degp (H(0, ·), Ω, y) . (Additivity) Suppose the Ω = Ω1 ∪ Ω2 , where Ωi , i = 1, 2 are disjoint open subsets of U , and that F ∈ C 1 (U, Y ) is Ω-admissible, p ∈ U is a base point

3.7 Extensions

203

of F . Then F is Ωi -admissible for i = 1, 2. Moreover, if y ∈ Y \F (∂Ω), then y ∈ Y \F (∂Ωi ), i = 1, 2, and degp (F, Ω, y) = degp (F, Ω1 , y) + degp (F, Ω2 , y) . (Normality) Let Ω be an open subset of X, ∀p ∈ X, and y ∈ / ∂Ω. Then / Ω. degp (id, Ω, y) = 1 if y ∈ Ω, = 0 if y ∈ Obviously, the excision property and the Kronecker existence theorem hold as well. The generalized degree has been used to extend Rabinowitz global bifurcation theorem [PR1], and is applied to the study of bifurcation problems for semi-linear elliptic equations on Rn (see [JLS]).

4 Minimization Methods

The calculus of variations studies the optimal shape, time, velocity, energy, volume or gain etc. under certain conditions. Laws in astronomy, mechanics, physics, all natural sciences and engineering technologies, as well as in economic behavior obey variational principles. The main object of the calculus of variations is to ﬁnd out the solutions governed by these principles. Tracing back to Fermat, who postulated that light follows a path of least possible time, this is a subject in ﬁnding the minimizers of a given functional. Starting from the brothers Johann and Jakob Bernoulli and L. Euler, the calculus of variations has a long history, and renews itself according to the developments of mathematics and other sciences. The problem is formulated as follows: Assume that f : Rn × RN × RnN → 1 R is a continuous function, and that E is a set of N-vector functions. Let J be a functional deﬁned on E:

J(u) = f (x, u(x), ∇u(x))dx . Find u0 ∈ E, such that J(u0 ) = Min{J(u) | u ∈ E} . The central problems in the calculus of variations are the existence and the regularity of the minimizers. These are the 19th and the 20th problems among the 23 problems posed by Hilbert in his famous lecture delivered at International Congress of Mathematicians in 1900. This chapter is devoted to an introduction of the minimization method. We pay attention only to the existence of minimizers, but not to the regularity, although the latter is a very important and rich part of the theory of the calculus of variations. The direct method, studied in Sect. 4.2, is the core of the minimization method, in which w∗ -compactness and w∗ lower semicontinuity (w∗ l.s.c.) play crucial roles. A necessary and suﬃcient condition on the integrand f for the w∗ l.s.c. of the functional J on the Sobolev space W 1,p , p ∈ (1, ∞] is studied in Sect. 4.3.

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4 Minimization Methods

In the case when w∗ l.s.c. fails, either the minimizing sequence does not converge or it does not converge to a minimizer. The Young measure and the relaxation functional are introduced in Sect. 4.4. In the spaces W 1,1 and L1 the closed balls are no longer w∗ compact. Instead, we consider the BV space and the Hardy space, respectively. They are studied in Sect. 4.5. Two interesting applications are given in Sect. 4.6. One is on the phase transitions and the other is the segmentation in the image processing. The concentration phenomenon, which happens in many problems from geometry to physics, concerns the lack of compactness. We give a brief introduction to the method of managing this phenomenon in Sect. 4.7. The minimax method dealing with saddle points is brieﬂy introduced in Sect. 4.8. With the aid of the Ekeland variational principle and the Palais– Smale condition, it is studied in the spirit of the minimization method. Section 4.1 is an introduction, where various variational principles and their reductions are introduced.

4.1 Variational Principles Let X be a real Banach space, and U ⊂ X be an open set. A point x0 ∈ U is called a local maximum (or minimum) point of f : U → R1 , if f (x) f (x0 ) (or f (x) f (x0 ))

∀x ∈ Bε (x0 ) ⊂ U ,

for some ε > 0. If further, f is G-diﬀerentiable at x0 , then df (x0 , h) =

d f (x0 + th)t=0 = θ ∀h ∈ X , dt

or simply df (x0 ) = θ .

(4.1)

Moreover, if f has second-order G-derivatives at x0 , then d2 f (x0 )(h, h) 0

(or 0)

∀h ∈ X .

In particular, if X is a Hilbert space, and f ∈ C 2 then d2 f (x0 ) is a selfadjoint operator. We conclude that d2 f (x0 ) is nonnegative (nonpositive) if x0 is a local minimum (or maximum) point. Conversely, x0 is a local minimum (or maximum) point if d2 f (x0 ) is positive (or negative) deﬁnite. 4.1.1 Constraint Problems Let X, Y be real Banach spaces, U ⊂ X be an open set. Suppose that f : U → R1 , g : U → Y are C 1 mappings. Let

4.1 Variational Principles

207

M = {x ∈ U g(x) = θ} . Find the necessary condition for min f (x) .

x∈M

(4.2)

Theorem 4.1.1 (Ljusternik) Suppose that x0 ∈ M solves (4.2), and that Im g (x0 ) is closed. Then ∃(λ, y ∗ ) ∈ R1 × Y ∗ such that (λ, y ∗ ) = (0, θ), and λf (x0 ) + g (x0 )∗ y ∗ = θ .

(4.3)

Furthermore, if Im g (x0 ) = Y , then λ = 0. Proof. In the case where Y1 = Im g (x0 ) Y , the conclusion (4.3) is trivial; one may choose λ = 0 and y ∗ ∈ Y1⊥ := {z ∗ ∈ Y ∗ | z ∗ , z = 0 ∀z ∈ Y1 }. We assume Y1 = Y . The tangent space Tx0 (M ) of M at x0 is as follows: Tx0 (M ) ˙ = h} . = {h ∈ X ∃ε > 0, ∃v ∈ C 1 ((−ε, ε), X), x0 + v(t) ∈ M, v(0) = θ, v(0) We want to prove that Tx0 (M ) = ker g (x0 ). In order to avoid technical complication, we make an additional assumption: Either g (x0 ) is a Fredholm operator, or X is a Hilbert space. The assumption is superﬂuous, because a modiﬁed IFT has been studied in [De] (pp. 334) to improve the proof. Indeed, from g(x0 + v(t)) = θ , it follows that g (x0 )h =

d g(x0 + v(t)) t=0 = 0 ∀ h ∈ Tx0 (M ) , dt

i.e., Tx0 (M ) ⊂ ker g (x0 ). On the other hand, if h ∈ ker g (x0 ), one solves the equation: g(x0 + th + w(t)) = θ , for w ∈ C 1 ((−ε, ε), X1 ), w(0) = θ, where X1 is the complement of ker g (x0 ) and ε > 0 is small. Since g(x0 ) = θ, g (x0 ) : X1 → Y is an isomorphism, one may apply the IFT to obtain such a solution w. Setting v(t) = th + w(t), we have v(0) = θ and v(0) ˙ = h + w(0). ˙ From ˙ =θ, g (x0 )(h + w(0)) it follows that w(0) ˙ ∈ ker g (x0 ), but w(0) ˙ ∈ X1 which implies that w(0) ˙ = θ, i.e., v(0) ˙ = h. Now, ∀ h ∈ Tx0 (M ), ∃v(t) satisfying v(0) = θ, v˙ = h and x0 + v(t) ∈ M , so that f (x0 + v(t)) f (x0 ) ,

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4 Minimization Methods

which implies that

f (x0 ), h = 0 ,

i.e. f (x0 ) ∈ ker g (x0 )⊥ . By the closed range theorem, ∃y ∗ ∈ Y ∗ such that −f (x0 ) = g (x0 )∗ y ∗ . This completes the proof. Corollary 4.1.2 Suppose that g1 , . . . , gm : U → R1 are C 1 functions, and that x0 solves (4.2) with M = {x ∈ U gi (x) = 0, i = 1, 2, . . . , m} . If {gi (x0 )}m 1 is linearly independent, then ∃ λ1 , . . . , λm such that f (x0 ) +

m

λi gi (x0 ) = 0 .

i=1

Moreover, we may also consider inequality constraints: Given f, g1 , . . . , gm , h1 , . . . , hl ∈ C 1 (X, R1 ) ﬁnd min {f (x)|gi (x) = 0, i = 1, . . . , m; hj (x) ≤ 0, j = 1, . . . , l} . In the same manner, we ﬁnd the necessary condition of an extremum point x0 :

∃ λ1 , . . . , λm ∈ R1 , ∃ µ0 , µ1 , . . . , µl ≥ 0, but not all zero ,

such that µ0 f (x0 )+

m i=1

λi gi (x0 )+

l

µj hj (x0 ) = θ, and µj hj (x0 ) = 0, ∀ j = 1, . . . , l .

j=1

This is called the Kuhn–Tucker condition in the mathematical programming. In particular, if all the functions hj , j = 1, . . . , l are convex, which implies that the set C = {x ∈ X|hj (x) ≤ 0, j = 1, . . . , l} is convex, then a necessary condition for an extremum x0 ∈ C is the following variational inequality: 9 8 m λi gi (x0 ), y − x0 ≥ 0, ∀y ∈ C . f (x0 ) + 1

Remark 4.1.3 λ1 , . . . , λm ; µ0 , µ1 , . . . , µl , are called the Lagrangian multipliers.

4.1 Variational Principles

209

4.1.2 Euler–Lagrange Equation In case f is a functional of the following form:

f (u) = ϕ(x, u(x), ∇u(x))dx

Ω ⊂ Rn ,

(4.4)

Ω

where ϕ : Ω × RN × RnN → R1 is a C 2 function, let us turn to the 1st and 2nd variations of the functional (4.4). ∀u ∈ C 1 (Ω) f (u), v

n N N ∂ϕ(x, u(x), ∇u(x)) ∂vi ∂ϕ(x, u(x), ∇u(x)) + vi , = ∂ξαi ∂xα i=1 ∂ui Ω i=1 α=1 and

N n ∂ 2 ϕ(x, u(x), ∇u(x)) ∂vi ∂vk f (u), v ⊗ v = ∂xα ∂xβ ∂ξαi ∂ξβk Ω i,k=1 α,β=1

+2

n N i,k=1

N ∂ 2 ϕ ∂vi ∂2ϕ · v + vi vk k i ∂ξα ∂uk ∂xα ∂ui ∂uk α=1

,

i,k=1

∀v ∈ C 1 (Ω, RN ). Thus the Euler–Lagrange equation under certain boundary conditions reads as: n ∂ϕ(x, u(x), ∇u(x)) ∂ ∂ϕ(x, u(x), ∇u(x)) = 0, i = 1, 2, . . . , N . + − i ∂x ∂ξ ∂ui α α α=1 It is a second-order diﬀerential system. If the functional f is only deﬁned on a closed convex subset C of the function space C 1 (Ω), and u is a minimizer in C, then we only have a variational inequality:

n N ∂ϕ(x, u(x), ∇u(x)) ∂(vi − ui ) ∂ϕ(x, u(x), ∇u(x)) + (vi − ui ) ∂ξαi ∂xα ∂ui Ω i=1 α=1 ≥0, for all v ∈ C. In case N = 1, the Euler–Lagrange equation reads as −divϕξ (x, u(x), ∇u(x)) + ϕu (x, u(x), ∇u(x)) = 0 . This is a diﬀerential equation. We give here a few examples.

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4 Minimization Methods

1. (Geodesics) Let U be an open set of Rn , and let gij : U → R1 , ∀i, j = 1, 2, . . . , n be symmetric and positive deﬁnite. For u = (u1 , . . . , un ) ∈ C 1 ([0, 1], U ), one deﬁnes ϕ(u, ξ) =

n

gij (u)ξi ξj , ∀ ξ = (ξ1 , . . . , ξn ) ∈ Rn .

i,j=1

Then the Euler–Lagrange equation is the following diﬀerential system: j k d2 u i i du du = 0, + Γ j,k dt2 dt dt

where Γij,k =

∀i = 1, 2, . . . , n, t ∈ [0, 1] ,

1 [∂j glk + ∂k glj − ∂l gjk ]g li , 2

and (g li ) is the inverse matrix of (gil ), i.e., gij g ik = δjk . 2. (Poisson equation) Let ϕ(x, u, ξ) = c(x)u + 12 |ξ|2 . The Euler–Lagrange equation is the Poisson equation: u = c(x)

in Ω .

3. (Hamiltonian systems) For H ∈ C 1 (R1 × Rn × Rn , R1 ), the following ODE system x˙ = −Hp (t, x, p), p˙ = Hx (t, x, p), with (t, x, p) ∈ R1 × Rn × Rn , is called a Hamiltonian system, and H is called a Hamiltonian function. Sometimes we use the notations: z = (x, p) ,

and J=

0 I

−I 0

,

where I is the n × n unit matrix. The above system then has a simple form: z˙ = J grad H(t, z) . J grad sometimes is also called a symplectic gradient. The 2π-periodic solution can be seen as a critical point of the following functional:

2π 1 f (z) = z, J z

˙ R2n + H(t, z) 2 0 on the space C 1 ([0, 2π], R2n ). In other words, the Euler–Lagrange equation of this functional is the Hamiltonian system.

4.1 Variational Principles

211

1

4. (Minimal surface) Let u ∈ C 1 (Ω, R1 ), and ϕ(ξ) = [1 + |ξ|2 ] 2 . Then the Euler–Lagrange equation reads as ∇u =0. div 1 [1 + |∇u|2 ] 2 5. (Obstacle problem revisit) Let Ω ⊂ Rn be a bounded domain, g ∈ L2 (Ω), ψ ∈ H 1 (Ω). Find a minimizer of the problem: Min{f (u) | u ∈ E}, where

1 2 |∇u| − g · u , f (u) = 2 Ω and E = {u ∈ H01 (Ω) | u(x) ≤ ψ(x) a.e. x ∈ Ω } . The variational inequality reads as

∇u · ∇(v − u) ≥ g(v − u), ∀ v ∈ E . Ω

Ω

Now we derive the necessary condition on ϕ for a minimizer of f under Dirichlet boundary conditions. If u is a minimizer, then by Taylor expansion, f (u), v ⊗ v 0,

∀v ∈ W01,∞ (Ω, RN ) .

(4.5)

Let ρ(t) = 1 + t for t ∈ [−1, 0], and ρ(t) = 1 − t for t ∈ : [0, 1]. ∀x0 = n (x0,1 . . . , x0,n ) ∈ Ω, ∃ε > 0 such that the cube centered at x0 : α=1 [x0,α − , x0,α + ] ⊂ Ω. ∀λ = (λ1 , . . . , λn ) ∈ Sn−1 and ∀ξ = (ξ1 , . . . , ξN ) ∈ RN , we deﬁne α λ (xα − x0,α ) vε (x) = Πnα=1 ρ ·ξ , ε where ρ is understood to be 0 outside (−1, 1). Substituting vε into (4.5), and letting ε → 0, we obtain n N ∂ 2 ϕ(x0 , u(x0 ), ∇u(x0 )) α β λ λ ξi ξk ≥ 0 ∀(λ, ξ) ∈ (Rn \{θ}) × RN . i ∂ξ k ∂ξ α β i,k=1 α,β=1

This is called the Legendre–Hardamard condition, which can be rewritten in a compact form: (λ ⊗ ξ)T ∂ 2 ϕ(x, u(x), ∇u(x))(λ ⊗ ξ) 0 ,

(4.6)

where λ ⊗ ξ denotes the rank-one n × N matrix. In the next section, this condition on ϕ is called rank-one convexity. Thus, rank-one convexity is the necessary condition for ϕ at a minimizer. In particular, if N = 1, condition (4.6) is reduced to n ∂ 2 ϕ(x, u(x), ∇u(x)) j l λ λ 0, ∂ξj ∂ξl

j,l=1

∀λ ∈ Rn \{θ}, ∀x ∈ Ω .

(4.7)

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4 Minimization Methods

However, if the strict inequality in (4.7) holds for all λ ∈ Rn \{θ}, i.e., ϕ is strictly convex with respect to ξ. This is the ellipticity condition. To the vectorial case, the condition: N n

∂2ϕ (u, u(x), ∇u(x))ξαi ξβk > 0, ∀ ξ = {ξαi } ∈ Mn×N , with |ξ| = 0, i ∂ξ k ∂ξ α β i,k=1 α,β=1 is called the strong ellipticity. 4.1.3 Dual Variational Principle The following notion was introduced by Fenchel, and is very important in convex analysis. Deﬁnition 4.1.4 Suppose that X is a real Banach space and that f : X → R1 ∪ {+∞} is proper. The conjugate function of f, f ∗ : X ∗ → R1 ∪ {+∞} is deﬁned by f ∗ (x∗ ) = sup { x∗ , x − f (x)} . x∈X

The notion was initiated by Young’s inequality: if f (x) = p1 |x|p , 1 < p <

∞, then f ∗ (x) = p1 |x|p , where p1 + p1 = 1. In particular, if f (x) = 12 Ax, x Rn , where A is positive deﬁnite, then ∗ f (p) = 12 p, A−1 p Rn . The following propositions hold: 1. f ∗ is convex and l.s.c. Moreover, it is proper, if f is proper l.s.c. and convex. Proof. Only the properness of f ∗ needs to be proved, i.e., one should ﬁnd x∗0 ∈ X ∗ such that f ∗ (x∗0 ) < +∞. We consider the closed convex set epi(f ) = {(x, t) ∈ X × R1 | f (x) t}. Since f is proper, ∃x0 ∈ X such that f (x0 ) < +∞. One chooses t0 < f (x0 ), then (x0 , t0 ) ∈ epi(f ). By the Ascoli separation theorem, ∃(x∗0 , λ) ∈ X ∗ × R1 , ∃α ∈ R1 , satisfying x∗0 , x + λt > α > x∗0 , x0 + λt0 , In particular,

∀(x, t) ∈ epi(f ) .

x∗0 , x0 + λf (x0 ) > x∗0 , x0 + λt0 .

It follows that λ > 0, and then < ; α 1 − x∗0 , x − f (x) < − λ λ

∀x ∈ D(f ) ,

(4.8)

4.1 Variational Principles

i.e., f

∗

x∗ − 0 λ

0, then x∗0 , x + λf (x) > α It follows that

∀ x ∈ dom(f ) .

∗ x α f∗ − 0 − . λ λ

However, by the deﬁnition of f ∗∗ , ; ∗ < ∗ x0 x ∗∗ ∗ f (x0 ) − , x0 − f − 0 . λ λ It follows that

x∗0 , x0 + λf ∗∗ (x0 ) α .

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4 Minimization Methods

This contradicts (4.8), in which t0 = f ∗∗ (x0 ). In the case where f (x0 ) < +∞, from (4.8), λ > 0. It remains to verify that f ∗∗ (x0 ) = +∞ if f (x0 ) = +∞ and λ = 0. Again from (4.8), ∃ > 0 such that x∗0 , x − x0 ≥ ∀x ∈ dom(f ) . Since f ∗ is proper, there is an x∗1 ∈ X ∗ such that f ∗ (x∗1 ) < +∞, and x∗1 , x − f (x) − f ∗ (x∗1 ) ≤ 0 ∀x ∈ dom(f ) . Putting them together, ∀n ∈ N, we have x∗1 − nx∗0 , x + n x∗0 , x0 + n − f (x) − f ∗ (x∗1 ) ≤ 0, ∀x ∈ dom(f ) , it follows that f ∗ (x∗1 − nx∗0 ) + n x∗0 , x0 + n − f ∗ (x∗1 ) ≤ 0 , or n + x∗1 , x0 − f ∗ (x∗1 ) ≤ x∗1 − nx∗0 , x0 − f ∗ (x∗1 − nx∗0 ) ≤ f ∗∗ (x0 ) . Letting n → ∞, we obtain f ∗∗ (x0 ) = +∞. Therefore, f = f ∗∗ .

Corollary 4.1.6 For a proper, l.s.c., convex function f , x∗ ∈ ∂f (x) ⇐⇒ x ∈ ∂f ∗ (x∗ ). Proof. It is a direct consequence of proposition 4 and Theorem 4.1.5.

In the case where both ∂f and ∂f ∗ are single valued, Corollary 4.1.6 means that they are mutually inverse. In this sense, the conjugate function f ∗ of f is called the Legendre transform of f . Recall the variational formulation of the Hamiltonian system, which describes the motion of particle systems. The functional

2π 1 z, J z

˙ R2n + H(t, z) f (z) = 2 0 is very indeﬁnite. One cannot pose the minimization problem. However, in some cases, the Legendre transform may help. (1) Assume that H(t, x, p) is proper, l.s.c., and convex in the variables p, we deﬁne the Legendre transform of H with respect to p, i.e., L(t, x, q) = sup{ p, q Rn − H(t, x, p)} . p

It is called the Lagrangian. In other words, ∀(x, t) as a function of q, L is the conjugate function of the function H(t, x, p). From proposition 4 and the Hamiltonian system, one has

4.1 Variational Principles

215

−x, ˙ p Rn = L(t, x, −x) ˙ + H(t, x, p) . If both H and L are diﬀerentiable, then, Lx (t, x, −x) ˙ = −Hx (t, x, p) . But from Corollary 4.1.6, x˙ = −Hp (t, x, p) ⇔ p = Lq (t, x, −x) ˙ . This shows that (x, x) ˙ satisﬁes the system: d Lq (t, x, −x) ˙ + Lx (t, x, −x) ˙ =0. dt

(4.9)

However, (4.9) is the Euler–Lagrange equation of the functional:

2π

L(t, x, −x)dt ˙ .

I(x) = 0

The system (4.9) is called the associate Lagrange system. For example, in a system of particles with generalized coordinates q = (q1 , . . . , qn ), the kinetic energy of the system is a positive deﬁnite quadratic aij qi qj , where aij = aij (t, q), and the potential energy is a form: T = 12 continuous function bounded from below: U = U (q). The total energy H = T + U is called the Hamiltonian, and L = T − U is called the Lagrangian. In this case, the functional I is bounded from below. (2) Assume that H(t, x, p) is strictly convex in z = (x, p), then the Legendre transform of H with respect to z reads as G(t, w) = sup{ z, w R2n − H(t, z)} . z

Assume Hz (t, θ) = θ. We study nontrivial 2π-periodic solutions of the Hamiltonian system. Let us consider a constraint variational problem:

2π G(t, −J w)dt ˙ , I(w) = 0

with

g(w) = 0

2π

w, J w

˙ R2n dt = c = 0 ,

where c is a parameter. The Euler–Lagrange equation reads as Gw (t, −J w) ˙ = λw . Claim: λ = 0. If not, provided by w = Hz (t, z) ⇔ z = Gw (t, w) ,

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4 Minimization Methods

we have −J w˙ = Hz (t, θ) = θ. This contradicts c = 0. Let z = λw. Again by duality, we have z˙ = λJHz (t, z) . One can adjust c such that λ = 1. For example, if H(t, z) is bounded by two quadratic functions: 0 < m|z|2 ≤ H(t, z) ≤ M |z|2 , then G(t, w) is also. Thus the functional I is bounded from below.

4.2 Direct Method 4.2.1 Fundamental Principle Given a topological space X and a function f : X → R1 ∪ {+∞}, which is bounded from below, we seek the minimizers of f on X, if they exist. It is natural to require the l.s.c. of f (i.e., ∀t ∈ R1 , the level set ft := {x ∈ X| f (x) t} is closed) and certain compactness on X. However, if X is not compact, the coercive condition on f should be assumed as a replacement: f is proper and ∀t ∈ R1 ∃ a compact subset Kt ⊂ X such that the level set ft ⊂ Kt . Indeed, we have: Theorem 4.2.1 If f : X → R1 ∪ {+∞} is l.s.c. and coercive, then it attains a minimizer on X. In particular, f is bounded from below. Proof. Since f is coercive, it is proper. Then −∞ ≤ m := inf f < +∞. By the assumption, ∀t > m, the set ft is compact. Therefore by the ﬁnite intersection property, ∩ ft = Ø. A point x0 in the intersection achieves f (x0 ) = m. Since t>m

f does not assume the value −∞, m > −∞, and then f is bounded from below. However, in analysis, people prefer to use a minimizing sequence (i.e., a sequence {xj } ⊂ X such that f (xj ) → m := inf f ) to approach the minimizer. In this case, it is required that f is sequentially lower semi-continuous (s.l.s.c., for short) and we assume that Kt is sequentially compact, ∀t ∈ R1 . These two notions are deﬁned as follows: f is s.l.s.c. ⇐⇒ limf (zj ) f (z) if zj → z . and K is sequentially compact ⇐⇒ any sequence {zj } ⊂ K contains a subsequence zj → z ∈ K . Namely,

4.2 Direct Method

217

Theorem 4.2.2 If f : X → R1 ∪ {+∞} is s.l.s.c., and that ∀t ∈ R1 ∃ a sequentially compact set Kt ⊂ X such that ft ⊂ Kt , then f attains a minimizer in X. Recall the Eberlein–Schmulian theorem, for weakly closed subsets of a Banach space, weakly compact = sequentially weakly compact. According to the Banach–Alaoglu theorem, every w∗ -closed norm bounded set in the dual space of a Banach space is w∗ compact. Thus, for every weakly closed subset of a reﬂexive Banach space, weakly closed plus norm bounded = weakly compact = sequentially weakly compact. In this case, the coerciveness of f is equivalent to that ∀t ∈ R1 the level set ft of f is bounded in norm, or equivalently, that f is proper and f (x) → +∞ as x → ∞. However, how do we verify the weakly* lower semi-continuity (w∗ l.s.c., in short) or sequentially weakly* lower semi-continuity (s.w∗ l.s.c., in short) of f? We notice that in a Banach space, closed convex set = weakly closed convex set (Hahn–Banach theorem) = sequentially weakly closed convex set (Mazur theorem). Thus, if f is convex, then l.s.c. = w.l.s.c. = s.w.l.s.c. Combining the above discussions, both Theorem 4.2.1 and Theorem 4.2.2 imply Corollary 2.3.7 as a special case. Because of the importance of this statement, we rewrite it as follow: Theorem 4.2.3 Let X be a reﬂexive Banach space and let E ⊂ X be a weakly (or weakly sequentially) closed nonempty subset. If f : E → R1 ∪ {+∞} is a l.s.c., convex and coercive function, then f has a minimizer on E. In fact, Theorem 4.2.3 is a general principle in the proof of the existence of a minimizer. We shall present several classical examples to show how it works. 4.2.2 Examples Example 1. (Dirichlet problem for Poisson equation) Given f ∈ L2 (Ω), where Ω ⊂ Rn is a bounded domain, ﬁnd a minimizer of the following functional on H01 (Ω):

1 2 |∇u| − f · u dx . J(u) = 2 Ω

218

4 Minimization Methods

Obviously, J is convex and l.s.c.. We verify the coerciveness: From the Poincar´e inequality, we have a constant C > 0 such that

|u|2 dx C |∇u|2 . Ω

Ω

1 Thus, we may assume uH01 = ( Ω |∇u|2 ) 2 . Then ∃C1 > 0, such that

1 J(u) |∇u|2 − f 2 · u 2 2 Ω

1 1 2 |∇u| − |∇u|2 − C1 f 22 2 Ω 4 Ω 1 = u 2H 1 −C1 f 22 → +∞ 0 4 as u H01 → +∞. It is well known that the Euler–Lagrange equation of J is the Poisson equation with Dirichlet boundary conditions: −u = f in Ω , u=0 on ∂Ω . Example 2. (Harmonic map) Let Ω ⊂ Rm be a bounded open domain with smooth boundary ∂Ω. A map u = (u1 , . . . , un+1 ) : Ω → S n ⊂ Rn+1 , where S n is the unit sphere, is called harmonic if −uk = uk |∇u|2 where |∇u|2 =

n+1

in Ω, k = 1, 2, . . . , n + 1 ,

|∇uk |2 .

k=1 1

Given ϕ = (ϕ , . . . , ϕn+1 ) : ∂Ω → S n , ﬁnd a harmonic map u with prescribed boundary condition u|∂Ω = ϕ. Let us consider a subset of the Banach space X = H 1 (Ω, Rn+1 ): M = {u ∈ X| u|∂Ω = ϕ, u(x) ∈ S n a.e., x ∈ Ω} and deﬁne the functional 1 E(u) = 2

|∇u|2 dx . Ω

Firstly, M is a weakly sequentially closed set. Indeed, if uj u in H 1 (Ω, ), then modulo a subsequence {uj } we have uj → u in L2 (Ω, Rn+1 ) and R then uj (x) → u(x) a.e. From uj ∈ M ∀j, we have u ∈ M . Obviously, E is l.s.c., convex and coercive. n+1

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219

We are then able to apply Theorem 4.2.3 to obtain the existence of a minimizer u∗ ∈ M . What is the Euler–Lagrange equation for E? From

∇u · ∇v = 0 , dE(u, v) = Ω

∀v ∈

H01 (Ω, Rn+1 )

satisfying v(x) ∈ Tu∗ (x) S n

we obtain

a.e. in Ω ,

(u∗ )T (x) = 0 a.e. in Ω ,

where (u∗ )T (x) is the tangential projection of u∗ (x) at u(x). Noticing that the normal projection of u(x) at u(x) reads as (u)N (x) = u(x) · u(x) , and then by diﬀerentiation twice of the constraint |u(x)|2 = 1, we obtain u(x) · u(x) = −|∇u(x)|2 , from which follows

−u∗ (x) = u∗ (x)|∇u∗ (x)|2 .

However, we have only proved that u∗ ∈ H 1 (Ω, Rn+1 ). The rest of the problem is about the regularity of u∗ . For m = 2, the harmonic map u∗ with minimal energy is smooth if ϕ is, according to a result due to Morrey [Mo 2], but for m > 2, generally speaking, there is no such regularity (see Schoen and Uhlenbeck [ScU 1, ScU 2], Lin [Lin]. The deﬁnition and an existence result for harmonic maps between two Riemannian manifolds can be found in Eells and Sampson [ES]. The regularity problem for harmonic maps has attracted many authors. As a special case for elliptic systems, this regularity problem is related to the Hilbert 20th problem. It is proved in Helein [Hel 1] that for m = 2 any harmonic map is regular, but for m = 3 Riviere [Ri] showed that there exists a non-minimal energy harmonic map discontinuous everywhere. Example 3. Nonlinear eigenvalue problem Let Ω ⊂ Rn be a bounded open domain with smooth boundary, and let φ ∈ C(Ω × R1 , R1 ) satisfy q < 2∗ =

(1) |φ(x, t)| C(1 + |t|q−1 ), (2) tφ(x, t) > 0 for t = 0;

2n n−2

if n 3,

then ∃c0 > 0 such that ∀c ∈ (0, c0 ] the equation −u(x) = λφ(x, u(x)) x ∈ Ω , u|∂Ω = 0 , has a solution (λc , uc ) satisfying

uc (x) φ(x, t)dtdx = c . Ω

0

220

4 Minimization Methods

t Proof. Set X = H01 (Ω), g(u) = Ω Φ(x, u(x))dx, where Φ(x, t) = 0 φ(x, s)dx = O(|t|q ), Mc = g −1 (c), and f (u) = 12 Ω |∇u|2 . Since Φ(x, t) 0 but not identical to 0, we have g(u) 0 but not identical to 0. We have c0 > 0 such that g −1 (c0 ) = Ø; let u0 ∈ g −1 (c0 ), t → g(tu0 ) is continuous on [0, 1], and the function ranges over [0, c0 ]. Therefore g −1 (c) = Ø ∀c ∈ [0, c0 ]. Since the embedding H01 (Ω) → Lq (Ω) is compact, g is completely continuous, i.e., uj u∗ =⇒ g(uj ) → g(u∗ ). Thus Mc is a sequential weakly closed subset. Obviously, f is l.s.c., convex and coercive, so we have proved the existence of a minimum point uc . It remains to verify that g (u∗ ) = φ(x, u∗ (x)) = θ. In fact, θ ∈ Mc for c = 0, then u∗ = θ, therefore, φ(x, u∗ (x)) = θ. As a corollary, we consider the nontrivial solution of the equation: −u = |u|q−2 u in Ω, 2 < q < 2∗ u|∂Ω = 0 . Let (λ0 , u0 ) ∈ R1 × H01 (Ω) be a solution of the nonlinear eigenvalue problem: −u = λ|u|q−2 u u|∂Ω = 0 ,

with

|u0 |q > 0 . Ω

Let

|∇u0 |2 /

λ0 = Ω

Setting

|u0 |q > 0 . Ω

1

u∗ = λ0q−2 u0 , we obtain

−u∗ = |u∗ |q−2 u∗ u∗ |∂Ω = 0 .

in Ω

Remark 4.2.4 If −∆ is replaced by −∆ + I, then the same conclusion holds. Example 4. (Prescribing constant mean curvature problem) Given a constant H > 0 and a curve γ : S 1 → R3 . Suppose diam (γ) = R C2

with HR < 1. Find a disc-type surface u : D2 −→ R3 satisfying u = 2Hux ∧ uy in D u|∂D = γ , where

(4.10)

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221

u = (u1 , u2 , u3 ) and

2 u x ux ∧ uy = 3 ux

u2y u3x , u3y u1x

u3y u1x , u1y u2x

u1y . u2y

This is a geometric problem, in which H is a prescribed constant mean curvature, and γ is a prescribed boundary value. Write the problem in its variational form; deﬁne

4H 2 J(u) = |∇u| + u · (ux ∧ uy ) on H 1 ∩ L∞ (D, R3 ) . 3 D D Since

J (u), v = 2 −u + 2H(ux ∧ uy ), v ,

∀v ∈ C0∞ (D, R3 ). The Euler–Lagrange equation is exactly (4.10). In appearance, inf J(u) = −∞, there is no minimum. However, if we add a constraint condition: u L∞ (D,R3 ) ≤ R , where R < R , and HR < 1, then by |ux ∧ uy | |ux ||uy |

we have J(u)

D

1 (|ux |2 + |uy |2 ) , 2

2 |∇u|2 − |∇u|2 3

=

1 3

|∇u|2 . D

Moreover, on the convex set CR = {u ∈ H 1 (D, R3 )| u|∂D = γ, u L∞ ≤ R } , which is sequentially weakly closed in H 1 (D, R3 ), J is nonnegative and coercive. We verify the w.s.l.s.c. of J on CR under H 1 norm. Suppose uj u∗ (H 1 (D, R3 )), with uj ∈ CR ; we have a subsequence, for which we do not change the subscripts: ⎧ ∗ 2 3 ⎪ ⎨uj → u , strongly in L (D, R ) , uj,x u∗x , weakly in L2 (D, R3 ) , ⎪ ⎩ uj,y u∗y , weakly in L2 (D, R3 ) . and

uj (x) → u∗ (x)

a.e. in D .

Rewrite uj · (ujx ∧ ujy ) − u∗ · (u∗x ∧ u∗y ) = vj · (u∗x ∧ u∗y ) + uj · (vjx ∧ u∗y + u∗x ∧ vjy ) + uj · (vjx ∧ vjy ) ,

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4 Minimization Methods

where vj = uj − u∗ . Noticing |uj ∧ u∗y | R |u∗y | ,

we obtain

|uj ∧ u∗y − u∗ ∧ u∗y |2 → 0 ,

D

provided by Lebesgue’s dominance theorem. Therefore

∗ uj · (vjx ∧ uy ) = − vjx · (uj ∧ u∗y ) → 0 . D

D

Similarly

uj · (u∗x ∧ vjy ) → 0 .

D

1 |∇u∗ |2 , 2 D D and |vj | 2R , again by Lebesgue’s dominance theorem, we have

vj · (u∗x ∧ u∗y ) → 0 .

Since

|u∗x ∧ vy∗ |

D

Hence, ∗

4H uj · (vjx ∧ vjy ) + ◦(1) |∇vj |2 + 2∇vj ∇u∗ + 3 D

1 |∇vj |2 + ◦(1) ; 3 D

J(uj ) − J(u ) =

it follows that

limj→∞ J(uj ) J(u∗ ) .

How do we drop the constraint? By deﬁnition, if u∗ ∈ CR achieves the minimum point of J on CR , then u∗ satisﬁes the variational inequality:

∇u∗ ∇v + 2Hv · (u∗x ∧ u∗y ) 0, ∀v = w − u∗ , (4.11) D

where w ∈ CR . In particular, for any η ∈ C0∞ (D, R1 ), η ≥ 0, one chooses ε > 0 such that (1 − εη)u∗ ∈ CR . Substituting v = −εηu∗ into (4.11), we have 1 − |u∗ |2 + |∇u∗ |2 + 2Hu∗ · (u∗x ∧ u∗y ) 0 in D . 2 It follows that −|u∗ |2 0, in D , |u∗ | = |γ| on ∂D . Thus, |u∗ |2 is subharmonic. Applying the maximum principle, we have u∗ L∞ < R < R , i.e., u∗ is strictly in the interior of CR under the L∞ (D, R3 ) norm. Therefore we may choose arbitrarily v ∈ C0∞ (D, R3 ) with v L∞ < R − R as variations. Thus u∗ is a weak solution of equation (4.10).

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223

4.2.3 The Prescribing Gaussian Curvature Problem and the Schwarz Symmetric Rearrangement We turn to studying a problem arising in diﬀerential geometry. Let (M, g0 ) be a smooth compact two-dimensional Riemannian manifold with metric g0 ; let k(x) be its Gaussian curvature. Given a function K(x) on M , does there exist a metric g, which is pointwisely conformal to g0 , such that K is the Gaussian curvature with respect to g? Setting a function u on M satisfying g = e2u g0 , the problem is reduced to solving the following PDE: u = k − Ke2u on M , where is the Laplace–Beltrami operator with respect to g0 . According to the Gauss–Bonnet formula:

KdVg = 2πχ(M ) ,

(4.12)

(4.13)

M

where χ(M ) is the Euler characteristic of M , and Vg is the volume form with respect to g. The curvature K is restricted by the topology χ(M ); e.g., if χ(M ) > 0 (or < 0) then max K > 0 (or min K < 0 resp.), and if χ(M ) = 0, then either K ≡ 0 or K changes sign. A result in the case where χ(M ) < 0 has been studied in Sect. 2.1, Example 5. Namely: Statement. Assume k(x) < 0 and K(x) < 0 ∀x ∈ M .

(4.14)

Then (4.12) possesses a solution. However, (4.14) is not a necessary condition; the problem remains open if one merely assumes min K < 0. In contrast, the case χ(M ) = 0 was completely solved by a variational argument. A necessary condition for the solvability of (4.12) with χ(M ) = 0 reads as either K ≡ 0 , (4.15) or K changes sign and M Ke2v dVg0 < 0 , where v is the solution of v = k, v =

1 Vol(M )

v=0.

(4.16)

M

In fact, if u is a solution of (4.12), and let w = u − v, then −w = Ke2(w+v) .

(4.17)

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4 Minimization Methods

(4.15) follows from

M

Ke2v dVg0 = −

we−2w dVg0

M

= −2 M

|∇w|2 e−2w dVg0 < 0 .

Conversely, the condition (4.15) is also suﬃcient. Without loss of generality, we assume K ≡ 0 (otherwise, this is a linear problem). Let us consider a variational problem associated with (4.17):

1 |∇w|2 dVg0 J(w) = 2 M deﬁned on M = {w ∈ H 1 (M )| g1 (w) = g2 (w) = 0} ,

where g1 (w) =

M

wdVg0 ,

and g2 (w) =

M

Ke2(w+v) dVg0 .

If M is a nonempty weakly closed Banach manifold, and if w is a local minimum point, then there exist Lagrange multipliers λ, µ ∈ R1 such that

∇w · ∇ϕ + λϕ + 2µKe2(w+v) ϕ = 0 , M

∀ϕ ∈ H (M ), i.e., 1

−w + λ + 2µKe2(w+v) = 0 .

From g2 (w) = 0, it follows that λ = 0; and according to (4.15) and integration by parts, we have µ < 0. Let us choose r = 12 log(−2µ), then u = v +w +r solves (4.12). We introduce the notion of the Schwarz symmetric rearrangement of a function and discuss its main properties, which will be often used in analysis. For a nonnegative measurable function u deﬁned on a n-dimensional measurable set Ω with m(Ω) < ∞, the Schwarz symmetric rearrangement of u is deﬁned to be the following: u∗ (y) = sup {t ≥ 0 | |y|n ≤ Cn−1 m{x ∈ Ω | u(x) ≥ t}} , where Cn is the volume of the unit ball in Rn . It possesses the following properties: 1. Let u∗ (y) = g(|y|) ∀ y ∈ Ω∗ := BR (θ), where Rn = nonincreasing and g(R) = 0.

1 Cn m(Ω).

Then g is

4.2 Direct Method

225

2. Let ut ={x ∈ Ω | u(x) ≥ t} ∀ t ≥ 0. Then m(ut ) = m(u∗t ) ∀ t ≥ 0. 3. Ω up = Ω∗ u∗p ∀ p ∈ [1, ∞). 4. (Faber–Krahn inequality, [Fa], [Krh]) If u ∈ H 1 (Ω), where Ω ⊂ Rn is a bounded domain, then

|∇u∗ |2 ≤ |∇u|2 . Ω∗

Ω

Proof. Let H n−1 (E) denote the (n − 1)-dimensional Hausdorﬀ measure. From 2, the isoperimetric inequality, and the rotational invariance of u∗ , we have H

n−1

(u

−1

2

2

(t)) =

≤

1 u−1 (t)

H

n−1

(u

∗−1

2

2

(t)) =

1

u−1 (t)

|∇u|

∗ 2

=

u∗−1 (t)

and

u∗−1

u−1

1 |∇u|

|∇u |

u∗−1 (t)

,

1 |∇u∗ |

,

H n−1 (u∗−1 (t)) ≤ H n−1 (u−1 (t)) .

By the co-area formula and 2, it follows that

dm(ut ) dm(u∗t ) 1 1 − = = =− . ∗| |∇u| dt dt |∇u u−1 (t) u∗−1 (t)

Thus

u∗−1 (t)

∗

|∇u | ≤

Again by the co-area formula

∞

|∇u∗ |2 = |∇u∗ |dt ≤ Ω∗

0

u∗−1 (t)

|∇u| .

u−1 (t)

∞

0

u−1 (t)

|∇u|dt =

|∇u|2 . Ω

We now turn to the variational problem. Lemma 4.2.5 (Trudinger) Assume that Ω is a bounded planar domain with smooth boundary. Then ∀u ∈ H01 (Ω) with u 2 := Ω |∇u|2 ≤ 1, ∀β < 2 4π, ∃γ = γ(β) such that Ω eβu ≤ γ. Proof. Since |u| ≤ u , we may assume u ≥ 0. Let u∗ (y) be the Schwarz symmetric rearrangement of u. Thus u∗ (y) is radially symmetric, nonincreasing and satisﬁes |{y ∈ R2 | u∗ (y) ≥ t}| = |{x ∈ Ω| u(x) ≥ t}| ∀t ∈ R1 .

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4 Minimization Methods 1

Let u∗ (y) = g(|y|), and R = (π −1 |Ω|) 2 . We have g(R) = 0. By changing variables: 6 r 72 and f (s) = g(r) , e−s = R we have f (0) = 0, f (s) = − R2 e− 2 g (r) and

s |f (σ)|dσ |f (s)| ≤ s

0

√ ≤ s

s

|f (σ)| dσ 2

12

0

12 = R s 2 ≤ |g (r)| rdr 2 0 =

s = |∇u∗ |2 4π BR = 12

s 2 ≤ |∇u| , 4π Ω provided by the Faber–Krahn inequality. Thus

exp[βu2 ] = exp[β(u∗ )2 ] Ω

BR

R

exp[βg(r)2 ]rdr

= 2π 0

= |Ω|

exp[βf (s)2 − s]ds

0

≤

∞

1−

β 4π

−1 |Ω| .

Corollary 4.2.6 There are constants β > 0, and γ > 0 such that

exp[βu2 ]dVg0 ≤ γ , M

for all u ∈ H 1 (M ) with u :=

1 Vol(M )

M

u = 0 and u 2 =

M

|∇u|2 dVg0 = 1. n

Proof. By the use of the partition of unity, we have u = j=1 χj u, where χj ≡ 1. the support of the function χj ≥ 0 ∀j is contained in a chart, and Since u = 0, from Poincar´e’s inequality, there is a constant c > 0 such that 1 |u|2 := ( M |u|2 ) 2 ≤ c u , and then χj u ≤ ∇χj ∞ |u|2 + |χj |∞ u ≤ c1 u for some constant c1 . We obtain from Lemma 4.2.5:

4.2 Direct Method

exp[βu2 ] ≤ M

227

n 1 exp[n2 β(χj u)2 ] ≤ γ n M j=1

for suﬃciently small β. Corollary 4.2.7 There exist constants β > 0, and γ > 0 such that

e2u dVg0 ≤ γ exp[β −1 u 2 +2u] ∀u ∈ H 1 (M ) .

(4.18)

M

Proof. Let a = u, v = a−1 (u − u), then v = 0 and v = 1. From 2av ≤ 2 βv 2 + aβ and Corollary 4.2.6, we obtain

M

e2(u−u) dVg0 ≤ γ exp(β −1 u 2 ) .

The proof is complete.

Lemma 4.2.8 If un u0 in H 1 (M ), then after a subsequence, eun → eu0 in L2 (M ). Proof. We have

|eun − eu0 |22 = ≤

e2u0 |e(un −u0 ) − 1|2 e2u0 |un − u0 |2 (1 + e2(un −u0 ) )

≤ 2|e2u0 |4 (1 + |e2(un −u0 ) |4 )|un − u0 |24 . On account of Corollary 4.2.7 the ﬁrst two factors are bounded. Then by the compactness of Sobolev embedding, one has a subsequence such that |un − u0 |4 → 0. Corollary 4.2.9 The g2 is C 1 and is weakly continuous on H 1 (M ) functional 2(w+v) ϕ ∀ϕ ∈ H 1 (M ). with g2 (w), ϕ = 2 Ke It remains to verify the following: (1) M is a nonempty weakly closed Banach manifold, (2) J|M is weakly lower semi-continuous, (3) J|M is coercive. Indeed, (2), (3) are trivial and the nonemptyness of M is easy to verify. The weak closedness of M follows from Corollary 4.2.9. Since the functions 1 and Ke2(w+v) are linear independent ∀w ∈ M, M is a submanifold of H 1 (M ). Namely, we have:

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4 Minimization Methods

Statement. (M. S. Berger) Assume χ(M ) = 0. (4.12) is solvable if and only if K satisﬁes (4.15). For χ(M ) > 0, topologically M = S 2 or RP 2 . As we have seen in (4.13), a necessary condition of the solvability of the problem (4.12) is max K > 0. If we follow the above procedure, let v be the solution of −v = k − k, v = 0 . We turn to the equation as before: −w = Ke2(w+v) − k , and introduce a functional 1 J(w) = 2

(4.19)

M

(|∇w|2 + 2kw)dVg0

deﬁned on M = {w ∈ H 1 (M )| g2 (w) = k} . If w is a local minimum point of J, then there exists µ ∈ R1 such that −w = 2µKe2(w+v) − k . From w ∈ M, it follows that µ = 1/2, i.e., (4.19) is the Euler–Lagrange equation for J. From the Gauss–Bonnet formula, k¯ > 0, therefore if max K > 0, M = ∅. Similarly, one can show that M is a weakly closed Banach manifold, and that J is weakly lower semi-continuous. If J were coercive, then (4.12) would be solvable. But this is not true in general. In fact, set w ! = w − w, we have

! 2w = log kVol(M ) − log Ke2(v+w) dVg0 , M

and

1 ! |∇w|2 + k Vol(M ) log kVol(M ) − k Vol(M ) log Ke2(v+w) 2 M M

1 4π ≥ |∇w|2 + const. 1− 2 β M

J(w) ≥

due to (4.18) and the Gauss–Bonnet formula

k Vol(M ) = kdVg0 = 4π . M

If β > 4π, then the coerciveness follows. But Moser [Mos 4] proved that the best constant β = 4π for M = S 2 with the canonical metric g0 . This is the point where our argument breaks down. However, for some special K, e.g., if K is even, we shall improve the estimate in (4.18). In this case, the sphere S 2 is reduced to the real projective space RP 2 geometrically.

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229

Lemma 4.2.10 (Aubin) For a function w ∈ H 1 (S 2 ), if there exist fj ∈ C 1 (S 2 ), j = 1, 2 . . . , k and α > 0, satisfying

fj e2w dVg0 = 0, j = 1, 2, . . . , k , S2

and

k

|fj (x)| ≥ α ∀x ∈ S 2 ,

j=1

then ∀ > 0 ∃C > 0 such that

1 2w 2 e ≤ C exp |∇w| + 2w ∀w ∈ H 1 (S 2 ) . 8π − S 2 S2 Proof. Set

α 2 , Ω± j = x ∈ S | ± fj (x) ≥ k

1 2 and gj± , h± j ∈ C (S ) satisfying − + ± ± suppgj+ ∩ supph− j = suppgj ∩ supphj = ∅, 0 ≤ gj , h ≤ 1 , ± ± h± j (x) = 1 ∀x ∈ Ωj , and gj (x) = 1 if ± fj (x) ≥ 0 ∀j .

Then and then

− S 2 = ∪kj=1 (Ω+ j ∪ Ωj ) ,

S2

e2w ≤

k j=1

Ω+ j

+

Ω− j

Without loss of generality, one many assume that e2w , j = 1, 2, . . . . Ω±

e2w . Ω+ 1

e2w is the largest among

j

− First, ∀w ∈ H 1 (S 2 ), if h+ 1 w ≤ g1 w , then ∀1 > 0, ∃C1 > 0, such that + − 2 2 2 2 h+ 1 w ≤ h1 w + g1 w

≤ w 2 +C1 ( w |w|2 + |w|22 ) ≤ (1 + 1 ) w 2 +C1 |w|22 . Following Lemma 4.2.5 and Corollary 4.2.7,

+ 1 2 2 e2h1 w ≤ C1 exp h+ w +2C |w| 2 2 1 4π − 1 S2 1 + 1 ≤ C1 exp w2 + C3 (1 )|w|22 . 2(4π − 1 )

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4 Minimization Methods

Otherwise, we have 2 2 g1− w 2 ≤ g1− w 2 + h+ 1w ≤ (1 + 1 ) w 2 +C1 |w|22 ,

and

2g1− w

e S2

1 − 2 2 ≤ C1 exp g1 w +2C2 |w|2 4π − 1 1 + 1 2 2 w + C3 (1 )|w|2 . ≤ C1 exp 2(4π − 1 )

Now let u = w − w. ∀0 > 0, ∃C4 = C4 (0 ) such that |u|1 ≤ 0 u 2 +C4 .

(4.20)

∀2 > 0 small, we choose a = a(2 ) > 0 such that |{x ∈ S 2 | u(x) ≥ a}| = 2 . Therefore, we obtain a≤

0 1 1 |u|1 ≤ u 2 + C4 , 2 2 2

(4.21)

from (4.20), and 1

1

|(u − a)+ |22 ≤ 22 |(u − a)+ |24 ≤ C5 22 u 2 .

(4.22)

Setting 0 = 12 22 , and choosing 1 , 2 > 0 so small that 1 1 + 1 1 + 2 + C3 (1 )C5 22 < , 2(4π − 1 ) 8π −

+ Substituting w = (u − a)+ in the estimates of the integrals: S 2 e2h1 w − + + − and S 2 e2g1 w , since S 2 e2h1 u ≤ e2a S 2 e2(h1 u−a)+ and S 2 e2g1 u ≤ e2a S 2 − 1 e2(g1 u−a)+ , the right-hand sides become B exp[ 8π− u2 ]. Next, for the speciﬁed w in the assumption, we have

+ e2u ≤ 2k e2u ≤ 2k e2h1 u . S2

Ω+ 1

S2

In particular, according to f1 e2u = 0, we also have

k k e2u ≤ (f1 )+ e2u = (f1 )− e2u + α α 2 2 Ω1 S S

4.3 Quasi-Convexity

≤ Max{|fi | | i = 1, . . . , k}

k α

231

−

e2g1 u . S2

In both cases,

S2

i.e.,

e2u ≤ C exp

S2

e2w ≤ C exp

1 u 2 8π −

.

(4.23)

1 w 2 +2w . 8π −

Now, we return to the case (M, g0 ) = (S 2 , g0 ) where g0 is the canonical metric, but assuming K(x) = K(−x), ∀x ∈ S 2 . Let us deﬁne Me = {w ∈ H 1 (S 2 )| g2 (w) = k and w(x) = w(−x)} . Then, again, Me is a nonempty weakly closed Banach manifold. The functional

1 (|∇w|2 + 2kw)dVg0 J(w) = 2 M is again weakly lower semi-continuous. Since ∀w ∈ Me

xi e2w(x) dVg0 = 0 i = 1, 2, 3 , S2

where xi are the Euclidean coordinates, when we embed S 2 into R3 canonically, Lemma 4.2.10 is applicable, and then the coerciveness of J on Me follows. Thus, we have: Statement (J. Moser) For M = RP 2 with the canonical metric g0 , (4.12) is solvable if and only if max K > 0. Remark 4.2.11 Under certain symmetric conditions on K other than the evenness, Hong [Hon 2] obtained the existence of solutions for equation (4.12). Remark 4.2.12 The best constant β = 4π is due to Moser; there are several diﬀerent proofs and extensions, see Adams [Ad], Carleson and Chang [CC] and Ding and Tian [DiT].

4.3 Quasi-Convexity In the calculus of variations, we pay attention to the following functional:

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4 Minimization Methods

f (x, u(x), ∇u(x))dx ,

J(u) =

(4.24)

Ω

where f : Ω × RN × RnN → R1 is a Caratheodory funciton, and Ω ⊂ Rn is an open domain. As we have seen, the w.l.s.c. (or w∗ l.s.c.) condition plays an important role in the proof of the existence of a minimizer. 4.3.1 Weak Continuity and Quasi-Convexity From the abstract theory, we know that an l.s.c. function is w.l.s.c. and s.w.l.s.c. if it is convex. But to the above integral functional, what is the relationship between the s.w.l.s.c. (or s.w∗ .l.s.c.) with the convexity of the function f with respect to its variables (x, u, ξ)? To this end we would like to understand better the weak convergence in the spaces W 1,p (Ω, Rn ), 1 p ∞. Perhaps the simplest example of a weakly convergent but not strongly convergent sequence in our mind is {sin(mt)}∞ 1 in L2 ([0, 2π]). The phenomenon is caused by the large oscillations of the sequence of functions. It can be extended as follows: n

Lemma 4.3.1 Let D =

Π (aj , bj ) be a rectangle in Rn , and let ϕ ∈

j=1

Lp (D), 1 p ∞, which is extended periodically to Rn . Let ϕm (x) = ϕ(mx), ∀m ∈ N, and

1 ϕ= ϕ(x)dx . m(D) D ∗

Then ϕm ϕ in Lp (D) 1 < p < ∞ and ϕm ϕ in L∞ (D). Proof. First, we may assume ϕ = 0. Otherwise, we consider ϕ ! = ϕ−ϕ instead. Second,

1 ϕm pp = |ϕ(mx)|p dx = n |ϕ(y)|p dy = ϕ pp , ∀1 p ∞ . m D m·D Third, we deﬁne a signed set function Φ(E) = E ϕ for any measurable set E. It is σ-additive, and satisﬁes Φ(x + D) = 0, ∀x ∈ Rn . n

Now, ∀ rectangles Q = Π (ci , di ), by cancelling the nonoverlapping transi=1

lations of D in kQ, we have the estimate:

ϕk χQ = ϕk | = 1 |Φ(kQ) n |ϕ| . k kn D Q D Thus, for simple functions of the form ξ = αi χQi , Qi ∩ Qj = Ø i = j, we have

ϕk · ξ → 0 as k → ∞ . D

4.3 Quasi-Convexity

233

As 1 < p ∞, the simple functions of the above form consist of a dense p . ∀f ∈ Lp (D), ∃ξ a simple function as above such subset of Lp (D), p = p−1 that f − ξ p < ε/(2 ϕ p ). Therefore

ϕk · f ϕk p f − ξ p + ϕk · ξ < ε D

D

for k large enough.

Remark 4.3.2 Lemma 4.3.1 also holds for p = 1. In fact, the only problem in the above proof for p = 1 is that the simple functions ξ cannot approximate any f ∈ L∞ in L∞ norm. But, one can choose ξ such that f − ξ 1 < ε, with |αi | 2 f ∞ ∀i. Now, ∀λ > 0, let Ek,λ = {x ∈ D| |ϕk (x)| λ}, ∀k. Since ϕ1 = ϕ ∈ L1 (D), ∀ε > 0, ∃λ = λ(ε) such that E1,λ |ϕ| < ε. From the deﬁnition of ϕk , so is Ek,λ |ϕk | < ε. Again,

ϕk · f |ϕk (f − ξ)| + ϕk · ξ = I + II D

D

D

where II → 0 as k → ∞, and

I= |ϕk (f − ξ)|

D |ϕk ||f − ξ| + = Ek,λ

|ϕk (f − ξ)|

D\Ek,λ

3 f ∞ ·

|ϕk | + λ f − ξ 1 . Ek,λ

Again, we have proved that ϕk ϕ in L1 (D). Corollary 4.3.3 Let Ω = (0, 1), 0 < λ < 1, α, β ∈ R1 , and α if x ∈ (0, λ) ϕ(x) = β if x ∈ (λ, 1) , then ϕk (∗ )λα + (1 − λ)β in Lp , p ∈ [1, ∞), (p = ∞, resp.). Now, we turn to studying the s.w.l.s.c. (or s.w∗ .l.s.c.) of J in W 1,p (Ω), 1 ≤ p ≤ ∞. For simplicity, we assume D = [0, 1]n , and f = f (ξ). ∀k ∈ N , let Dlk be a sub-cube of D with length 2−k on each side and centered at ckl = 2−k (y1l + 1 1 l l l k 2 , . . . , yn + 2 ), where (y1 , . . . , yn ) runs over the lattice points (0, 1, . . . , 2 − kn + 2 1)n , l = 1, . . . , 2kn . Then D = l=1 Dlk , ∀k ∈ N . ∀v ∈ C0∞ (D, RN ). Let us deﬁne 1 (4.25) wk (x) = k v(2k (x − ckl )), ∀x ∈ Dlk , ∀l = 1, . . . , 2kn . 2 Then

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4 Minimization Methods

∇wk (x) = ∇v(2k (x − ckl )) ∀x ∈ Dlk , ∀l = 1, . . . , 2kn and

wk → 0 L∞ (D) , ∇wk (∗ )0 in Lp (D) p ∈ [1, ∞)(p = ∞ resp.) .

provided by Lemma 4.3.1. Now we arrive at:

Lemma 4.3.4 Let Ω ⊂ Rn be a domain, and let J(u) = Ω f (∇u). If J is s.w.l.s.c. on W 1,p (Ω, RN ) 1 p < ∞ (s.w∗ l.s.c. on W 1,∞ (Ω, RN )), then for ∀A ∈ M n×N , ∀ cube D ⊂ D ⊂ Ω, we have

1 f (A) f (A + ∇v) ∀v ∈ W01,∞ (D, RN ) , m(D) D where M n×N denotes the n × N matrix space. Proof. We choose a sequence in the Banach space X = W 1,p (Ω, RN ) as follows: uk (x) = Ax + wk (x) k = 1, 2, . . . , where the sequence {wk } is deﬁned in (4.25) and equals to zero outside D. Then it weakly converges to u(x) = Ax (w∗ -converges as p = ∞,). It follows that m(Ω)f (A) = J(u) lim inf J(uk ) . k→∞

But

f (A + ∇wk (x))dx +

J(uk ) =

f (A)dx Ω\D

D

2 kn

=

Dlk

l=1

f (A + ∇v(2k (x − ckl )))dx + f (A)m(Ω\D)

f (A + ∇v(x))dx + f (A)m(Ω\D) .

= D

This is our conclusion. According to Lemma 4.3.4, we introduce the following:

Deﬁnition 4.3.5 A Borel measurable and locally integrable function f : RnN → R1 is called quasi-convex (in the Morrey sense), if for ∀A ∈ M n×N , ∀v ∈ W01,∞ (D, RN ), and for any cube D ⊂ R , the inequality

m(D)f (A) f (A + ∇v(x))dx (4.26) D

holds. In the sequel of this chapter, we shall specify the terminology “quasiconvexity” to be quasi-convexity in the Morrey sense.

4.3 Quasi-Convexity

235

Remark 4.3.6 The cube D in the above deﬁnition can be replaced by any bounded domains, from a simple scaling and covering argument. Let us study the relationship between the convexity and the quasiconvexity. On the one hand, according to Jessen’s inequality, it is easy to see that if f is convex, then

1 1 f (A + ∇v(x)) dx f (A + ∇v(x) dx) = f (A) ; m(D) D m(D) D i.e., convexity =⇒ quasi-convexity. On the other hand, ∀B, C ∈ M n×N , with rank (B −C) = 1, ∀λ ∈ (0, 1), let A = λB + (1 − λ)C. After suitable translation and rotation, we may assume B = (1 − λ)a ⊗ e1 and C = −λa ⊗ e1 for some a ∈ RN and e1 ∈ Rn . Let us introduce a 1-periodic sawtooth function: (1 − λ)t, t ∈ [0, λ] , ϕ(t) = −λ(t − 1), t ∈ [λ, 1] . For ∀x ∈ D = [0, 1]n , let uk (x) = ak−1 ϕ(kx1 ) , then

(1 − λ) {kx1 } ∈ (0, λ) ∇uk (x) = a ⊗ e1 −λ {kx1 } ∈ (λ, 1) ,

where {y} denotes the fractional part of y ∈ R1 ; and let vk (x) = a min {k −1 ϕ(kx1 ), dist (x, ∂D)} , where dist (x, ∂D) = inf { sup xi − yi | y = (y1 , . . . , yn ) ∈ ∂D}. Then we 1in

have vk |∂D = 0 and there is a constant K > 0 such that |vk (x) − vk (y)| ≤ x − y. Therefore vk ∈ W01,p (D, RN ). Furthermore, we have m{x ∈ D| ∇uk (x) = ∇vk (x)} → 0 . as k → ∞. Let us divide D into two disjoint parts D1 and D2 , where D1 = {x ∈ D|∇uk (x) = (1 − λ)a ⊗ e1 } and D2 = {x ∈ D|∇uk = −λa ⊗ e1 }. Thus D = D1 ∪ D2 . If f is quasi-convex, then

f (λB + (1 − λ)C + ∇vk ) . m(D)f (λB + (1 − λ)C) ≤ D

On the account of the local integrability of f and the absolute continuity of the integral, we have

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4 Minimization Methods

f (λB + (1 − λ)C + ∇vk ) = lim

lim

f (λB + (1 − λ)C + ∇uk )

f (B) + f (C) = lim

D

D

D1

D2

= λm(D)f (B) + (1 − λ)m(D)f (C) . Finally, we obtain: f (λB + (1 − λ)C) λf (B) + (1 − λ)f (C) , i.e., f is convex along segments connecting two matrices with rank-1 diﬀerence. Deﬁnition 4.3.7 A function f : Mn×N → R1 ∪ {+∞} is said to be rank one convex, if f (λB + (1 − λ)C) λf (B) + (1 − λ)f (C) ∀λ ∈ [0, 1], ∀B, C ∈ Mn×N with rank {B − C} 1. As a direct consequence, we see that either n = 1 or N = 1, rank one convex = convex. In summary, we have proved: convex =⇒ quasi-convex =⇒ rank one convex. In particular, either n = 1 or N = 1, quasi-convex = convex. A natural question: are there any counterexamples of these reverse implications? 1. Quasi-convex in Morrey sense =⇒ convex. For A ∈ Mn×n , det A is quasi-convex in Morrey sense, but not convex. It is easily seen that det A is not convex, even for n = 2. Let us show that it is quasi-convex in Morrey sense. More precisely, we have

1 det (A + ∇v) = det (A) ∀D ⊂ Ω, ∀v ∈ C0∞ (D, Rn ) . m(D) D For simplifying the computation, we only verify the cases n 3. Higherdimension cases can be shown by induction. For n = 2, let v1 a11 a12 , v= ∈ C0∞ (D, R2 ) . A= a21 a22 v2 From det (∇v) = ∂x1 (v1 ∂x2 v2 ) − ∂x2 (v1 ∂x1 v2 ) , we have

det (∇v) = 0 , D

and then

4.3 Quasi-Convexity

237

1 det (A + ∇v) m(D) D

1 = [det (A) + a11 ∂x2 v2 + a22 ∂x1 v1 − a12 ∂x1 v2 − a21 ∂x2 v1 + det(∇v)] m(D) D = det (A) . For n = 3, let A = (aij )1i,j3 , adj2 (A) = (Aij )1i,j3 where Aij is the (ij) minor of A, det (A + ∇v) = (A + ∇v)1 , (adj2 (A + ∇v))1

= A1 , (adj2 (A + ∇v))1 + (∇v)1 , (adj2 (A + ∇v))1

3 ∂v1 (adj2 (A + ∇v))1i , A1i (adj2 (A + ∇v))1i + = ∂xi i=1 where B 1 = (b11 , . . . , b1n ) is the ﬁrst row vector of the matrix B = (bij )1i,jn , and , is the scalar product in Rn . By the conclusion for case n = 2, we have 1 m(D) i=1 3

A1i (adj2 (A + ∇v))1i D

1 1 = A (adj2 (A))1i m(D) i=1 i 3

= det (A) . Since 3 ∂ (adj2 (A + ∇v))1i ∂x i i=1 ∂v2 ∂v2 ∂v2 ∂ ∂x3 ∂ ∂x2 ∂x3 = + ∂x1 ∂v3 ∂v3 ∂x2 ∂v3 ∂x2

∂x3

∂x3

∂v2 ∂x1 ∂v3 ∂x1

∂v2 ∂ ∂x1 + ∂x3 ∂v3 ∂x1

∂v2 ∂x2 ∂v3 ∂x2

=0, after integration by parts, it follows that

3 ∂v1 (adj2 (A + ∇v))1i = 0 . ∂x i D i=1 This proves the conclusion. 2. Rank one convex =⇒ quasi-convex. There is a counterexample due to Sverak [Sv 1]. 4.3.2 Morrey Theorem The importance of the notion of quasi-convexity is due to the following:

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4 Minimization Methods

Theorem 4.3.8 Suppose that f : M n×N → R1 is continuous and quasiconvex. If the following growth condition holds: |f (A)| α(1 + |A|), (Cp )

as p = 1 ,

− α(1 + |A|q ) f (A) α(1 + |A|p ), as 1 q < p < ∞ , |f (A)| η(|A|), as p = +∞ ,

where η is continuous and increasing, and α > 0, ∀A ∈ M n×N , then for every bounded open domain Ω ⊂ Rn ,

J(u) = f (∇u)dx Ω

is s.w.l.s.c. in W 1,p (Ω, RN ) (s.w∗ .l.s.c. in W 1,∞ (Ω, RN )). Before going to the proof, we need the piecewise aﬃne function approximation in Sobolev spaces. Deﬁnition 4.3.9 (Triangulation) Let Ω ⊂ Rn be a bounded domain with piecewise aﬃne boundary. A triangulation τ˜ of Ω is a collection of ﬁnitely many n-simplices {Ki |i = 1, 2, . . . , I}, such that ∀i =+j, Ki ∩ Kj is either empty or equal to a p-simplex, 0 ≤ p ≤ n − 1, and Ω = 0≤i≤I Ki . We call hτ˜ = max{diam(Ki )|i = 1, . . . , n} the mesh size of τ˜. To an n-simplex, K = {p0 , p1 , . . . , pn }, (in which, pi ∈ Rn , i = 0, 1, . . . , n, and {pi − p0 |i = 1, 2, . . . , n} are linearly independent). We deﬁne n + 1 aﬃne functions {λ0 , λ1 , . . . , λn } such that λi (pj ) = δij , i, j = 0, 1, . . . , n . ∀v ∈ C(K), we deﬁne a piecewise aﬃne function by the following interpolation formula: n v˜K (x) = v(pi )λi (x) . (4.27) i=0

One shows that ∀v ∈ C (K), ∀x ∈ K, 2

∇v − ∇˜ vK ∞ ≤

n2 (n + 1) h2K vC 2 (K) , 2 ρK

(4.28)

where hK = diam K, and ρK = sup{2R|BR (x) ⊂ K, x ∈ K}. In fact, let pi = (p1i , . . . , pni ), i = 0, 1, . . . , n. We consider the functions vj (x) = xj , j = 1, 2, . . . , n, where x = (x1 , . . . , xn ), and v0 (x) ≡ 1 respectively. We obtain: xj =

n 0

pji λi (x), j = 1, 2, . . . , n, and

n 0

λi (x) = 1 .

4.3 Quasi-Convexity

239

Diﬀerentiating these formulas, it follows that n i=0

pji

∂λi = δkj , j, k = 1, 2, . . . , n, ∂xk

n ∂λi = 0, k = 1, 2, . . . , n . ∂xk i=0

Thus by Taylor expansion, ∂λi (x) ∂˜ vK (x) = v(pi ) ∂xk ∂xk i=0 n 1 2 2 ∂λi (x) = v(x) + ∇v(x)(pi − x) + ∇ v(ξx )(pi − x) 2 ∂xk i=0 n

∂v(x) ∂λi (x) + Ri (x) , k ∂x ∂xk i=0 n

=

where ξx ∈ K, and Ri (x) = 12 ∇2 v(ξx )(pi − x)2 . Since |Ri (x)| ≤

n2 vC 2 h2K ∀i , 2

and |∇λi | ≤ ρ1K . We obtain the desired estimate (4.28). A triangulation τ˜ is called regular, if ∃ a constant C > 0, such that C, ∀K ∈ τ˜.

hK ρK

≤

Lemma 4.3.10 Suppose that Ω ⊂ Rn is a bounded domain with piecewise aﬃne boundary. Then for ∀u ∈ W 1,p (Ω, RN ), ∀ > 0, ∃ a piecewise aﬃne function v on Ω, satisfying: 1. u − v1,p < , 2. |∇v(x)| ≤ |∇u(x)|, and |v(x)| ≤ |u(x)| a.e. Proof. With the aid of the extension operator and the molliﬁer, we have a function u ˜ ∈ C 2 (Ω, RN ), such that 1. ˜ u − u1,p < 3 , 2. |˜ u(x)| ≤ |u(x)|, |∇˜ u(x)| ≤ |∇u(x)| a.e., 3. ˜ uC 2 (Ω) ≤ Cu1,p , where C > 0 is a constant depending on Ω. To any regular triangulation τ˜ of Ω, we deﬁne a linear operator π from W 1,p (Ω, RN ) into itself as follows: (πu)(x) = u ˜K (x) ∀x ∈ K, ∀K ∈ τ˜ , where u ˜K is deﬁned as in equation (4.27).

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4 Minimization Methods

From |(πu)(x)| ≤ |u(x)|, and |∇(πu)(x)| ≤ |∇u(x)| a.e., it follows that π ≤ 1. On account of (4.28) ˜ u − πu ˜1,p is small as hτ˜ > 0 is small. Therefore, ˜1,p + ˜ u − πu ˜1,p + π˜ u − πu1,p < . u − πu1,p ≤ u − u

Set v = πu. This is what we need.

Now, let us return to the proof of Theorem 4.3.8 for the case p = ∞. Other cases are similar, so are omitted. Proof. Suppose u ∈ W 1,∞ (Ω, RN ) and uj ∗ u in W 1,∞ , we shall prove that

lim inf f (∇uj ) ≥ f (∇u) . Ω

Ω

We may assume |∇uj |, |∇u| ≤ C, M = sup{f (ξ)||ξ| ≤ 3C + 1}. ∀ > 0∃δ > as |ξ −ξ | ≤ δ and |ξ|, |ξ | ≤ 3C. One chooses 0 such that |f (ξ)−f (ξ )| ≤ 6m(Ω) . There Ω1 ⊂ Ω1 ⊂ Ω, in which ∂Ω1 is piecewise aﬃne with m(Ω\Ω1 ) < 8M k k 1,∞ (Ω, RN ) exists a regular triangulation of Ω1 : τk = {D1 , . . . , DIk }, ∃vk ∈ W such that vk is a piecewise aﬃne function on Ω1 with vk = u on ∂Ω, and vk − u1,∞ < . Let vk = Aki ∀x ∈ Dik . This can be realized by considering a domain Ω2 : Ω1 ⊂ Ω2 ⊂ Ω2 ⊂ Ω, with a piecewise aﬃne boundary ∂Ω2 , which is parallel to ∂Ω1 . We deﬁne vk in Ω1 according to Lemma 4.3.10, vk (x) = u(x), ∀x ∈ Ω\Ω2 , and linearly interpolate u|∂Ω2 and vk |∂Ω1 in Ω2 \Ω1 . According to Lemma 4.3.10, we may choose τk with hτk → 0 such that ∇vk → ∇u in measure. Let ujk = vk + uj − u, then ujk ∗ vk in W 1,∞ as j → ∞; and |∇ujk | ≤ 3C. One has k0 ∈ N , as k > k0 ,

(4.29) |f (∇vk ) − f (∇u)| < 6 Ω

and

|f (∇ujk ) − f (∇uj )| < Ω

. 6

(4.30)

Fixing k > k0 , let pki be the center of Dik , ∀r ∈ (0, 1); we consider a similar +I subsimplex Cik = r(Dik − {pki }) + pki , i = 1, 2, . . . , Ik , and Ω1 = 1k Cik . Deﬁne a smooth function ρ on Ω1 , such that ρ(x) = 1 ∀x ∈ Ω1 , ρ(x) = 0 ∀x ∈ +Ik k 1 ∂Di , and |∇ρ| ≤ C(τ˜k , r), a constant depending on τ˜k and r. Deﬁne vjk = vk + ρ(ujk − vk ) = vk + ρ(uj − u); we have

f (∇ujk ) = f (∇vjk ) + f (∇ujk ) Ω1

Ω1

Ω1 \Ω1

f (∇vjk ) +

= Ω1

= A+B+C .

Ω1 \Ω1

[f (∇ujk ) − f (∇vjk )]

4.3 Quasi-Convexity

241

On account of the quasi-convexity of f ; we have

f (∇vjk ) =

A= Ω1

Ik

i=1

≥

Ik

Dik

f (Aki + ∇[ρ(ujk − u)])

f (Aki )m(Dik )

i=1 f (∇vk ) .

= Ω1

It is easily seen that

B= Ω1 \Ω1

f (∇ujk ) ≥ −M (1 − rn )m(Ω1 ) .

As to C, from uj − u∞ → 0 as j → ∞, we obtain ∇vjk ∞ ≤ ∇vk ∞ + ∇uj ∞ + ∇u∞ + ∇ρ∞ uj − u∞ ≤ 3C + 1 , as j > j0 (τ˜k , r). Thus

Ω1 \Ω1

f (∇vjk ) ≤ M (1 − rn )m(Ω1 ) .

Putting them together, we have

f (∇ujk ) ≥ f (∇vk ) − 2M m(Ω1 )(1 − rn ), Ω1

(4.31)

Ω1

as j > j0 (k, r). Combining (4.29), (4.30) and (4.31), and suitably choosing r ∈ (0, 1), we have

f (∇uj ) ≥ f (∇ujk ) + f (∇ujk ) − 6 Ω Ω\Ω1 Ω1

f (∇vk ) + [f (∇ujk ) − f (∇vk )] − − 2M (1 − rn )m(Ω) ≥ 6 Ω Ω\Ω1

≥ f (∇u) − , Ω

as j > j0 (k, r). Since > 0 is arbitrarily small, we have proved the s.w∗ .l.s.c. of J.

Remark 4.3.11 Theorem 4.3.8 is initially due to Morrey under some additional conditions. It was reﬁned by Meyers, and greatly improved by Acerbi and Fusco [AF] and Marcellini [Mar]. In the ﬁnal version, f can be a Caratheodory function: Ω × RN × Mn×N → R1 , and (Cp ) is replaced by the following:

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4 Minimization Methods

|f (x, u, A)| α(1 + |u| + |A|)

as p = 1 ,

−α(1 + |u|r + |A|q ) f (x, u, A) α(1 + |u|r + |A|p ) as 1 < p n , np where 1 q < p, and r ∈ [1, n−p ),

−α(1 + |A|q ) f (x, u, A) α(1 + |A|p ) as n < p < ∞ , where 1 ≤ q < p, and f (x, u, A) η(x, |u|, |A|) as p = ∞ , where η is an increasing function in each of its argument. Theorem 4.3.12 Suppose that f : Mn×N → R1 is continuous, quasi-convex and satisfying c|A|p f (A) C(1 + |A|p )

for C > c > 0 ,

for 1 < p < ∞. Then for ∀v ∈ W 1,p (Ω, RN ), the functional

f (∇u) J(u) = Ω

achieves its minimum on E = Wv1,p := {u ∈ W 1,p (Ω, RN )| u|∂Ω = v|∂Ω }. Proof. In fact, J is s.w.l.s.c. and coercive, and E is weakly closed.

4.3.3 Nonlinear Elasticity The body of a given elastic material occupies a bounded domain Ω in R3 ; it is called the reference conﬁguration. External force gives the material a de∂ui formation u : Ω → R3 . The gradient of u, ∇u = ( ∂x )1≤i,α≤3 is called the α deformation gradient. To the hyper-elastic materials, a stored energy density → R1 is introduced with certain symmetric properfunction W : Ω × M3×3 + ties (e.g., it is left invariant under rotations and right invariant under certain denotes all 3 × 3 matrices with positive deterisotropic groups), where M3×3 + minants. The elastic energy reads as

W (x, ∇u)dx . I(u) = Ω

The basic assumption of the variational approach to this problem is that the observed deformations correspond to minimizers of the elastic energy. After the reductions from these invariances, the stored energy density is, in fact, dependent on the determinant, the minors and the eigenvalues of the deformation gradient ∇u only. A function f : Mn×N → R1 ∪ {+∞} is called poly-convex if there exists g : Rτ (n,N ) → R1 convex such that

4.3 Quasi-Convexity

243

f (A) = g(T (A)) where τ (n, N ) =

n∧N s=1

and T : M

n×N

→R

τ (n,N )

n!N ! , (s!)2 (n − s)!(N − s)!

is of the form:

T (A) = (A, adj2 A, . . . , adjn∧N A) , and adjs A denotes the matrix of all s × s minors of A. Examples. Let A = (aij ) ∈ Mn×n , then A 2 =

n

|aij |2 ,

i,j=1

det (A) , and adj2 A p p ≥ 1 , are poly-convex functions. What are the relationships between poly-convexity and other convexities? We claim that convex =⇒ poly-convex =⇒ quasi-convex. In fact, the ﬁrst implication is trivial. To verify the second implication, let us recall

1 det (B + ∇v) = det (B), ∀B ∈ Mn×n , ∀D ⊂ Ω, ∀v ∈ C0∞ (D, Rn ) . m(D) D It implies that ∀A ∈ Mn×n ,

1 adjs (A + ∇v) = adjs (A) m(D) D and then

1 m(D)

2sn,

T (A + ∇v) = T (A) . D

Thus if f is poly-convex, and f (A) = g(T (A)), where g is convex, then by Jessen’s inequality

1 1 f (A + ∇v) = g(T (A + ∇v)) m(D) D m(D) D

1 g T (A + ∇v) m(D) D

244

4 Minimization Methods

= g(T (A)) = f (A) , i.e., f is quasi-convex. In summary, we have the following implications: convex =⇒ poly-convex =⇒ quasi-convex =⇒ rank-one convex . One may ask: Are these implications strict? The following example is due to Dacorogna and Marcellini [DM]: A ∈ M2×2 , f (A) = A 4 − γ A 2 det A, then 4√ 2, 3 f is poly-convex ⇐⇒ |γ| 2 , f is quasi-convex ⇐⇒ |γ| 2 + ε, for some ε > 0 , 4 f is rank-one convex ⇐⇒ |γ| √ . 3

f is convex ⇐⇒ |γ|

It is not known whether 2 + ε = √43 . In addition to the counterexample due to Sverak [Sv 1], all the reverse implications are false.

4.4 Relaxation and Young Measure Not all variational integrands are quasi-convex. In this case a minimizing sequence may not converge to a minimizer, there may even be no minimizer! For example, let Ω = (0, 1), we consider the functional

J(u) =

1

[u2 (x) + (u (x)2 − 1)2 ]dx

(4.32)

0

on the Sobolev space W01,4 (Ω). The integrand is not quasi-convex. It is easily seen that inf J = 0. Indeed, on the one hand, we deﬁne a sequence of sawtooth functions: if x ∈ [ nk , 2k+1 x − nk 2n ] , un (x) = k+1 2k+1 k+1 if x ∈ [ n , n ] . −x + n From |un (x)| = 1 a.e., and |un | J(un )

1 2n ,

we conclude that

1 →0. 4n2

On the other hand, J 0. Then {un } is a minimizing sequence with inf J = 0.

4.4 Relaxation and Young Measure

245

However, un θ in W01,4 (Ω), because

1

un

·ϕ=−

0

1

un · ϕ → 0, ∀ϕ ∈ C0∞ (Ω), as n → 0 .

0

But J(θ) = 1. Moreover, the functional does not have a minimizer in W01.4 (Ω). Because, if u0 ∈ W01,4 (Ω) achieves J(u0 ) = 0, then u0 (x) = 0, a.e., and u0 (x) = 0, a.e., which imply that J(u0 ) = 1. This is impossible. Nevertheless, to this kind of problem, the minimizing sequences must provide some useful information in understanding the variational problems. There are several ways to studying these minimizing sequences: (1) Relaxation method: Introduce a relaxed functional with a quasi-convex integrand. It shares a common minimizing sequence with J. (2) Young measure: Extend the working Sobolev space to a measure space so that the minimizing sequences converge in some sense to a measure. (3) Change the working space to something other than the Sobolev space W 1,p (Ω, RN ). 4.4.1 Relaxations In the ﬁrst aspect, let us recall the relationship between s.w∗ .l.s.c. and the quasi-convexity. We are inspired to study various convex envelopes for nonconvex functions. Deﬁnition 4.4.1 Let f : Mn×N → R1 ∪ {+∞}, we call Cf = sup {g f | g convex} , P f = sup {g f | g poly-convex} , Qf = sup {g f | g quasi-convex} , Rf = sup {g f | g rank-one convex} , the convex, poly-convex, quasi-convex and rank-one convex envelope of f , respectively. Obviously, (1) Cf, P f, Qf and Rf are convex, poly-convex, quasi-convex and rank-one convex functions, respectively. (2) Cf P f Qf Rf f . In particular, if n = 1 or N = 1, then Cf = P f = Qf = Rf . By deﬁnition, f Cf f ∗∗ , the biconjugate function of f . But one can show that if f is proper, then Cf = f ∗∗ . Since Qf is quasi-convex, in addition to some growth condition, the associated functional

246

4 Minimization Methods

! J(u) =

(Qf )(x, u(x), ∇u(x))

(4.33)

Ω

is s.w∗ .l.s.c. in a certain Sovolev space W 1,p (Ω, RN ), 1 < p ≤ ∞. The following theorem is very basic in this aspect. Theorem 4.4.2 Suppose that f : Mn×N → R1 is measurable and locally integrable, and satisﬁes C2 |A|p f (A) C1 (1 + |A|p )

(4.34)

for some C1 > C2 > 0 and p ∈ (1, ∞). For ∀v ∈ W 1,p (Ω, RN ), deﬁne Wv1,p = {u ∈ W 1,p (Ω, RN )| u|∂Ω = v|∂Ω }, then ! u ∈ Wv1,p } . inf {J(u)| u ∈ Wv1,p } = inf {J(u)| Moreover, a function u0 ∈ Wv1,p is a minimizer of J! if and only if it is a cluster point (with respect to the weak topology of W 1,p ) of a minimizing sequence for J. Before going to the proof of Theorem 4.4.2, we give a characterization of the quasi-convex envelope. Lemma 4.4.3 If f : Mn×N → R1 is locally bounded and Borel measurable, then

1 1,∞ N f (A + ∇ϕ(x))dx| ϕ ∈ W0 (D, R ) , (4.35) (Qf )(A) = inf m(D) D where D ⊂ Rn is a bounded domain with m(∂D) = 0. Proof. We denote the RHS of (4.35) by (qf )(A, D). By deﬁnition, f ≤ g implies that qf ≤ qg; and if f is quasi-convex, then qf = f . We want to show: (1) (qf )(A, D) is independent of D, then we denote it by (qf )(A). (2) (qf )(A) is quasi-convex. We assume these two conclusions at this moment, then by the deﬁnition of the quasi-convex envelope, (qf )(A) (q(Qf ))(A) = (Qf )(A) ∀A ∈ M n×N . From (2), qf is quasi-convex and qf f . Since Qf is the largest quasi-convex function f , it follows that qf Qf . Therefore qf = Qf .

4.4 Relaxation and Young Measure

247

1. Veriﬁcation of (1). One observes that for any two bounded domains D1 and D2 , if after translation and scaling, D1 is transformed into D2 (denoted by D1 ∼ D2 ), then (qf )(A, D1 ) = (qf )(A, D2 ) . This proves that (qf )(A, D) are equal for all D ∼ D1 . Moreover, if D1 ⊂ D2 , then (qf )(A, D2 ) ≤ (qf )(A, D1 ). Now for any two bounded domains D1 and D2 in Rn , we may assume that D1 is the unit cube. And for any ε > 0, we ﬁnd {Ej | ∼ D1 , j = 1, 2, . . . , m}, m

with Ei ∩ Ej = Ø i = j and m(D2 \ ∪ Ej ) < ε. On each Ej one may ﬁnd a j=1

piecewise aﬃne function ϕεj ∈ W01,∞ (Ej , Rn ), such that

f (A + ∇ϕεj (x))dx (ε + (qf )(A, D1 ))m(Ej ) , Ej

and let ϕε (x) =

⎧ ⎪ ⎨ϕεj (x) x ∈ Ej ⎪ ⎩0

Thus

m

ε f (A + ∇ϕ (x))dx D2

j=1

m

x ∈ D2 \ ∪ Ej . j=1

f (A +

∇ϕεj (x))dx

+

Ej

(ε + (qf )(A, D1 ))

m

f (A)dx.

D2 \ ∪ Ej j=1

m

m m(Ej ) + f (A)m D2 \ ∪ Ej .

j=1

j=1

Since ε > 0 is arbitrary, it follows that (qf )(A, D2 ) (qf )(A, D1 ) . The positions of D1 and D2 are symmetric, thus (qf )(A, D1 ) = (qf )(A, D2 ) , i.e., (qf )(A, D) is independent of D, and will be written as (qf )(A). 2. Veriﬁcation of (2). First, we verify:

1 (qf )(A + ∇ψ(x))dx (qf )(A) , m(D) D for all piecewise aﬃne ψ ∈ W01,∞ (D, RN ), where D is the unit cube. In fact, m

let D = ∪ Di , where Di , i = 1, . . . , m are simplexes with Dj◦ ∩ Di◦ = Ø ∀i = j i=1

248

4 Minimization Methods

and ∇ψ|Di = Bi ∀i. By the deﬁnition of qf , for ∀ε > 0, ∃ϕi ∈ W01,∞ (Di , RN ) such that

1 f (A+∇ψ+∇ϕi )−ε, ∀x ∈ Di . (qf )(A+∇ψ(x)) = (qf )(A+Bi ) m(Di ) Di One extends ϕi to be zero outside Di , and deﬁne ϕ = ψ + ϕ ∈ W01,∞ (D, RN ), and

(qf )(A + ∇ψ(x))dx = D

m

i=1 Di m

i=1

m i=1

ϕi . Then

(qf )(A + Bi ) f (A + ∇ϕ) − εm(D)

Di

f (A + ∇ϕ) − εm(D)

= D

m(D)[(qf )(A) − ε] . Since ε > 0 is arbitrary, it follows that

1 (qf )(A + ∇ψ)dx| ψ ∈ W01,∞ (D, RN ) inf m(D) D piecewise linear (qf )(A) .

(4.36)

By the same procedure as in Sect. 4.3, we conclude that (qf ) is rank-one convex. Therefore qf is continuous, and in fact, is locally Lipschitzian (Theorem 2.2.1, Sect. 2.2). Since piecewise aﬃne functions are dense in W01,∞ (D, RN ), (4.36) is extended to all ψ ∈ W01,∞ (D, RN ). Thus qf is quasi-convex. Proof of Theorem 4.4.2. ! inf {J}. We only want to show that any Proof. 1. Since Qf f , inf {J} ! minimizing sequence of J is indeed a minimizing sequence of J. Thus it is 1,∞ (Ω, RN ) such suﬃcient to prove that ∀u ∈ Wv1,∞ (Ω, RN ) ∃{ul }∞ 1 ⊂ Wv that ! as l → ∞ . J(ul ) → J(u) Due to the assumption (4.34) on f and the density of piecewise aﬃne functions in W01,∞ (Ω, RN ), we may assume that u is piecewise aﬃne in a smaller domain Ω0 ⊂ Ω0 ⊂ Ω. By a standard argument, it is suﬃcient to replace Ω by Ω0 . m 2. Let Ω0 = ∪ Di , where Di , i = 1, . . . , m are simplexes with Di◦ ∩ Dj◦ = i=1

(j)

i ⊂ Di Ø i = j and let ∇u|Di = Bi ∀i. Then we choose small cubes {Ci }pj=1

(j)◦

with Ci

(k)◦

∩ Ci

= Ø as j = k, and

4.4 Relaxation and Young Measure

pi

(j) Di \ ∪ C i j=1

(1 + |∇u(x)|)p dx

249

ε . mC1 (j)

We are going to construct a sequence ul on each cube C = Ci . By Lemma 4.4.3, ∃ϕl ∈ W01,∞ (C, RN ) such that for ∀x ∈ C

1 1 f (∇u(x) + ∇ϕl (x))dx (Qf )(∇u(x)) + . (4.37) (Qf )(∇u(x)) m(C) C l Since C is a cube and ϕl ∈ W01,∞ (C, RN ), we may extend ϕl periodically to RN . Let 1 ul (x) = u(x) + ϕl (lx), ∀x ∈ C . l Then ul |∂C = u|∂C , and

f (∇ul (x))dx = f (Bi + ∇ϕl (lx))dx C C

1 f (Bi + ∇ϕl (y))dy = n l lC

= f (Bi + ∇ϕl (y))dy . (4.38) C

Combining (4.37) and (4.38), we obtain

(j) f (∇ul (x))dx → (Qf )(Bi )m(Ci ) , (j)

Ci

as l → ∞. 3. “Putting them together” we deﬁne ⎧ m pi (j) ⎪ ⎨u x ∈ Ω0 \ ∪ ∪ Ci , i=1 j=1 ul (x) = m pi (j) ⎪ ⎩ul (x) x ∈ ∪ ∪ Ci . i=1 j=1

It follows that

f (∇ul (x))dx =

Ω0

m pi

(j)

f (∇u(x))dx

Ω0 \ ∪ ∪ Ci i=1j=1

+

pi m

(j)

m(Ci )(Qf )(Bi ) + o(1)

i=1 j=1

(Qf )(∇u(x))dx + o(1) .

= Ω0

After a diagonal process, we obtain

250

4 Minimization Methods

! J(ul ) → J(u) as l → ∞ . ! then by the above argument, we obtain a 4. If u0 is a minimizer of J, minimizing sequence {ul } of J, with ul |∂Ω = u0 |∂Ω . Combining the ﬁrst inequality of (4.34) with the second inequality of (4.37), we obtain

p |∇ul | f (∇ul ) C2 Ω

Ω

! 0 ) + m(Ω) J(u J(u0 ) + m(Ω) . Therefore there exists a weakly convergent subsequence {uli } in Wv1,p . Recalling the construction of {ul }, we have ul (x) → u0 (x), a.e. Therefore, uli u0 in Wv1,p . ! l ) J(ul ) → Conversely, if {ul } is a minimizing sequence of J, then J(u ! ! inf J = inf J, i.e., {ul } is also a minimizing sequence of J. Again by the ﬁrst inequality of (4.34), there exists a subsequence {uli } that weakly converges to ! u0 in W01,p . Since J! is s.w.l.s.c., u0 is a minimizer of J. Now, let us return to the example we met in the beginning of this section. The function f (u, ξ) = u2 + (ξ 2 − 1)2 is obviously not convex in ξ. Since in the case where N = 1, convex = quasiconvex, the convex envelope of f (u, ξ) is the following: u2 + (ξ 2 − 1)2 if |ξ| 1 Qf (u, ξ) = 0 if |ξ| < 1 , and then ! J(u) =

1

Qf (u(x), u (x))dx .

0

In fact, the minimizing sequence introduced previously weakly converges ! to 0 in W 1,4 ([0, 1]), and u = 0 is exactly the minimizer of J. Since the initial functional J is in the lack of weakly lower semi-continuity, we then minimize another functional J! instead, which is s.w.l.s.c. and shares a common minimizing sequence with the original one. This passage is called relaxation. The solution of J! is called a generalized solution of J. Noticing that the representation of the quasi-convex envelope in Lemma 4.4.3 can be seen as an average over ﬁne scale oscillations, we give a physical interpretation of this phenomenon: J represents the microscopic energy, while J! is the macroscopic. There are models in elastic crystals, in which this phenomenon occurs. The quasi convex envelope of a non-convex function is not easy to compute. People have developed various methods in approaching the computations. Readers who are interested in this problem are referred to the book of

4.4 Relaxation and Young Measure

251

Dacorogna [Dac]. In particular, to the interesting example on the two well problem: K = {A, −A}, the quasi convex envelope of the square distance function Qdist(P, K)2 has been expressed explicitly by K. Zhang [Zh 2]. 4.4.2 Young Measure In a variational problem, there may be many minimizing sequences. If minimizers exist, then the minimizers describe the common feature of these sequences. Otherwise, the sequences do not converge to a minimizer as an element of the given function space. In his study of generalized solutions of optimal control problems, L. C. Young [Yo] introduced a family of measures in certain measure space as a replacement of the minimizer. The idea is as follow: Let Ω ⊂ Rn be a measurable set, and let zj : Ω → RN , j = 1, 2, . . . , , be a sequence of measurable functions. To each zj , we relate a family of probability measures νxj = δzj (x) on RN , ∀x ∈ Ω. Instead of the convergence of zj , we consider the convergence of the family of probability measures: {νxj }; i.e., whether there exists a family of measures νx on RN , ∀x ∈ Ω, such that for ∀f ∈ C0 (RN ), the space of continuous functions with compact support,

j f (y)dνx (y) → f (y)dνx (y) , RN

RN

as j → ∞. The family of measures {νx |x ∈ Ω} is a Young measure. However, the function zj is well deﬁned in a.e. sense, what is the meaning of the measure: δzj (x) ? What is the exact meaning of the family of measures {νx |x ∈ Ω} ? In order to clarify the meaning, we appeal to the slicing measures. Let µ be a ﬁnite nonnegative Radon measure on Rn+N . Denote by σ the canonical projection of µ onto Rn : σ(E) = µ(E × RN ) ∀ Borel set E ⊂ Rn . Lemma 4.4.4 For σ − a.e. point x ∈ Rn , ∃ a Radon probability measure νx on RN such that for each bounded continuous function f , 1. x → RN f (x, y)dνx (y) is σ − measurable, 2. Rn ×RN f (x, y)dµ(x, y) = Rn ( RN f (x, y)dνx (y))dσ(x). Proof. Recall that the dual space of C0 (RN ), the space of continuous functions with compact support, is the signed measure space M(RN ). The family of measures νx on RN , ∀x ∈ Rn , should be determined with the aid of functions on C0 (RN ). Since C0 (RN ) is separable, there is a countable dense subset {gk }∞ 1 . Now, ∀ Borel sets E ⊂ Rn , we deﬁne

γk (E) = gk (y)dµ(x, y), ∀k . E×RN

Obviously, γk is absolutely continuous with respect to σ. There exists a σnull set N such that the derivatives exist for ∀x ∈ / N:

252

4 Minimization Methods

Dσ γk (x) = lim

r→0

γk (Br (x)) , ∀k , σ(Br (x))

and then x → Dσ γk (x) are bounded and σ measurable. Moreover, for all Borel sets E ⊂ Rn ,

gk (y)dµ(x, y) = γk (E) = Dσ γk (x)dσ(x), ∀k . E×RN

(4.39)

(4.40)

E

N N Since {gk }∞ 1 is dense in C0 (R ), ∀g ∈ C0 (R ), we have a subsequence gkj → g N / N , Γx (g) = limj→∞ Dσ γkj (x) exists, and uniformly on R . Thus for ∀x ∈

|Γx (g)| ≤ Cµ gC0 (RN ) ,

(4.41)

where Cµ = µ(Rn × RN ). Since |γk (Br (x)) − γl (Br (x))| ≤ gk − gl C0 (RN ) σ(Br (x)) , the limit Γx (g) does not depend on the choice of the subsequence converging to g. Also, the map: g → Γx (g) is linear and positive. Provided by the Riesz representation theorem, ∃νx ∈ M(RN ), which is nonnegative, such that for ∀g ∈ C0 (RN ):

Γx (g) =

RN

g(y)dνx (y), ∀x ∈ /N .

By taking limits, we see νx (RN ) = Γx (1) = 1. Therefore, it is a probability measure. As a function of x on RN \N , x → Γx (g) is bounded and σ-measurable, because it is the pointwise limit of a sequence of σ-measurable functions Dσ γkj (x). Therefore, the conclusion (1) is obtained by approximation. Following (4.40), by taking limits we obtain

g(y)dµ(x, y) = Γx (g)dσ(x) = g(y)dνx (y) dσ(x) . (4.42) E×RN

E

E

From (4.42) ∀h ∈ C0 (Rn ), we have

g(y)h(x)dµ(x, y) = Rn ×RN

Since C0 (Rn ) mation.

Rn

>

RN

RN

g(y)dνx (y) dσ(x) .

(4.43)

C0 (RN ) is dense in C0 (Rn+N ), (2) is proved by approxi

Remark 4.4.5 By standard measure theoretic argument, one can show that the statements of Lemma 4.4.4 hold, if the bounded continuous function f is replaced by f ∈ L1 (Rn × RN , µ). Moreover, we have f (x, ·) ∈ L1 (RN , νx ) for a.e., x ∈ Rn , and the function x → RN f (x, y)dνx (y) is in L1 (Rn , σ).

4.4 Relaxation and Young Measure

253

Amap ν : Rn → M(RN ) is called weakly ∗ σ-measurable, if the functions x → RN f (y)dνx (y) are σ-measurable for all f ∈ C0 (RN ) on Rn . In the context of the duality between M(RN ) and C0 (RN ), the map ν : n R → M(RN ), obtained in Lemma 4.4.4 is w∗ σ-measurable. After the preparations, we are able to identify an L∞ (Ω, RN ) function with a w∗ -measurable map ν : Ω → M(RN ). Suppose that Ω ⊂ Rn is a Lebesgue measurable set with a ﬁnite Lebesgue measure. For ∀z ∈ L∞ (Ω, RN ), one deﬁnes a measure µ on Rn × RN , provided by the duality:

f (x, y)dµ(x, y) = f (x, z(x))dx, ∀f ∈ C0 (Rn × RN ) , (4.44) Rn ×RN

Ω

and then a family of measures on RN : δz(x) for a.e. x ∈ Ω. Since the righthand side of (4.44) is linear and positive in f , and

f (x, z(x))dx ≤ m(Ω)f C , 0 Ω

the existence of µ follows from the Riesz representation theorem. Obviously, the canonical projection of µ onto Rn is the Lebesgue measure restricted on the measurable set Ω, and the w∗ -measurable map ν : Ω → M(RN ) satisﬁes νx = δz(x) for a.e. x ∈ Ω. Now we turn to the fundamental theorem for Young measures. Theorem 4.4.6 Let Ω ⊂ Rn be a bounded open domain and let {zk } ⊂ L∞ (Ω, RN ) be a bounded sequence. Then there exist a subsequence {zkj } and a w∗ -measurable map ν : Ω → M(RN ) such that 1. νx is a Radon probability measure for a.e. x ∈ Ω. 2. For ∀g ∈ C0 (RN ), one has

g(zkj ) ∗ g(x) := νx , g = g(y)dνx (y), in L∞ (Ω) . RN

3. Let K ⊂ RN be closed, and let dist(zkj , K) → 0 in measure as j → ∞. Then supp νx ⊂ K for a.e. x ∈ Ω. Proof. According to (4.44), ∀zk , ∃µk on Rn × RN , for which the canonical projection onto Rn is the Lebesgue measure restricted to Ω. Since µk (Rn × RN ) ≤ Ln (Ω), there exist a measure µ on Rn × RN and a subsequence such that µkj ∗ µ, according to the Banach–Alaoglu theorem. We claim that the canonical projection σ of µ into Rn is the Lebesgue measure Ln restricted on Ω. On one hand, if U ⊂ Ω is open, then by the w∗ -convergence: σ(U ) = µ(U × RN ) ≤ lim inf µkj (U × RN ) = Ln (U ) .

254

4 Minimization Methods

According to the Radon–Nikodym theorem, there exists a ∈ L1 (Ω, Ln ) such that σ = aLn . On the other hand, for any compact subset K ⊂ Ω, we show that σ(K) ≥ Ln (K). Since {zk } is bounded in L∞ (Ω, RN ), ∃R > 0, such that supp(µkj ), and then supp(µ) are contained in Ω × BR (θ).∀ > 0, ∃f ∈ C0 (Rn × RN ), with f = 1 on K × BR (θ) and f ≥ 0, such that

f (x, y)dµ(x, y) < µ(K × BR (θ)) + . Rn ×RN

Thus, σ(K) = µ(K × RN ) = µ(K × BR (θ))

> f (x, y)dµ(x, y) − Rn ×RN

= lim f (x, y)dµkj (x, y) − j→∞

Rn ×RN

> Ln (K) − . Since > 0 is arbitrary, we have proved that σ(K) ≥ Ln (K) .

(4.45)

Therefore a = 1, a.e., and σ is the restriction of the Lebesgue measure on Ω. The ﬁrst conclusion follows directly from Lemma 4.4.4. We turn to proving the second conclusion. ∀h ∈ C(Ω), and ∀g ∈ C0 (RN ), ∃ ˜ ∈ C0 (Rn ), from (4.44), we have an extension of h, h

˜ h(x)g(y)dµ kj (x, y) Rn+N

h(x)g(zkj (x))dx . = Ω

By taking limits, and from (4.43), we have

˜ h(x)g(zkj (x))dx → h(x)g(y)dµ(x, y) Ω Rn ×RN

= h(x) g(y)dνx (y) dx . RN

Ω

Since C(Ω) is dense in L1 (Ω) and g(zkj ) is bounded in L∞ (Ω), the above equality implies that

g(y)dν(y) =< ν, g > in L∞ (Ω, RN ) , g(zkj ) ∗ RN

4.4 Relaxation and Young Measure

255

i.e., (2) holds. We prove the third conclusion. ∀g ∈ C0 (RN \K), ∀ > 0, ∃C > 0, such that |g(y)| ≤ + C dist(y, K), ∀y ∈ RN , which implies that |g(zk (x))| ≤ + C dist(zk (x), K), a.e., x ∈ Ω . From dist(zkj (x), K) → 0 in measure, we have | νx , g | ≤ , a.e. x ∈ Ω Since > 0 is arbitrary, supp(νx ) ⊂ K, a.e. x ∈ Ω.

Deﬁnition 4.4.7 The w∗ -measurable map ν : Ω → M(RN ), deﬁned in Theorem 4.4.6, is called a Young measure generated by the sequence {zkj } ⊂ L∞ (Ω, RN ). Remark 4.4.8 The conclusion (2) has a probability interpretation: In the limit, g(zkj ) takes the value g(y) with probability νx (y) at x. Therefore the Young measure can be used to describe the local phase proportions in an inﬁnitesimally ﬁne mixture. Corollary 4.4.9 Assume that ν is the Young measure associated with the sequence {zk } and that zk → z in measure. Then νx = δz(x) a.e. Proof. ∀g ∈ C0 (RN ), g(zk ) → g(z) in measure. But by Theorem 4.4.6, g(zk ) ∗ g = ν, g . Therefore, νx , g = g(z(x)) = δz(x) , g , a.e . Remark 4.4.10 In Theorem 4.4.6, the L∞ -bounded sequence {zk } ⊂ L∞ (Ω, RN ) can be replaced by the L1 bounded sequence {zk } ⊂ L1 (Ω, RN ). In this case, νx is Ln *Ω a.e. a probability measure with

|y|dνx (y)dx ≤ lim inf zk L1 . Ω

k→∞

RN

Again we claim that σ is the restriction of the Lebesgue measure on Ω. In fact, only (4.45) should be modiﬁed: ∀R > 0 σ(K) ≥ µ(K × BR (θ)) ≥ lim sup µk (K × BR (θ)) k→∞

≥ lim sup Ln ({x ∈ K | |zk (x)| ≤ R}) k→∞

256

4 Minimization Methods

≥ Ln (K) −

1 sup zk L1 . R k

As R → ∞, it follows that σ(K) ≥ Ln (K). Finally, we use Remark 4.4.5, and choose a sequence of positive functions fk (x, y) ↑ |y| pointwisely; the monotone convergence theorem implies the conclusion. Corollary 4.4.11 Assume that {zk } ⊂ L∞ (Ω, RN ), satisﬁes zk → z, a.e., and that {wk } ⊂ L∞ (Ω, RM ) generates the Young >measure ν. Then {zk , wk } : Ω → RN +M generates the Young measure δz(x) νx , x ∈ Ω. Proof. ∀ϕ ∈ C0 (RN ), ∀ψ ∈ C0 (RM ), ∀η ∈ L1 (Ω), by deﬁnition and the Lebesgue’s dominance theorem, we have ϕ(zk ) → ϕ(z), a.e. , ηϕ(zk ) → ηϕ(z) in L1 (Ω) , ψ(wk ) ∗ ψ = ν, ψ in L∞ (Ω) . This implies that

? η(ϕ ψ)(zk , wk )dx = ηϕ(zk )ψ(wk )dx → η(x)ϕ(z(x)) νx , ψ dx . Ω

Ω

Ω

Then,

? ? ? ν, ϕ ψ in L∞ (Ω) . (ϕ ψ)(zk , wk ) ∗ δz > Since C0 (RN ) C0 (RM ) is dense in C0 (RN +M ), it follows that ? f (zk , wk ) ∗ δz ν, f in L∞ (Ω) , for all f ∈ C0 (RN +M ).

Example 4.4.12 Let ϕ be the function deﬁned in Corollary 4.3.3, and ϕk (x) = ϕ(kx), ∀k ∈ N . It is known that ϕk

∗

1

ϕ(x)dx = λα + (1 − λ)β .

0

Similarly, ∀f ∈ C(R1 ), we have f ◦ ϕk

∗

1

f (ϕ(x))dx = λf (α) + (1 − λ)f (β) .

0

Therefore the sequence {ϕk } is associated with a Young measure: ν = λδα + (1 − λ)δβ ; i.e., νx = λδ(x − α) + (1 − λ)δ(x − β), ∀x ∈ R1 .

4.4 Relaxation and Young Measure

257

Example 4.4.13 Let {un } be the sequence of sawtooth functions: x − nk if x ∈ [ nk , 2k+1 n ] un (x) = k+1 2k+1 k+1 −x + n if x ∈ [ n , n ] , which is a minimizing sequence of the functional J (see equation 4.32). Let 1 if x ∈ [ nk , 2k+1 n ], zn (x) = un (x) = −1 if x ∈ [ 2k+1 n ]. Let ν be the Young measure associated with to the sequence zk . Conclusion: ν = 12 (δ−1 + δ+1 ). In fact, g(zn (x))

∗

νx , g ,

for all g ∈ C0 (R ). Let us take g(y) = min{(y 2 − 1)2 , 1}. Since J(un ) → 0, νx , g = 0 a.e. This implies that supp(νx ) ⊂ {−1, 1}, i.e., νx = λ(x)δ−1 + (1 − λ(x))δ+1 . Also, zn ∗ 0 in L∞ (R1 ) and the relation g1 (zn (x)) ∗ νx , g1 in ∞ L (R1 ), holds for all g1 ∈ C0 (R1 ) with g1 (y) = y, as |y| < 2. Thus 1

λ(x)δ−1 + (1 − λ(x))δ+1 , g1 = 0 . This implies that 1 − 2λ(x) = 0, or λ(x) = 12 . Example 4.4.14 Let B, C ∈ M n×N satisfy rank(B − C) = 1. Assume ∃λ ∈ (0, 1), such that λB + (1 − λ)C => 0. It is known that without loss of generality, > one may assume B = (1 − λ)a e1 , and C = −λa e1 for some a ∈ RN and e1 ∈ Rn . Let (1 − λ)t if t ∈ [0, λ] , ϕ(t) = −λ(t − 1) if t ∈ [λ, 1] . ∀x = (x1 , . . . , xn ) ∈ D := [0, 1]n , let uk (x) = ak−1 ϕ(kx1 ), k = 1, 2, . . . , and zk = ∇uk . Then {zk } is associated with the Young measure ν = λδB + (1 − λ)δC , where δB and δC are the probability measures concentrated at the matrices B and C, respectively. In fact,

B if {kx1 } ∈ (0, λ) , zk (x) = C if {kx1 } ∈ (λ, 1) ,

so is dist(zk , {B, C}) = 0. By Theorem 4.4.6, the probability measure ν satisﬁes supp(νx ) ⊂ {B, C} a.e. Therefore, ∃µ(x) ∈ [0, 1] such that νx =

258

4 Minimization Methods

µ(x)δB + (1 − µ(x))δC , and ∀g ∈ C0 (M n×N ), g(zkj ) ∗ ν, g for some subsequence > {zkj }. In particular, we take g(y) = y in a ball centered at θ, containing a e1 . Again, from zk ∗ 0, we obtain µ(x)B + (1 − µ(x))C = 0, i.e., µ(x) = λ. Therefore, ν = λδB + (1 − λ)δC . A w∗ -measurable map ν : Ω → M(M n×N ) is called a gradient Young measure, if ∃{uj } ⊂ W 1,∞ (Ω, RN ) such that uj

∗

u in W 1,∞ (Ω, RN ), and δ∇uj

∗

ν.

A central problem for the gradient Young measure is the following: Given a set K ⊂ M n×N , how do we characterize all W 1,∞ gradient Young measures ν such that supp(νx ) ⊂ K for a.e. x ∈ Ω? Indeed, letting ν be the Young measure generated by {uj }, the range of {∇uj } must be contained in a big ball BR (θ) of M n×N . Then by taking K = B R (θ), and according to (3) of Theorem 4.4.6, it follows that: (1). There exists a compact set K ⊂ M n×N such that supp(νx ) ⊂ K, a.e. If we choose g(y) = y on BR (θ) and g ∈ C0 (M n×N ), then g(∇uj ) = ∇uj ∗ ∇u in L∞ (Ω, M n×N ). From the second conclusion of Theorem 4.4.6, we have g(∇uj ) ∗ ν, g in L∞ (Ω, M n×N ). Since g is arbitrary outside the ball BR (θ), we may simply write ν, g as ν, id . Thus, (2) νx , id = ∇u(x) a.e. Finally, assume that g : M n×N → R1 is continuous and quasi-convex, then by the Morrey theorem, for all open sets U ⊂ Ω, we have

g(∇uk (x))dx ≥ g(∇u(x))dx . lim inf k→∞

U

U

Note that g(∇uk (x)) ∗ νx , g , and g(∇u(x)) = g( νx , id ) ; we obtain

νx , g dx ≥ U

g( νx , id )dx . U

Since U ⊂ Ω is arbitrary, we arrive at (3) νx , g ≥ g( νx , id ) a.e. In fact, all these propositions together characterize the W 1,∞ gradient Young measure, i.e., the converse is also true. Namely, one has the following: Theorem 4.4.15 A w∗ -measurable map ν : Ω → M(M n×N ) is a gradient Young measure if and only if νx ≥ 0 a.e., ∃ a compact set K ⊂ M n×N and ∃u ∈ W 1,∞ (Ω, RN ) such that 1. supp(νx ) ⊂ K, a.e. 2. νx , id = ∇u(x), a.e. 3. νx , g ≥ g( νx , id ) a.e., ∀ continuous quasi-convex g: M n×N → R1 . The suﬃcient part of the proof can be found in [KP 1].

4.4 Relaxation and Young Measure

259

Remark 4.4.16 In the modeling of microstructure, the reason people use W 1,∞ sequences approaching a compact set K is based on a truncation lemma due to Kewei Zhang [Zh 1]: Let K be a compact set in M n×N , and let K∞ = 1,1 (Ω, RN ) sup{|A| | A ∈ K}. Let Ω ⊂ Rn be a bounded domain. If {uj } ⊂ Wloc satisﬁes dist (∇uj , K) → 0 in L1 (Ω), and uj → u in L1loc (Ω), then there ex1,1 ists a sequence {vj } ⊂ Wloc (Ω, RN ) such that |∇vj | ≤ cn,N K∞ , |{x ∈ Ω | uj (x) = vj (x)}| → 0, and vj = u near ∂Ω. As an application, we extend that Morrey theorem to the case where the integrand is a function depending on (x, u, ∇u). Corollary 4.4.17 Suppose that f : Ω × RN × M n×N → R1 is a nonnegative Caratheodory function, and that uj u in W 1,∞ (Ω, RN ) generates a gradient Young measure ν. Then

f (x, uj (x), ∇uj (x))dx ≥ f (x, u(x), ξ)dνx (ξ) dx lim inf j→∞

Ω

Ω

Mn×N

If further, ∀(x, u) ∈ Ω × RN , ξ → f (x, u, ξ) is quasi-convex, then

lim inf f (x, uj (x), ∇uj (x))dx ≥ f (x, u(x), ∇u(x))dx , j→∞

Ω

Ω

i.e., the functional J(u) = RN ).

Ω

f (x, u(x), ∇u(x))dx is s.w∗ .l.s.c. on W 1,∞ (Ω,

Proof. According to Corollary 4.4.13, the sequence {zj = (uj , ∇uj )} generates > the Young measure δu ν. Since the range of {zj } are contained in a big ball BR (θ) ⊂ RN × M n×N , we may assume that f = 0 outside BR (θ). Since f is a Caratheodory function, by the Scorza–Dragoni–Vainberg theorem (a version of the Luzin theorem with parameters, see also Vainberg [Va], there exists an increasing sequence of compact sets Ck ⊂ Ω such that m(Ω\Ck ) → 0, and f |Ck ×RN ×Mn×N is continuous. Let gk (x, ·) = χCk (x)f (x, ·) : Ω → C0 (RN × Mn×N ). We obtain

f (x, uj (x), ∇uj (x))dx ≥ gk (x, uj (x), ∇uj (x))dx Ω

Ω ? νx , gk (x, ·) dx, as j → ∞ . → δu(x) Ω

But,

RHS =

δu(x) Ck

?

= Ck

M n×N

νx , f (x, ·, ·) dx f (x, u(x), ξ)dνx (ξ) dx

260

4 Minimization Methods

→ Ω

f (x, u(x), ξ)dνx (ξ) dx, as k → ∞ .

M n×N

This proves the ﬁrst conclusion. If further, f is quasi-convex in ξ, then from the conclusions (2) and (3) of Theorem 4.4.15, we have

f (x, u(x), ξ)dνx (ξ) ≥ f (x, u(x), ∇u(x)), a.e. x ∈ Ω . Mn×N

The s.w∗ .l.s.c. of J is proved.

Remark 4.4.18 A gradient Young measure ν, if it is constant up to a null set, is called homogeneous. This is a useful notion in studying oscillations of sequences because it isolates the oscillation near a point in a domain. Examples 4.4.13 and 4.4.14 are homogeneous.

4.5 Other Function Spaces We have seen that the w∗ -compactness plays an important role in the calculus of variations. That is why we always choose Sobolev spaces W m,p , 1 < p ≤ ∞ as working spaces. The case p = 1 is diﬀerent, even if to the Lebesgue space L1 , the Banach–Alaoglu theorem cannot be applied. In fact, the positive deltatype sequence does not weakly * converge to any L1 function. In the extremal case, we study some other function spaces as replacements. In this section, the BV space, the Hardy space and some other related spaces are introduced in the applications to variational problems. 4.5.1 BV Space We study the minimal surface problem. Let Ω ⊂ Rn be a bounded open domain; a hypersurface u ∈ C 1 (Ω, R1 ) with prescribed boundary condition has the area:

(1 + |∇u|2 ) . A(u) = Ω

This leads to the following variational problem: Min{A(u)} in the above set of hypersurfaces. One might expect that the Sobolev space W 1,1 would be a working space for the direct method. Unfortunately, the closed ball in this space is not weakly closed, and so a minimizing sequence may fail to converge. We are introduced to the space of bounded variations, which is widely used in the so-called free discontinuous boundary problems, in which the boundary of the domain is to be determined in the extremum problems. Recall that a continuous linear functional deﬁned on the space of continuous functions on [0, 1] can be represented by a BV function u as follows:

4.5 Other Function Spaces

1

φ(t)du(t), ∀φ ∈ C([0, 1]) .

L(φ) =

261

(4.46)

0

If φ ∈ C01 ([0, 1]), then

L(φ) = −

1

u(t)φ (t)dt .

(4.47)

0

We shall extend this relation to higher dimensional space. Let Ω ⊂ Rn be an open domain, and let C0 (Ω) be the space of continuous functions of compact support with respect to Ω. First, we note that L is a linear continuous functional on C0 (Ω) if and only if there exists a Radon measure µ and a µ-measurable function ν on Ω with |ν(x)| = 1, a.e., such that

φ(x)ν(x)dµ(x), φ ∈ C0 (Ω) . L(φ) = Ω

Second, for each compact subset K ⊂ Ω, let LK = L|C(K) , then LK := sup{L(f ) | f C(K) ≤ 1} = µ(K) . Motivated by (4.46), and (4.47), one deﬁnes the counterpart of a BV function u in higher dimensional spaces, as being a function u ∈ L1 (Ω) such that the functional

u divφ L(φ) := Ω

C01 (Ω, Rn ),

and can be extended continuously to is deﬁned for all φ ∈ C0 (Ω, Rn ). Then by Riesz representation theorem, there exist Radon measures ρi , and ρi measurable functions τi (x) on Ω, with |τi (x)| ≤ 1, ρi − a.e., i i = 1, . . . , n, and let µ = Σni=1 ρi , νi = τi dρ dµ , i = 1, . . . , n. such that

n u divφ = − Σ1 φi τi dρi = − (φ · ν)dµ , (4.48) Ω

Ω

Ω

where ν(x) = (ν1 (x), . . . , νn (x)) with ν(x)∞ ≤ 1 µ− a.e. We deﬁne · := · Rn . Deﬁnition 4.5.1 Let Ω ⊂ Rn be open. A function u ∈ L1 (Ω) is said to be of bounded variations, if the total variation of u on Ω is:

u divφ | φ ∈ C01 (Ω, Rn ), φ(x) ≤ 1, for a.e. x ∈ Ω < ∞ . Du(Ω) := sup Ω

The space BV (Ω) consists of all bounded variation functions on Ω with norm: uBV = uL1 + Du(Ω) .

262

4 Minimization Methods

Since .Rn and .∞ are equivalent, for u ∈ BV (Ω), the total variation Du can be regarded as a measure µ: √ 1 √ µ(Ω) ≤ Du(Ω) ≤ nµ(Ω) . n

(4.49)

Moreover, the distributional derivative Du makes sense, it is related to the Radon measure µ and vector measurable function ν(x). Example 4.5.2 If u ∈ W 1,1 (Ω), then u ∈ BV (Ω), and uBV = uW 1,1 . In fact, we shall verify that Du(Ω) = Ω |∇u|dx. On one hand,

u divφ dx| = | ∇u · φ dx| ≤ |∇u|dx | Ω

Ω

Ω

with φ(x) ≤ 1 ∀ x ∈ Ω, it follows that Du(Ω) ≤ for all φ ∈ |∇u|dx. Ω On the other hand, ∀u ∈ W 1,1 (Ω), ∀ > 0, ∃φ ∈ C01 (Ω, Rn ) with φ (x) ≤ 1, ∀x ∈ Ω satisfying:

|∇u| dx ≤ ∇u · φ dx + = u · divφ dx + ≤ Du(Ω) + . C01 (Ω, Rn )

Ω

Ω

Since > 0 is arbitrary, we obtain

Ω

Ω

|∇u|dx ≤ Du(Ω).

Example 4.5.3 Let S ⊂ Rn be a C ∞ compact (n − 1)-dimensional hypersurface with the induced metric, and let Hn−1 be the (n−1)-dimensional Hausdorﬀ measure in Rn . The area of S is Hn−1 (S). Let Ω be the body bounded by S, and let χΩ be the characteristic function of Ω. Then χΩ ∈ BV (Rn ), and DχΩ (Rn ) = Hn−1 (S) . Indeed, by the Gauss formula, ∀φ ∈ C01 (Rn , Rn )

χΩ divφdx = divφdx = n(x) · φ(x)dHn−1 , Rn

Ω

S

where n(x) is the unit exterior normal. Thus DχΩ (Rn ) ≤ Hn−1 (S) . On the other hand, one extends n to be a C ∞ vector ﬁeld V over Rn with V (x) ≤ 1 ∀ x ∈ Rn . This can be done by a partition of unity. Then, ∀ρ ∈ C0∞ (Rn , R1 ) with |ρ(x)| ≤ 1, ∀x ∈ Rn , let φ = ρV , we have

4.5 Other Function Spaces

263

ρdHn−1 .

χΩ divφ dx = Rn

S

Thus, DχΩ (Rn )

= sup χΩ divφdx | φ ∈ C0∞ (Rn , Rn ), with φ(x) ≤ 1, ∀x ∈ Rn n R ≥ sup ρdHn−1 | ρ ∈ C0∞ (Rn , R1 ), |ρ(x)| ≤ 1, ∀x ∈ Rn S

=H

n−1

(S) .

These two examples show that W 1,1 (Ω) is strictly contained in BV (Ω), since for n = 1, the characteristic function χ[0,1] ∈ BV (R1 ) but not in W 1,1 . This leads us to extend the deﬁnition of the co-dimensional one area to the boundary of more general domains. Deﬁnition 4.5.4 Let E be a Borel set in an open domain Ω ⊂ Rn . We call ∂E(Ω) = DχE (Ω) the perimeter of E in Ω. As a function space with the norm uBV = uL1 + Du(Ω), BV (Ω) is a Banach space. Only the completeness remains to be veriﬁed. Lemma 4.5.5 (Lower semi-continuity) If {uj } ⊂ BV (Ω), and uj → u in L1 ; then for every open U ⊂ Ω Du(U ) ≤ lim inf Duj (U ) . j→∞

(4.50)

If further, sup{Duj (Ω) | j ∈ N } < ∞, then u ∈ BV (Ω). Proof. ∀φ ∈ C01 (U, Rn ) with φ(x) ≤ 1, one has

u divφ dx = lim uj divφ dx ≤ lim inf Duj (U ) . U

j→∞

U

j→∞

(4.50) is proved. Theorem 4.5.6 BV (Ω) is complete.

Proof. For a Cauchy sequence {uj } in the BV norm, it is obvious that uj → u in L1 . By the previous lemma, Du(Ω) < ∞, and then u ∈ BV (Ω). It remains to show that D(uj − u)(Ω) → 0. Again, from the lower semi-continuity lemma, ∀ > 0, ∃j0 ∈ N such that D(uj − u)(Ω) ≤ lim inf D(uj − uk )(Ω) < , as j ≥ j0 . k→∞

264

4 Minimization Methods

Now we consider the possibility of C ∞ approximation of the BV functions. Since the W 1,1 norm equals the BV norm for C 1 functions, and C ∞ is dense in W 1,1 , we can only have: Theorem 4.5.7 Let Ω be an open domain of Rn . Then for ∀u ∈ BV (Ω), ∃uj ∈ BV (Ω) ∩ C ∞ (Ω) such that 1. uj → u in L1 (Ω), 2. Duj → Du in the sense of Radon measure. In particular, Duj (Ω) → Du(Ω). We omit the proof, but refer to Giusti [Gi], p. 14. Theorem 4.5.8 (Compactness) Let Ω be a bounded open domain of Rn . Any sequence {uj } ⊂ BV (Ω) with uj BV ≤ M < ∞ possesses a convergent subsequence in the L1 norm, and the limit u ∈ BV (Ω), with uBV ≤ M . Proof. We take a sequence {vj } ⊂ BV (Ω) ∩ C ∞ (Ω) such that uj − vj L1 < 1 1 1,1 = j , and Dvj (Ω) ≤ M + j . From Example 4.5.2, we have vj W Dvj BV . According to the Rellich–Kondrachev compactness theorem, there is a subsequence vjk L1 -converges to u. From the lower semi-continuity lemma, Du(Ω) ≤ M , and u ∈ BV (Ω). Obviously, ujk → u in L1 . Besides the above preparations, in order to study the minimal surface problem stated at the beginning of the subsection we have to deﬁne the trace of BV functions. The following theorem can be found in Giusti [Gi] and Evans and Gariepy [EG]: Theorem 4.5.9 (Trace) Let Ω ⊂ Rn be a bounded domain with Lipschitzian boundary ∂Ω. Then the trace operator T : BV (Ω) → L1 (∂Ω, Hn−1 ) is bounded, and we have

u div φ = − (φ · ν)dµ + T u(φ · n)dHn−1 , ∀φ ∈ C01 (Ω, Rn ), Ω

Ω

∂Ω

where µ is the Radon measure, ν is the µ-measurable vector function with ν(x) ≤ 1, a.e. with respect to µ, and n is the unit normal vector ﬁeld over ∂Ω, which is well deﬁned almost everywhere. We omit the proof and turn to the existence of the minimal surface problem. Theorem 4.5.10 (Nonparametric minimal surface) Let Ω ⊂ Rn be a bounded Lipschitzian domain, and let u0 ∈ BV (Ω). Then there exists a minimizer of the problem: Min{A(u) | u ∈ BV (Ω), T u = T u0 }, where A(u) =

Ω

1 + |Du|2

4.5 Other Function Spaces

265

(φ0 + udivφ)dx | (φ0 , φ) ∈ C01 (Ω, R1+n ) with |φ0 (x)| + φ(x) ≤ 1 ∀x ∈ Ω .

:= sup

Ω

Proof. One considers the set: XM := {u ∈ BV (Ω) | T u = T u0 , and Du(Ω) ≤ M }, for any M > 0. For suﬃciently large M > 0, XM = ∅ since we have Du(Ω) ≤ A(u) ≤ Du(Ω) + mes(Ω) . Combining with Theorems 4.5.7 and 4.5.8, XM is compact in L1 -topology. By the same proof as for Lemma 4.5.5, A is l.s.c. on XM in L1 -topology. The proof follows from the general principle of the direct method, i.e., Theorem 4.2.1. The problem stated above is called the nonparametric minimal surface problem, there is another formulation called parametric minimal surface problem: Again, let Ω be a bounded open domain in Rn , and let L be a set of ﬁnite perimeters. Deﬁne F = {F is Borel measurable | F \Ω = L\Ω} . Theorem 4.5.11 (De Giorgi) The problem Min{DχF (Rn ) | F ∈ F} has a solution. Proof. Since Ω is bounded, ∃R > 0 such that Ω ⊂ BR (θ). If F \Ω = L\Ω, then F \BR (θ) = L\BR (θ). It follows that DχF (Rn ) = DχF (BR (θ)) + DχL (Rn \BR (θ)) . Thus the problem is reduced to Min{DχF (BR (θ)) | F ∈ F} . The functional J(u) := Du(BR (θ)) is bounded from below on the set X := {u = χF | Du(BR (θ)) ≤ ∂χL (BR (θ))}. We shall verify that X is a compact subset under L1 -topology. According to Theorem 4.5.8, it remains to verify that any limit point u of X is a characteristic function of a measurable set of ﬁnite perimeter. In fact, it is the a.e. limit of a sequence of X. Since J is l.s.c. on X under L1 -topology, there exists a minimizer u. Although we have proved the existence of minimal surfaces, the solutions we obtained are very weak. Thus, in order to verify that the weak solutions are geometric minimal surfaces, the diﬃculties lie in the regularity, cf E. Giusti [EG] and De Giorgi [DG 1].

266

4 Minimization Methods

4.5.2 Hardy Space and BMO Space It is known that a bounded sequence in L1 space may not have a w∗ -convergent subsequence. Therefore L1 is not a good space for the variational method on which we work. Comparing with the Lp space p ∈ (1, ∞), the latter has many important properties in analysis: the uniform convexity, the reﬂexivity, the smoothness of the norm, the boundedness of singular integral operators in Calderon–Zygmund theory, as well as the Lp estimates in the elliptic theory, which L1 space does not have. The Hardy space H1 is in some sense a replacement of L1 . It is deﬁned as follows: Deﬁnition 4.5.12 (Hardy space) H1 (RN ) = {f ∈ L1 (RN ) | Rj f ∈ L1 (RN ), j = 1, 2, . . . , N } . 1

where Rj = ∂j (−∆)− 2 is the Riesz transform, j = 1, 2, . . . , N . The norm is deﬁned to be N f H1 = f L1 + Rj f L1 . 1

There are many equivalent characterizations of the Hardy space. The fol lowing is one of them. ∀h ∈ C0∞ (RN ), with h ≥ 0, and h = 1. ∀t > 0 let ht (x) = t1N h( xt ). For every distribution f , one deﬁnes f ∗ (x) = sup |(ht ∗ f )(x)| . t>0

If f ∗ ∈ L1 , then from the Lebesgue dominance theorem, we have f ∈ L1 and f L1 ≤ f ∗ L1 . By deﬁnition f ∗ depends on h, but it is proved that f ∈ H1 if and only if f ∗ ∈ L1 for any such h, and f = f ∗ L1 is an equivalent norm. H1 is a Banach space. In order to study the H1 space, we need some knowledge on harmonic analysis, in particular, the notion of Hardy-Littlewood maximal functions: ∀f ∈ L1loc , the function

1 |f (y)|dy M (f )(x) = sup r>0 |Br (x)| Br (x) is called the maximal function of f . It possesses the following properties: 1. M (cf ) = |c|M (f ), ∀c ∈ R1 , and M (f1 + f2 ) ≤ M (f1 ) + M (f2 ), ∀f1 , f2 ∈ L1loc . 2. |f (x)| ≤ M (f )(x) a.e. 3. If M (f ) ∈ L1 (Rn ), then f = 0 a.e.

4.5 Other Function Spaces

267

Proof. If |f | > 0 on a positive measure set E, then we may assume that ∃R > 0, ∃ > 0 such that E ⊂ BR (θ), |f (x)| ≥ , ∀x ∈ E. Thus, M (f )(x) ≥ CN |E|(|x| + R)−N , where CN is a constant depending on N only. It contradicts M (f ) ∈ L1 (RN ). 4. The maximal function operator is of weak (1, 1) type, i.e., there exists a constant C > 0 such that |Eλ | ≤

C f L1 ∀λ > 0, f ∈ L1 (RN ) , λ

where Eλ = {x ∈ RN | M (f )(x) > λ}. Proof. For ∀x ∈ Eλ , ∃rx > 0 such that

|f |dx ≥ λ|Brx (x)| . Brx (x)

Thus, {Brx (x) | x ∈ Eλ } is a covering of Eλ . According to the Vitali covering theorem, there exist a family of countable disjoint balls {Brxj (xj )} and a constant cN such that

|Eλ | ≤ cN |Brxj (xj )| ≤ cN λ−1 |f |dx ≤ cN λ−1 f L1 . j

j

Brx (xj ) j

5. For p ∈ (1, ∞), if f ∈ Lp (RN ), then M (f ) ∈ Lp (RN ) and there exists a constant Cp > 0 such that M (f )Lp ≤ Cp f Lp . Proof. Since M is of (∞, ∞) type, in combination with property 4, the conclusion follows from the interpolation theory. 6. (Kolmogorov’s inequality) For any δ ∈ (0, 1), there is a constant C = Cδ > 0 such that

1 C δ |E|1−δ f δL1 , ∀ measurable E ⊂ RN . M (f )δ dx ≤ 1−δ E Proof. In fact from property 4,

∞

M (f )δ dx = δ |{x | M (f )(x) > λ} ∩ E|λδ−1 dλ E 0 t ∞ + |{x | M (f )(x) > λ} ∩ E|λδ−1 dλ =δ 0

t

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4 Minimization Methods

≤ |E|t + Cδ δ

∞

λδ−2 dλf L1

t

≤ |E|tδ +

Cδ δ−1 t f L1 /; , 1−δ

where C is the weak (1, 1) norm of M (f ). Setting t = desired inequality.

C f L1 |E|

, we obtain the

We investigate the behavior of H1 functions. 1. There exists a constant C1 > 0 such that f ∗ ≤ C1 M (f ). If f ≥ 0, then M (f ) ≤ C2 f ∗ for some constant C2 . Thus if f ≥ 0 and f ∈ H1 , then f = 0. 2. If f ∈ H1 , then RN f dx = 0. Proof. Let φ ∈ C ∞ (RN ) with suppφ ⊂ B1 (θ) and φ(θ) = 1. ∀x ∈ RN , ∀s ≥ |x|, one deﬁnes h(y) = cφ(s−1 x − 2y), where c is the normalized constant such that RN h(y)dy = 1. Then f ∗ (x) = sup |(ht ∗ f )(x)| t>0

x−y ≥ (2s)−N | h f (y)dy| 2s RN

6y7 φ = (2s)−N c| f (y)dy| , s RN since the last integral tends to RN f (y)dy as s → ∞. If RN f (y)dy = 0, then we would have lim inf |x|→∞ f ∗ (x)|x|N > 0, which contradicts f ∗ ∈ L1 (RN ). 3. If f ∈ L∞ (RN ) with compact suppf and RN f (x)dx = 0, then f ∈ H1 . Proof. Assume suppf ⊂ BR (θ). Since

[ht (x − y) − ht (θ)]f (y)dy |(ht ∗ f )(x)| = N R 1 = (x − y) · ∇ht (s(x − y))dsf (y)dy RN 0

≤C t−(N +1) dyf ∞ , |x−y|≤t, |y|≤R

therefore f ∗ (x) ≤ CR|x|−N −1 f ∞ . The conclusion follows. 1

The dual space of H is the BMO space, which is deﬁned as follows: ∀f ∈ L1loc (RN ), ∀x ∈ RN , ∀r > 0, let

1 f r (x) = f (y)dy , |Br (x)| Br (x) and deﬁne

4.5 Other Function Spaces

269

Deﬁnition 4.5.13 (BMO space) f BMO = sup sup x∈RN r>0

1 |Br (x)|

|f (y) − f r (x)|dy . Br (x)

BMO(RN ) is the space of functions such that f BMO < ∞. The quantity f BMO is a semi-norm; in fact, f BMO = 0 if and only if f = constant. After modulo constants, BMO is a Banach space. There is an equivalent BMO semi-norm:

1 |f (y) − c|dy . f = sup sup inf x∈RN r>0 c |Br (x)| Br (x) In fact, on one hand it is obvious f ≤ f BMO , and on the other hand, fBr (x) |f (y) − f r (x)|dy ≤ fBr (x) |f (y) − c|dy + |f r (x) − c| ≤ 2f , where f denotes the average. The following simple properties are easily seen from the deﬁnition. 1. L∞ ⊂ BMO ⊂ L1loc . 2. The function log|x| ∈ BMO. In fact, the scaling transformation: ∀λ > 0, f (x) → f (λx) maps BMO functions to BMO functions, and preserves the semi-norm. Under the scaling, log|x| is changed by adding a constant. The veriﬁcation is then reduced to

|log|x||dx ≤ C, or |log|x| − log |x0 ||dx ≤ C , B

B

where B is unit ball centered at x0 , The ﬁrst inequality is true for |x0 | ≤ 1 and the second for |x0 | ≥ 1. These hold by elementary calculation. 3. If f ∈ BMO,then |f | ∈ BMO, and then f± ∈ BMO. 1 denote the average; one has Let fΩ = |Ω| Ω fBr (x) ||f (y)| − |c||dy ≤ fBr (x) |f (y) − c|dy , which implies that |f | ≤ f . In the case where (x, r) is not speciﬁed, we deﬁne Q = Br (x) and mQ (f ) = f r (x). 4. If f ∈ L1loc (RN ) and if there exists C > 0 such that mQ (f ) − essinfQ f ≤ C ∀ Q, then f ∈ BMO and f BMO ≤ 2C. Proof. Since Q [f (y) − mQ (f )]dy = 0, fQ |f (y) − mQ (f )|dy = 2fQ max[mQ (f ) − f (y), 0]dy ≤ 2fQ [mQ (f ) − essinfQ f ]dy ≤ 2C .

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4 Minimization Methods

5. Let f ∈ L1loc (RN ) with f ≥ 0. If there exists a constant C > 0 such that mQ (f ) ≤ CessinfQ f, ∀ Q, then log f ∈ BMO. Proof. Taking logarithms and using Jessen’s inequality, we have mQ (log f ) = fQ log f dy ≤ log fQ f dy = log mQ (f ) ≤ log C + essinfQ log f .

The conclusion follows from property 4.

6. Assume f ∈ L1loc (RN ) and that M (f )(x) = ∞ a.e. ∀ δ ∈ (0, 1), set w = M (f )δ . Then there exists a constant C = Cδ > 0 independent of Q, such that mQ (w) ≤ CinfQ w, ∀ Q. Proof. Fixing Q, we may assume Q = BR (θ) without loss of generality. Decompose f = f1 + f2 , where f1 = χQ · f with Q = 2Q. Let wi = M (fi )δ , i = 1, 2. Since M (f ) ≤ M (f1 ) + M (f2 ), there is a constant c depending on δ, such that w ≤ c(w1 + w2 ). Now we apply Kolmogorov’s inequality to w1 : mQ (w1 ) = fQ M (f1 )δ (y)dy ≤

Cδ |Q|1−δ f1 δL1 |Q|

= 2N Cδ mQ (f )δ ≤ C1 infQ w ≤ C1 infQ w where C1 is independent of Q. Next we turn to estimating w2 . It remains to verify that there exists a constant C2 > 0 such that M (f2 )(x) ≤ C2 M (f2 )(y), ∀ x, y ∈ Q . Indeed, it implies that w2 (x) ≤ C2 infQ w2 ≤ C2 infQ w ∀x ∈ Q, then mQ (w2 ) ≤ C2 infQ w. Combining these two estimates together, we obtain mQ (w) ≤ c(mQ (w1 ) + mQ (w2 )) ≤ CinfQ w . Now we return to estimate M (f 2 ). Notice that for any r > 0, only when Br (x) ∩ (Q )c = Ø, i.e., r ≥ R, Br (x) |f2 (z)|dz can be nonzero. From fBr (x) |f2 (z)|dz ≤

|Br+R (y)| f |f2 (z)|dz ≤ 2N M (f2 )(y) , Br (x) Br+R (y)

it follows that M (f2 )(x) ≤ 2N M (f2 )(y).

Combining properties 5 and 6, we obtain: 7. If f ∈ L1loc (RN ) and M (f )(x) = ∞ a.e., then log M (f ) ∈ BMO and log M (f )BMO ≤ cN , a constant depending on N only.

4.5 Other Function Spaces

271

Deﬁnition 4.5.14 (VMO space) The space VMO(RN ) is the closure of C0∞ (RN ) in BMO. It is proved (see Stein [Ste 2]): that (VMO)∗ = H1 , (H1 )∗ = BMO, in the following sense, ∀f ∈ BMO, the following integral makes sense:

1 ∞ g=0 l(g) = f · gdx, ∀g ∈ Hα := g ∈ L | supp (g) is compact, and (see property 3 of H1 functions), and has a unique bounded extension to H1 : |l(g)| ≤ Cf BMO gH1 for some constant C > 0. Conversely, ∀l ∈ (H1 )∗ , ∃f ∈ BMO, such that l(g) = f · g ∀g ∈ H1α , with f BMO ≤ Cl(H1 )∗ . Since H1 is the dual of a separable Banach space VMO, the weak-* topology on H1 is well deﬁned, and then the Banach–Alaoglu theorem is applicable. Namely: If {fj } is a bounded sequence in H1 , then there is a subsequence {fj }, which converges in the distribution sense to f ∈ H1 , with f H1 ≤ lim inf j→∞ fj H1 . 4.5.3 Compensation Compactness If we intend to use the space H1 in variational problems, then the prices we have to pay are: (1) To verify the H1 boundedness of nonlinear quantities appearing in the integrand of the functional along a minimizing sequence, and (2) such nonlinear quantities are sequentially weakly ∗ l.s.c. There are few examples of typical nonlinear quantities, which are in the space L1 with some additional compensation conditions so that they fall into H1 and preserve the w∗ -continuity. Example 1. (div-curl) Suppose {En } ⊂ Lp (RN , RN ), divEn = 0 in the distri bution sense, and {Bn } ⊂ Lp (RN , RN ), curlBn = 0 in the distribution sense, where p ∈ (1, ∞), p1 + p1 = 1. Assume that both {En }, {Bn } are bounded in their own spaces respectively, then En · Bn is bounded in L1 (RN , R1 ). The compensation conditions divEn = 0, curlBn = 0 play a role in the weak*convergence. Example 2. (Jacobian) Suppose u ∈ Lqloc (RN , RN ), for some q ∈ (1, ∞), in which ∇u ∈ LN (RN , RN ×N ). Then det(∇u) ∈ L1 (RN , R1 ). The special structure of the Jacobian is also a kind of compensation condition. First we verify that the compensation condition implises the nonlinear quantities are indeed in H 1 , and then these nonlinear quantities are w∗ continuous along weakly convergent sequences.

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4 Minimization Methods

Theorem 4.5.15 Suppose that E ∈ Lp (RN , RN ), B ∈ Lp (RN , RN ), with p ∈ (1, ∞), p1 + p1 = 1. If curl B = 0 and divE = 0, then E · B ∈ H1 and ∃C > 0 a constant such that E · BH1 ≤ CEp Bp . Proof. Taking α, β satisfying α1 + β1 = 1 + nonnegative, with h = 0, we claim that

1 N,α

∈ (1, p), β ∈ (1, p ), ∀h ∈ C0∞ 1

1

|{ht ∗ (E · B)}(x)| ≤ C(fBt (x) |E|α ) α (fBt (x) |B|β ) β , 1 denotes the average. where fB = |B| B

Indeed, From curl B = 0, we have ω ∈ W 1,p (RN , R1 ) such that ∇ω = B. divE = 0 yields div(ωE) = ωdivE + E · ∇ω = E · B . By suitably choosing h such that supp(h) ⊂ B1 (θ), we have

1 x−y |ht ∗ (E · B)(x)| = ∇h E(y)ω(y)dy N +1 t t

1 x − y ∇h = E(y){ω − fBt (x) ω}dy N +1 t t α 1 α 1 1 α α ≤ C(fBt (x) |E| ) fBt (x) |ω − fBt (x) ω| t 1

1

≤ C(fBt (x) |E|α ) α (fBt (x) |∇ω|β ) β from the Sobolev–Poincar´e inequality. With the aid of the maximal function of f ∈ L1loc (Rn ):

1 |f (y)|dy . M (f )(x) = sup r>0 m(Br (x)) Br (x) The above inequality can be rewritten as 1

1

sup ht ∗ (E · B)(x) ≤ C(M (|E|α )(x)) α (M (|B|β )(x)) β . t>0

Again by Holder inequality and the Lp boundedness of the maximal function, we have

1 E · BH = sup |ht ∗ (E · B)(x)|dx t>0

≤C

α

p α

(M (|E| )) dx

p1

pβ 1 dx) p (M (|B| ) β

≤ CEp Bp .

4.5 Other Function Spaces

273

The second example can be reduced to the ﬁrst one, since ∂i u1 A1i = ∇u1 · σ , det(∇u) = i 1

2

where u = (u , u , . . . , u ), and σ = (A11 , A12 , . . . , A1N ) is the N -vector consisting of all N − 1 minors of det(∇u) with respect to the ﬁrst row. Deﬁne N E = σ and B = ∇u1 , then E ∈ L N −1 (RN , RN ), satisfying divE = 0, and B ∈ LN (RN , RN ), satisfying curlB = 0. According to Theorem 4.5.15, det(∇u) ∈ H1 and there exists a constant C > 0 such that det(∇u)H1 ≤ C∇uN LN . N

Apart from the w∗ compactness, compensation compactness theory identiﬁes classes of such nonlinear quantities, which are sequentially weakly ∗ continuous. Generally speaking, weak convergence may not preserve nonlinear quantities, e.g., if un u, vn v, then we cannot conclude that un vn uv. This can be seen from the example un = vn = sin(nπx) in L2 [0, 1]. Both un and vn 0, but un vn = 12 (1 − cos(2nπx)) 12 . However, luckily we have: Theorem 4.5.16 Suppose that {Ej }, {Bj } are two bounded sequences in L2 (RN ) satisfying div(Ej ) = 0 and curl(Bj ) = 0 in the distribution sense. If further, Ej E, Bj B in L2 (RN ), then Ej · Bj E · B in the distribution sense, and then Ej · Bj ∗ E · B in H1 . Proof. If B is curl free, we claim that there exists a distribution w such that ˆ ∇w = B. This can be shown by Fourier transformations. Let B(ξ) be the ˆk −ξk B ˆi = 0, ∀ i, k = Fourier transform of B, since curlB = 0 implies that ξi B N ˆ ξ B 1, 2, . . . , N . Let w be the Fourier inverse transform of 1|ξ|2k k . Obviously, ∇w = B. Moreover, if B has a bounded support: suppB ⊂ BR (θ)◦ , then we have w ∈ L2 and suppw ⊂ BR (θ)◦ . Now let wj be the distribution such that Bj = ∇wj . For a given φ ∈ C0∞ (RN ), we want to prove that

Ej · Bj φdx → E · Bφdx as j → ∞ . RN

RN

In this case, we may assume all the supports supp wj ⊂ BR (θ)◦ j = 1, 2, . . . , for some R > 0. Thus {wj } ⊂ H 1 (BR (θ)) and is bounded. Modulo a subsequence, we may assume wj → w in L2 (BR (θ)). Thus Ej · Bj , φ = div(wj Ej ) , φ

= − Ej , wj (∇φ)

→ − E , w(∇φ)

= div(wE) , φ = E · B , φ

where in taking limits, we used Ej E and wj → w in L2 (BR (θ)). Since C0∞ is dense in VMO space, Ej · Bj ∗ E · B in H1 .

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4 Minimization Methods

Remark 4.5.17 The above theorem has the following generalization: Let Ω ⊂ RN be an open domain. Assume that {Ej }, {Bj } are two bounded sequences in L2 (Ω, RN ) such that 1. {div Ej } lies in a compact subset of W −1,2 (Ω), 2. {curl Bj } lies in a compact subset of W −1,2 (Ω, MN ×N ). If further, Ej E and Bj B in L2 (Ω, RN ), then Ej · Bj E · B in the distribution sense. Similarly, one has: Theorem 4.5.18 Assume that {uj } is bounded in W 1,N (RN , RN ), and uj u. Then det(∇uj ) det(∇u) in the distribution sense, so is div(∇uj ) ∗ div(∇u) in H1 (Rn ). Proof. Again we use the previous special decomposition: ∂k u1 A1k = ∇u1 · σ = ∇(u1 · σ) , det(∇u) = k

where σ = (A11 , A12 , . . . , A1N ). Since divσ = 0, for ∀φ ∈ C0∞ (RN ), one has det(∇uj ) , φ = − u1j · σj , ∇φ → − u1 · σ , ∇φ = det(∇u) , φ . One can deﬁne a Hardy space on a domain Ω in RN . A distribution f on Ω is said to be in H1loc (Ω) if for each compact set K ⊂ Ω, there is an > 0 so that

( sup |ht ∗ f (x)|)dx < ∞ , K 0 0, we deﬁne gR (y) = y if 0 ≤ y ≤ R, = 2R − y if R ≤ y ≤ 2R, = 0 if y > 2R or y < 0. Obviously, gR ∈ C0 (R1 ), and |yχ[0,R] (y)| ≤ gR (y) ≤ |y|. According to Remark 4.4.10, we have

uj dx ≤ lim sup gR (y)dνj (x, y) lim sup {uj ≤R}

j→∞

Ω×R1

j→∞

= Ω

R1

Ω

R1

gR (y)dνx (y)dx

≤

|y|dνx (y)dx < ∞ .

We choose sequences nj ≥ 2j , kj ≥ j such that

lim sup vj dx ≤ |y|dνx (y)dx, j→∞

{vj ≤nj }

Ω

where vj = ukj . ∀k ∈ N , let Ek =

+

j≥k {vj

(4.51)

R1

> nj }. We shall verify:

1. |Ek | → 0 as k → ∞. 2. For ∀k ∈ N , {vj χFk } is uniformly absolutely continuous in j, where Fk = Ω \ Ek . By deﬁnition we have the estimates: |Ek | ≤

|{vj > nj }| ≤ sup vj L1 j

j≥k

Thus (1) is veriﬁed. We claim that

1 1 ≤ k−1 sup vj L1 . nj 2 j j≥k

lim sup lim sup R→∞

j→∞

Fk ∩{vj >R}

vj dx = 0 .

(4.52)

If it is proved, then for ∀ > 0, let R0 > 0, j0 > 0 be such that Fk ∩{vj >R0 } vj < N measurable set U with |U | < 2R 0 , we have 2 as j > j0 . For any L

U ∩Fk

vj dx ≤

Fk ∩{vj >R0 }

vj dx +

U ∩Fk ∩{vj ≤R0 }

vj dx ≤

+ R0 |U | < , 2

for ∀j ≥ j0 , i.e., vj is uniformly absolutely continuous on Fk ∀k.

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4 Minimization Methods

Now we return to proving (4.52). Since for ∀ Borel sets E ∈ B(Ω), ∀φ ∈ C0 (R1 ),

φ(vj (x))dx = φ(y)dνx (y)dx , lim j→∞

E∩{vj ≤nj }

E

R1

Choosing φj ↑ |y|, it follows that

vj (x)dx ≥ lim inf j→∞

E∩{vj ≤nj }

E

R1

|y|dνx (y)dx ,

(4.53)

because for j ≥ k, Fk ⊂ {vj ≤ nj }, Fk ∪ ({vj ≤ nj } ∩ Ek ) = {vj ≤ nj }. Combining (4.51) and (4.53) with E = Ek , we have

lim sup vj (x)dx ≤ |y|dνx (y)dx . (4.54) j→∞

Fk

Fk

R1

Thus,

lim sup lim sup R→∞

Fk ∩{vj >R}

j→∞

−

= lim sup lim sup R→∞

vj (x)dx

j→∞

Fk

Fk ∩{vj ≤R}

vj (x)dx

|y|dνx (y)dx − ≤ lim sup lim sup j→∞ R→∞ Fk R1 Fk

= lim sup (|y| − gR (y))dνx (y)dx = 0 . R→∞

Fk

gR (vj (x) dx

R1

Deﬁnition 4.5.20 Let {uj } ⊂ L1 (Ω) be bounded; we say that uj converges to u ∈ L1 (Ω) in the sense of the bitting lemma, if for all > 0 there exists a measurable subset E ⊂ Ω such that |E| < and uj u in L1 (Ω\E). What is the relationship between H1 −w∗ convergence and the convergence in the sense of the bitting lemma? Lemma 4.5.21 Let {uj } ∗ u in H1 (RN ). For each R > 0, let v ∈ L1 (BR (θ)) be such that there exists a subsequence nj for which unj converges to v in the sense of the bitting lemma in BR (θ), then u = v a.e. on BR (θ). Proof. We want to show that for ∀φ ∈ C0∞ (RN ), uφdx = vφdx. Assume suppφ ⊂ BR (θ) for suitable R > 0, vj = unj ∀j. By the assumption, for all > 0 there exists E such that |E| < and vj v on L1 (Ω\E). For all λ > 0 deﬁne wλ = (1 + λ log M (χE ))+ , where M (f ) is the maximal function of f . Provided by the propositions (3) and (7) of BMO, wλ ∈ BMO, and log M (χE ) ≤ cN , and then wλ BMO ≤ Cλ, where C is a constant independent of λ and .

4.5 Other Function Spaces

277

Since χE ≤ M (χE ) ≤ 1 a.e., it follows χE ≤ wλ ≤ 1 a.e. According to the proposition (4) of the maximal function, we have 1 1 1 1 |{wλ > 0}| = M (χE ) > exp − ≤ C exp λ χE L ≤ Cε exp λ , λ and then

unj φdx = RN

vφdx

RN

+ RN

unj φwλ dx +

RN

(unj − v)φ(1 − wλ )dx −

RN

vφwλ dx

= I1 + I2 + I3 + I4 , where

I1 =

vφdx, RN

I2 ≤ unj H1 wλ φBMO ≤ Cwλ φBMO ,

(unj − v)(1 − wλ )φdx → 0, as j → ∞, I3 = Ω\E

I4 ≤ |v||φ|dx ≤ φL∞ |v| . {wλ >0}

BR (θ)∩{wλ >0}

The limit of I3 is due to the assumption that uj converges weakly to v in L1 (Ω\E). We turn to estimate I2 . ∀r > 0, ∀x ∈ RN , let Q = Br (x), we have

1 fQ dz |φ(z) − φ(y)|dy fQ |φwλ − fQ φwλ |dz ≤ φL∞ wλ BMO + |Q| {wλ >0} C(λ + R|{wλ > 0}|) if r < R , ≤ C(λ + R1N |{wλ > 0}|) if r ≥ R . ∀η > 0 ﬁxing λ > 0 such that Cλ < η2 at ﬁrst, and then we choose > 0 so small such that the summation of I4 and C(R + R−N )|{wλ > 0}| < η2 , we obtain the desired conclusion. Combining Lemmas 4.5.19 and 4.5.21, we have: Theorem 4.5.22 Assume that uj ∗ u in H1 (RN ). Then for all R > 0 there exists a subsequence {unj } converging to u on BR (θ) in the sense of the bitting lemma. As an application to the calculus of variations, we study the polyconvex functionals in Sect. 4.3.3. Recall that a function f : Ω × M n×n → R1 is called polyconvex, if f (x, A) = g(x, T (A)), where g : Ω×Rτn → R1 is a Caratheodory

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4 Minimization Methods

function, for a.e. x ∈ Ω, ξ → g(x, ξ) is convex, T = {adjs }n1 : M n×n → Rτn , and 2 n n! . τn = s!(n − s)! 1 Theorem 4.5.23 Let Ω be an open bounded domain in Rn ; f (x, A) = g(x, T (A)) is polyconvex, satisfying C|A|n ≤ g(x, T (A)) ∀A ∈ M n×n , C > 0 . Then the functional

f (x, ∇u)dx

J(u) = Ω

has a minimizer in W01,n (Ω, Rn ). Proof. Let c = inf u∈W 1,n J(u), and {uj } be a minimizing sequence, i.e., 0 J(uj ) → c. By the coercive condition, ∇uj Ln (Ω,M n×n ) is bounded, and then uj W 1,n (Ω,Rn ) is bounded. Modulo a subsequence, we have uj n

n!

2

u in W 1,n (Ω, Rn ), adjs (∇uj ) adjs (∇u), in L s (Ω, R( s!(n−s)! ) ), 1 ≤ s < n. Since we can extend uj to be zero outside Ω, provided by Theorem 4.5.18, we have det(∇uj ) ∗ det(∇u) in H1 (Rn ). For ∀ > 0, we ﬁnd δ > 0 such that if |E| < δ then E g(x, T (u(x)))dx < . According to Theorem 4.5.22, there exists a measurable set E ⊂ Ω such that |E| < δ and det(∇(uj )) det(∇u) in L1 (Ω\E). Obviously, we also 2 n! have adjs (∇uj ) adjs (∇u), 1 ≤ s < n in L1 (Ω\E, R( s!(n−s)! ) ). Since a.e., x ∈ Ω, ξ → g(x, ξ) is convex, we have

g(x, T (∇uj (x)))dx ≥ lim inf g(x, T (∇uj (x)))dx lim inf j→∞ j→∞ Ω Ω\E

≥ g(x, T (∇u(x)))dx Ω\E

≥ g(x, T (∇u(x)))dx − . Ω

Since > 0 is arbitrary, we obtain

g(x, T (∇uj (x)))dx ≥ g(x, T (∇u(x)))dx . lim inf j→∞

Ω

Thus u ∈ W01,n (Ω, Rn ) is a minimizer.

Ω

Comparing this result with Theorem 4.3.12, the growth condition for polyconvex functions is dropped. It is useful in the elasticity theory. Another advantage in using Hardy space is on regularity. As we mentioned above, the Calderon–Zygmund theory can be applied to the Hardy space. Once some nonlinear expression of the minimizer is proved in H1 , the Euler

4.6 Free Discontinuous Problems

279

Lagrange equation is applied to obtain an a priori estimate, and gain better regularity of the solution, cf. Evans [Ev 2], Helein [Hel 1]. The compensation compactness method has been successfully applied to elasticity and hyperbolic systems by Ball [Bal 1], Di Perna [Di], Tartar [Tar 1] etc.

4.6 Free Discontinuous Problems 4.6.1 Γ-convergence Γ-convergence has been studied extensively by De Giorgi’s school. The idea is to use a family of functionals J , depending on a parameter and approaching the given functional J in a variational problem, such that the minimizers v of J have a limit point v which is a minimizer of J. Deﬁnition 4.6.1 Let X be a metric space and let Fn : X → [0, ∞] be a sequence of functionals. We say that Fn Γ-converges to F on X as n → ∞, written as Γ − limn→∞ Fn = F , if (1) ∀u ∈ X, ∀{un } such that un → u in X one has lim inf n→∞ Fn (un ) ≥ F (u), (2) ∀u ∈ X, ∃{un } such that un → u in X and lim supn→∞ Fn (un ) ≤ F (u). The importance of the notion lies in the following theorems. Theorem 4.6.2 If F = Γ − limn→∞ Fn , then F is l.s.c. Proof. We prove by contradiction. If F is not l.s.c., then ∃xn → x, such that F (x) > limn→∞ F (xn ). However, by deﬁnition, ∀n, ∃{xnm } ⊂ X, xnm → xn , satisfying Fn (xnm ) → F (xn ). We may assume that limn→∞ F (xn ) > −∞ and F (x) < ∞. Let δ = 14 (F (x) − limn→∞ F (xn )) > 0, then, ∀n ∈ N , ∃mn ∈ N such that |Fn (xnmn ) − F (xn )| < δ, mn → ∞, and xnmn → x .

(4.55)

On one hand, by Γ-convergence, F (x) ≤ lim inf Fn (xnmn ) . n→∞

On the other hand, by the deﬁnition of δ, Fn (xnmn ) > F (x) − δ for n large, we have (4.56) F (xn ) < F (x) − 3δ . It follows that Fn (xnmn ) − F (xn ) > 2δ, which contradicts (4.55).

Theorem 4.6.3 Suppose that Γ − limn→∞ Fn = F and that vn minimizes Fn over X ∀ n ∈ N . If v is a limiting point of vn , then v is a minimizer of F and F (v) = lim inf n→∞ Fn (vn ).

280

4 Minimization Methods

Proof. By (2) of the deﬁnition of Γ-convergence, for every u ∈ X, there exists {un } ⊂ X such that un → u, and lim Fn (un ) = F (u) .

n→∞

Since vn is a minimizer of Fn , we have Fn (vn ) ≤ Fn (un ). Therefore (1) of the deﬁnition of Γ-convergence yields F (u) ≥ lim sup Fn (un ) ≥ lim inf Fn (vn ) ≥ F (v) , n→∞

n→∞

(4.57)

as vn → v in X. This proves the theorem.

We note that Γ-convergence is diﬀerent from pointwise convergence. Example 4.6.4

then

⎧ 1 ⎪ ⎨1 if x ≥ n , Fn (x) = nx if x ∈ [− n1 , n1 ] , ⎪ ⎩ −1 if x ≤ − n1 1, if x > 0 , Γ − lim Fn (x) = n→∞ −1 if x ≤ 0 .

But pointwisely, Fn (x) → sgn(x) . The diﬀerence is at x = 0. Example 4.6.5 Γ − limn→∞ sin nx = −1. 4.6.2 A Phase Transition Problem We present here an example due to Cahn and Hilliard on phase transitions showing how Γ-convergence is applied to variational problems. In a container Ω ⊂ R3 ﬁlled with two immiscible and incompressible ﬂuids, the two ﬂuids arrange themselves in order to minimize the area of the interface which separates these two phases. Let u be a function, which takes the value 0 on the set occupied by the ﬁrst ﬂuid, and the value 1 on the set occupied by the second. Let V = u be the total volume of the second ﬂuid. Obviously one has 0 < V < m(Ω). Let S(u) be the interface, i.e., the singular set of u. The equilibrium conﬁguration is obtained by minimizing the energy F (u) := σH2 (S(u)), where σ is the surface tension between the two ﬂuids, and H2 is the two dimensional Hausdorﬀ measure. On the macroscopic level, we allow for a mixture of two ﬂuids; let u : Ω → [0, 1] be the average density of the second ﬂuid, and let W be a double-well potential, i.e., a continuous positive function that vanishes only at 0 and 1.

4.6 Free Discontinuous Problems

281

Cahn and Hilliard established a model: ∀ > 0, we consider a minimizer u of the following energy functional:

1 2 |∇u| + W (u), with V = u. (4.58) E (u) = Ω Ω Ω Under the volume constraint u cannot be a constant; but as → 0, for a.e. x ∈ Ω, u (x) must tend to either 0 or 1. Does u tend to a minimizer of F ? The following Modica–Mortola theorem gives an answer to this question. Before going to the proof, we need a few lemmas. Lemma 4.6.6 Suppose that um → u in a metric space (X, d) satisfying lim supm→∞ F (um ) ≤ F (u), and that ∀m, ∃um,n → um as n → ∞ such that lim supn→∞ Fn (um,n ) ≤ F (um ). Then ∃mn → ∞, such that for vn = umn n we have vn → u and lim supn→∞ Fn (vn ) ≤ F (u). 1 Proof. We may assume d(u, um ) < m . Deﬁne 1 1 mn = max m |, d(umn , u) < , Fm (umn ) ≤ F (um ) + . m m

It is easy to verify that mn → ∞. We set vn = umn n , then vn satisﬁes the requirements. The following lemma is just the one-dimensional version of the general theorem. Lemma 4.6.7

2 1 1 [|γ (t)| + W (γ(t))]dt | γ ∈ C (R ), γ(−∞) = 0, γ(∞) = 1 min R1 1

=2

W (t)dt .

0

Proof. We write the functional to be minimized as J(γ), and the right-hand integral as σ. According to the Cauchy–Schwarz inequality,

1

W (γ(t))γ (t)dt = 2 W (s)ds = σ . J(γ) ≥ 2 R1

0

√ The equality holds if and only if γ = W (γ). Since W is continuous and positive between the two zeroes 0 and 1, the ﬁrst-order equation has a global solution γ on R1 , with γ(−∞) = 0, γ(∞) = 1.

Let U ⊂ Rn be an open set with nonempty boundary ∂U . One deﬁnes the signed distance function as follows: dist(x, ∂U ), if x ∈ U, d(x) = −dist(x, ∂U ), if x ∈ / U. Thus, d is Lipschitzian continuous with Lipschitz constant 1.

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4 Minimization Methods

Lemma 4.6.8 Assume that ∂U ∈ C 2 is bounded. ∀t ∈ R1 we set Σt = {x ∈ Rn | d(x) = t}. Then ∃ > 0 such that for |s| < , Σs is a hypersurface of C 2 class and lims→0 Hn−1 (Σs ) = Hn−1 (∂U ). Moreover, |∇d(x)| = 1 . Proof. ∀x0 ∈ ∂U , let n(x0 ) be the unit outer normal vector of ∂U at x0 , and let Tx0 (∂U ) be the tangent hyperplane at x0 . Rotate the coordinates such that the ﬁrst n−1 coordinate axes lie on Tx0 (∂U ) and the xn coordinate axis points in the direction −n(x0 ). In a neighborhood V (x0 ) of x0 , ∂U can be represented by xn = f (x ) where x = (x1 , . . . , xn−1 ) with f (x0 ) = 0, ∇f (x0 ) = θ. There exists a tubular neighborhood Ξ of ∂U , such that ∀x ∈ Ξ, there is a unique representation ξ = π(x) ∈ ∂U such that x = π(x) − n(x)d(x), where n(x) = n(π(x)), and that ξ is the unique point satisfying |x − ξ| = d(x) is a neighborhood of x. There exists > 0 such that Σs ⊂ Ξ, ∀|s| < . We claim that: |∇d(x)| = 1, and ∇d(x) = −n(ξ) . Thus ∀x ∈ Σs , s = d(x), we have a local representation: x = F (x , s) := (x , f (x )) − n(x , f (x ))s. By the use of the implicit function theorem, one can verify that F is invertible as > 0 is small. This implies that d(x) = d(ξ − sn(ξ)) = s. Therefore, ∇d(x)n(ξ) = −1. Since d has Lipschitz constant 1, the claim is proved. Using the Gauss–Green formula, we have

div(∇d(x)) = ∇d(x)n(x)dHn−1 + ∇d(x)n(x)dHn−1 . 0 0, let E (u) = Ω |∇u|2 + 1 Ω W (u) if u ∈ W 1,2 ∩ X , F (u) := (4.59) +∞, otherwise , and

σDχE (Ω) if u = χE , F (u) = +∞, otherwise ,

(4.60)

1 where σ = 2 0 W (t)dt, and E is a set of ﬁnite perimetrics. Then Γ − lim F = F in X, and every sequence of minimizers of F is subconvergent in X to a minimizer of F.

4.6 Free Discontinuous Problems

283

Proof. 1. We claim that ∀u ∈ X and for every un → u in X with n → 0, if ∃M > 0 such that

1 2 W (un ) ≤ M , n |∇un | + n then W (u) ≤ lim inf n→∞ W (un ) = 0, provided by Fatou’s lemma. From the double well assumption on W u = χE for some Borel set E. 2. We begin with the veriﬁcation of (1) of the deﬁnition of Γ-convergence. We may assume without loss of generality that lim inf →0 F (u ) is bounded. According to step 1, u = χE for some Borel set E. From the Cauchy–Schwarz inequality and the l.s.c. of the total variations, we have

1 2 |∇u | + W (u ) dx lim inf F (u ) = lim inf →0 →0 Ω

≥ lim inf 2 W (u )|∇u |dx →0

Ω |∇(Ψ(u ))|dx = lim inf →0

Ω

= lim inf DΨ(u )(Ω) →0

≥ DΨ(u)(Ω) , √ where Ψ is the primitive of 2 W satisfying Ψ(0) = 0. One has Ψ(u) = Ψ(χE ) = Ψ(1)χE = σχE . Thus, DΨ(u) = F (u), and then (1) is veriﬁed. 3. We now verify (2) of the deﬁnition of Γ-convergence. By the deﬁnition of F , it is suﬃcient to verify (2) for u = χE . Suitably modifying the approximation theorem of BV functions (Theorem 4.5.7), we ﬁnd approximate functions of the form: j χΩi , with uj − uL1 → 0 , uj = i

Ωji , i

= 1, 2, . . . , j = 1, 2, . . . , are disjoint bounded open sets with where smooth boundary ∂Ωji . On account of Lemma 4.6.6, the veriﬁcation is reduced to verifying (2) for characteristic functions of open sets with bounded smooth boundaries. Let u = χΩ , let d be the signed distance function of ∂Ω, and let γ be a minimizer obtained in Lemma 4.6.7. ∀ > 0, we deﬁne u (x) = γ( d(x) ). From Lemma 4.6.8 and the co-area formula, we have

2 d 1 γ +W γ d F (u ) = dx Ω

1 g(t)[γ (t/)2 + W (γ(t/))]dt = R1

g(t)[γ (t)2 + W (γ(t))]dt . = R1

284

4 Minimization Methods

where g(t) = Hn−1 (Σt ). Lemma 4.6.8 yields g(t) → Hn−1 (∂Ω) as → 0. Thus

[|γ (t)|2 + W (γ(t))]dt = F (u) . lim sup F (u ) ≤ DχE (Ω) R1

4. Finally we verify that if u is a minimizer of F , then {u } is precompact. In fact, one may assume that F (u ) is bounded, and then by step 1, so is DΨ(u )(Ω). Thus |Ψ(u )BV = Ψ(u )L1 + DΨ(u )(Ω) is bounded. According to the compactness property of BV functions, it is precompact in L1 (Ω). Since Ψ admits a continuous inverse, u is precompact in L1 . 4.6.3 Segmentation and Mumford–Shah Problem In image segmentation, we want to detect the edge of an image from a picture. Given a bounded domain Ω ⊂ R2 and an image represented by a function g ∈ L2 (Ω), ﬁnd a closed set K ⊂ Ω with ﬁnite one-dimensional Hausdorﬀ measure H1 (K), and a function u ∈ H 1 (Ω\K), which minimize the following cost functional:

|∇u|2 dx + µ |u − g|2 dx + λH1 (K) , E(K, u) = Ω\K

Ω\K

where λ, µ > 0 are parameters. This is the Mumford–Shah problem. The original image g is thereby estimated by u which is smooth in Ω except for the discontinuous set K (in fact, in H 1 (Ω\K)), and the latter is the estimated edge. This is a variational problem with unknown (K, u). Obviously, the diﬃculty lies in K, which is an object we have never met. Tentatively, suppose K is given. Since E is quadratic in u, so is convex and coercive, we obtain a minimizer u(K) with mΩ\K := E(K, u(K)). The problem is then reduced to minimizing the functional: J(K) = mΩ\K + λH1 (K) .

(4.61)

Since the discontinuous set K is only assumed to be Hausdorﬀ measurable, the problem is more complicated and requires further and deeper prerequisites. For pedagogical purposes, we follow Nordstr¨ om [No] and restrict ourselves to assuming that only the union of ﬁnitely many C 1 -curves γ is the admissible candidate for K. Given ξ ∈ C 1 ([0, 1], R2 ) the image of ξ is a C 1 -curve (may be self intersection). Thus the union of N − C 1 -curves γ is deﬁned by ξ = (ξ1 , . . . , ξN ) ∈ C 1 ([0, 1], R2 )N , with N 5 γ = ξ[0, 1] := ξn [0, 1] . n=1

4.6 Free Discontinuous Problems

285

Let us recall the notion of a domain with minimally smooth boundary (cf. Stein [Ste 1]). An open set O ⊂ Rn is said to have a minimally smooth boundary if ∃ε > 0, ∃ an integer N , a positive M > 0, and a sequence of open sets U1 , U2 , . . . , such that (i) If x ∈ ∂O, then Bε (x) ⊂ Ui for some i, (ii) No point of Rn is contained in more than N of Ui ’s, (iii) ∀i, ∃ a special Lipschitz domain Di whose bound does not exceed M so that Ui ∩ O = Ui ∩ Di . For example, if ∂O ∈ C 1 , then ∂O is minimally smooth. The introduction of this notion is due to the following extension theorem: Theorem 4.6.10 Let O ⊂ Rn be a domain with minimally smooth boundary. Then ∃ a linear operator T mapping functions on O to functions on Rn such that (1) T u|O = u, (2) T u W l,p (Rn ) Cl,p u W l,p (O) for some constant Cl,p , ∀l ∈ N , p ∈ [1, ∞]. Deﬁnition 4.6.11 An N − C 1 -curve γ is said to be an admissible image segmentation of Ω if ∀ connected components G of Ω\γ, ∀ε > 0, ∃ a domain with minimally smooth boundary Oε such that (1) Oε ⊂ G, (2) m(G − Oε ) < ε. We denote by EN the set of all N − C 1 -curve admissible image segmentations of Ω endowed with the C 1 ([0, 1], R2 )N topology. Thus for ∀γ ∈ EN ,

1

˙ |ξ(t)|dt :=

H1 (γ) = 0

N

n=1

1

|ξ˙n (t)|dt,

where γ = ξ[0, 1] =

0

N 5

ξn [0, 1] ,

n=1

i.e., the length of the piecewise C 1 -curve γ. Now, let us make some preparations. To simplify the notations, we assume λ = µ = 1. For any bounded open domain G, we deﬁne

2 |∇u| + |u − g|2 on H 1 (G) . IG (u) = G

G

It is known that IG has an unique minimizer uG . Let mG = IG (uG ) =

min

u∈H 1 (G)

IG .

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4 Minimization Methods

By simple calculations,

|g|2 − g · uG .

mG = G

Moreover, we have the following properties: (uG ) = 0, i.e., (1) IG

∇uG ∇ϕ + (uG − g)ϕ = 0 ∀ϕ ∈ H 1 (G) .

(2)

G

G

|∇uG | + |uG | 2

2

G

|g| . 2

G

Lemma 4.6.12 Let G be a connected component of Ω\γ where γ is an admissible image segmentation of Ω. Then for ∀ε > 0, ∃ a domain with minimally smooth boundary O such that |mO − mG | < ε .

Proof. By the deﬁnition of the admissibility for γ, ∀ > 0, ∃ a domain with minimally smooth boundary O ⊂ O ⊂ G such that m(G\O) < . Letting uO be the minimizer of IO , we have

(|g|2 − guG ) − (|g|2 − guO ) mG − mO = O

G 2 (|g| − guG ) + g(uO − uG ) . = O

G\O

Noticing that

∇uG ∇! uO + uG u !O G\O G\O

= g! uO − guG O

G

= g! uO + g(uO − uG ) ,

G\O

where u !O = T uO , we have

g(uO − uG ) = O

G\O

O

(∇uG ∇! uO + uG u !O ) −

g! uO . G\O

Applying the Cauchy–Schwarz inequality, the ﬁrst integral uO H 1 . By (2) and Theorem 4.6.10 ∃C0 > 0 such that uG H 1 (G\O) ! ! uO H 1 ≤ C0 uO H 1 (O) ≤ C0 gL2 (O) ≤ C0 gL2 (G) .

≤

4.6 Free Discontinuous Problems

Therefore ∃C > 0 a constant such that

287

12

|mG − mO | C g L2 (G)

|g| + 2

uG 2H 1 (G\O)

.

G\O

Since m(G\O) can be suﬃciently small, the conclusion follows.

Lemma 4.6.13 The function J restricted on EN is l.s.c. Proof. The second term H1 (γ) of J is trivially continuous in C 1 -topology, so is l.s.c. It is suﬃcient to verify the l.s.c. of mΩ\γ with respect to γ, i.e., for ∀γ0 ∈ EN , ∀ε > 0, ∃δ > 0 such that dist (γ, γ0 ) < δ implies that mΩ\γ > mΩ\γ0 − ε. Let E be the collection of all connected components of Ω\γ0 , then E is countable, say E = {Gi }∞ 1 . From ∞

i=1

|g|2 =

|g|2 g 2L2 (Ω) , Ω\γ

Gi

there exists K > 0 such that

∞ i=K+1

|g|2

0 be such that dist (Oi , ∂Gi ) δ > 0; we choose a 2δ neighborhood K + Oi ⊂ Ω\γ as dist (γ, γ0 ) < 2δ , and we have of γ0 , then i=1

mΩ\γ = IΩ\γ (uΩ\γ )

K

IOi (uΩ\γ )

i=1

K

mOi

i=1

−ε + mG i . 2 i=1 K

+K Let u(x) = uGi as x ∈ Gi , i = 1, 2, . . . , K and u(x) = 0 as x ∈ E\ i=1 Gi , +K +∞ then Ω\(γ0 ∪ i=1 Gi ) = i=K+1 Gi , we obtain from inequality (4.62) mΩ\γ0 ≤ IΩ\γ0 (u) =

K i=1

IGi (uGi ) + IΩ\(γ0 ∪+ k

i=1

Gi ) (0)

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4 Minimization Methods

≤

K

mG i +

i=1

ε . 2

This proves that mΩ\γ > mΩ\γ0 − ε . Now for ∀N ∈ N , ∀α ∈ (0, 1), ∀ρ > 0, ∀ω > 0, let us deﬁne CN (α, ρ, ω) =

γ = ξ[0, 1] ∈ EN | ξ C 1 +

1

˙ − ξ(s)| ˙ |ξ(t) |t − s|α s,t∈[0,1],s=t sup

˙ dt ≥ ω ξ(t)

≤ ρ, 0

endowed with the C 1 [0, 1]N topology, and let C = {Ø} ∪

∞ 5

CN (α, ρ, ω) .

N =1

We arrive at: Theorem 4.6.14 ∃γ0 ∈ C, ∃u0 ∈ H 1 (Ω\γ0 ), such that E(γ0 , u0 ) = Min(γ,u)∈(C×H 1 (Ω\γ)) E(γ, u) .

Proof. The lower bound ω > 0 forces us to minimize E on CN (α, ρ, ω) for some ﬁnite N , and the latter is a closed compact set in C 1 [0, 1]N . Our conclusion follows from the general principle of calculus of variations. The simpliﬁed model of Mumford–Shah problem is restricted to the space of possible discontinuous sets consisting of those corresponding to line drawings. The Mumford–Shah functional has been studied by De Giorgi and Ambrosio [DG 2], [Amb], [DA]. In the variational problem, the working space is an SBV space, a subspace of BV (Ω), which consists of those BV functions in which the Cantor part of their derivatives vanishes. The distributional derivative of a BV function u can be decomposed into three parts: Du = Da u + Dj u + Dc u, where Da , Dj , and Dc are the absolutely continuous part, jump part, and the Cantor part, resp. The advantage of the subspace SBV (Ω) is in the characterization of its w∗ -compactness. The reader is referred to Ambrosio [Amb]. Mumford and Shah conjectured that if (K, u) is an optimal essential pair of J, then K is locally in Ω the union of ﬁnitely many C 1,1 embedded curves, see Morel [Mor].

4.7 Concentration Compactness

289

4.7 Concentration Compactness The loss of compactness breaks down the standard variational techniques. However, most problems arising in geometry (prescribing scalar curvature problem, Yamabe problem, minimal surfaces, ﬁxed points and intersections in symplectic geometry etc.) and in physics (N -body problem, Yang–Mills equation, nonlinear Schr¨ odinger equations, gravity theory etc.) are in this realm. Semilinear elliptic PDEs in Rn and those with the critical Sobolev exponent are two typical examples on this topic. They have been studied extensively in the last two decades. If Ω ⊂ Rn is a bounded domain, then a bounded sequence {uj } in H 1 (Ω) has a weakly convergent subsequence, and then a strongly convergent subsequence in Lq (Ω) as q ∈ [2, 2∗ ), according to the compactness of the embedding H 1 (Ω) → Lq (Ω). This argument is frequently used. (See Example 3 in Sect. 4.2.) However, when Ω is unbounded, the argument does not work. This can be seen by the following example: Let Ω = Rn , {xj } be a sequence that tends to inﬁnity. Given a nonzero u0 ∈ C0∞ , let uj (x) = u0 (x − xj ). Then ||uj ||H 1 = ||u0 ||H 1 and |uj |q = |u0 |q . Therefore, uj (x) → 0 a.e., and then uj 0 in H 1 (Rn ), but uj cannot Lq converge to 0. The reason is that the associated measures

|uj |2 ∀ Lebesgue measurable set E ⊂ Rn (4.63) µj (E) = E

leak out at inﬁnity. 4.7.1 Concentration Function Relating to a measure µ, P. Levy introduced the concentration function Q(r) = sup µ(Br (x)), r ≥ 0 . x∈Rn

We have the following: Lemma 4.7.1 Suppose that {uj } ⊂ H 1 (Rn ) is a bounded sequence. If ∃ R > 0 such that limj→+∞ Qj (R) = 0, where Qj is the concentration functions with respect to µj deﬁned in equation (4.63), then there exists a subsequence {ujk } such that ujk → 0 in Lq (Rn ), ∀ q ∈ (2, 2∗ ). Proof. Due to the interpolation inequality, we have: θ ∗ ||u||Lq (BR (x)) ≤ ||u||1−θ L2 (BR (x)) ||u||L2 (BR (x))

∀ x ∈ Rn , where θ =

(q−2)n 2q .

4 In the case where θq ≥ 2, we have q ≥ 2 + and a constant C > 0 such n that

290

4 Minimization Methods

(1−θ)q

2 |u|q ≤ C||u||L2 (BR (x)) ||u||θq−2 H 1 (BR (x)) uH 1 (BR (x)) BR (x)

(1−θ)q θq−2 ≤ C( sup ||u||L2 (BR (x)) )||u||H 1 (Rn ) (|u|2 + |∇u|2 ) x∈Rn

BR (x)

from the Sobolev embedding theorem. We ﬁnd a locally ﬁnite covering, i.e., ∃ {xk } and ∃ ∈ N such that ∀ x ∈ Rn , ∃ at most balls of BR (xk ) to which x belongs. Then

|u|q ≤ |u|q Rn

BR (xk )

k

≤ C||u||θq H 1 (Rn )

(1−θ)q

sup ||u||L2 (BR (x))

.

x∈Rn

By the assumptions that uj H 1 is bounded and modulo a subsequence supx∈Rn BR (x) |uj |2 → 0, it follows that Rn |uj |q → 0. 4 In the case where θq < 2, we have 2 < q < 2 + := q0 . Since θq0 = 2, n ||ujk ||Lq0 (Rn ) → 0. Letting 1q = λ2 + (1−λ) q0 , we obtain, by interpolation, that →0. ||ujk ||Lq (Rn ) ≤ |ujk |λ2 |ujk |1−λ q0 The lemma shows that the concentration function plays an important role in the compactness argument. It will be seen by the following: Corollary 4.7.2 (Strauss) Let Hr1 (Rn ) be the subspace of H 1 (Rn ) consisting of radial symmetric functions. The embedding Hr1 (Rn ) → Lp (Rn ), 2 < p < 2∗ , n ≥ 2 is compact. Proof. ∀x ∈ Rn , ∀R > 0, let m(x, R) be the largest number of disjoint balls with radius R and the centers lie on the same sphere with radius |x| centered at θ. It is easily seen that m(x, R) → ∞ as x → ∞. By deﬁnition ∀u ∈ L2 (Rn ), ∀r > 0,

|u|2 ≤ m(x, r)−1 u2L2 .

Br (x)

If {uj } is a bounded sequence in Hr1 (Rn ), then ∀ > 0, ∃R > 0 such that

|uj |2 | |x| ≥ R

sup

0, ∃ R > 0, ∃ {xj } ⊂ Rn such that µj (BR (xj )) ≥ 1−. 1 In fact, by deﬁnition, ∃ R0 > 0 s.t. Q(R0 ) > . Let xj ∈ Rn satisfying 2 1 Qj (R0 ) ≤ µj (BR0 (xj )) + . For ∈ (0, 12 ), we choose R1 > 0 such that j 1 Q(R1 ) > 1 − , and let yj satisfy Qj (R1 ) ≤ µj (BR1 (yj )) + . Then j µj (BR1 (yj )) + µj (BR0 (xj )) ≥ Qj (R1 ) + Qj (R0 ) −

2 > 1, for j large . j

Therefore BR1 (yj ) ∩ BR0 (xj ) = ∅, which implies BR1 (yj ) ⊂ B2R1 +R0 (xj ) and 1 1 µj (B2R1 +R0 (xj )) ≥ µj (BR1 (yj )) ≥ Qj (R1 ) − > 1 − − for j large. This j j is the conclusion. For λ ∈ (0, 1), ∀ > 0, ∃ R0 > 0, ∃ {xj } such that Qj (R0 ) ≥ µj (BR0 (xj )) ≥ λ − for j large. Also, we may ﬁnd a sequence Rj → ∞ such that Qj (R0 ) ≤ Qj (Rj ) < λ + for j large . Thus λ − < µj (BR0 (xj )) ≤ µj (BRj (xj )) ≤ λ + . Now for any given R > R0 , we may assume Rj > R for all j and let µ1j = µj |BR0 (xj ) and µ2j = µj |Rn \BRj (xj ) . Then 0 ≤ µ1j + µ2j ≤ µj , supp µ1j ⊂ BR0 (xj ), supp µ2j ⊂ Rn \ BRj (xj ) ⊂ Rn \ BR (xj ), and so |λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )| ≤ |λ − µj (BR0 (xj ))| + |λ − µj (BRj (xj )| < 2 . In summary, we have:

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4 Minimization Methods

Theorem 4.7.3 (Concentration compactness principle) Suppose that {µj } ⊂ M (Rn ) is a sequence of probability measures. Then one of the following three conclusions holds: 1. (Compactness) ∃ {xj } ⊂ Rn such that ∀ > 0, ∃ R > 0 with µj (BR (xj )) ≥ 1 − for all j. 2. (Vanishing) ∀ R > 0, limj→∞ (supx∈Rn µj (BR (x))) = 0. 3. (Dichotomy) ∃ λ ∈ (0, 1) such that ∀ > 0, ∃ R > 0 and ∃ {xj } with the property: Given R > R, there exist positive measures µ1j and µ2j such that 0 ≤ µ1j + µ2j ≤ µj , supp µ1j ⊂ BR (xj ), supp µ2j ⊂ Rn \ BR (xj ), and lim supj→∞ {|λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )|} ≤ . We shall present an example showing how this principle is applied. In contrast with Example 3 in Sect. 4.2, we consider the same equation but on the whole space Rn : Find a nontrivial solution u ∈ H 1 (Rn ) satisfying −u + u = |u|q−2 u, 2 < q < 2∗ . If we follow the steps in the proof of that example, then it can be stated as a minimizing problem for the functional:

J(u) = (|∇u|2 + |u|2 ) Rn

subject to the constraint:

M = u ∈ H 1 (Rn ) |

|u| = 1 . q

Rn

It is important to note that both J and M are translation invariant. From the translation invariance, we intend to prove that, for any minimizing sequence {uj }, after suitable translation, the new minimizing sequence {vj } is Lq subconvergent. Then the limit v0 ∈ M . Let Sq = inf J . M

In light of the concentration compactness principle, let µj = |uj |q dx. If one can exclude Cases 2 and 3, then ∃{xj } ⊂ Rn , ∀ > 0, ∃ R > 0, such that

1= |uj (x − xj )|q dx ≥ µj (BR (xj )) ≥ 1 − . Rn

Let vj (x) = uj (x − xj ). Again {vj } is a minimizing sequence, vj H 1 (Rn ) is bounded, after a subsequence vj v0 , and then vj → v0 strongly in Lq (BR (θ)).

|v0 (x)|q ≥ 1 − . Therefore,

Then we have 1 ≥ BR (0)

v0 ∈ M . Moreover, we have vj → v0 in Lq (Rn ) strongly.

Rn

|v0 |q = 1, i.e.,

4.7 Concentration Compactness

293

• To exclude the vanishing case: Suppose we had R > 0 such that

lim inf sup |uj |q = 0 .

j→∞ x∈Rn

|uj |2 = 0. According to the Lemma 4.7.1, uj → 0

Then lim inf sup

j→∞ x∈Rn

BR (x)

BR (x) ∗

in Lp (Rn ) ∀ p ∈ (2, 2 ). In particular, when we take p = q, it contradicts |uj |q = 1. • To exclude the dichotomy case: Suppose ∃λ ∈ (0, 1) such that ∀ > 0, ∃ R > 0, ∃ {xj }, and positive measures µ1j and µ2j such that 0 ≤ µ1j + µ2j ≤ µj , supp µ1j ⊂ BR (xj ), supp µ2j ⊂ Rn \ B2R (xj ), and limj→∞ {|λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )|} ≤ . Choosing j → 0, ∃ Rj > 0, after a subsequence, we have supp µ1j ⊂ BRj (xj ), supp µ2j ⊂ Rn \ B2Rj (xj ), and limj→∞ {|λ − µ1j (Rn )| + |(1 − λ) − µ2j (Rn )|} = 0. In fact, we may assume Rj → ∞. Let φ ∈ C0∞ (B2 (θ)) such that φ = 1 x − xj ). We write vj = uj φj + uj (1 − φj ). Then in B1 (θ) and let φj (x) = φ( Rj J(uj ) = J(uj φj ) + J(uj (1 − φj )) + Bj , where Bj is the interaction term:

Bj = 2 ∇(uj φj )∇(uj (1 − φj )) + u2j φj (1 − φj ) . By simple estimation ∀ > 0, 1 Bj ≥ − − Ej , where

|uj ∇φj |2 ≤

Ej = 2 B2Rj (xj )\BRj (xj )

C Rj2

|uj |2 . B2Rj (xj )\BRj (xj )

Fixing > 0 arbitrarily small, since Ej → 0 as j → ∞, we have 1 J(uj ) ≥ Sq (|uj φj |2q + |uj (1 − φj )|2q ) − − Ej 2 2 1 ≥ Sq (µ1j (BRj (xj )) q + µ2j (Rn \ B2Rj (xj )) q ) − − Ej 2 2 q q ≥ Sq (λ + (1 − λ) ) − + o(1) . 2

2

But λ q + (1 − λ) q > 1 as λ ∈ (0, 1). This is a contradiction. Namely we obtain: Theorem 4.7.4 (P.L. Lions) For any minimizing sequence {uj } ⊂ H 1 (Rn ) of the functional J(u) = Rn (|∇u|2 + |u|2 ) subject to |u|q = 1 for q ∈ (2, 2∗ ),

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4 Minimization Methods

there exists a sequence {xj } ⊂ Rn such that vj (x) = uj (x − xj ) Lq -strongly converges to a minimizer of J. Example 1. Assume a ∈ C(Rn ) satisﬁes a(x) → a∞ as |x| → ∞. One studies the nontrivial solution u ∈ H 1 (Rn ) of the equation: −∆u + a(x)u = |u|q−2 u, 2 < q < 2∗ . Deﬁne

I(u) = Rn

(|∇u|2 + a(x)|u|2 ), I∞ (u) =

Rn

(|∇u|2 + a∞ |u|2 ) ,

and M (u) := |u|qq =

|u|q . Rn

We deﬁne S = inf{I(u) | M (u) = 1} and S∞ = inf{I∞ (u) | M (u) = 1} , and conclude: If S < S∞ , then there is a nontrivial solution of the above equation. Before going to the proof, we need a lemma (see [BL]): Lemma 4.7.5 (Brezis–Lieb) Suppose that Ω ⊂ Rn and {uj } ⊂ Lp (Ω), p ∈ [1, ∞) If {uj } is bounded in Lp (Ω), and uj → u a.e., on Ω. Then lim (uj pp − uj − upp ) = upp .

j→∞

Proof. According to Fatou’s lemma, |u|pp ≤ lim inf j→∞ uj pp < ∞. We begin with an elementary inequality: ∀ > 0, there exists C > 0 such that ∀x, y ∈ R1 , ||x + y|p − |x|p | ≤ |x|p + C |y|p . Let vj, = (| |uj |p − |uj − u|p − |u|p | − |uj − u|p )+ , then vj, ≤ (1 + C )|u|p , and vj, → 0 a.e. Then we can use Lebesgue’s dominance theorem to conclude: Ω vj, → 0 as j → ∞. Since | |uj |p − |uj − u|p − |u|p | ≤ vj, + |uj − u|p ,

we have

| |uj |p − ||uj − u|p − |u|p | ≤ C ,

lim sup j→∞

where C = sup{uj −

upp

Ω

< ∞. Letting → 0, the lemma is proved.

4.7 Concentration Compactness

295

Now we turn to the proof of the conclusion in Example 1. In fact, by the previous discussion, after a subsequence, any minimizing sequence {uj } ⊂ H 1 (Rn ) ∩ M −1 (1) is weakly H 1 convergent to some u0 ∈ H 1 (Rn ), so it is suﬃcient to show that u0 ∈ M −1 (1). Let vj = uj − u0 , we have vj 0, in H 1 (Rn ) and vj → 0 in L2loc , and then vj → 0 a.e. Since we have: 1. I(uj ) = I(u0 ) + I(vj ) + o(1). 2. |uj |qq = |u0 |qq + |vj|qq + o(1) (from Lemma (4.7.6)) 3. a(x)|vj |2 = a∞ |vj |2 + o(1), (from the estimate

|a(x) − a∞ ||vj (x)|2 ≤ |a(x) − a∞ ||vj |2 + C |x|≥R

|x|≤R

|vj |2 ,

for suitably chosen R > 0, where C = 2 Max{a(x)}). It follows that I(vj ) = I∞ (vj ) + o(1). Let |u0 |qq = λ. Obviously λ ∈ [0, 1]; we are going to exclude the case that λ ∈ [0, 1). Suppose not, either λ ∈ (0, 1), we would have S ≥ S|u0 |2q + S∞ |vj |2q + o(1) 2

2

= Sλ q + S∞ (1 − λ) q + o(1) 2

2

≥ S(λ q + (1 − λ) q ) + o(1) > S , or, λ = 0, then S > S∞ . These are all impossible. 4.7.2 The Critical Sobolev Exponent and the Best Constants The following best constant plays an important role in many variational problems arising in geometry and analysis: |∇u|2 Rn , S= inf 2 u∈D 1,2 \{θ} ( n |u|2∗ ) 2∗ R

1 where D1,2 is the closure of C0∞ (Rn ) under the norm: u = ( Rn |∇u|2 ) 2 . We intend to ﬁgure out the precise value of S and reduce it to a variational problem. Let

|∇u|2 , I(u) = u2 := Rn

and ∗

M (u) = |u|22∗ =

Rn

∗

|u(x)|2 dx .

We study the minimization problem: Min{I(u) | M (u) = 1} .

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4 Minimization Methods

Both I and M are not only translation invariant, but also invariant under the scaling transformation: 6x7 n−2 Tθ : u(x) → θ− 2 u , ∀θ > 0 . θ By the same idea as in the previous subsection, one may use the concentration compactness principle to show that after translation and scaling, the ∗ minimizing sequence is again convergent in L2 (Rn ), and then a minimizer does exist. Let u0 be a minimizer; obviously |u0 | is also. Therefore we may assume the minimizer u0 is nonnegative. The Euler–Lagrange equation of the variational problem reads as −∆u = λ|u|2

∗

−2

u in Rn ,

(4.64)

where λ > 0 is the Lagrange multiplier. After suitably adjusting a factor one may assume λ = 1. By a moving plane argument due to Gidas, Ni and Nirenberg, [GNN] (see also Yanyan Li [Li 2]), u0 is radially symmetric: u0 (x) = 1 . Moreover, g is nonincreasing. g(|x|), for some nonnegative function on R+ Plugging g into equation (4.64), it becomes an ODE: −(rn−1 g ) = Srn−1 g 2

∗

−1

.

However, we shall present here a more direct proof due to Lieb. Lemma 4.7.6 If the problem Min{I(u) | M (u) = 1} possesses a minimizer, then it has a radially symmetric and nonincreasing minimizer u0 ; i.e., there is a nonincreasing function g deﬁned on R1+ such that u0 (x) = g(|x|). Proof. We may restrict ourselves to nonnegative minimizers, because |u| ≤ u, and |u|q = uq . Let u∗ be the Schwarz rearrangement of u (see Lemma 4.2.5), then u∗ is nonincreasing, radial symmetric and satisﬁes m(ut ) = m(u∗t ), ∀t ∈ R1 , where ut = {x ∈ Rn | u(x) ≥ t}. Thus,

∞

∞ q q |t| dm(ut ) = |t|q dm(u∗t ) = |u∗ |qq . |u|q = −∞

−∞

According to the Faber–Krahn inequality, it follows that

∗ 2 |∇u | ≤ |∇u|2 . Rn

Rn

Combining Corollary 4.7.2 with Lemma 4.7.6, one proves that the following equation possesses a positive solution in H 1 (Rn ) directly: −∆u + u = |u|q−2 u for 2 < q < 2∗ . Lemma 4.7.7 The minimizer of the problem in lemma 4.7.6 is achieved.

4.7 Concentration Compactness

297

Proof. 1. By changing variables, let F (αt) = eαt g(et ), where α = n2 − 1, and r = et ; we have

∞

∞

∞ ∗ 1 ∞ 2∗ F dt = g(r)2 rn−1 dr, α (F − F )2 dt = g (r)2 rn−1 dr . α −∞ 0 −∞ 0 ∞ 2 n−1 ∞ 2∗ n−1 Setting f (g) = 0 g (r) r dr, φ(g) = 0 g(r) r dr, and E0 = {g ∈ −1 1 1 (R+ ) | g ≥ 0, φ(g) = ωn−1 }, E0 = {g ∈ E0 | g is nonincreasing}, where Hloc n

ωn−1 =

nπ 2 Γ( n 2 +1)

is the area of the unit sphere S n−1 , we have:

Min{I(u) | M (u) = 1} ≤ Min{f (u) | u ∈ E0 } ≤ Min{f (u) | u ∈ E0 } ≤ Min{I(u) | M (u) = 1} . Our problem is reduced to ﬁnd: (4.65) Min{f (u) | u ∈ E0 } . ∞ ∗ ∞ 2. Let f1 (F ) = −∞ (F 2 + F 2 )dt, φ1 (F ) = −∞ F 2 dt, and E1 = {F ∈ −1 }. Then the problem (4.65) is equivalent to the probH 1 (R1 ) | φ1 (F ) = αωn−1 lem: Min{f1 (F ) | F ∈ E1 } . In fact, on the one hand, if F solves the latter, then F (t) → 0 as t → ±∞, One has

∞

∞ 2 (F − F ) dt = (F 2 + F 2 )dt . (4.66) −∞

−∞

On the other hand, functions with compact supports consist of a dense subset of E0 , and for g with compact support F (t) is zero as t becomes large. Moreover, F (t) ≤ g(0)et , and then F (t) → 0 as t → ±∞. Again equation (4.66) holds. 1 }. Again, by 3. Let E1 = {F ∈ E1 | F is even and nonincreasing on R+ symmetric rearrangement, we have Min{f1 (F ) | F ∈ E1 } = Min{f1 (F ) | F ∈ E1 } . Let {Fj } be a minimizing sequence in E1 , the sequence is weakly conver∗ α . gent to some F0 ∈ E1 . We are going to verify that |F0 |22∗ = ωn−1 ∗

Since Fj → F0 , a.e., it is suﬃcient to ﬁnd a dominant function in L2 (R1+ ). From Fj (t) → 0 as t → ∞, we have

∞

∞ 2 Fj (t) = −2 Fj Fj dt ≤ ( Fj2 + Fj2 )dt ≤ C . 0

t

Since Fj is nonincreasing on [0, ∞), and

t tFj (t)2 ≤ Fj2 ≤ C , 0 − 12

the function M (t) = Min{1, t (R1+ ).

∗

} is a dominant function for Fj in L2

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4 Minimization Methods

Now we can ﬁnd the minimizer explicitly. Following the above notations, we conclude that the function F0 satisﬁes the ordinary diﬀerential equation: ⎧ −2 2∗ −1 ⎪ , ⎨−F + F = Sα F F (0) = 0 , ⎪ ⎩ F (∞) = 0 . where S is the best constant, α = n2 − 1. The unique solution (see Lieb [Lieb], Aubin [Au 1] and Talenti [Tal]) reads as = α n(n − 2) sech(α−1 t) . F0 = 4S or

= g0 (r) = r−α F0 (α log r) =

n(n − 2) 1 S 1 + r2

n2 −1 .

By direct computation, we obtain S = n(n − 2)π

Γ(n) " # Γ n2

− n2 .

Therefore we have proved: Theorem 4.7.8 The functions u(x) =

(n(n − 2)θ) (θ2 + |x − y|2 )2

n−2 4

, ∀θ > 0, ∀ y ∈ Rn

(4.67)

are minimizers of the problem S = Min{u22 | |u|2∗ = 1}. In fact, all minimizers for S are of the form (4.67), see Caﬀarelli, Gidas, and Spruck [CGS]. A simple proof via the moving plane method [GNN] was given by W. Chen and C. Li, Yanyan Li, see [ChL 2], [Li 2]. Example 2. Find a nontrivial solution u ∈ H01 (Ω) of the following equation, where Ω ⊂ Rn is a bounded domain: −∆u − λu = |u|2

∗

−2

u in Ω .

(4.68)

where we assume λ < λ1 , the ﬁrst eigenvalue of −∆. Again, the minimization problem Min{Iλ (u) | |u|2∗ = 1} is considered, where Iλ (u) = u2 − λ|u|22 . Let Sλ (Ω) = inf {Iλ (u) | u ∈ H01 (Ω), |u|2∗ = 1} .

4.7 Concentration Compactness

299

Lemma 4.7.9 S0 (Ω) = S . Proof. 1. For Ω1 ⊂ Ω2 , by deﬁnition, S0 (Ω2 ) ≤ S0 (Ω1 ). 2. For ∀R > 0, let Ωj = BjR (θ), j = 1, 2. We claim that S0 (Ω1 ) = S0 (Ω2 ) This is due to the scaling invariance of the functional S0 (Ω). 3. By the translation invariance, we conclude that for all Ω, S0 (Ω) = S0 (B1 (θ)). This proves the lemma. Following the steps in Example 1, there exists a minimizing sequence {uj } with |uj |2∗ = 1 and uj u0 in H01 (Ω). Let vj = uj − u0 , we have. 1. Iλ (uj ) = Iλ (u0 ) + Iλ (vj ) + o(1). ∗ ∗ ∗ 2. |uj |22∗ = |u0 |22∗ + |vj |22∗ + o(1). ∗

Let |u0 |22∗ = µ ∈ [0, 1]. It remains to verify that µ = 1. We shall exclude the cases µ = 0, and µ ∈ (0, 1). In fact, if µ ∈ (0, 1), then Sλ ≥ Sλ (|u0 |22∗ + |vj |22∗ ) + o(1) 2

2

= Sλ (µ 2∗ + (1 − µ) 2∗ ) + o(1) > Sλ + o(1) . This is impossible. If µ = 0, then u0 = θ, i.e., uj θ in H01 (Ω), and then |uj |2 → 0. Thus Iλ (uj ) = |∇uj |22 + o(1) 2

≥ S|uj |22∗∗ + o(1) = S + o(1) . Now, we need the following: Lemma 4.7.10 (Brezis–Nirenberg [BN 1] For n ≥ 4, ∀λ > 0, Sλ (Ω) :=

inf

u∈H01 (Ω)\{θ}

Iλ (u) <S. |u|22∗

(4.69)

Proof. One may assume θ ∈ Ω. Let ϕ ∈ C0∞ (Ω), and ϕ(x) = 1 in a small neighborhood of θ. For ε > 0, we deﬁne vε (x) = Then ∇vε (x) =

(ε2

(ε2

ϕ(x) . n + |x|2 ) 2 −1

∇ϕ(x) (n − 2)ϕx − 2 n n . −1 2 2 + |x| ) (ε + |x|2 ) 2

We have constants K1 , K2 , K3 > 0 such that

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4 Minimization Methods

vε 22

= (n − 2)

2 Ω

|x|2 dx + O(1) = (n − 2)2 (ε2 + |x|2 )n

Rn

|x|2 dx + O(1) (ε2 + |x|2 )n

K1 = n−2 + O(1) , ε

∗ |ϕ|2 dx dx K2 2∗ = + O(1) = n + O(1) , |v|2∗ = 2 + |x|2 )n 2 2 n (ε (ε + |x| ) ε Ω Rn and

|vε |22 = Ω

|ϕ|2 dx = (ε2 + |x|2 )n−2

as n ≥ 5, and

Rn

(ε2

dx K3 + O(1) = n−4 + O(1) , 2 n−2 + |x| ) ε

|vε |22 = K3 | log ε| + O(1) ,

as n = 4. In summary, we have

Sλ ≤

⎧ K1 −λK3 ε2 ⎪ if n ≥ 5 , ⎪ 1− 2 ⎨ n K2

K1 −λK3 ε2 log |ε| ⎪ ⎪ if n = 4 . 1 ⎩ 2 K2

Since K3 > 0 and

K1

1− 2 n

K2

= S for small ε > 0, Sλ < S.

Provided by Lemma 4.7.10, the vanishing case is excluded. We have proved the existence of a nontrivial solution. Remark 4.7.11 In fact, the restriction λ < λ1 is not necessary. Provided by minimax methods, which we shall study in the next subsection, one can remove this restriction. Also the case n = 3 has been discussed, see Brezis–Nirenberg [BN 1]. Remark 4.7.12 (Yamabe problem) Let (M, g0 ) be a connected compact ndimensional Riemannian manifold. One asks: Does there exist a metric g pointwise conformal to g0 such that the scalar curvature R with respect to g is a constant? 4

Set g = u n−2 g0 , where u ∈ C ∞ (M ) with u > 0. It is reduced to the following PDE: ∗ 4(n − 1) ∆g0 u + Rg0 u = Ru2 −1 , − n−2 where ∆g0 is the Laplacian Beltrami operator, and Rg0 is the scalar curvature with respect to g0 . Then it is transferred to the variational problem: n−2 |∇g0 u|2 + 4(n−1) R g0 u 2 M . λ(M ) = min 2 u∈H 1 (M ) ( M |u|2∗ ) 2∗

4.8 Minimax Methods

301

This is a problem very similar to Example 2. Again, λ(M ) ≤ S. If λ(M ) < S, then the problem is solvable by the previous method. However, it is easily seen that if (M, g0 ) = (S n , gˆ0 ), where gˆ0 is the canonical metric, then λ(S n ) = S. The Yamabe problem has been solved by Aubin [Au 1] and Schoen [Sco]. They proved: Theorem 4.7.13 For n ≥ 3, if (M, g0 ) is a compact connected Riemannian manifold, which is not conformally equivalent to (S n , gˆ0 ), then λ(M ) < S. Then Yamabe problem is solvable.

4.8 Minimax Methods 4.8.1 Ekeland Variational Principle It is well known that the direct method does not work in the lack of compactness (i.e., the coerciveness). Without coerciveness, only approximate minimizers can be found. Let us recall the Ekeland variational principle, which we have derived in Chap. 2 as an equivalence of Caristi ﬁxed-point theorem. Due to the importance of this principle, we present here a direct proof. Theorem 4.8.1 (Ekeland) Let (X, d) be a complete metric space, and let / + ∞. If f is bounded from below and l.s.c., and f : X → R1 ∪ {+∞}, but f ≡ if ∃ ε > 0, ∃xε ∈ X satisfying f (xε ) < inf X f + ε. Then ∃ yε ∈ X such that 1. f (yε ) f (xε ), 2. d(xε , yε ) 1, 3. f (x) > f (yε ) − εd(yε , x),

∀x = yε .

Proof. Note that the required yε is the minimum of the function f (x) + εd(yε , x), while the new function contains yε itself. Accordingly, we deﬁne a sequence approaching yε . Choose u0 = xε . Suppose that un is already chosen. Set Sn = {w ∈ X f (w) f (un ) − εd(w, un )} . Obviously Sn = ∅. We choose un+1 ∈ Sn satisfying 1 f (un+1 ) − inf f f (un ) − inf f , Sn Sn 2

(4.70)

n = 0, 1, 2, . . . . We want to show that {un } is a Cauchy sequence. Indeed, εd(um , un ) f (un ) − f (um ),

∀m > n .

(4.71)

Since f is bounded below, f (un ) − f (um ) → 0 as m, n → ∞, it follows un → u∗ ∈ X. From the l.s.c. of f and (4.70), we have

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4 Minimization Methods

f (u∗ ) lim f (un ) lim inf f . n→∞ Sn

n→∞

(4.72)

We are going to verify that yε = u∗ satisﬁes the conclusions 1–3. In fact, since from (4.71) f (un ) is nonincreasing, conclusion 1 is trivially true. Again by (4.71), we have εd(xε , yε ) = εd(u0 , u∗ ) f (xε ) − f (u∗ ) f (xε ) − inf f Sn

0, ∃ yε ∈ X such that f (x) > f (yε ) − εd(x, yε ), ∀x = yε . Let us introduce the following notion, which plays an important role in the calculus of variations in the large. Deﬁnition 4.8.3 Let X be a Banach space, and f ∈ C 1 (X, R1 ). If any se quence {xj }∞ 1 ⊂ X along which f (xj ) → c and f (xj ) → θ implies a convergent subsequence. Then f is said to satisfy the (P S)c condition. If f satisﬁes the (P S)c condition, for ∀c, then it is said to satisfy the (P S) condition. Let us return to the minimizing problem: Corollary 4.8.4 Let X be a Banach space, and let f ∈ C 1 (X, R1 ) be bounded from below. If (P S)c holds with c = inf f , then f possesses a minimum. X

Proof. According to Theorem 4.8.1, ∀n 0 ∃ xn ∈ X satisfying f (x) > f (xn ) − n1 x − xn f (xn ) < c + n1 which implies that f (xn ) → c and f (xn ) → θ. Therefore we obtain a convergent subsequence xnj → x∗ , according to (P S)c . So is f (x∗ ) = c = inf f . X

4.8 Minimax Methods

303

4.8.2 Minimax Principle In this section, we are concerned with those critical points that are saddle points but not minima. The following geometric intuition suggests a minimax consideration. A valley is surrounded by mountains. Starting from a point x1 on the ground outside these mountains, we intend to get into a place x0 in the valley. A path l one would like to take, is a path along which the highest point is lower than those at neighboring paths. The highest point on this path is indeed a saddle point of the height function. Theorem 4.8.5 (Ambrosetti–Rabinowitz [AR]) Let X be a Banach space, / Ω. and f ∈ C 1 (X, R1 ). Suppose that Ω ⊂ X is an open set, x0 ∈ Ω, and x1 ∈ Set Γ = {l ∈ C([0, 1], X) l(i) = xi , i = 0, 1} , and c = inf max f ◦ l(t) . l∈Γ t∈[0,1]

If 1. α = inf f (x) > max{f (x0 ), f (x1 )}, x∈∂Ω

2. (P S)c holds for f ; then c α is a critical point. We shall not prove this theorem at this moment, but introduce a more general notion. Deﬁnition 4.8.6 Let X be a Banach space. Let Q ⊂ X be a compact manifold with boundary ∂Q and let S ⊂ X be a closed subset of X. ∂Q is said linking with S, if 1. ∂Q ∩ S = ∅, 2. ∀ϕ : Q → X continuous with ϕ|∂Ω = id |∂Ω , we have ϕ(Q) ∩ S = ∅. Example 4.8.7 (Mountain pass) Let Ω, x0 , x1 be as in Theorem 4.8.5. Set Q = the segment {λx0 + (1 − λ)x1 | λ ∈ [0, 1] }, and S = ∂Ω. Then, ∂Q = {x0 , x1 } and S link. Example 4.8.8 Let X1 be a ﬁnite-dimensional linear subspace of the Banach space X, and let X2 be its complement: X = X1 + X2 . Let Q = BR ∩ X1 , S = X2 , where BR is the ball with radius R > 0 centered at θ. Then ∂Q and S link. Indeed, ∀ϕ : Q → X continuous with ϕ∂Q = id ∂Ω , we want to show: ϕ(Q) ∩ X2 = ∅.

304

4 Minimization Methods

Let P be the projection onto X1 . It is suﬃcient to show that P ◦ϕ : Q → X1 has a zero. Obviously deg(P ◦ ϕ, Q, θ) = deg(id, Q, θ) = 1 . The conclusion follows from the Brouwer degree theory. Theorem 4.8.9 Let X be a Banach space, and let f ∈ C 1 (X, R1 ). Assume that Q ⊂ X is a compact manifold with boundary ∂Q which links with a closed subset S ⊂ X. Set Γ = {ϕ ∈ C(Q, X) ϕ|∂Q = id |∂Q } , and c = inf max f ◦ ϕ(ξ) . ϕ∈Γ ξ∈Q

If ∃ α < β such that sup f (x) α < β inf f (x) , x∈S

x∈∂Q

and if (P S)c holds, then c ( β) is a critical value. Proof. Let d be the distance on C(Q, X). Then (Γ, d) is a metric space. Let J(ϕ) = max f ◦ ϕ(ξ) . ξ∈Q

Invoke the assumption that ∂Q and S link, J inf f . Moreover, J is locally S

Lipschitzian, since J(ϕ1 ) − J(ϕ2 ) max[f ◦ ϕ1 (ξ) − f ◦ ϕ2 (ξ)] ξ∈Q

max f (θϕ1 (ξ) + (1 − θ)ϕ2 (ξ))d(ϕ1 , ϕ2 ) . θ∈[0,1] ξ∈Q

It follows that

J(ϕ1 ) − J(ϕ2 ) Cd(ϕ1 , ϕ2 ) ,

where C is a constant depending on ϕ1 , ϕ2 . We apply the Ekeland variational principle to J, and obtain a sequence {ϕn } ⊂ Γ satisfying 1 , n

(4.74)

1 d(ϕ, ϕn ) , n

(4.75)

c J(ϕn ) < c + and J(ϕ) J(ϕn ) − n = 1, 2, 3, . . . . Set

M(ϕ) = {ξ ∈ Q f ◦ ϕ(ξ) = J(ϕ)} .

4.8 Minimax Methods

305

Obviously M(ϕ) is compact. We claim that M(ϕ) ⊂ int(Q). Indeed, since ∂Q and S link, if ∃ ξ0 ∈ M(ϕ) ∩ ∂Q, then f ◦ ϕ(ξ0 ) = max f ◦ ϕ(ξ) inf f β . S

ξ∈Q

But, f ◦ ϕ(ξ0 ) = f (ξ0 ) sup f (x) α . x∈∂Q

This is a contradiction. Set Γ0 = {ψ ∈ C(Q, X) ψ|∂Q = θ} . It is a linear closed subspace of X. Let · be the norm of C(Q, X). ∀h ∈ Γ0 , with h = 1, ∀λj ↓ 0, ∀ξj ∈ M(ϕn + λj h), we have λ−1 j [ f ◦ (ϕn + λj h)(ξj ) − f ◦ ϕn (ξj ) ] −

1 , n

(4.76)

from (4.75). Since {ξj } ⊂ Q, we obtain a convergent subsequence ξj → ηn∗ ∈ M(ϕn ), which depends on ϕn , λj and h. After taking limits, we have f ◦ ϕn (ηn∗ ), h(ηn∗ ) −

1 . n

(4.77)

We want to show that ∃ ηn ∈ M(ϕn ) such that f ◦ ϕn (ηn ), u −

1 , n

(4.78)

∀u ∈ X with u = 1. If not, ∀η ∈ M(ϕn ), ∃ vη ∈ X with vη = 1, satisfying f ◦ ϕn (η), vη < −

1 , n

then there exists a neighborhood of η, Oη ⊂ int(Q) such that 1 f ◦ ϕn (ξ), vη < − , n

∀ξ ∈ Oη .

Since M(ϕn )+is compact, there is a ﬁnite covering. Let m be the least number m of covering: i=1 Oηi ⊃ M(ϕn ). We obtain the associate {vηi }m 1 , vηi = 1 satisfying 1 ∀ξ ∈ Oηi , f ◦ ϕn (ξ), vηi < − n i = 1, 2, . . . , m. Construct a partition of unity subject to {Oηi }m 1 : 0 i 1, sup i ⊂ Oηi , i = 1, . . . , m, and m i=1

i (ξ) ≡ 1,

∀ξ ∈ M(ϕn ) .

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4 Minimization Methods

We set v = v(ξ) =

m

i (ξ)vηi .

i=1

Thus v ∈ Γ0 and v 1. But, ∃ ξ ∗ ∈ M(ϕn ) such that there is only one i0 satisfying ξ ∗ ∈ Oηi0 . Therefore v = 1. We obtain 1 f ◦ ϕn (ξ), v(ξ) < − , ∀ξ ∈ M(ϕn ) . n This contradicts (4.77). Thus (4.78) holds. Setting xn = ϕn (ηn ), we have f (xn ) → θ . Combining with (4.74), we have also f (xn ) → c . The (P S)c condition implies that c is a critical value.

The above proof is based on Shi [Shi] and Aubin and Ekeland [AE]. The following example due to Brezis and Nirenberg [BN 3] asserts that the (P S)c condition is crucial in Theorem 4.5.5. Example. The function ϕ(x, y) = x2 + (1 − x)3 y 2 deﬁned on R2 does have a mountain surround (0, 0): ϕ(x, y) ≥ c > 0 on the circle x2 + y 2 = 14 , ϕ(0, 0) = 0, and ϕ(4, 1) = −11. But by direct computation it has only one critical point: (0, 0). 4.8.3 Applications The mountain pass lemma and the related minimax principles are widely used in the study of diﬀerential equations. We are satisﬁed to give few examples. Example 1. We study the following forced resonance problem. Given h ∈ L1 ([0, π]) satisfying

π h(t) sin tdt = 0 .

(4.79)

0

Assume that g ∈ C(R1 ) is T -periodic, T > 0, and satisﬁes:

T g(t)dt = 0 .

(4.80)

0

Find a solution of the nonlinear BVP: x ¨(t) + x(t) = h(t) + g(x(t)) t ∈ (0, π) x(0) = x(π) = 0 . First we introduce a generalized Riemann-Lebesgue lemma.

(4.81)

4.8 Minimax Methods

307

Lemma 4.8.10 Under the above assumptions of g, if un → u in C([0, π]), αn → ∞, u ∈ C 1 ([0, π]) and k ∈ L1 ([0, π]), then

π g(un (t) + αn sin t)k(t)dt → 0 . lim n→∞

0

Proof. Since the sequence is bounded in C([0, π]), and since g(un (t) + αn sin nt) − g(u(t) + αn sin t) uniformly converges to 0, it is suﬃcient to prove that

π g(u(t) + αn sin t) · χE (t)dt = 0 ,

lim

n→∞

0

∀E = (a, b) ⊂ (0, π), where χE is the characteristic function of E. We may assume αn → +∞. We write vn (t) = αn−1 u(t) + sin t, and let G(t) be a primitive of g. Then G is T -periodic, according to (4.80). At ﬁrst, we assume π2 ∈ E. Then for n large, s = vn (t) is strictly increasing (decreasing), if b < π2 (or a > π2 resp.). We have t = vn−1 (s). Let δ = dist (E, π2 ), ∃ a constant Cδ depending on δ > 0, such that 1 Cδ a.e. By changing variables, |v (v −1 (s))| n

n

b vn (b) ds g(αn vn (t))dt = g(αn s) −1 a vn (a) vn (vn (s))

Cδ |G(αn vn (b)) − G(αn vn (a))| . αn

(4.82)

Since G is continuous and periodic, the RHS of the above inequality tends to zero as n → ∞. Second, we consider the case π2 ∈ E. Since g is bounded, provided by the absolute continuity of the integral of k, ∀ε > 0, ∃δ > 0 such that

π/2+δ ε (4.83) k(t)g(αn vn (t))dt < . π/2−δ 2 Combining (4.82) with (4.83), the conclusion follows.

Let us reformulate (4.81) in a variational version. Again let G be a primiT tive of g satisfying 0 G(t)dt = 0. Setting X = H01 ([0, π]),

π

π 1 2 2 (u˙ − u ) + hu dt, N (u) = I(u) = G(u(t))dt , 2 0 0 and J(u) = I(u) + N (u) . The functionals I and J are not coercive. In fact, letting un (t) = n sin t, we have I(un ) = 0, and from |G(t)| C, a constant, we have |N (un )| Cπ, and then |J(un )| Cπ.

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4 Minimization Methods

Now, let us decompose X into H1 ⊕ span {e1 }, where e1 = sin t, and 1 H1 = e⊥ 1 , i.e., ∀u ∈ H0 ([0, π]) one has the orthogonal decomposition: u = u1 + αe1 ,

where

π

u(t) sin tdt .

α= 0

Then we have: (1) I is s.w.l.s.c. and coercive on H1 . Moreover, from (4.79), I(u) = I(u1 ). (2) N is bounded and weakly continuous on X, and according to Lemma 4.8.10, N (u1 + αe1 ) → 0 as α → ∞, ∀u1 ∈ H1 . (3) J is bounded from below on X and coercive on H1 . Let

m∗ = inf J(u) . u∈X

Lemma 4.8.11 If m∗ is not a minimum of J, then ∃{u1j } ⊂ H1 and αj → ∞ such that J(u1j + αj e1 ) → m∗ . Moreover, u1j u1 in H1 , which is a solution of x ¨ + x = h, and m∗ = I(u1 ) = inf I(u1 ), u1 ∈ H1 . Proof. 1. We choose a minimizing sequence {uj } of J, and decompose it on H1 ⊕ span {e1 }: uj = u1j + αj e1 . We claim that αj → ∞. Indeed, invoke the coerciveness of I on H1 and I(u1j ) = I(uj ) = J(uj ) − N (uj ) < m∗ + 1 + Cπ , {u1j } is bounded in H1 . Thus after a subsequence u1j u1 in H1 , which implies u1j → u1 in C([0, π]). If {αj } is bounded, then after a subsequence αj → α, we have N (u1j + αj e1 ) → N (u) , where u = u1 + αe1 . We have lim inf I(uj ) = lim I(u1j ) I(u1 ) = I(u) , j→∞

j→∞

and m∗ = lim J(uj ) = lim (I(uj ) + N (uj )) I(u) + N (u) = J(u) . j→∞

j→∞

Therefore, u ¯ is a minimizer of J. This is a contradiction. The claim is proved. 2. ∀α ∈ R1 , ∀u1 ∈ H1 , one has m∗ ≤ J(u1 + αe1 ) = I(u1 ) + N (u1 + αe1 ) .

4.8 Minimax Methods

309

Letting α → ∞, provided by Lemma 4.8.10, N (u1 + αe1 ) → 0, thus m∗ I(u1 ) = I(¯ u1 ),

∀u1 ∈ H1 .

(4.84)

But to the minimizing sequence {uj } we have I(u1 ) lim I(u1j ) j→∞

= lim (J(uj ) − N (u1j + αj e)) j→∞

= lim J(uj ) = m∗ . j→∞

(4.85)

Combining (4.84) and (4.85), we conclude that m∗ = I(u1 ) is the minimum of I on H1 , and u1 is the associated minimizer, which satisﬁes I (u1 ) = 0, i.e., u1 is a solution of x ¨+x=h, x(0) = x(π) = 0 . Lemma 4.8.12 ∀c > m∗ , J satisﬁes (P S)c . Proof. Assume that {uj } is a (P S)c sequence of J, i.e., J(uj ) → c and J (uj ) → θ in X . From the coerciveness of I on H1 , and I(u1j ) = J(uj ) − N (u1j + αj e1 ) c + 1 + cπ , after a subsequence, we have u1j u1 in H1 . We are going to verify that after a subsequence αj → α. For otherwise, αj → ∞, we would have g(uj ) ∗ θ in L∞ ([0, π]), according to Lemma 4.8.10. d 2 ) + 1 has a compact inverse K on Noticing that the diﬀerential operator ( dt H1 , from J (uj ) → θ, we have u1j = Kh + KP g(uj ) + o(1) ,

(4.86)

where P is the orthogonal projection from X onto H1 . Thus u1j → Kh = u1 in H1 strongly. It follows from 2 in the proof of Lemma 4.8.11, c = lim J(uj ) j→∞

= lim (I(u1j ) + N (u1j + αj e1 )) j→∞

= lim I(u1j ) j→∞

= I(u1 ) = m∗ .

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4 Minimization Methods

This is a contradiction. Therefore αj → α. Substituting in (4.86), we obtain a convergent sequence: uj = u1j + αj e1 → Kh + KP g(u1 + αe1 ) + αe1 . Conclusion. The problem (4.81) has at least one solution. Proof. If m∗ is attainable at u∗ , then u∗ is a minimizer, which is a solution of the Euler Lagrange equation (4.81) for J. Otherwise, let b = inf J(u1 ) . u1 ∈H1

Note that J|H1 is s.w.l.s.c. and coercive, ∃u∗1 ∈ H1 such that b = J(u∗1 ). Therefore b > m∗ . Since lim J(u1 + αe1 ) = I(u1 ) = m∗ ,

α→±∞

we may choose a > 0 large enough such that J(u1 ± ae1 ) < b .

Deﬁne Γ=

l ∈ C([0, 1], X)| l

1 1 ± 2 2

= u1 ± ae1

,

and c=

inf

max J(l(t)) .

l∈Γ t∈[0,1]

Since for ∀l ∈ Γ, l[0, 1] ∩ H1 = Ø, we have c b > m∗ . Thus (P S)c holds. According to the mountain pass theorem there is a critical point u∗ with J(u∗ ) = c. Again u∗ solves (4.81). Example 2. We present here an application to the superlinear elliptic boundary problem, see Ambrosetti and Rabinowitz [AR]. Let Ω ⊂ Rn be a bounded domain with smooth boundary. Given a function f ∈ C(Ω × R1 ) satisfying the following conditions: n+2 ) (when n ≥ 3) [F1 ] There exist positive constants C1 , C2 and α ∈ (1, n−2 such that |f (x, t)| ≤ C1 + C2 |t|α . [F2 ] ∃θ ∈ (0, 12 ), ∃ a constant M > 0 such that

t

F (x, s)ds ≤ θtf (x, t), as |t| ≥ M .

0 < F (x, t) := 0

4.8 Minimax Methods

311

[F3 ] lim sup t→0

f (x, t) < λ1 , uniformly in x ∈ Ω . t

[F4 ] f (x, t) > λ1 , uniformly in x ∈ Ω . t where λ1 is the ﬁrst eigenvalue of the Laplacian −∆ on Ω with Dirichlet boundary conditions. lim inf t→∞

Theorem 4.8.13 Under the assumptions F1 , F2 , F3 , F4 , the problem −∆u = f (x, u), x ∈ Ω , (4.87) u = 0, x ∈ ∂Ω , has a nontrivial solution. Proof. We consider the functional on H01 (Ω):

1 |∇u|2 − F (x, u) dx . J(u) = Ω 2

(4.88)

From the growth condition F1 , J is C 1 . Noticing J(θ) = 0, we claim that there is a mountain surrounding θ and / Bρ (θ) with J(u0 ) = 0. Indeed, from F3 , we have ∃α > 0, ∃ρ > 0 and u0 ∈ > 0, δ > 0 such that f (x, t) ≤ λ1 − , as 0 < |t| < δ , t which implies that F (x, t) ≤

1 (λ1 − )t2 as |t| < δ . 2

Combining with F1 , we have a constant C3 such that F (x, t) ≤

1 (λ1 − )t2 + C3 |t|p , 2

where p = α + 1, thus

1 F (x, u(x))dx ≤ u2H 1 + C4 upLp . 1− 2 λ 1 Ω This proves that J(u) ≥

u2H 1 − C4 upLp . 2λ1

Therefore ∃α > 0, ∃ρ > 0 such that J|∂Bρ (θ) ≥ α.

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4 Minimization Methods

Now, we take u0 = sφ1 , where φ1 > 0 is the normalized ﬁrst eigenfunction, and s > 0 is to be determined. Setting

λ1 2 s − F (x, sφ1 (x))dx , g(s) = J(sφ1 ) = 2 Ω From F4 , g(s) → −∞ as s → +∞. Therefore there exists s0 > 0 satisfying g(s0 ) = 0, set s = s0 , u0 is as required. It remains to verify the Palais-Smale condition. Suppose that a sequence {uj } ⊂ H01 (Ω) satisﬁes J(uj ) → c, J (uj ) → θ; we shall show that it possesses a convergent subsequence. Firstly, {uj } is bounded. In fact, ∃C3 > 0 such that

1 −C3 ≤ uj 2H 1 − F (x, uj (x))dx ≤ C3 . 2 Ω According to F2 ,

uj (x)f (x, uj (x))dx ≤ − F (x, uj (x))dx + C4 . , −θ Ω

it follows that

Ω

1 − θ uj 2H 1 + θJ (uj )(uj ) ≤ C5 . 2

Since J (uj ) → θ, this proves that {uj } is bounded in H01 (Ω). Next, we show the existence of a convergent subsequence. Noticing that 2n , from the Sobolev embedding theorem H 1 (Ω) → Lp (Ω) is bounded, p < n−2 so that uj Lp is bounded, and then after a subsequence, {f (x, uj (x))} is weakly convergent in Lp (Ω), p1 + p1 = 1. However, (−∆)−1 : Lp (Ω) → H01 (Ω) is compact. Therefore, (−∆)−1 f (·, uj ) is strongly convergent in H01 (Ω). Since J (uj ) = uj − (−∆)−1 f (·, uj ) → θ ,

uj converges strongly.

It is interesting to compare the assumptions F1 to F4 with those in Sect. 3.6, Example 4. They are very similar. But the diﬀerences lie as follows: The partial diﬀerential operator in the cited example is not necessarily a divergent form, in which case the variational method cannot be applied; while in the above example, the solution is not necessarily positive. Moreover, one may even change the assumption F3 (see below); in these cases the a priori estimate, which the degree theoretic argument relies on, either fails or holds under other assumptions. Example 3. (Ni [Ni]) We make the conditions [F3 ] [F4 ]

f (x, t) = o(t) as t → 0 uniformlyin x ∈ Ω . tf (x, t) ≥ 0, ∀(x, t) ∈ Ω × R1 .

4.8 Minimax Methods

313

Theorem 4.8.14 Under the assumptions F1 , F2 , F3 , F4 , assume a ∈ C(Ω) is positive. Then the following equation possesses a nontrivial solution: −∆u(x) = a(x)u(x) + f (x, u(x)), x ∈ Ω , (4.89) u(x) = 0. x ∈ ∂Ω . Proof. We consider the functional on H01 (Ω):

1 [|∇u|2 − au2 ] − F (x, u) dx , J(u) = 2 Ω For which the Euler–Lagrange equation is (4.89). In the same manner we verify the Palais–Smale condition. We claim that there are linking sets, which separate values of J. Indeed, let 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ 1 < λk+1 ≤ · · · be the eigenvalues and {φ1 , . . . , φk , φk+1 , . . .} be the eigenfunctions of the eigenvalue problem: −∆u = λa · u, u ∈ H01 (Ω) . Taking Ek = span{φ1 , . . . , φk }, we have the orthogonal decomposition: ˆ E. H01 (Ω) = Ek ˆ we have On E,

|∇u|2 dx ≥ λk+1 au2 dx , Ω

therefore

From F3 ,

Ω

1 1 2 J(u) ≥ |∇u| − F (x, u) dx . 1− 2 λk+1 Ω

F (x, u(x))dx = o(u2 ) as u → 0 . Ω

Therefore there exist ρ > 0, α > 0 such that ˆ. J(u) ≥ α as u ∈ ∂Bρ (θ) ∩ E From F4 , F (x, t) ≥ 0 ∀(x, t) ∈ Ω × R1 , it implies that

1 J(u) ≤ [|∇u|2 − a(x)u2 ]dx ≤ 0, ∀u ∈ Ek . 2 Ω From F2 , we have

1

F (x, t) ≥ C1 |t| θ − C2 ,

where C1 , C2 are positive constants. Since θ < 12 ,

1 2 au dx − C1 |u| θ dx + C2 |Ω| → −∞ , J(u) ≤ (λk+1 − 1) Ω

Ω

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4 Minimization Methods

as u ∈ Ek+1 := span{φ1 , . . . , φk , φk+1 } and uH 1 → ∞. There exists R > ρ such that J(u) ≤ 0 as u ∈ ∂BR (θ) ∩ Ek+1 . ˆ and ∂Q = (BR (θ) ∩ Ek ) ∪ (∂BR (θ) ∩ {(x1 , . . . , Setting S = ∂Bρ (θ) ∩ E xk+1 ) ∈ Ek+1 | xk+1 ≥ 0}). Then S and ∂Q link, and they separate values of J. Theorem 4.8.9 yields the existence of a critical value c ≥ α > 0, which implies the existence of a nontrivial solution of equation (4.89).

5 Topological and Variational Methods

This chapter is devoted to the critical point theory and Conley index theory. A typical problem from geometry in the calculus of variations is to ﬁnd geodesics between two points q0 , q1 on a given Riemannian manifold (M, g). A path on M connecting q0 and q1 is denoted by γ : [0, 1] → M with γ(i) = qi , i = 0, 1. Let N = {γ ∈ C 1−0 ([0, 1], M ) | γ(i) = qi , i = 0, 1} , and deﬁne the energy functional:

I(γ) = 0

1

dγ 2 dt , dt

where · is the scalar product induced by the metric g, or in the local coordinates: dγ 2 dγ dγ j , = Σi,j gij (γ(t)) dt dt dt and γ = (γ 1 , . . . , γ p ), p = dim M. Then the Euler–Lagrange equation reads as the geodesic equation: 2 d dγ j dγ k , i = 1, 2, . . . , p . γ i = Σj,k Γijk (γ) dt dt dt 2

Geometrically, a curve satisfying the geodesic equation has the feature that tangents along it are parallel. Generally speaking, there are many geodesics joining q0 and q1 , which are critical points of the energy functional. However, besides those minimal in length, how do we reach the others? Inﬂuenced by the pioneering work of Birkhoﬀ (1917) [Bi] on closed geodesics, two global methods have appeared: The minimax method and the Morse theory. Both are based on topological arguments. The minimax method, as an outgrowth of the max-min characterization of the eigenvalues of Laplacian with Dirichlet data, was successfully developed

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5 Topological and Variational Methods

by Ljusternik and Schnirelmann (1934)[LjS 1]. The method provided a proof of the existence of at least three closed geodesics on a closed surface with genus zero, see Ljusternik and Schnirelmann [Lj, LjS 1], Ballmann [Bm], Klinenberg [Kl]. In the 1950s Krasnosel’ski [Kr 1] introduced the notion of genus, which is a topological index related to the group Z2 , in the study of the nonlinear eigenvalue problem for a class of integral equations. An important step towards the recent development of critical point theory is due to Palais (1966) [Pa 1, Pa 2]. Two major contributions are: (1) The Ljusternik Schnirelmann theory was extended to inﬁnite-dimensional manifolds. (2) Homotopically stable families were used in the minimax principle. The reviva of the study of the minimax method began with the works of Ambrosetti and Rabinowitz (1974)[AR] and Rabinowitz [Ra 3]. The ﬁrst paper deals with functionals unbounded from below, and provides many applications in the study of nonlinear elliptic equations, Hamiltonian systems, and problems from geometry and mathematical physics. The second paper provides a proof of the existence of a closed orbit of an autonomous Hamiltonian system on a star-shaped hypersurface in R2n (independently, see Weinstein [Wei 1]). This is the ﬁrst global result on this problem and is a breakthrough to the Weinstein conjecture. In parallell, in the 1930s, Morse developed a theory which reveals the relationship between the critical points of a nondegenerate function and the topology of the underlying compact manifold. Although the nondegeneracy condition and the compactness assumption in his theory do not meet in most variational problems, he succeeded in applying his theory in the study of closed geodesics and unstable minimal surfaces, see Morse [Mo], Milnor [Mi 1], Bott[Bo 1], Morse and Tompkins [MT], Schiﬀman [Sch]. Moreover, Morse theory became a basic tool in computing the homology of compact manifolds. The work of Smale on the solution of the Poincar´e conjecture for n ≥ 5 pushes the development of the theory to a new peak, see Smale [Sm 1], Milnor [Mi 3]. In the 1950s and 1960s Rothe [Ro 1–3], Palais [Pa 3], and Smale [Sm 2] extended the Morse theory to inﬁnite-dimensional manifolds by using the Palais–Smale condition. Later, Marino and Prodi [MP 1, MP 2] and Gromoll and Meyer [GM] endeavored in weakening the restriction of the nondegeneracy. In applying Morse theory to diﬀerential equations, most functionals are unbounded from below, and the nondegenerate assumption should be removed; to this end, critical groups for isolated critical points are introduced and the relative homology is involved, by which the interconnection with the minimax principle has been revealed, see K. C. Chang [Ch 4,5,7]. Conley extended the Morse theory to ﬂows on a compact space without a variational structure. The Conley index for isolated invariant sets with respect to the ﬂow is a homotopy invariant. A Morse decomposition is also extended. With the aid of Conley’s theory, Floer established his homology theory, which is an important tool in the symplectic geometry.

5.1 Morse Theory

317

In this chapter we shall introduce these theories as tools in the study of multiple solutions for diﬀerential equations. Morse theory is introduced in Sect. 5.1, we shall present it in a way comparing with the Leray–Schauder degree. After the basic theorems, we shall focus on the computation of critical groups for isolated critical points, in particular, those obtained by various minimax methods. Section 5.2 is on the minimax principles, which include the Ljusternik– Schnirelmann category theory, cap length estimates, Krasnosel’ski’s genus and other index theories. Various extensions can be found in Rabinowitz [Ra 4], [Ra 5]. The central result is the multiplicity theorem. Section 5.3 is deals with an application to the Weinstein conjecture, see Viterbo [Vi 1]. We return to the prescribed Gauss curvature problem on S 2 in Sect. 5.4. We shall apply the Morse theory with boundary conditions to attack the problem. We introduce the deﬁnitions of the isolated invariant set, the index pair, and the Conley index in metric space without compactness. The fundamental properties of the Conley index, in particular, the homotopy invariance and the Morse decomposition are studied. Examples and the relationship with the Morse theory are also presented.

5.1 Morse Theory 5.1.1 Introduction Morse revealed a deep relation between the critical points of any nondegenerate function and the topology of the underlying compact manifold M . Let f ∈ C 1 (M ). ∀a ∈ R1 , the set fa = {x ∈ M |f (x) ≤ a} is called a level set. We denote the set of all critical points by K. Here the terminology “critical point” is coincident with that for general diﬀerential mappings, which was introduced in Sect. 1.3.5, i.e., K = {x ∈ M | f (x) = θ}. A real value {c} is called critical if f −1 (c) ∩ K = ∅; otherwise, it is called regular. According to the Sard–Smale theorem, the set of critical values is of the ﬁrst category. A C 2 function f is called nondegenerate if it has only nondegenerate critical points. A critical point p is called nondegenerate if the Hessian f (p) at this point has a bounded inverse (“boundedness” is assumed if M is an inﬁnite-dimensional Hilbert–Riemannian manifold). As an application of the Sard–Smale theorem, the set of nondegenerate functions is dense in C 2 (M ). To a nondegenerate critical point p, we call the dimension of the subspace of negative eigenvectors of f (p) the Morse index, and denote it by ind(f (p)). The basic idea in the Morse theory is to relate the local behavior of a nondegenerate critical point with the variations of the topological structures of the level sets fa . More precisely, if f −1 ∩ K = {p} and ind(f (p)) = j, then fc+ ∼ fc− ∪ B j ,

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5 Topological and Variational Methods

where B j is the j-ball, for > 0 small, “∪” means “attached by” and ∼ means homotopical equivalence. Since M is compact, the nondegenerate function f can only have ﬁnitely many critical points {p1 , . . . , pl }. Then the space M is homotopically equivalent to the space by gluing ﬁnitely many balls with dimensions {ind(f (pi ))|i = 1, . . . , l} according to their levels. The above consideration is based on the following Morse lemma. We assign a Riemannian structure M . Lemma 5.1.1 (Morse lemma) Suppose that f ∈ C 2 (M, R1 ) and that p is a nondegenerate critical point; then there exists a neighborhood U of p and a local diﬀeomorphism Φ : U → Tp (M ) with Φ(p) = θ, such that 1 f ◦ Φ−1 (ξ) = f (p) + (f (p)ξ, ξ)p 2

∀ξ ∈ Φ(U ) ,

where (, )p is the Riemannian structure at p. From the homology point of view, for a pair of topological spaces (A, B) with B ⊂ A, let Hq (A, B, G) be the qth relative homology group with coeﬃcient group G, q = 0, 1, 2, . . . . They are homotopically invariant. According to the Morse lemma, after a linear homeomorphism, roughly speaking, we may assume that f is a quadratic function in a neighborhood B (θ) of θ in Tp (M ) of the form: 1 f (x) = c + ( x+ 2 − x− 2 ) , 2 where x = x+ + x− ∈ H± , dimH− = j, and Rn = H+ ⊕ H− is an orthogonal decomposition, where n = dimM . To characterize homologically the nondegenerate critical point p with Morse index j, we have Hq (fc+ , fc− , G) = Hq (B j , G) = δqj G

∀q ∈ N .

Since p satisﬁes f (p) = θ, it is a nondegenerate zero of the mapping f : M −→ T ∗ (M ), the index of the Brouwer degree at p is well deﬁned and

i(f , p) = sgn det (f (p)) = (−1)j =

n

(−1)q rankHq (B j , G) .

q=0

If further, p is the only critical point in f −1 [c − , c + ], then as a local characterization of f at p, Hq (fc+ , fc− , G), q = 0, 1, . . . , n, is better than the Brouwer index. We are encouraged by using the homotopy invariance {Hq (fb , fa ; G)}nq=0 (for a < b) as a replacement for deg(f , f −1 (a, b), θ). Unfortunately there is no additivity (but only the subadditivity, see later). In the degree theory, the excision property and then the Kronecker existence,

5.1 Morse Theory

319

which makes it useful in the study of ﬁxed points, are based on the additivity; while for relative homology groups, the excision property is related to a deformation argument. In this context, we shall establish the counterpart of the Kronecker existence theorem by deformation as follows: Theorem 5.1.2 (Nontrivial interval theorem) If ∃q ∈ N and ∃a < b such that Hq (fb , fa ; G) is nontrivial, then K ∩ f −1 [a, b] = ∅. One proves the theorem by contradiction, i.e., if there is no critical value in the interval [a, b], then fb ∼ fa . In fact, a homotopy between fb and fa can be easily constructed by the negative gradient ﬂow: x(t) ˙ = −f (x(t)) ∀x ∈ f −1 [a, b] . Noticing that along the ﬂow line f is strictly decreasing and that f −1 [a, b] is compact, the proof is left to readers as an exercise. We shall present Morse theory in this book as a topological tool in the study of the existence and the multiplicity of solutions of certain nonlinear diﬀerential equations with variational structures. To our purpose the two assumptions: the compactness of manifolds and the nondegeneracy of functions, are too restricted. In dealing with inﬁnitedimensional manifolds, a certain kind of compactness is assumed on the function f , e.g., the Palais–Smale condition. Under this condition, deformation theorems are derived, which implies the noncritical interval theorem. Let us use singular relative homology groups with an Abelian coeﬃcient group G, H∗ (X, Y ; G) to describe the topological diﬀerence between the topological spaces X and Y , with Y ⊂ X. Namely, if H∗ (fb , fa ; G) is not trivial, then K ∩ f −1 [a, b] = ∅. This is the main result in Sect. 5.1.2. Instead of the nondegeneracy, we study a certain kind of isolatedness of critical points. A series of critical groups is introduced to replace the Morse index in describing the local behavior of isolated critical points. The basic properties and computations of critical groups are studied, e.g., the shifting theorem and that for mountain pass points. The local theory is described in Sect. 5.1.3. The Morse relation, which is the subadditivity of topological invariances Hq (fb , fa ; G), q = 0, 1, 2, . . . , links, on one hand, the global invariants: H∗ (fb , fa ; G), ∗ = 0, 1, . . . , and on the other hand, the local invariants: the critical groups for isolated critical points in f −1 [a, b]. It is applied to the estimation of the critical groups for isolated critical points obtained by minimax principles. It is also used to set up the relationship between the Leray– Schauder index and the critical groups for an isolated zero of a potential compact vector ﬁeld, by which, we see that for a potential compact vector ﬁeld, critical groups provide more information than the Leray–Schauder index. This is the contents of Sect. 5.1.4. 5.1.2 Deformation Theorem The following terminologies are used:

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5 Topological and Variational Methods

Let X be a topological space. A deformation of X is a continuous map η : X × [0, 1] → X such that η(·, 0) = id. Deﬁnition 5.1.3 (Deformation retract) Given a topological pair Y ⊂ X. A continuous map r : X → Y is called a deformation retract, if r ◦ i = idY and i ◦ r ∼ idX , where i : Y → X is the injection. In this case Y is called a deformation retraction of X. Deﬁnition 5.1.4 (Strong deformation retract) A deformation retract r is called a strong deformation retract, if there exists a deformation η : X × [0, 1] → X, such that η(·, t)|Y = idY ∀t ∈ [0, 1] and η(·, 1) = i ◦ r. Then Y is called a strong deformation retraction of X. According to the homotopy invariance of the relative singular homology group H∗ (X, Y ; G) (where G is the coeﬃcient group), if Y is a strong deformation retraction of X, then H∗ (X, Y ; G) = 0. In the following, M is assumed to be a Banach–Finsler manifold with a Finsler structure · . But for pedagogical reasons, all of the following theorems are proved only on Banach spaces. Readers who are familiar with the background material on inﬁnite-dimensional manifolds may complete the proofs themselves or may refer to [Ch 1]. As we have seen in Sect. 5.1.1, the deformation is constructed by negative gradient ﬂow on Riemannian manifolds. However, if the C 1 function f is deﬁned on a Banach space M , f (x) ∈ M ∗ , the dual space of M , then the gradient ﬂow does not make sense. We introduce: Deﬁnition 5.1.5 (Pseudo-gradient vector ﬁeld) Let M be a Banach–Finsler manifold and f ∈ C 1 (M, R1 ). Let K be the critical set of f . A vector ﬁeld X over M is called a pseudo-gradient if (1) X(p) < 2|f (p)| and 1 := M \K, (2) f (p), X(p) > |f (p)|2 , ∀p ∈ M where , is the duality between T ∗ (M ) and T (M ), · and | · | stand for the Finsler structures on T (M ) and T ∗ (M ), respectively. Our main results of this section are the following: Theorem 5.1.6 (Noncritical interval theorem) If f ∈ C 1 (M, R1 ) satisﬁes (PS)c ∀c ∈ [a, b] and if K ∩ f −1 [a, b] = ∅ , then fa is a strong deformation retraction of fb . As a direct consequence, Theorem 5.1.2 is extended to Banach–Finsler manifolds for C 1 functions satisfying the (P S)c condition ∀c ∈ [a, b]. Theorem 5.1.7 (Second deformation theorem) If f ∈ C 1 (M, R1 ) satisﬁes (PS)c ∀c ∈ [a, b] and M is C 2 . If K ∩ f −1 (a, b] = ∅ and the connected components of K ∩ f −1 (a) are only isolated points, then fa is a strong deformation retraction of fb . We need the following lemmas:

5.1 Morse Theory

321

Lemma 5.1.8 (The existence of a pseudo-gradient vector ﬁeld) ∀f ∈ C 1 (M, R1 ), there exists a continuous pseudo-gradient vector ﬁeld of f on 1 = M \K. M 1, ∃ξ(p0 ) ∈ Tp (M ) such that ξ(p ) = 1 and f (p ), ξ(p ) > Proof. ∀p0 ∈ M 0 0 0 0 2 3 3 |f (p0 )|. Setting Xp0 = 2 |f (p0 )|ξ(p0 ), we have Xp0 < 2|f (p0 )|, and f (p0 ), Xp0 > |f (p)|2

∀p ∈ Vp0 .

1 is metrizable, it is paracompact. There is a locally ﬁnite partition of Since M 1. Let unity {ηβ |β ∈ B}, with supp ηβ ⊂ Vp0 for some p0 = p0 (β) ∈ M X(p) =

ηβ (p)Xp0 (β) .

β∈B

This is a required pseudo-gradient vector ﬁeld.

Lemma 5.1.9 If f satisﬁes (PS)c ∀c ∈ [a, b], and if K ∩ f −1 [a, b] = ∅, then ∃, δ > 0 such that |f (x)| ≥

∀x ∈ f −1 [a − δ, b + δ] .

Proof. If not, then there exists a sequence {xn } ⊂ f −1 [a − n1 , b + n1 ] such that |f (xn )| < n1 . We ﬁnd a subsequence {xn } such that f (xn ) → c ∈ [a, b] and f (xn ) → θ. According to (PS)c , we ﬁnd x∗ ∈ f −1 (c) ∩ K. This is a contradiction. Proof. (Proof of the nontrivial interval theorem) The deformation is constructed by the ﬂow, which deforms the level sets: σ(t) ˙ = −X(σ(t)) σ(0) = x0 ∈ f −1 [a, b] . 1. It is a decreasing ﬂow: d f (σ(t)) = f (σ(t)), σ(t)

˙ ≤ −|f (σ(t))|2 . dt 2. The ﬂow does not stop until it arrives at fa . Indeed, if the maximal existence time for the initial data x0 is Tx0 and f (σ(Tx0 )−0) > a, from Lemma 5.1.9:

t f (σ(ι)), σ(ι) dι ˙ < −2 t ∀ t < Tx0 , a − b ≤ f (σ(t)) − f (x0 ) = 0

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5 Topological and Variational Methods

then Tx0

0 such that lim f ◦ σ(t, x) = a. t→Tx −0

1. We claim that

lim σ(t, x) does exist and then

t→Tx −0

f (σ(Tx − 0, x)) = a . Indeed, (PS)a implies that Ka is compact. Either one of the following cases holds: (a) inf dist(σ(t, x), Ka ) > 0 , t∈[0,Tx )

(b) inf dist(σ(t, x), Ka ) = 0 . t∈[0,Tx )

In case (a), again by (PS)c ∀c ∈ [a, b], ∃α > 0 such that inf t∈[0,Tx )

f (σ(t, x)) ≥ α .

5.1 Morse Theory

323

Thus

dist(σ(t2 , x), σ(t1 , x)) ≤

t2

t1 t2

≤

t1

Since Tx is ﬁnite,

dσ dt dt dt ≤ X(σ(t, x))

t2

t1

|t2 − t1 | dt ≤ . f (σ(t, x)) α

lim σ(t, x) = z.

t→Tx −0

In case (b), we shall prove that ∃z ∈ Ka such that lim σ(t, x) = z .

t→Tx −0

First, we claim that lim dist(σ(t, x), Ka ) = 0 .

t→Tx −0

If not, ∃0 > 0, ∃ti → Tx − 0, such that dist(σ(ti , x), Ka ) ≥ 0 . By the assumption (b), ∃ti → Tx − 0 such that lim dist(σ(ti , x), Ka ) = 0 .

i→∞

Thus we have two sequences t∗i < t∗∗ i both converging to Tx such that 0 , 2 ∗∗ dist(σ(ti , x), Ka ) = 0 , dist(σ(t∗i , x), Ka ) =

and

σ(t, x) ∈ (Ka ) \(Ka )◦0 ∀t ∈ [t∗i , t∗∗ i ], 0

2

where (Ka )δ denotes the δ-neighborhood of Ka . Again by (PS)c ∀c ∈ [a, b], we have inf

∗∗ t∈[t∗ i ,ti ]

f (σ(t, x)) ≥ α > 0 .

Therefore 0 ∗ ≤ dist(σ(t∗∗ i , x), σ(ti , x)) 2

t∗∗ i dσ dt ≤ ∗ dt tt 1 ≤ |t∗∗ − t∗i | → 0 . α i

(5.1)

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5 Topological and Variational Methods

This is a contradiction, (5.1) is proved. It follows that lim f (σ(t, x)) = θ , t→Tx −0

from the compactness of Ka . The (PS)a condition then implies that the limit set A of the orbit {σ(t, x)|t ∈ [0, Tx )} is nonempty and that, for each sequence ti → tx−0 , there ti , x) is convergent. exists a subsequence ! ti such that σ(! Next we prove that A is a compact connected subset of Ka , and then a single point, according to the assumption that the connected components of Ka are isolated points. The compactness of A is obvious. We only want to prove the connectedness. If not, ∃ open subsets O and O such that O ∩ O = ø, A = (O ∩ A) ∪ (O ∩ A) , and O ∩ A = ø, O ∩ A = ø. Choosing z ∈ O ∩ A, z ∈ O ∩ A, ∃ti → Tx − 0, ti → Tx − 0 satisfying σ(ti , x) → z, σ(ti , x) → z . For large i, we have σ(ti , x) ∈ O, σ(ti , x) ∈ O , so ∃t∗i ∈ [ti , ti ] (or [ti , ti ]) such that σ(t∗i , x) ∈ O ∪ O . Denote the limit of σ(t∗i , x) by z ∗ . (It exists because of (PS)a .) So z ∗ ∈ A, but z ∗ ∈ O ∪ O . This is a contradiction. We conclude that A = {z} ⊂ Ka . In both cases, claim 1 is proved. 2. We shall prove the continuity of the function Tx . As in the proof of Theorem 5.1.6, if σ(Tx0 − 0, x0 ) ∈ Ka , then the function Tx is continuous at x0 . So we restrict ourselves to the case z = σ(Tx0 −0, x0 ) ∈ Ka . If Tx is not continuous at such a x0 , then ∃0 > 0, ∃xn → x0 such that |Txn − Tx0 | ≥ 0 , so there exists a subsequence either Txn ≤ Tx0 − 0 , or Txn ≥ Tx0 + 0 . Since

f ◦ σ(Tx − , x) − f ◦ σ(t, x) =

Tx −

f (σ(t, x))dt

t

1 ≤ − (Tx − − t) , 4

5.1 Morse Theory

325

we have

1 f ◦ σ(t, x) ≥ a + (Tx − − t) . 4 But for any ﬁxed > 0, according to the continuous dependence of the initial data ODE, dist(σ(Tx0 − , xn ), σ(Tx0 − , x0 )) → 0 . If Txn ≥ Tx0 + 0 , then we have f ◦ σ(Tx0 − , x − 0) = lim f ◦ σ(Tx0 − , xn ) n→∞ 1 ≥ lim a + (Txn − Tx0 + ) n→∞ 4 1 ≥ a + (0 + ) . 4 Letting → 0, we obtain a≥a+

0 . 4

This is a contradiction. Similarly, we prove that Txn ≤ Tx0 − 0 is impossible. Therefore, Tx is continuous. 3. Finally, we deﬁne the deformation retract as follows: ⎧ if (t, x) ∈ [0, 1] × fa , ⎨x if (t, x) ∈ [0, 1) × (fb \(fa ∪ Kb )) , η(t, x) = σ(Tx t, x) ⎩ σ(Tx − 0, x) if (t, x) ∈ {1} × (fb \(fa ∪ Kb )) . Claim. Only the continuity of η has to be veriﬁed. Four cases are distinguished: (a) (b) (c) (d)

◦

(t, x) ∈ [0, 1] × f a , (t, x) ∈ [0, 1) × (f −1 (a, b]\Kb ), (t, x) ∈ {1} × (f −1 (a, b]\Kb ), (t, x) ∈ [0, 1] × f −1 (a).

Only cases (c) and (d) have to be veriﬁed. Since their proofs are similar, we only give the veriﬁcation for (c). If η is discontinuous at (1, x0 ), then ∃ > 0, ∃tn → Tx0 − 0, and ∃xn → x0 such that dist(σ(tn , xn ), σ(Tx0 − 0, x0 )) ≥ . Let z = σ(Tx0 − 0, x0 )(∈ Ka ), and let ◦

F1 = {z}, F2 = (M \B (z)) ∩ Ka . Both F1 and F2 are compact subsets of Ka . Provided by the assumption of Ka , and Lemma 3.5.2, we have compact subsets K1 , K2 ⊂ Ka such that

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5 Topological and Variational Methods

K1 ∩ K2 = ∅, Fi ⊂ Ki , i = 1, 2, and K1 ∪ K2 = Ka . Obviously, we may take ◦

◦

K1 ⊂ B (z). Let N = K2 ∪ (M \B (z)), then α = dist(N, K1 ) > 0 . The continuity of the ﬂow as well as of the arrival time Tx implies that ∃δ > 0 such that α dist(σ(t, x0 ), σ(Tx0 − 0, x0 )) < 8 if t ∈ [Tx0 − δ, Tx0 ], and ∃δ1 > 0 such that Tx > Tx0 − δ if x ∈ Bδ (x0 ). For t ∈ [Tx0 − δ, Tx0 ) ∩ [Tx0 − δ, Tx ), x ∈ Bδ1 (x0 ), ∃t ∈ (0, δ) such that α dist(σ(t, x), σ(t, x0 )) < . 8 In summary, for such a t, and for any x ∈ Bδt (x0 ), α neighborhood of K1 . σ(t, x) ∈ (K1 ) α4 , the 4 For large n, xn ∈ Bδt (x0 ), let tn be such a t, satisfying σ(tn , xn ) ∈ (K1 ) α4 . Reducing δ, and repeating the above procedure, we obtain subsequences {xn , tn , tn } such that ⎧ ⎨ tn , tn → Tx0 − 0 σ(tn , xn ) ∈ (K1 ) α4 ⎩ σ(tn , xn ) ∈ B (z) . We may assume tn < tn ; then we have tn , tn such that tn ≤ tn < tn ≤ tn and σ(tn , xn ) ∈ ∂[(K1 ) α4 ] σ(tn , xn ) ∈ ∂[(N ) α4 ] σ(t, xn ) ∈ (K1 ) α4 ∪ (N ) α4 ∀t ∈ [tn , tn ] , n = 1, 2, . . . . According to (PS)c , ∀c ∈ [a, b] β = inf { f (x) |x ∈ f −1 [a, b)\((K1 ) α4 ∪ (N ) α4 )} > 0 . Therefore α ≤ dist(σ(tn , xn ), σ(tn , xn )) ≤ 2

tn

tn

1 dσ dt ≤ |tn − tn | → 0 . dt β

This is a contradiction. The proof is completed.

5.1 Morse Theory

327

Remark 5.1.10 A ﬂow on a metric space M is a continuous map η : M × R1 → M possessing the following properties: (i) η(x, 0) = x ∀x ∈ M , (ii) η(η(x, t1 ), t2 ) = η(x, t1 + t2 )

∀t1 , t2 ∈ R1 , ∀x ∈ M .

A set A ⊂ M is said to be (positively) invariant with respect to η, if η(x, t) ∈ A

1 ∀x ∈ A, ∀t ∈ R1 (∀t ∈ R+ ).

! = ∪ η(A, t) (A+ = ∪ η(A, t)) is the By deﬁnition, for any subset A, A 1 1 t∈R

t∈R+

smallest η- (positively) invariant set containing A. With these terminologies, it is easy to extend Theorems 5.1.6 and 5.1.7 to pseudo-gradient ﬂow invariant sets. Namely, let η be a pseudo-gradient ﬂow for f ∈ C 1 (M, R1 ) satisfying (PS)c ∀c ∈ [a, b], and let A be a η-positively invariant set. If K ∩ f −1 (a, b] ∩ A = ∅, and the connected components of K ∩ f −1 (a) ∩ A are only isolated points, then fa ∩ A is a strong deformation retraction of fb ∩ A. 5.1.3 Critical Groups Deﬁnition 5.1.11 Let f be a C 1 function deﬁned on M , let p be an isolated critical point f , and let c = f (p). Cq (f, p) = Hq (fc ∩ U, (fc \{p}) ∩ U ; G) is called the qth critical group of f at p, q = 0, 1, 2, . . . , where U is an isolated neighborhood of p, i.e., K ∩ U = {p}. According to the excision property of the singular homology theory, the critical groups are well deﬁned, i.e., they do not depend on the special choice of U . From the deﬁnition, we have: Example 1. If p is an isolated minimum point of f , then Cq (f, p) = δq0 · G . Under some additional conditions, the converse of the above statement is true. Namely: If f ∈ C 2−0 (M, R1 ) satisﬁes (PS), and if p is an isolated critical point, which is not a local minimum point, then C0 (f, p) = 0 . See [Ch 5].

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5 Topological and Variational Methods

Example 2. If M is n-dimensional, and p is an isolated local maximum point of f , then Cq (f, p) = δqn · G . In the following, we assume that M is a Hilbert–Riemannian manifold. The Morse lemma, which is a special case of the following more general decomposition theorem – the splitting theorem – proved later, is a cornerstone in studying the local behavior of a nondegenerate critical point. Deﬁnition 5.1.12 Let p be a nondegenerate critical point of f , we call the dimension of the negative space corresponding to the spectral decomposition of f (p), the Morse index of p, and denote it by ind(f, p). Example 3. Suppose that f ∈ C 2 (M, R1 ) and p is a nondegenerate critical point of f with Morse index j; then Cq (f, p) = δqj · G . Proof. According to the Morse lemma, after a linear homeomorphism, we may assume that f is a quadratic function on a Hilbert space H of the form: f (x) =

1 ( x+ 2 − x− 2 ) , 2

where x = x+ + x− , x± ∈ H± , and H = H+ ⊕ H− is an orthogonal decomposition. Let U = B be the -ball centered at θ: B ∩ f0 = {x ∈ B | x+ ≤ x− } . Deﬁne η(x, t) = x− + tx+

∀(x, t) ∈ (B ∩ f0 ) × [0, 1] .

It is a strong deformation retract from (B ∩ f0 , B ∩ (f0 \{θ})) to (H− ∩ B , (H− \{θ}) ∩ B ). Thus Cq (f, q) ∼ = Hq (H− ∩ B , (H− \{θ}) ∩ B ) ∼ = Hq (B j , S j−1 ) = δqj · G , if j < +∞. Nevertheless, for j = +∞, since S ∞ is contractible, again, we have Cq (f, p) ∼ =0. The conclusion is proved.

Theorem 5.1.13 (Splitting) Suppose that U is a neighborhood of θ in a Hilbert space H, and that f ∈ C 2 (U, R1 ). Assume that θ is the only critical point of f , and that A = f (θ) is a Fredholm operator. Then there exist a ball B ⊂ U centered at θ, an origin-preserving local homeomorphism ϕ deﬁned on B, and a C 1 map h : B ∩ N → N ⊥ , where N = ker(A) such that

5.1 Morse Theory

f ◦ ϕ(z + y) =

1 (Az, z) + f (y + h(y)) 2

329

∀x ∈ B ,

where y ∈ N and z ∈ N ⊥ , with x = y + z. Proof. • Decomposing H into N ⊕ N ⊥ , let P be the orthogonal projection onto N ⊥ . We solve the equation for ﬁxing y ∈ B ∩ N : P f (y + z) = θ1

(θ1 and θ2 are the origins in N ⊥ and N resp.) .

Since f (θ1 + θ2 ) = θ1 , and f (θ1 + θ2 ) = A, by the implicit function theorem, there exist a ball B and h : B ∩ N → N ⊥ satisfying P f (y + h(y)) = θ1

∀y ∈ B ∩ N .

• Setting u = z − h(y), and letting F (u, y) = f (z + y) − f (h(y) + y) , F2 (u) = 12 (Au, u) , we have F (θ1 , y) = 0 Fu (θ1 , y) = P f (y + h(y)) = θ1 Fu (θ1 , θ2 ) = P f (θ) = A|N ⊥

, , .

Deﬁne ξ : (u, y) → u0 ∈ F2−1 ◦ F (u, y) ∩ {η(u, t)||t| F (u, y) > F2 (η(u, t)) holds for = 2 u ≤ t ≤ u . 1− 1− C We conclude the existence and the uniqueness of the function t(u, y), with = 2 |t(u, y)| ≤ 1 − 1 − u . C Remark 5.1.14 In the case N = {0}, the Morse lemma is a consequence of the above theorem. It remains to verify that ϕ is a diﬀeomophism. We omit the veriﬁcation, in fact, a weaker version is suﬃcient to prove Example 3. Remark 5.1.15 In the study of nonlinear PDEs, sometimes we work on Banach spaces rather than Hilbert spaces. We note that the above theorem can be applied under the following assumptions: Let X be a Banach space, embedded continuously into a Hilbert space H as a dense linear subspace. Assume: (A1) There are Banach spaces as follows: X = Xn

Xn−1 K

...

Yn−1

G

K

G

...

X0 = H

X1

Y1

Fig. 5.2.

(A2) f ∈ C 2 (H, R1 ) has a derivative of the form f (x) = x − K · G(x) ,

K

G Y0

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5 Topological and Variational Methods

where K : Yi → Xi+1 is linear and continuous, and G : Xi → Yi is continuous, i = 0, 1, . . . , n − 1. Let A = I − K · G (θ), and let H∗ , ∗ = −, 0, +, be the negative, null, and positive spaces resp., according to the spectral decomposition of A. (A3) H0 ⊂ X. A map is called regular, if it maps X into X. We claim that the splitting theorem remains true for the Banach space X if assumptions (A1 ), (A2 ) and (A3 ) hold. To this end, we continue to use the notations ϕ, ξ, η, h and B of Theorem 5.1.13, and prove that ϕ is regular. It is suﬃcient to show that the map ψ : (u, y) −→ (ξ(u, y), y) is regular, and that Im h ⊂ X. Indeed, the ﬂow η is a reparametrization of the following ﬂow ζ: ˙ ζ(s) = −ζ(s) + K · G (θ)ζ(s) ζ(0) = u . Or equivalently, −t

t

ζ(t) = e u + K · G (θ)

et−s ζ(s)ds . 0

By a bootstrap procedure, we have ζ ∈ X whenever u ∈ X ∩ B. This shows the regularity of ψ. Again, as an element in H, z = h(y) satisﬁes the equation: P f (y + z) = θ , i.e., h(y) = −P K · G(y + h(y)) + (I − P )h(y) . Noticing that (I − P )H = H0 ⊂ X, P is regular. By the same reason, P Xi ⊂ Xi , i = 0, 1, . . . , n. After a bootstrap iteration, it follows that Im h ⊂ X. Theorem 5.1.16 Under the assumptions of the splitting theorem, (A1), (A2) and (A3), we have Cq (f, θ) = Cq (f |X , θ) .

Proof. We use ϕ and B as in the splitting theorem, and choose U = ϕ(B). Deﬁne a deformation η : U × [0, 1] → U as follows: η(x, t) = ϕ(tx+ + x− ) , where x+ + x− = ϕ−1 (x), and x+ ∈ H+ , x− ∈ H− ⊕ H0 . Then η is a deformation retract fc ∩ U → fc ∩ U− , where U− = ϕ(B ∩ (H− ⊕ H0 )). Therefore C∗ (f, θ) = H∗ (fc ∩ U, fc ∩ U \{θ})

5.1 Morse Theory

333

= H∗ (fc ∩ U− , fc ∩ U− \{θ}) , where c = f (θ). Since ϕ is regular, we also have C∗ (fˆ, θ) = H∗ (fˆc ∩ U− , fˆc ∩ U− \{θ}) , where fˆ = f |X . However, fc ∩ U− = fˆc ∩ U− from (A3). This proves our conclusion. As a consequence of the splitting theorem, we have: Theorem 5.1.17 (Shifting) Assume that the Morse index of f at an isolated critical point p is j < +∞. Under the assumptions of the splitting theorem, we have Cq (f, p) = Cq−j (f!, θ) , where f!(y) = f (y + h(y)) as in the splitting theorem. Proof. We may assume that f is of the form f (z + y) = f1 (z) + f (y + h(y)), where f1 (z) = 12 ( z+ 2 − z− 2 ), z = z+ +z− , (z+ , z− ) ∈ H+ ⊕H− = N ⊥ , and y ∈ N , where the notations are inherited from the splitting theorem. Since dimN < ∞, it is easy to construct a new function f2 such that f2 equals f (y + h(y)) in a neighborhood of θ in N , and f2 satisﬁes the (PS) condition. Let Ui be an isolated neighborhood of θi for fi in the space Yi , i = 1, 2, where Y1 = N ⊥ and Y2 = N . Suppose Ui ⊂ fi−1 [−, ], i = 1, 2, for some !i ∩ f −1 [−, ], then Wi is again an isolated neighborhood > 0. Let Wi = U i !i is the smallest η-invariant set containing Ui , i = 1, 2. of θi for fi , where U We have a strong deformation retract ψ deforming (f2 )ε to (f2 )0 , according to the second deformation theorem. One constructs a deformation as follows: ϕ(z, y, t) = z− + (1 − t)z+ + ψ(y, t) . Accordingly, C∗ (f, p) ∼ = H∗ (f0 ∩ (W1 × W2 ), (f0 ∩ (W1 × W2 )\{(θ1 , θ2 )}); G) ∼ = H∗ (((f1 )0 ) ∩ W1 ) × ((f2 )0 ∩ W2 ) , ((f1 )0 ) ∩ W1 ) × ((f2 )0 ∩ W2 )\{(θ1 , θ2 )}; G) ∼ = H∗ ((f1 )0 ∩ W1 , (f1 )0 ∩ W1 \{θ1 }; G) ⊗H∗ ((f2 )0 ∩ W2 , (f2 )0 ∩ W2 \{θ2 }; G) ∼ C∗ (f1 , θ1 ) ⊗ C∗ (f2 , θ2 ) = Provided by the Kunneth formula, where we omit the veriﬁcation of the pair ((((f1 )0 ∩ W1 )\{θ1 }) × ((f2 )0 ∩ W2 ), ((f1 )0 ∩ W1 ) × ((f2 )0 ∩ W2 )\{θ2 }) is an excision couple. By the use of Example 3, we obtain Cq (f, p) = Cq−j (f2 , θ) . Since f2 (y) = f (y + h(y)) = f!(y), this proves our conclusion.

334

5 Topological and Variational Methods

Corollary 5.1.18 Under the assumption of the shifting theorem with dimN = k, if p is: (1) a local minimum point of f!, then Cq (f, p) = δqj · G; (2) a local maximum point of f!, then Cq (f, p) = δq(j+k) · G; (3) neither a local minimum point nor a local maximum point, then Cq (f, p) = 0 for q ≤ j and q ≥ j + k . Deﬁnition 5.1.19 (Mountain pass point) An isolated critical point p of f is called a mountain pass point if C1 (f, p) = 0. Theorem 5.1.20 Suppose that f ∈ C 2 (M, R1 ) has a mountain pass point p, and that f (p) is a Fredholm operator satisfying the condition: (Φ) f (p) ≥ 0 and 0 ∈ σ(f (p)) =⇒ dim ker f (p) = 1 . Then Cq (f, p) = G · δq1 .

Proof. Let j = ind(f (p)). If p is nondegenerate, then by Example 3: Cq (f, p) = δqj ·G; we have j = 1. The conclusion follows. Otherwise, from the shifting theorem, Cq (f, p) = Cq−j (f!, p); we obtain j ≤ 1. In the case where j = 1, C0 (f!, θ) = 0. θ is a local minimum of f , from Example 1, Cq (f, p) = Cq−1 (f!, θ) = δq1 · G. In the case where j = 0, C1 (f!, θ) = 0. Now, dim ker (f (p)) = 1, θ is a local maximum. From Example 2, Cq (f, p) = Cq (f!, θ) = δq1 · G. The proof is complete. 5.1.4 Global Theory For a function f ∈ C 1 (M, R1 ) satisfying the (PS)c ∀c ∈ [a, b], where a, b are regular values, we know from the nontrivial interval theorem that the nontriviality of H∗ (fb , fa ; G) implies the existence of a critical point in f −1 (a, b). In this subsection we shall show that the topological invariance H∗ (fb , fa ; G) possesses the homotopy invariance and the subadditivity as follows. Theorem 5.1.21 Suppose that {fσ ∈ C 1 (M, R1 )|σ ∈ [0, 1]} is a family of functions satisfying the (PS) condition. Assume that a(σ) and b(σ) are two continuous functions deﬁned on [0, 1] with a(σ) < b(σ), and that both a(σ) and b(σ) are regular values of fσ , ∀σ ∈ [0, 1]. Assume that σ → fσ is continuous in C 1 (M ) topology. Then the homology group H∗ ((fσ )b(σ) , (fσ )a(σ) ; G) is independent of σ.

5.1 Morse Theory

335

Proof. For simplifying notations, ∀σ0 , σ1 ∈ [0, 1], we write fσi , a(σi ), b(σi ), (a(σi ), b(σi )) as fi , ai , bi and Ki , respectively, i = 0, 1. and K(fσi ) ∩ fσ−1 i As |σ1 − σ0 | is small, we have c < d such that f0 (K0 ) ⊂ (c, d) ⊂ [c, d] ⊂ (a0 , b0 ) ∩ (a1 , b1 ) . The ﬁrst inclusion is due to the (PS) condition for f0 . By the continuity of f0 , we have δ > 0 such that f0 ((K0 )δ ) ⊂ (c, d) , where (K0 )δ is the δ-neighborhood of K0 . Again by the (PS) condition of f0 , ∃ = (δ) > 0 such that f0 (x) ≥ ∀x ∈ f0−1 [a0 , b0 ]\(K0 )δ . As |σ1 − σ0 | is small, we have K1 ⊂ (K0 )δ , f1 ((K0 )δ ) ⊂ (c, d) and f1−1 [c, d] ⊂ f0−1 (a, b). Thus fi (Kj ) ⊂ (c, d), i, j = 0, 1. Again as |σ1 − σ0 | is small, we can construct a pseudo-gradient vector ﬁeld for f1 , which coincides with that of f0 in ((f1 )b1 ∩ (f0 )b0 )\(K0 )δ . Thus the pseudo-gradient ﬂow deﬁnes strong deformation retracts ((f1 )b1 , (f0 )c ∩ (f1 )c ) → ((f0 )d ∩(f1 )d , (f0 )c ∩(f1 )c ) and (f1 )c → (f0 )c ∩(f1 )c , according to the nontrivial interval theorem. In the same manner we have strong deformation retracts ((f0 )b0 , (f0 )c ∩ (f1 )c ) → ((f0 )d ∩ (f1 )d , (f0 )c ∩ (f1 )c ) and (f0 )c → (f0 )c ∩ (f1 )c . Therefore the exactness of the homology sequence yields: H∗ ((f0 )b0 , (f0 )c ; G) = H∗ ((f1 )b1 , (f1 )c ; G) . Again by the nontrivial interval theorem, there are strong deformation retracts (f0 )c → (f0 )a0 , (f1 )c → (f1 )a1 . We arrive at H∗ ((f0 )b0 , (f0 )a0 ; G) = H∗ ((f1 )b1 , (f1 )a1 ; G) .

Remark 5.1.22 One may use other homology theories instead, e.g., the sin¯ ∗ (A, B; F ), gular cohomology H ∗ (A, B; F ) or Alexander–Spanier cohomology H with a coeﬃcient ﬁeld F , etc. Since they share most common properties but have only a few diﬀerences, we use special homology theory for special problems. The homotopy invariance H∗ (fb , fa ; G) can be localized on a pseudogradient ﬂow invariant set. For details see Chang and Ghoussoub [CG]. We present here a special case, which will be used later. Let f ∈ C 1 (M, R1 ) satisfy the (PS) condition, and let a < b be regular values. Since K ∩ f −1 [a, b] is compact, it has a bounded neighborhood U ⊂ ! be the η-invariant set for a pseudo-gradient ﬂow η for f . f −1 [a, b]. Let U ! ∩ f −1 [a, b] is bounded. Lemma 5.1.23 The set W = U

336

5 Topological and Variational Methods

Proof. From the (PS) condition, we have > 0 such that f (x) ≥

∀x ∈ f −1 [a, b]\U .

∀y ∈ W \U , there exist x ∈ ∂U and t > 0 (or t < 0) such that y = η(x, t) and ! \U ∀s > 0 (∀s < 0 resp.). Let X be the pseudo-gradient vector η(x, s) ∈ U ﬁeld with respect to η. Thus

t 2 f (η(x, s)) 2 ds t≤ 0

t f (η(x, s)), X(η(x, s)) ds ≤ 0

t d f (η(x, s))ds = f (x) − f (y) ≤ b − a =− ds 0 as t > 0. We obtain the estimate 0 < t ≤ b−a 2 ; similarly, for t < 0, by which, we have the estimate

t y − x = η(x, t) − x ≤ η(x, ˙ s) ds 0

≤2 t

t

f (η(x, s)) 2 ds

1/2

0

≤ 2(t(b − a))1/2 ≤

2(b − a) .

Therefore W is bounded.

Since the regular set is open, we have regular values c, d such that [a, b] ⊂ ! ∩ f −1 [c, d]. (c, d). Let W = U Corollary 5.1.24 With the above notations, the singular homology groups ! , fa ∩ U ! ; G) are invariant under small C 1 (W ) perturbation of f , H∗ (fb ∩ U i.e., if g − f C 1 (W ) is small and both f and g satisfy the (PS) condition, then ! , fa ∩ U ! ; G) = H∗ (gb ∩ U ! , ga ∩ U ! ; G) . H∗ (fb ∩ U

Corollary 5.1.25 With the (PS) condition, the critical groups C∗ (f, p) for an isolated critical point p are invariant under C 1 topology on any bounded neighborhood of p. Now we turn to the subadditivity. Let S be an integer-valued function on certain pairs of spaces. S is called subadditive if S(X, Z) ≤ S(X, Y ) + S(Y, Z), whenever Z ⊂ Y ⊂ X .

5.1 Morse Theory

337

Example. ∀q ∈ N, rank Hq (X, Y ; G) is subadditive. In fact, by examining the exact sequence: j∗

i

∂

∗ Hq (X, Y ; G) →∗ Hq−1 (Y, Z; G) → · · · · · · → Hq (Y, Z; G) → Hq (X, Z; G) →

we obtain rank Hq (X, Z; G) = rank Im i∗ + rank ker i∗ = rank Im i∗ + rank Im j∗ ≤ rank Hq (X, Y ; G) + rank Hq (Y, Z; G) . Applying this to the triple (fc , fb , fa ) with regular values c > b > a, we have rank Hq (fc , fa ; G) ≤ rank Hq (fb , fa ; G) + rank Hq (fc , fb ; G) . If further, we write q (X, Y ) = rank Im i∗ , q (x, z) = rank Im j∗ and q−1 (Y, Z) = rank Im ∂∗ , then rank Hq (X, Y ) = q (X, Y ) + q−1 (Y, Z) , rank Hq (X, Z) = q (X, Z) + q (X, Y ) , rank Hq (Y, Z) = q (Y, Z) + q (X, Z) . Summing them up, we obtain P (t; X, Y ) + P (t; Y, Z) = P (t; X, Z) + (1 + t)Q(t; Y, Z) , where P (t; X, Y ) =

∞

tq rank Hq (X, Y ; G), and Q(t; Y, Z) =

q=0

∞

tq q (Y, Z)

q=0

are formal series with nonnegative coeﬃcients. If there is a multiple X0 ⊂ X1 ⊂ X2 ⊂ · · · ⊂ Xn , we have n

P (t, Xj , Xj−1 ) = P (t, Xn , X0 ) + (1 + t)Q(t) .

(5.2)

j=1

where Q is a formal series with nonnegative coeﬃcients. Thus, ∀ t ≥ 0, P (t; X, Y ) is subadditive, and P (−1, X, Y ) is additive. We turn to establishing the relationship between the singular relative homology groups H∗ (fb , fa , G) and the critical groups C∗ (f, p). This is the Morse relation. Suppose that f ∈ C 1 (M, R1 ) has only isolated critical values, and that each of them corresponds to a ﬁnite number of critical points; say · · · < c−2 < c−1 < c0 < c1 < c2 < · · ·

338

5 Topological and Variational Methods

are critical values with i K ∩ f −1 (ci ) = {zji }m j=1 , i = 0, ±1, ±2, . . . .

One chooses 0 < i < max {ci+1 − ci , ci − ci−1 }, i = 0, ±1, ±2, . . . . Deﬁnition 5.1.26 For a pair of regular values a < b, we call Mq (a, b) = rank Hq (fci +i , fci −i ; G) aδ, and x0 ∈ H will be determined later. Let β = inf { f (x) x ∈ Bδ \Bδ/2 } , The (PS) condition of f yields β > 0. We choose x0 ∈ H such that 0 0 fc+ would be a strong deformation retract of H. Since m is the minimum, fm− = ∅, it follows that χ(fc+ , fm− ) = χ(H) = 1. According to the Morse equality, we would have 1 + (−1)j = 1 .

This is a contradiction.

Example 1. Let Ω ⊂ Rn be a bounded domain with smooth boundary; we study the problem: −u = g(u) in Ω , (5.5) u=0 on ∂Ω . Let λ1 < λ2 ≤ · · · be the eigenvalues of − with the Dirichlet condition, and let ci , i = 1, 2, 3 be various constants. Assume (g1) g ∈ C 1 (R1 ) with g(0) = 0. (g2) |g (t)| ≤

c1 (1 + |t|p−1 ), p < no restriction

n+2 n−2

t (g3) G(t) := 0 g(s)ds ≤ c2 |t|2 + c3 , where c2 < λ1 /2. (g4) ∃i ≥ 1 such that λi < g (0) ≤ λi+1 .

if n ≥ 3 , if n ≤ 2 .

Theorem 5.1.36 The problem (5.5) under assumptions (g1)–(g4) has at least three distinct solutions.

344

5 Topological and Variational Methods

Proof. Deﬁne

J(u) = Ω

1 |∇u|2 − G(u) on H01 (Ω) . 2

According to (g1) and (g2), J is C 2 . (g3) implies that J is bounded below. g(0) = 0 means that θ is a critical point of J. The veriﬁcation of the (PS) condition follows from (g2) and (g3). In fact, (g3) implies that a (PS) sequence must be bounded in H 1 , and (g2) implies that |g(t)| ≤ c4 (1+|t|p ), the compact embedding is applied. Since J (θ) = id − g (0)(−∆)−1 , (g4) implies that θ is nondegenerate with ﬁnite index ind(J (θ)) = i ≥ 1. 2. Bifurcation We know from Theorem 3.5.1 that on a Banach space, for a compact vector ﬁeld with parameter λ : F (x, λ) = x − λT x + N (x, λ), where T is compact, N (x, λ) = ◦( x ) uniformly in λ, the bifurcation point occurs at eigenvalues λ with odd multiplicity. Generally speaking, the odd multiplicity condition cannot be removed. However, if ∀ λ ∈ R1 , F (·, λ) is a potential operator, i.e., the diﬀerential of a certain functional with a parameter λ, we have: Theorem 5.1.37 Suppose that H is a Hilbert space, and that f ∈ C 2 (H, R1 ) satisﬁes f (θ) = θ. If λ0 is an isolated eigenvalue of the self-adjoint operator f (θ) with ﬁnite multiplicity, then (θ, λ0 ) is a bifurcation point of the equation: F (u, λ) = f (u) − λu = θ .

(5.6)

Let f (u) = 12 (Lu, u) + g(u), where g(u) = o(u2 ), then L = f (θ), and f (u) = Lu + G(u), with G(u) = g (u), G(u) = ◦( u ) as u → 0. The proof depends upon the Lyapunov–Schmidt reduction. Let X = ker (L − λ0 I), with dim X = n; and let P, P ⊥ be the orthogonal projections onto X and X ⊥ , respectively. Then (5.6) is equivalent to a pair of equations

λ0 x + P G(x + x⊥ ) = λx , Lx⊥ + P ⊥ G(x + x⊥ ) = λx⊥ ,

(5.7) (5.8)

where u = x + x⊥ , x ∈ X, x⊥ ∈ X ⊥ . According to the IFT, equation (5.8) is uniquely solvable in a small bounded neighborhood O of (λ0 , θ) ∈ R1 × X, say x⊥ = ϕ(λ, x) for (λ, x) ∈ O, where ϕ ∈ C 1 (O, X ⊥ ). Substituting x⊥ = ϕ(λ, x) into (5.7), we have (5.9) λ0 x + P G(x + ϕ(λ, x)) = λx , which is again an Euler–Lagrange equation on the ﬁnite-dimensional space X. Indeed, let

5.1 Morse Theory

345

λ ( x 2 + ϕ(λ, x) 2 ) 2 λ 1 1 = (λ0 − λ) x 2 + (Lϕ, ϕ) − ϕ 2 +g(x + ϕ) . 2 2 2

Jλ (x) = f (x + ϕ(λ, x)) −

It is easy to verify that (5.9) is the Euler–Lagrange equation for Jλ , and that ϕ(λ, x) = ◦( x ) as x → θ. The problem is reduced to ﬁnding the critical points of Jλ near x = θ for ﬁxed λ near λ0 , where Jλ ∈ C 1 (Ω1 , R1 ), Ω1 is a neighborhood of θ in X. If (θ, λ0 ) were not a bifurcation point, then there would be a small neighborhood Ω1 × (λ0 − δ, λ0 + δ), δ > 0 such that θ is the unique critical point in Ω1 for Jλ as |λ − λ0 | < δ. However, as λ < λ0 , θ is a local minimum point, Cq (Jλ , θ) = δq0 G, but as λ > λ0 , θ is a local maximum point, Cq (Jλ , θ) = δqn G. This contradicts the invariance of critical groups under C 1 -perturbation on bounded neighborhoods of θ. Note: From the degree point of view, there is no diﬀerence between λ > λ0 and λ < λ0 , except n is odd. Compare with Krasnoselski’s theorem (Theorem 3.5.1). 3. Superlinear Elliptic Problem We turn to the problem studied previously in Sect. 3.6 and Sect. 4.8. Example 2.

−u = g(x, u) in Ω , u = 0 on ∂Ω ,

(5.10)

where Ω ⊂ Rn is a bounded domain with smooth boundary. Assume: n+2 (g1) |g(x, t)| ≤ C(1 + |t|α ) α < n−2 if n ≥ 3. (g2) ∃θ > 2, ∃M > 0 such that 0 < θG(x, t) ≤ tg(x, t) ∀x ∈ Ω, for |t| ≥ M , t where G(x, t) = 0 g(x, s)ds. (g3) g ∈ C 1 (Ω × R1 ) with g(x, 0) = gt (x, 0) = 0.

Deﬁne J(u) =

Ω

1 |∇u|2 − G(x, u(x)) dx . 2

Lemma 5.1.38 (Z.Q. Wang) Under the assumptions of (g1), (g2) and (g3), there exists a constant A > 0, such that Ja S ∞ , the unit sphere in H01 (Ω) for −a > A. Proof. By (g2 ), we have a constant C > 0 such that G(x, t) ≥ C(|t|θ − 1) ∀t, |t| ≥ M .

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5 Topological and Variational Methods

Thus ∀u ∈ S ∞ ,

J(tu) → −∞ as t → +∞ .

First we want to show that ∃A > 0 such that ∀a < −A, if J(tu) ≤ a, then d dt J(tu) < 0. In fact, set |g(x, t)| + 1 . A = 2M |Ω| max (x,t)∈Ω×[−M,M ]

If J(tu) =

2

t 2

−

Ω

G(x, tu(x))dx ≤ a, then

d J(tu) = (J (tu), u) dt

= t− u(x) · g(x, tu(x))dx Ω

2 1 ≤ G(x, tu(x))dx − tu(x) · g(x, tu(x))dx + a t 2 Ω Ω

1 1 2 ≤ − tu(x)g(x, tu(x))dx + (A − 1) + a t θ 2 |tu(x)|≥M

1 1 2 θ θ − ≤ |t| |u(x)| dx − 1 < 0 , Cθ t θ 2 |tu(x)|≥M as a < −A. The implicit function theorem is employed to obtain a unique T (u) ∈ C(S ∞ , R1 ) such that J(T (u)u) = a ∀u ∈ S ∞ . Next, we claim that T (u) possesses a positive lower bound > 0. In fact, 2 by (g3), g(x, 0) = gt (x, 0) = 0, J(tu) = t2 − ◦(t2 ) ∀u ∈ S ∞ as |t| is small. The conclusion follows. Finally, let us deﬁne a deformation retract η : [0, 1] × (H\B (θ)) → H\B (θ), where H = H01 (Ω) and B (θ) is the -ball with center θ, by η(s, u) = (1 − s)u + sT (u)u ∀u ∈ H\B (θ) . This proves that H\B (θ) Ja , i.e., Ja S ∞ .

Indeed, J satisﬁes the (PS) condition (see e.g., Sect. 4.8.3, Example 2). It is easy to give a diﬀerent proof of the conclusion we obtained previously, i.e., there is a nontrivial solution of (5.10). Indeed, θ is a critical point of J, which is an isolated minimum. Therefore Cq (J, θ) = δq0 G. It is known that S ∞ is contractible. The pair (H01 (Ω), Ja ) must be trivial, i.e., Hq (H01 (Ω), Ja ; G) = 0 ∀q .

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If there were no other critical points, then the Morse relation would be 1 = 0. This is impossible. In fact, from Sect. 4.8, Example 2, we have already known that there is a critical point via the mountain pass lemma, we shall see in the next section, equation (5.10) possesses at least three nontrivial solutions.

5.2 Minimax Principles (Revisited) The minimax method was initiated by G. Birkhoﬀ in his pioneering work on closed geodesics. It was successfully developed by Ljusternik and Schnirelmann in proving that on a closed surface with genus zero there are at least three closed geodesics. The method has been well developed by M. A. Krasnoselski, P. S. Palais, A. Ambrosetti and P. Rabinowitz etc. 5.2.1 A Minimax Principle We have studied saddle points by the minimax method in Sect. 4.8. In this section we shall introduce a general minimax principle based on the deformation theorem, which contains various types of this method as special cases. Deﬁnition 5.2.1 For a given function f ∈ C(M, R1 ) satisfying the (PS) condition on a Banach–Finsler manifold M , for a given a ∈ R1 , let Φa (f ) = {φ = η(1, ·) | η ∈ C([0, 1] × M, M ) satisfy η(t, x) = x, ∀ (t, x) ∈ ({0} × M ) ∪ ([0, 1] × fa )}. A family of subsets F of M is called Φa (f )-invariant if ∀A ∈ F φ(A) ∈ F, whenever φ = Φa (f ). Example 1. (Linking) Let X be a Banach space, let Q ⊂ X be a compact manifold with boundary ∂Q and let S ⊂ X be a closed subset of X. Assume ∂Q and S link (see Deﬁnition 4.8.6). Let f ∈ C 1 (X, R1 ) satisfy the (PS) condition. Assume a = max∂Q f < inf S f . We set the family F = {ψ(Q)|ψ ∈ C(Q, X) with ψ|∂Q = id∂Q }. Then F is Φa (f )-invariant. In fact, for A = ψ(Q), one has φ(A) = φ ◦ ψ(Q), ∀φ ∈ Φa (f ), and then φ ◦ ψ|∂Q = id|∂Q , so is φ(A) ∈ F. Example 2. (Homology class) Let f ∈ C 1 (M, R1 ) satisfy the (PS) condition. For a pair of real numbers a < b, if [σ] ∈ Hq (fb , fa ; G) is a nontrivial q-relative homology class for some q ∈ N. Set F = {|σ||σ ∈ [σ] is a singular q-closed chain in (fb , fa ) with coeﬃcient group G} , where |σ| is the support of σ. Then F is Φa (f )-invariant. In fact, let φ = η(1, ·) where η ∈ C([0, 1] × M, M ) satisﬁes η(t, x) = x ∀ (t, x) ∈ ({0} × M ) ∪ ([0, 1] × fa ), then ∀σ ∈ [σ], φ(|σ|) = |φ(σ)|, and φ(σ) ∈ [σ].

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Theorem 5.2.2 (Minimax principle) Suppose that M is a smooth Banach– Finsler manifold and that f ∈ C 1 (M, R1 ) satisﬁes the (PS) condition. Let F be a Φa (f )-invariant family of subsets of M for some a ∈ R1 . Set c = inf sup f (x) . A∈F x∈A

(5.11)

If (1) c is ﬁnite, (2) a < c, then c is a critical value of f . Proof. It is proved by contradiction. If c is not a critical value, then there exists > 0 such that a < c − , f −1 [c − , c + ] ∩ K = ∅, provided by the (PS) condition. Taking A0 ∈ F such that sup f (x) < c + i.e., A0 ⊂ fc+ . x∈A0

According to the nontrivial interval theorem, ∃ a strong deformation retraction η : [0, 1] × M → M such that η(0, ·) = id, η(t, ·)|fc− = idfc− ∀ t ∈ [0, 1], and η(1, fc+ ) ⊂ fc− . Thus η(t, x) = x ∀ (t, x) ∈ ({0} × M ) ∪ ([0, 1] × fa ), and then φ := η(1, ·) ∈ Φa (f ). Since F is Φa (f )-invariant, it follows that inf sup f (x) < c − .

A∈F x∈A

This is a contradiction.

Thus, in Example 1, Since Q is compact, c < ∞. From the S and ∂Q link, we have c > a. Theorem 5.2.2 implies Theorem 4.8.9. Also, applying Theorem 5.2.2 to Example 2, c = inf sup f (x)

(5.12)

σ∈[σ]x∈|σ|

is a critical value of f . In fact, obviously c is ﬁnite. We verify that c is a critical value, if not, c ≤ a. ∀ > 0, ∃σ ∈ [σ], with |σ| ⊂ fc+ , according to the second deformation theorem, there exists η deforming fc+ to fc . Since η(1, σ) ∈ [σ] and |η(σ)| ⊂ fa , [σ] is trivial in (fb , fa ). A contradiction. What can we say about the critical point with critical value c deﬁned by equation (5.12)? Theorem 5.2.3 Suppose that M is a smooth Banach–Finsler manifold, and that f ∈ C 1 (M, R1 ) satisﬁes (PS)d ∀d ∈ [a, b], where a, b are regular values. If [σ] is a nontrivial class in Hq (fb , fa ; G), and c is deﬁned by (5.12). Then c ∈ (a, b) and K ∩ f −1 (c) = ∅. Moreover, if c is isolated and K ∩ f −1 (c) consists of isolated critical points, then ∃p ∈ K ∩ f −1 (c) such that Cq (f, p) = 0 . Proof. The ﬁrst part of the conclusion has been proved. Since c is an isolated critical value, i.e., ∃ > 0 such that f −1 [c − , c + ] ∩ K = K ∩ f −1 (c) and a < c − .

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It is suﬃcient to show Hq (fc− , fc+ ; G) = 0, for then by Theorem 5.1.27, there must be a p ∈ K ∩ f −1 (c) such that Cq (f, p) = 0. Now we prove it by contradiction. If Hq (fc+ , fc− ; G) = 0, then by the long exact sequence j∗

i

∗ Hq (fc+ , fa ; G) −→ Hq (fc+ , fc− ; G) −→ · · · , · · · −→ Hq (fc− , fa ; G) −→

we have Im i∗ = Hq (fc+ , fa ; G). This shows that the class [σ] is again non trivial in Hq (fc− , fa ; G). But this contradicts the deﬁnition of c. Combining Theorem 5.2.3 with the shifting theorem, we have: Corollary 5.2.4 Let M be a smooth Hilbert–Riemannian manifold. Suppose that f ∈ C 2 (M, R1 ) satisﬁes the assumptions of Theorem 5.2.3, and that f (x) is a Fredholm operator ∀x ∈ K ∩ f −1 (c). Then ∃p ∈ K ∩ f −1 (c) such that ind(f (p)) ≤ q ≤ ind(f (p)) + dim ker f (p). Return to the mountain pass point, we have: Corollary 5.2.5 Suppose that M is a smooth Hilbert–Riemannian manifold and that f ∈ C 1 (M, R1 ). If ∃p0 , p1 ∈ M such that c = inf sup f ◦ l(t) > max {f (p0 ), f (p1 )} , l∈Γt∈[0,1]

where Γ = {l ∈ C([0, 1], M )|l(i) = pi , i = 0, 1}. Assume (PS)c . If K ∩ fc consists of isolated points, then there exists a mountain pass point p ∈ K ∩ f −1 (c) (see Deﬁnition 5.1.19) i.e., C1 (f, p) = 0 .

(5.13)

5.2.2 Category and Ljusternik–Schnirelmann Multiplicity Theorem The notion of category was introduced by Ljusternik and Schnirelmann. It is a topological invariant used in the estimate of the lower bound of the number of critical points. Deﬁnition 5.2.6 Let M be a topological space, A ⊂ M be a closed subset. Set catM (A) = inf{m ∈ N ∪ {+∞} | ∃m contractible closed subsets of M : F1 , F2 , . . . , Fm such that A ⊂ ∪m i=1 Fi } . A set F is called contractible (in M ) if ∃η : [0, 1] × M → M such that η(0, ·) = idM and η(1, F ) = one-point set. Example 1. If C is a closed convex set in a Banach space X, then catX (C) = 1. In fact, pick any p ∈ C, and set

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5 Topological and Variational Methods

η(t, x) = (1 − t)x + tp . Thus C is contractible. Example 2. Let S n be the unit ball in Rn+1 . Then catS n (S n ) = 2. It is easy to see that catS n (S n ) ≤ 2, because the upper and the lower n = {(x , xn+1 ) ∈ Rn × R1 | x 2 +x2n+1 = 1, ±xn+1 ≥ hemispheres S± 0} are contractible. On the other hand, S n is not contractible in itself, i.e., catS n (S n ) > 1. This can be shown by contradiction. Suppose there were a continuous η : [0, 1]×S n → S n such that η(0, ·) = idS n and η(1, S n ) = p ∈ S n . Let A : B1n+1 (θ) → S n be deﬁned by: −p if x = θ , A(x) = x −η(1− x , x ) if x ∈ B1n+1 (θ)\{θ} , then A would be continuous. According to the Brouwer ﬁxed-point theorem, ∃p0 ∈ S n such that A(p0 ) = p0 , But by the deﬁnition of A, A(p0 ) = −p0 , thus p0 = θ. This is a contradiction. Example 3. S ∞ is contractible, so is catS ∞ (S ∞ ) = 1. (see Sect. 3.3).

CatT 2 (T 2) = 3 Fig. 5.3.

Example 4. Let T 2 be the two-dimensional torus, i.e., S 1 × S 1 . Then catT 2 (T 2 ) = 3. This can be shown by Fig. 5.3. The following fundamental properties for the category hold: catM (A) = 0 ⇔ A = ∅. (Monotonicity) A ⊂ B ⇒ catM (A) ≤ catM (B). (Subadditivity) catM (A ∪ B) ≤ catM (A) + catM (B). (Deformation nondecreasing) If η : [0, 1] × M → M is continuous such that η(0, ·) = idM , then catM (A) ≤ catM (η(1, A)). (5) (Continuity) If A is compact, then there is a closed neighborhood N of A such that A ⊂ int(N ) and catM (A) = catM (N ). (6) (Normality) capM ({p}) = 1 ∀p ∈ M .

(1) (2) (3) (4)

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351

In fact properties (1), (2), (3) and (6) follow from the deﬁnition directly. We shall prove (4) and (5) below. Proof. Proof of (4). Let B = η(1, A) and let catM (B) = m, i.e., ∃ closed −1 (1, Fi ), contractible sets F1 , . . . , Fm of M such that B ⊂ ∪m i=1 Fi . Let Gi = η then Gi is closed and contractible, because ∃ξ : [0, 1] × M → M such that ξ(0, . . .) = idM and ξ(1, Fi ) = pi ∈ M . Now let ϕ = ξ ◦ η : [0, 1] × M → M , we have ϕ(0, ·) = idM and ϕ(1, Gi ) = pi , i = 1, 2, . . . , m. Since A ⊂ ∪m i=1 Gi , property (4) is proved. In order to verify (5), we need a result on continuous extensions due to Hanner [Hann] and Palais [Pa 3]: Let X, Y be metrizable Banach manifolds, and let A ⊂ X be a closed subset. Then for any continuous map φ : ({0} × X) ∪ ([0, 1] × A) → Y , there is a continuous extension φ! : [0, 1] × X → Y . Proof. Since A is compact, catM (A) < ∞, i.e., ∃ closed contractible sets F1 , . . . , Fm such that A ⊂ ∪m i=1 Fi . Since ∃ϕi :: [0, 1] × M → M such that ϕi (0, . . .) = idM and ϕi (1, Fi ) = pi ∈ M , we apply the above continu!i : [0, 1] × M → M such that ous extension theorem to φi and obtain ϕ ϕ˜i |({θ}×M )∪([0,1]×Fi ) = ϕi |({θ}×M )∪([0,1]×Fi ) , i = 1, 2, . . . , m. Let Vi be a !−1 closed contractible neighborhood of pi , then Ui = ϕ i (1, Vi ) is closed and −1 m contractible, because ϕ˜i (0, Vi ) = Vi . Thus N = ∪i=1 Ui is a closed neighbor hood of A, we obtain m = catM (A) ≤ catM (N ) ≤ m. As a consequence of the additivity and normality, we have (7) If catM (A) = m, then A ≥ m, i.e., there are at least m distinct points in A. The main theorem of this subsection is the following multiplicity theorem: Theorem 5.2.7 (Ljusternik–Schnirelman theorem) Let M be a smooth Banach–Finsler manifold. Suppose that f ∈ C 1 (M, R1 ) is a function bounded from below, satisfying the (PS) condition. Then f has a least catM (M ) critical points. This theorem is based on the following version of the deformation theorem: Theorem 5.2.8 Let M be a smooth Banach–Finsler manifold. Suppose f ∈ C 1 (M, R1 ). Let N ⊂ N be two closed neighborhoods satisfying 7 δ, δ > 0 . 8 Suppose that there are positive constants b and , such that dist(N , ∂N ) ≥

df (x) ≥ b ∀x ∈ fc+ \(fc− ∪ N ) , 1 2 1 0 < < 3 min δb , δb . 4 8 Then for any 0 < < 2 , there exists η ∈ C([0, 1] × M, M ) satisfying

352

(1) (2) (3) (4) (5)

5 Topological and Variational Methods

η(0, ·) = id, η(t, ·)|Cf −1 [c−,c+] = id|Cf −1 [c−,c+] , η(t, ·) : M → M is a homeomorphism ∀t ∈ [0, 1], η(1, fc+ \N ) ⊂ fc− , f ◦ η(t, x) is nonincreasing in t ∀(t, x) ∈ [0, 1] × M .

Proof. Deﬁne a smooth function: 0 for s ∈ [c − , c + ] , p(s) = 1 for s ∈ [c − , c + ] , with 0 ≤ p(s) ≤ 1. Let A = M \(N ) δ , and B = N be two closed subsets. Let 8

g(x) =

dist(x, B) . dist(x, A) + dist(x, B)

We see 0 ≤ g ≤ 1, g = 0 on N and g = 1 outside (N ) δ . Deﬁne 8

q(s) =

1 1 s

0≤s≤1, s≥1.

Letting X be a pseudo-gradient vector ﬁeld of f , we deﬁne V (x) = −g(x)p(f (x))q( X(x) )X(x) . Thus V ∈ C 1−0 , and V (x) ≤ 1. Consider the ODE: σ(t) ˙ = V (σ(t)) σ(0) = x0 ∀x0 ∈ M . Since V is bounded, the global existence and uniqueness of the ﬂow σ(t) on R1 are known. Let η(t, x) = σ(t), with σ(0) = x . Then η ∈ C([0, 1] × M, M ) satisﬁes (1), (2), (3) and (5). It remains to verify (4). From (5), it is suﬃcient to verify (4) for x ∈ fc+ \(fc− ∪ N ). More precisely, we shall prove: 3 δ, x ≤ c − ∀x ∈ fc+ \(fc− ∪ N ) . f ◦η 4 Indeed, if not, ∃x ∈ f −1 [c − , c + ]\N such that 3 c − < f ◦ η(t, x) ≤ c + ∀t ∈ 0, δ , 4 it follows that p(f ◦ η(t, x)) = 1 .

5.2 Minimax Principles (Revisited)

353

Noticing that η(t, x) − η(0, x) ≤ t, we have dist(η(t, x), (N ) δ ) ≥ dist(η(0, x), (N ) δ ) − t 8 8 7 1 3 − − > δ=0, 8 8 4 so g ◦ η(t, x) = 1. Now, d f ◦ η(t, x) = f (η(t, x)), η(t, ˙ x)

dt = −q( X(η(t, x)) ) f (η(t, x)), X(η(t, x))

≤ −q( X(η(t, x)) ) f (η(t, x)) 2 . If X(η(t, x)) ≤ 1, then we have d f ◦ η(t, x) ≤ − f (η(t, x)) 2 ≤ −b2 . dt Otherwise, d f ◦ η(t, x) ≤ − f (η(t, x)) 2 / X(η(t, x)) dt 1 1 ≤ − f (η(t, x)) < − b . 2 2 In summary,

d 2 1 f ◦ η(t, x) ≤ − min b , b . dt 2

Thus, 3 2 1 f ◦ η(t, x) ≤ f ◦ η(0, x) − δ min b , b 4 2 1 2 1 ≤ c + − 3δ min b , b > 0 such that (1)–(5) hold. With the aid of category, we introduce families of subsets of M . Let Fk = {A| closed subset of M with catM (A) ≥ k}, ∀k ∈ N .

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5 Topological and Variational Methods

If f ∈ C 1 (M, R1 ) is bounded from below, then Fk is Φa (f )-invariant with a < inf M f , provided by the deformation nondecreasing property (4) of the category. According to minimax principle ck = inf sup f (x) A∈Fk x∈A

k = 1, 2, . . . ,

(5.14)

are critical values whenever f satisﬁes the (PS) condition. Since Fk+1 ⊂ Fk , we have ck ≤ ck+1 , k = 1, 2, . . .. Generally, we do not have ck < ck+1 . However, in order to get the multiplicity of critical points, we need: Lemma 5.2.10 Let M be a smooth Banach–Finsler manifold. Suppose that f ∈ C 1 (M, R1 ) is bounded from below and satisﬁes the (PS)c condition, where c = ck+1 = · · · = ck+m , are deﬁned by (5.14), then catM (Kc ) ≥ m. Proof. Since Kc is compact, provided by the continuity of the category, we have a closed neighborhood N of Kc such that catM (N ) = catM (Kc ). From the deﬁnition of c, ∀ > 0 ∃ a closed subset A ⊂ fc+ such that catM (A ) ≥ k + m. Applying ﬁrst deformation lemma, ∃η : [0, 1] × M → M satisfying (1)–(5) in Theorem 5.2.8. Let φ = η(1, ·), then ◦

◦

φ(A \N ) ⊂ φ(fc+ \N ) ⊂ fc− . It follows that ◦

k + m ≤ catM (A ) ≤ catM (A\N ) + catM (N ) ◦

≤ catM (φ(A\N )) + catM (Kc ) ≤ catM (fc− ) + catM (Kc ) ≤ k + catM (Kc ) . Therefore catM (Kc ) ≥ m . Proof. Proof of Theorem 5.2.7: Deﬁne ck , k = 1, 2, · · · catM (M ); by (5.14), we have c1 ≤ c2 ≤ . . . ≤ ck ≤ . . .. According to Lemma 5.2.10, the multiplicity of c = ck is dominated by catM (Kc ), and then by the number of critical points in Kc . Combining with the fact that distinct critical values correspond to distinct critical points, the conclusion is proved. 5.2.3 Cap Product It is natural to ask: If we have two singular homology classes [τ1 ], [τ2 ] ∈ H∗ (fb , fa ; G), both nontrivial, and if c1 , c2 are deﬁned in the same way as

5.2 Minimax Principles (Revisited)

355

in (5.12), are there two distinct critical points? Generally speaking, no, but with the aid of the notion of the cap product, the answer is yes if [τ1 ] and [τ2 ] are related by a cap product. Noticing that there is a ring structure on the cohomology groups. let X be a topological space and Cq (X, G) be the set of all singular q-chains in X with coeﬃcient-group G. Let C q (X, G) be the set of all singular q-cochains, i.e., Hom(Cq (X, G), G). It is a module, and then the duality [,] is a bilinear form over Cq (X, G) × C q (X, G). For a topological pair (X, Y ), let C q (X, Y ; G) = {c ∈ C q (X, G)|[σ, c] = 0 ∀σ ∈ Cq (Y, G)} . Under the duality, the coboundary operator δq : C q−1 (X, G) → C q (X, G) is deﬁned to be the dual of the boundary operator ∂q : Cq (X, G) → Cq−1 (X, G): [∂q σ, c] = [σ, δq c]

∀σ ∈ Cq (X, G), ∀c ∈ C q−1 (X, G) .

The qth singular cohomology group H q (X, G) = ker δq+1 /Imδq . q A cup product structure on the graded singular cochains ⊕∞ q=0 C (X, G) is deﬁned as follows: [σ, c ∪ d] = [σ ◦ λq , c] · [σ ◦ µq , d] ∀c ∈ C p (X, G), ∀d ∈ C q (X, G) and ∀σ ∈ Cp+q (X, G), where λp : p → p p p+q and µq : q → p+q read as: λp ( i=0 xi ei ) = i=0 xi ei , and q q µq ( i=0 xi ei ) = i=0 xi ei+p , and j is the standard j-simplex. It is well deﬁned on H ∗ (X, G), and makes the latter a graded algebra. The cup product is bilinear, associative, and possesses the unit element, i.e., the 0-cochain 1, which is deﬁned by [x, 1] = e, the unit element of G, ∀x ∈ X. The cap product is deﬁned to be the dual operator of the cup product. For a topological pair (X, Y ), one deﬁnes ∩ : Cp+q (X, Y ; G) × C p (X, G) → Cq (X, Y ; G) by [σ ∩c, d] = [σ, c∪d] ∀σ ∈ Cp+q (X, Y ; G), ∀c ∈ C p (X; G), ∀d ∈ C q (X, Y ; G). Since one has ∂(σ ∩ c) = (−1)p (∂σ ∩ c − σ ∩ δc)

∀σ ∈ Cp+q (X, Y ; G) ∀c ∈ C p (X, G) ,

the cap product is also induced to relative singular homology groups: ∩ : Hp+q (X, Y ; G) × H p (X, G) → Hq (X, Y ; G) . Deﬁnition 5.2.11 Let (X, Y ) be a topological pair. [τ1 ], [τ2 ] ∈ H∗ (X, Y ; G) are nontrivial classes. [τ1 ] is called subordinate to [τ2 ] if ∃ω ∈ H ∗ (X; G), dim ω > 0 such that [τ1 ] = [τ2 ] ∩ ω , where ∩ is the cap product. In this case we write [τ1 ] < [τ2 ].

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Lemma 5.2.12 Let B ⊂ X be a contractible set, and [ω] ∈ H q (X; G) with q > 0. Then ∃ˆ ω ∈ [ω] such that |ˆ ω | ∩ B = ∅. Proof. Since B is contractible, there is a closed contractible neighborhood N of B (property (5) of the category). For all closed singular q-chains σ, by adding a q + 1 singular chain τ with |τ | ⊂ N and |σ| ∩ N ⊂ |∂τ |, after subdivision, we ﬁnd a closed singular q-chain σN satisfying: (1) σ − σN = ∂τ, τ ∈ Cq+1 (X, G) and |τ | ⊂ N , (2) |σN | ∩ B = ∅.

σ

σN

τ B

N Fig. 5.4.

For any ω ∈ [ω], deﬁne ω ˆ by [σ, ω ˆ ] = [σN , ω] , then we have: (1) ω ˆ ∈ [ω]. Because [σ, ω ˆ ] = [σN , ω] = [σ, ω] − [∂τ, ω] = [σ, ω]

∀σ ∈ Cq (X, G) .

(2) |ˆ ω | ∩ B = ∅. Because ∀σ ∈ Cq (X, G) with |σ| ⊂ B, we have |σN | = ∅. Therefore [σ, ω ˆ ] = [σN , ω] = 0. Theorem 5.2.13 Let M be a smooth Banach–Finsler manifold. Suppose that f ∈ C 1 (M, R1 ) with regular values a < b. Assume that [τ1 ] < [τ2 ] < · · · < [τm ] are nontrivial homology classes in H∗ (fb , fa ; G). Let

5.2 Minimax Principles (Revisited)

357

ci = inf sup f (x), i = 1, 2, . . . , m . τi ∈[τi ]x∈|τi |

If c = c1 = c2 = · · · = cm , and if f satisﬁes the (P S)c condition, then we have catM (Kc ) ≥ m .

Proof. Since Kc is compact, we may choose neighborhoods N ⊂ N ⊂ N of Kc satisfying the following conditions: (1) catM (N ) = catM (Kc ); (2) there exist constants 1 < < < min {b − c, c − a} and η ∈ C(M, M ) such that η|fc− = idfc− η ∼ id , and

η(fc+ \N ) ⊂ fc− ,

according to the ﬁrst deformation theorem. We prove our theorem by contradiction. If catM (Kc ) ≤ m − 1, then there m are (m − 1) contractible closed sets {Bj }m 2 that cover N : ∪j=2 Bj ⊃ N . Since [τ1 ] < [τ2 ] < · · · < [τm ], there exist ω2 , ω3 , . . . , ωm ∈ H ∗ (fb , G), with dim ωj > 0 such that [τj−1 ] = [τj ] ∩ ωj ,

j = 2, 3, . . . , m .

ωj | ∩ Bj = Provided by Lemma 5.2.12, we always may choose ω ˆ j ∈ ωj , with |ˆ ∅, j = 2, 3, . . . , m. Again we choose τ ∈ [τm ] with |τ | ⊂ fc+ . Subdividing τ into τ = σ1 + σ2 + · · · + σm such that |σ1 | ⊂ fc+ \N , and |σj | ⊂ Bj , j = 2, 3, . . . , m , one has ω2 ∪ · · · ∪ ω ˆm) τ = τ ∩ (ˆ = σ1 ∩ (ˆ ω2 ∪ · · · ∪ ω ˆm) . Hence |τ | ⊂ fc+ \N and η(|τ |) ⊂ fc− . However, τ ∈ [τ1 ]; therefore, η(τ ) ∈ [τ1 ], which implies that c1 ≤ c − . This is a contradiction. Deﬁnition 5.2.14 For a pair of topological spaces (X, Y ), let L(X, Y ; G) = sup{l ∈ Z+ | ∃ nontrivial classes [τ1 ], [τ2 ], . . . , [τl ] ∈

358

5 Topological and Variational Methods

H∗ (X, Y ; G) such that [τ1 ] < · · · < [τl ]} . L(X, Y ; G) measures the length of the chain of subordinate nontrivial singular homology classes. It is directly related to the notion of cup length: CL(X, Y ; G) = sup{l ∈ Z+ | ∃c0 ∈ H ∗ (X, Y ; G), ∃c1 , . . . , cl ∈ H ∗ (X; G) such that dim(ci ) > 0, i = 1, 2, . . . , l, and c0 ∪ · · · ∪ cl = 0} . It is known from algebraic topology that catX (X) ≥ CL(X, ∅; G), i.e., the cup length provides a lower bound estimate for the category; we have catP n (P n ) = n + 1, catT n (T n ) = n + 1, where Pn is the n- dimensional real projective space, and T n is the n-torus. 5.2.4 Index Theorem Symmetric functions, or more generally, a function, which is invariant under some G-group action, may have more critical points. This is due to the fact that the underlying space quotient by the G-group action has more complicated topology. In this subsection we shall deal with this problem. Let us recall the proof of the Ljusterik–Schnirelmann multiplicity theorem, in which only the fundamental properties (1)–(6) of the category were used. We are inspired to extend an abstract theory based on these properties for G-group action invariant functions. Let M be a Banach–Finsler manifold with a compact group action G. Let Σ be the set of all G-invariant closed subsets of M , and H be the set of all G-equivariant continuous mappings from M into itself, i.e., h ∈ H if and only if h ∈ C(M, M ) and h ◦ g = g ◦ h ∀g ∈ G. Deﬁnition 5.2.15 An index (Σ, H, i) with respect to G is deﬁned by i : Σ → N ∪ {+∞}, which satisﬁes: i(A) = 0 ⇔ A = ∅, ∀A ∈ Σ. (Monotonicity) A ⊂ B ⇒ i(A) ≤ i(B), ∀A, B ∈ Σ. (Subadditivity) i(A ∪ B) ≤ i(A) + i(B), ∀A, B ∈ Σ. (Deformation nondecreasing) If η : [0, 1] × M → M satisﬁes η(t, ·) ∈ H ∀t ∈ [0, 1], and η(0, ·) = idM , then i(A) ≤ i(η(1, A)), ∀A ∈ Σ. (5) (Continuity) If A ∈ Σ is compact, then there is a neighborhood N ∈ Σ of A such that A ⊂ int(N ) and i(A) = i(N ). (6) (Normality) i([p]) = 1 ∀p ∈ FixG where [p] = {g · p| g ∈ G} and FixG = {x ∈ M | g · x = x ∀g ∈ G} is the ﬁxed-point set of G, if G = {e}.

(1) (2) (3) (4)

Example 1. If G = {e}, then category (Σ, H, catM ) is an index. Example 2. Recall the genus deﬁned in Sect. 3.3, where M is a Banach space, G = Z2 = {I, −I}, i.e., Ix = x, (−I)x = −x ∀x ∈ M ,. Thus Σ = the set of all closed symmetric subset, H is the set of odd continuous mappings.

5.2 Minimax Principles (Revisited)

Let

359

⎧ if A = ∅ , ⎨0 γ(A) = inf{k ∈ N| ∃ an odd map φ ∈ C(A, Rk \{θ})} , ⎩ +∞ if no such odd map .

Then genus (Σ, H, γ) is an index with respect to Z2 . (Veriﬁcations) By deﬁnition, (1) and (2) are trivial. (3) (Subadditivity) Set γ(A) = n, γ(B) = m; we may assume that both n, m < ∞. This means that ∃ϕ : A → Rn \{θ} ∃ψ : B → Rm \{θ} both are odd and continuous. By Tietze’s theorem ∃ϕ ! ∈ C(M, Rn ), ∃ψ! ∈ C(M, Rm ) ! B = ψ. Without loss of generality, we may assume that such that ϕ| ! A = ϕ, ψ| ! ! ϕ ! and ψ are odd. Deﬁne f (x) = (ϕ(x), ! ψ(x)), then f ∈ C(M, Rn+m ) is odd n+m \{θ}. This implies that γ(A ∪ B) ≤ γ(A) + γ(B). and f (A ∪ B) ⊂ R (4) (Deformation nondecreasing) Suppose η : [0, 1]×M → M is continuous and odd, and satisﬁes η(0, ·) = id with γ(η(1, A)) = n, i.e., ∃ϕ : η(1, A) → Rn \{θ} is continuous and odd. Let ψ = ϕ ◦ η(1, ·) then ψ : A → Rn \{θ} is continuous and odd. Therefore γ(A) ≤ n. (5) (Continuity) Let γ(A) = n. From (2), with no loss of generality, we may assume n < +∞. There exists an odd continuous ϕ : A → Rn \{θ}. By Tietze’s theorem, there exists an odd continuous mapping ϕ ! : M → Rn , with ϕ| ! A = ϕ. Since A is compact and θ ∈ ϕ(A), ! ∃δ > 0 such that θ ∈ ϕ(N ! ), where N = Aδ , the closure of the δ-neighborhood of A, i.e., ϕ ! : N → Rn \{θ} . Combining with the monotonicity, we obtain n = γ(A) ≤ γ(N ) ≤ n (6) (Normality) Note that FixG = {θ}, if p = θ, then [p] = {p, −p}. We deﬁne φ(±p) = ±1; it follows that γ([p]) = 1. Moreover, applying the Borsuk–Ulam theorem in Sect. 3.2, we have computed γ(S n−1 ) = n. Example 3. (S 1 -index) Let M be a Banach space, S 1 = {eiθ | θ ∈ [0, 2π]} be a compact Lie group. Let G be the isometry representation of S 1 , i.e., the group consists of homomorphisms T : S 1 → L(M, M ), i.e., T (eiθ ) are isometric operators, satisfying T (eiθ ) · T (eiϕ ) = T (ei(θ+ϕ) ) ∀θ, ϕ ∈ [0, 2π]. Set Σ = the set of all G − invariant closed subsets of M . H = {h ∈ C(M, M ), T (eiθ ) ◦ h = h ◦ T (eiθ ) ∀θ ∈ [0, 2π]} . ∀A ∈ Σ, deﬁne

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5 Topological and Variational Methods

⎧ 0 if A = ∅, ⎪ ⎪ ⎨ inf{k ∈ N| ∃φ ∈ C(A, Ck \{θ}), ∃n ∈ N satisfying γ(A) = φ(T (eiθ )x) = einθ φ(x) ∀(x, eiθ ) ∈ A × S 1 }, ⎪ ⎪ ⎩ +∞ if there is no such φ, where Ck is k-dimensional unitary space (complex k space). We shall verify that (Σ, H, γ) is an index with respect to S 1 . By deﬁnition, (1) and (2) are trivial. Before going on to verify other fundamental properties, we need: Lemma 5.2.16 For any A ∈ Σ, if φ ∈ C(A, Ck \{θ}) satisﬁes φ(T (eiθ )x) = einθ φ(x) ∀x ∈ A for some n ∈ N , ! A = φ and then ∃φ! : M → Ck satisfying φ| ! ! (eiθ )x) = einθ φ(x) ∀(x, eiθ ) ∈ M × S 1 . φ(T ˆ A = φ. Setting Proof. By Tietze’s theorem, ∃φˆ ∈ C(M, Ck ) satisfying φ| 1 ! φ(x) = 2π

2π

ˆ (eiϕ )x)dϕ , e−inϕ φ(T

0

we have inθ ! (eiθ )x) = e φ(T 2π

2π

ˆ (ei(θ+ϕ) )x)dϕ = einθ φ(x) ! e−in(θ+ϕ) φ(T ,

0

and ∀x ∈ A, by the assumption. 1 ! φ(x) = 2π

2π

φ(x)dx = φ(x) . 0

(Veriﬁcations) (3) (Subadditivity) ∀A, B ∈ Σ, let γ(A) = n, γ(B) = m. We may assume n, m < ∞. There exist ϕ : A → Cn \{θ}, ψ : B → Cm \{θ} and k, l ∈ N such that ϕ(T (eiθ )x) = eikθ ϕ(x) and ψ(T (eiθ )y) = eilθ ψ(y) ∀(x, y, eiθ ) ∈ A × B × S 1 . Deﬁne ! k) , f (x) = (ϕ(x) ! l , ψ(x) where the power map z → z p is deﬁned to be z = (z1 , . . . , zk ) → (z1p , . . . , zkp ). Thus f : A ∪ B → Cn+m is continuous and f (z) = θ as z ∈ A ∪ B. Moreover, ! (eiθ )x)k ) f (T (eiθ )x) = (ϕ(T ! (eiθ )x)l , ψ(T ! k) = eilkθ (ϕ(x) ! l , ψ(x) = eilkθ f (x) .

5.2 Minimax Principles (Revisited)

361

Therefore γ(A ∪ B) ≤ γ(A) + γ(B) . (4) (Deformation nondecreasing) Assume A ∈ Σ, η ∈ [0, 1] × M → M is continuous and satisfying η(0, ·) = id, η(t, ·) ∈ H, and γ(η(1, A)) = k. By deﬁnition, ∃ϕ : η(1, A) → Ck \{θ} continuous and satisfying ϕ(T (eiθ )η(1, x)) = einθ ϕ(η(1, x)) for some n ∈ N, ∀(x, eiθ ) ∈ A × S 1 . Now let ψ = ϕ ◦ (η(1, ·)). Then ψ : A → Ck \{θ} and ψ(T (eiθ )x) = ϕ(η(1, T (eiθ )x)) = ϕ(T (eiθ )η(1, x)) = einθ ϕ(η(1, x)) = einθ ψ(x) . Therefore γ(A) ≤ k. (5) (Continuity) Assume that A ∈ Σ is compact and γ(A) = k. Provided ˜ (eiθ )x) = by Lemma 5.2.16, ∃ a continuous φ! : M → Ck ∃n ∈ N satisfying φ(T ˜ ! A = φ and φ : A → Ck \{θ}. φ| einθ φ(x), Setting V = φ!−1 (Ck \{θ}), then V is an open neighborhood of A. Since A is compact, ∃δ > 0 such that the closure of the δ-neighborhood of A, N ⊂ V . Therefore φ! : N → Ck \{θ} . It follows that γ(N ) ≤ k. (6) (Normality) Let p ∈ FixG , i.e., ∃θ ∈ [0, 2π] such that T (eiθ )p = p. Since [p] = {T (eiθ )p| θ ∈ [0, 2π]} ∈ Σ and is not empty, γ([p]) ≥ 1. There exists θ0 > 0, which is the minimal period of T (eiθ )p, i.e., T (eiθ0 )p = p but T (eiθ )p = p ∀θ ∈ (0, θ0 ). n = 2π θ0 must be an integer. We deﬁne φ : [p] = {T (eiθ )p| θ ∈ [0, θ0 ]} → C1 \{θ} as follows: φ : T (eiθ )p → einθ . It is continuous, and φ(T (eiϕ )T (eiθ )p) = φ(T (ei(θ+ϕ) p)) = ein(θ+ϕ) = einϕ φ(T (eiθ )p) . Therefore γ([p]) = 1. We have proved that γ so deﬁned is an index. In particular, if M is a Hilbert space, and G is an unitary representation of S 1 , we have the following Borsuk–Ulam-type theorem for S 1 equivariant maps. Theorem 5.2.17 Under the above assumptions, suppose that V 2k is a 2kdimensional S 1 -invariant subspace, which is isomorphic to Ck . If V 2k ∩FixG = {θ}, then γ(V 2k ∩ ∂B1 (θ)) = k.

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5 Topological and Variational Methods

Proof. According to Stone’s representation theorem for one-parameter unitary groups, one may diagonalize {T (eiθ )| θ ∈ [0, 2π]} simultaneously: T (eiθ ) = diag{eiλ1 θ , . . . , eiλk θ } . Since T (eiθ ) is 2π-periodic, λj ∈ Z, j = 1, 2, . . . , k. By the assumption that there is no ﬁxed point of {T (eiθ )| θ ∈ [0, 2π]} in Ck \{θ}, therefore λj = 0, j = 1, 2, . . . , k. Firstly, we show γ(V 2k ∩ ∂B1 (θ)) ≤ k by constructing a continuous map ∧

∧ λ

φ : z = (z1 , . . . , zk ) → (z1λ1 , . . . , zk k ), where ∧ = |λ1 . . . λk |. Obviously, φ : Ck ∩ ∂B1 (θ) → Ck \{θ} continuously and φ(T (eiθ )z) = ei∧θ φ(z), ∀z ∈ Ck ∩ ∂B1 (θ). Next we are going to show that γ(Ck ∩ ∂B1 (θ)) ≥ k. Suppose γ(Ck ∩ ∂B1 (θ)) = k1 < k, i.e., ∃ a continuous φ : Ck ∩ ∂B1 (θ) → Ck1 \{θ}, and ∃n ∈ N such that φ(T (eiθ )z) = einθ φ(z). From Lemma 5.2.16, ! Ck ∩∂B (θ) = φ. Accordthere is an continuous extension φ! : Ck → Ck1 with φ| 1 ing to Theorem 3.3.10, ! B1 (θ), θ) = deg(φ, B1 (θ), θ) = 0 . deg(φ, Thus for y ∈ Ck with small y , one has ! B1 (θ), y) = 0 . deg(φ, In particular, we take y ∈ Ck \Ck1 , then φ!−1 (y)∩B1 (θ) = ∅, which contradicts ! k ) ⊂ Ck1 . φ(C The Ljusternik–Schnirelmann multiplicity theorem is extended as follows: Theorem 5.2.18 (Multiplicity theorem) Let M be a smooth Finsher–Banach manifold with a compact Lie group action G. Let (Σ, H, i) be an index with respect to G. Suppose that f ∈ C 1 (M, R1 ) satisﬁes (PS)c conditions, where c = ck+1 = · · · = ck+m is ﬁnite, and cn = inf sup f (x)

n = 1, 2, . . .

i(A)≥nx∈A

Then i(Kc ) ≥ m. Proof. The proof follows step by step that of Lemma 5.2.10. The only thing to be worked out is that the deformation constructed in Corollary 5.2.9 is G-equivariant. In fact, let η : [0,1] × M → M be the deformation obtained in Theorem 5.2.8. Then ηg (t, z) = G g −1 η(t, g · z)dµ is the G-equivariant deformation, where dµ is the right invariant Haar measure on compact continuous group G. We omit the veriﬁcations.

5.2 Minimax Principles (Revisited)

363

5.2.5 Applications Example 1. (An existence of three nontrivial solutions) Return to Sect. 5.1.5, Example 2, where we proved the existence of a nontrivial solution for the superlinear elliptic BVP both at inﬁnity and at zero. In fact, it was also studied in Sect. 4.8, Example 2, where we showed the existence of a mountain pass solution p. After Corollary 5.2.5, we know that C1 (f, p) = 0; if further, the condition (Φ) in Theorem 5.1.20 is fulﬁlled, then we have Cq (f, p) = δq1 G. We assume condition (Φ) at this moment, and conclude: Statement 5.2.19 Under assumptions (g1) (g2) and (g3) of Sect. 5.1.5 Example 2, equation (5.10) possesses at least three nontrivial solutions. Proof. We continue to use the notations in Sect. 5.1.5 Example 2. 1. From (g3), 1 (5.15) J(u) = u 2 + ◦ ( u 2 ) . 2 Therefore θ is a local minimum, and Cq (J, θ) = δq0 G. 2. We claim the existence of two nontrivial solutions. Let us deﬁne g(x, t) t ≥ 0 g+ (x, t) = 0 t≤0,

and J+ (u) =

Ω

1 2 |∇u| − G+ (x, u(x)) dx , 2

where

G+ (x, t) =

t

g+ (x, s)ds . 0

Similarly, one veriﬁes that J+ ∈ C 2 (H01 (Ω), R1 ) satisﬁes (PS), and that J+ (tϕ1 ) → −∞ as t → +∞ , where ϕ1 > 0 is the ﬁrst eigenvector of − with 0-Dirichlet data. On the other hand, ∃δ > 0 such that J+ |∂Bδ (θ) ≥

1 2 δ , 4

From (5.15). The mountain pass lemma is applied to obtain a critical point u+ ∈ H01 (Ω), with critical value c+ > 0, which satisﬁes −u+ = g+ (x, u+ ) u+ |∂Ω = 0 . By using the maximum principle, u+ ≥ 0, so it is again a critical point of J. Analogously, we deﬁne

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5 Topological and Variational Methods

g− (x, t) =

g(x, t) 0

t≤0 t>0,

and obtain a critical point u− ≤ 0, with critical value c− > 0. Since (Φ) is assumed, Cq (J± , u± ) = δq1 G. According to Theorem 5.1.16, we have ! Cq (J± , u± ) = Cq (J1 ± , u± ) = Cq (J, u± ) = Cq (J, u± ) , where J! = J|C01 . Therefore Cq (J, u± ) = δq1 G . 3. Suppose that there were no other critical points of J. The Morse type numbers over the pair (H01 (Ω), Ja ) would be M0 = 1, M1 = 2, Mq = 0, q ≥ 2 , but the Betti numbers read as βq = 0 ∀q = 0, 1, 2, . . . From Lemma 5.1.38: Hq (H01 (Ω), Ja ) Hq (H01 (Ω), S ∞ ) 0. This contradicts the Morse relation.

Now we return to verifying the condition (Φ): J (u) ≥ 0 and 0 ∈ σ(J (u)) imply dim ker(J (u)) = 1. It is based on the following: Theorem 5.2.20 (Manes–Micheletti) Suppose that m ∈ C(Ω) satisﬁes sup{m(x)| x ∈ Ω} > 0. Then the equation −u = λmu in Ω u|∂Ω = 0 admits a smallest positive eigenvalue λ1 , associated with a positive eigenfunction. Moreover, dim ker(− − λ1 m·) = 1 . Continuation of the proof of Statement 5.2.19 (Veriﬁcation of the condition (Φ)) Now

|∇v|2 − mv 2 ∀v ∈ H01 (Ω) , (5.16) (J (u)v, v) = Ω

where m(x) = g (x, u(x)). Assume that the right-hand side of equation (5.16) is nonnegative and that ∃φ ∈ H01 (Ω)\{θ} satisﬁes the equation −v − m · v = 0 .

(5.17)

5.2 Minimax Principles (Revisited)

365

We want to show that all solutions of equation (5.17) are multiples of φ. After Theorem 5.2.20, it is suﬃcient to show that λ = 1 is the smallest positive eigenvalue of (5.18) −v = λmv v ∈ H01 (Ω) . On the one hand, |∇φ|2 |∇v|2 inf ≤ =1, λ1 := mv 2 mφ2 v∈H01 \{θ} from Ω mφ2 = Ω |∇φ|2 = 0. On the other hand, provided by the nonnegativeness of J (u), |∇v|2 λ = Ω ≥1 mv 2 Ω for every λ ∈ σ(J (u)). Therefore the smallest positive eigenvalue λ1 of equation (5.18) equals 1. Condition (Φ) is veriﬁed. We turn to the proof of Theorem 5.2.20. Proof. 1. Let T : u → (−)−1 m · u be the self-adjoint compact operator on H01 (Ω). According to Courant’s max-min characterization, ± Ω mu2 −1 , n = 0, 1, 2, . . . ±λ±n = sup inf 2 V ∈Fn u∈V Ω |∇u| are eigenvalues of T , where Fn is the family of all n-dimensional linear subspaces of H01 (Ω). Thus, ±λ±1 ≤ ±λ±2 ≤ · · · , if they exist. 2. Under our assumption, λ1 > 0. Let ω be an eigenfunction. We claim that ω does not change sign. Otherwise, let ω+ = ω ∨ 0 and ω− = ω ∧ 0; they are not zero, then

2 2 2 mω = mω+ + mω− = α+ + α− , Ω Ω Ω

|∇ω|2 = |∇ω+ |2 + |∇ω− |2 = β+ + β− , Ω

Ω

Ω

2 > 0 and β± = Ω |∇ω± |2 . where α± = Ω mω± α+ +α− α+ +α− − − or αβ+ < max{ αβ++ , αβ− }. However, the latter Either β+ +β− = β+ = αβ− + +β− α+ +α− −1 case cannot occur, because λ1 = β+ +β− = supu Ω mu2 / Ω |∇u|2 . In the former case, both ω± are eigenfunctions with respect to λ1 and ±m · ω± ≥ 0. From the maximum principle, we obtain ±ω± (x) > 0 a.e. x ∈ Ω. This is impossible. 3. We show that dim ker(− − λ1 m·) = 1. Suppose that there are ω1 and ω2 ∈ H01 (Ω) satisfying

−ωi = λ1 mωi

i = 1, 2 .

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5 Topological and Variational Methods

Then by step 2, ∀α ∈ R1 , ω1 + αω2 does not change sign. Let A+ = {α ∈ R1 | ω1 + αω2 ≥ 0} and A− = {α ∈ R1 | ω1 + αω2 ≤ 0}. Then both A+ and A− are nonempty closed subsets and A+ ∪ A− = R1 . Let α0 ∈ A+ ∩ A− , then ω1 + α0 ω2 = 0, i.e., ω1 and ω2 are colinear. This proves dim ker(− − λ1 m·) = 1. Example 2. (Spectrum of the p-Laplacian) Let Ω ⊂ Rn be a bounded open domain. ∀p ∈ (1, ∞) we deﬁne the pLaplacian operator to be p u = div(|∇u|p−2 ∇u)

∀u ∈ W01,p (Ω) .

In particular, 2 is the Laplacian. λ is called an eigenvalue of −p if there exists a nontrivial weak solution u ∈ W01,p (Ω) of the equation: −p u = λ|u|p−2 u i.e.,

in Ω,

(5.19)

|∇u|p−2 ∇u∇ϕ = λ Ω

|u|p−2 u · ϕ

∀ϕ ∈ W01,p (Ω) ,

Ω

where p1 + p1 = 1. This is reduced to a constraint variational problem. Set

1 1 p |u| and g(u) = |∇u|p . f (u) = p Ω p Ω Let M = g −1 (p−1 ). Itis a nonempty closed submanifold of W01,p (Ω). In fact, ∀u ∈ M, g (u), u = Ω |∇u|p = 1, which implies that g (u) = θ. Let f! = f |M . We have

|u|p (p u) ∈ W −1,p (Ω) , df!(u) = |u|p−2 u + Ω

which satisﬁes df!(u), u = 0 ∀u ∈ M . Set J0 = (−2 )−1/2 and J1 w = |w|p −2 w/ w pp −2 , we have J0 w p = w −1,p , J0 w 1,p = w p , and J1 w p = w p .

Thus J = J0 ◦ J1 ◦ J0 : W −1,p (Ω) → W 1,p (Ω) is well deﬁned and satisﬁes: (1) J is continuous, (2) Jw, w = w 2−1,p ∀w ∈ W −1,p (Ω), (3) Jw 1,p = w −1,p ∀w ∈ W −1,p (Ω).

5.2 Minimax Principles (Revisited)

367

An odd pseudo-gradient vector ﬁeld for f! is deﬁned by X(u) = J · df!(u) − λu ∈ W 1,p (Ω) , where λ = J · df!(u), −p u . We claim that X(u) ∈ Tu M . ˙ = In fact, let s : (−1, +1) → M be a C 1 -curve with s(0) = u and s(0) X(u), then d g(s(t))|t=0 = X(u), g (u) = X(u), −p u = 0 . dt But we have X(u), df!(u) = J · df!(u), df!(u) = df!(u) 2−1,p . From

X(u) 1,p ≤ df!(u) −1,p + |λ| u 1,p ,

and |λ| ≤ J · df!(u) 1,p −p u −1,p ≤ df!(u) −1,p u 1,p = df!(u) −1,p , it follows

X(u) 1,p ≤ 2 df!(u) −1,p .

Thus X(u) is a pseudo-gradient vector ﬁeld for f!. Now we verify the (PS)c condition for f!, as c > 0. Suppose that {uj } is a sequence on M such that df!(uj ) → θ and f!(uj ) → c > 0. {uj } is bounded in W01,p (Ω), and then after a subsequence uj u0 ∈ W01,p (Ω). From df!(uj ) → θ (W −1,p (Ω)), and f!(uj ) → c one has −p uj → (pc)−1 |u0 |p−2 u0

W −1,p (Ω) .

From Sect. 2.6, Example 4, −p is a monotone operator with a continuous inverse: (−p )−1 : W −1,p (Ω) → W01,p (Ω), therefore uj strongly converges in W01,p (Ω). We notice that M is the unit sphere in W01,p (Ω), and that the function ! f is even. Quotient by a Z2 -group, M/Z2 is homeomorphic to the inﬁnitedimensional projective space P ∞ . Applying the Ljusternik–Schnirelmann theorem, we have: Statement 5.2.21 For 1 < p < ∞, the equation (5.19) has inﬁnitely many 1 × (W01,p (Ω)\{θ}), m = 1, 2, . . .. Moreover, pairs of solutions (λm , ±um ) ∈ R+ let σp be the spectrum of the p-Laplacian, i.e., the set of all λ associated with nontrivial solutions. Then σp is unbounded.

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5 Topological and Variational Methods

Proof. Since f! satisﬁes (PS)c , c > 0, we apply the Ljusternik–Schnirelmann theorem to −f!, i.e., let ck =

inf sup − f!(u)

k = 1, 2, . . .

γ(A)≥ku∈A

where A is a closed symmetric (with resp. to θ) subset of M and γ is the genus. All these ck are negativet critical values with ck ≤ ck+1 . We have λk = − c1k , k = 1, 2, . . .. We shall prove that σp is unbounded by contradiction. If not, then there exists > 0 such that ck < −. ∀k large. Let us choose a sequence of linear subspaces {Ej }∞ 1 , such that 1,p Ej ⊂ Ej+1 , dim Ej = j, and ∪∞ j=1 Ej = W0 (Ω) .

Set dk =

inf

sup

γ(A)≥ku∈A∩E ⊥

− f!(u) ,

k−1

Ek⊥

where denotes a complementary linear subspace of Ek . We need the following intersection result: Lemma 5.2.22 Let X be a Banach space and E ⊂ X be a j < ∞ dimensional linear subspace. If A is a symmetric closed set with γ(A) > j then A∩E ⊥ = ∅, where E ⊥ is any complementary subspace of E. the projection Proof. We prove it by contradiction. If A ∩ E ⊥ = ∅, one deﬁnes P : X → E, according to the direct sum composition: X = E E ⊥ . Deﬁne 1 1 a function ρ ∈ C(R+ , R+ ) to be ρ (t) = as t ≤ , and ρ (t) = t as t ≥ for > 0 and an odd deformation: η(t, x) = (1 − t)x + t

Px . ρ ( P x )

From A ∩ E ⊥ = ∅, we have > 0 such that P (A) ∩ B (θ) = ∅. Since η(1, x) =

Px , η(1, A) ⊂ E ∩ ∂B1 (θ) . ρ ( P x )

Provided by the monotonicity and the deformation nondecreasing, γ(A) ≤ γ(η(1, A)) ≤ γ(E ∩ ∂B1 (θ)) = j . This is a contradiction.

(Continuing the proof of Statement 5.2.21) According to Lemma 5.2.22, if ⊥ = ∅, therefore dk is well deﬁned, and dk ≤ ck , k = γ(A) ≥ k, then A ∩ Ek−1 1, 2, . . . .

5.2 Minimax Principles (Revisited)

369

We suppose ck < − ∀k large, then there exists a closed symmetric subset ⊥ such that −f!(uk ) < −, i.e., Ak ∈ , with γ(Ak ) ≥ k, and ∃uk ∈ Ak ∩ Ek−1 f!(uk ) > ∀k large. However, uk ∈ M implies that after a subsequence uk v in W01,p (Ω), and then uk → v in Lp (Ω), in particular, f!(uk ) → f (v), as k → ∞. But ⊥ , which implies uk θ in W01,p (Ω), and f (θ) = 0. This contradicts uk ∈ Ek−1 f!(uk ) > . Example 3. Let Ω ⊂ Rn be a bounded domain with smooth boundary. We consider the problem: −u(x) = λg(x, u(x)) in Ω (5.20) u(x) = 0 on ∂Ω where g ∈ C(Ω × R1 ) is odd in t, and satisﬁes: (1) ∃t0 > 0 such that g(x, t0 ) ≤ 0 ∀x ∈ Ω (2) ∃a ∈ C(Ω), a > 0, such that g(x, t) − a(x)t = ◦(t) as t → 0 uniformly in x ∈ Ω. One has the following abstract result: Theorem 5.2.23 Let X be a Banach space, and f ∈ C 1 (X, R1 ) be an even function satisfying the (PS) condition. Assume a < b and either f (θ) < a or f (θ) > b. If further, (1) there are an m-dimensional linear subspace E and ρ > 0 such that sup

f (x) ≤ b ,

x∈E∩∂Bρ (θ)

(2) there is a j-dimensional linear subspace F such that inf f (x) > a ,

x∈F ⊥

where F ⊥ is a complementary space of F , (3) m > j, then f has at least m − j pairs of distinct critical points. Proof. Let cn =

inf sup f (x)

n = 1, 2, . . .

γ(A)≥nx∈A

1. From condition (1) E ∩ ∂Bρ (θ) ⊂ fb , by the monotonicity of genus γ(fb ) ≥ γ(E ∩ ∂Bρ (θ)) = m. We obtain cm ≤ b. 2. We verify that cn ≥ a as n > j. If it is not true, then ∃n0 > j such that cn0 < a, i.e., ∃ a closed symmetric set A with γ(A) ≥ n0 such that

370

5 Topological and Variational Methods

f (x) < a ∀x ∈ A. Condition (2) implies that f (x) > a ∀x ∈ F ⊥ . But according to Lemma 5.2.22, A ∩ F ⊥ = ∅. This is a contradiction. Combining steps 1 and 2, we obtain critical values in [a, b] cj+1 ≤ · · · ≤ cm . From the multiplicity theorem, there are at least m − j pairs of critical points. Remark 5.2.24 Another proof of Theorem 5.2.23 based on the estimation of cup length can be found in [Ch 5]. Now we return to equation (5.20). First, we replace g by the function g(x, t) t ≤ t0 gˆ(x, t) = g(x, t0 ) t > t0 . From the maximum principle, we know that if u ∈ H01 (Ω) is a solution of the equation: −u(x) = λˆ g (x, u(x)) in Ω (5.21) u(x) = 0 on ∂Ω , then u(x) < t0 ∀x ∈ Ω, and then it is a solution of equation (5.20). Let us write p(x, t) = gˆ(x, t) − a(x)t t and P (x, t) = 0 p(x, s)ds. Consider the functional with a parameter λ:

Jλ (u) = Ω

1 |∇u|2 − λ 2

au2 + P (x, u(x)) dx . 2

Obviously, ∀λ ∈ R1 , Jλ is bounded from below. Since a > 0, the eigenvalue problem: −v = µa · v

v ∈ H01 (Ω)

has eigenvalues λ1 < λ2 ≤ · · · ≤ λk ≤ · · · . If λ > λk then there exists ρ > 0 such that Jλ |Ek ∩∂Bρ (θ) < 0, where Ek is the subspace spanned by eigenfunctions with eigenvalues µ ≤ λk . Omitting the veriﬁcation of the (PS) condition, we apply Theorem 5.2.23 and obtain: Theorem 5.2.25 Under assumptions (1) and (2) of Example (3), (5.20) possesses at least k distinct pairs of solutions for λ > λk .

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

371

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture For H ∈ C 1 (Rn × Rn , R1 ), the following ODE system: x˙ = −Hp (x, p) p˙ = Hx (x, p)

(5.22)

∀ (x, p) ∈ Rn × Rn , is called a Hamiltonian system, and H is called a Hamiltonian function. Sometimes we use the notations: z = (x, p) ,

and J=

0 I

−I 0

,

where I is the n × n unit matrix. then (5.22) has a simple form: −J z˙ = H (z) , or

z˙ = JH (z) .

(5.23)

J grad is called a symplectic gradient. For the Hamiltonian system two types of periodic solution problems are often examined. 1. For a given period T , ﬁnd a solution z satisfying (5.23) and z(t) = z(t+T ). 2. For a given energy surface, i.e., a real number c, ﬁnd a periodic solution z of (5.23) lying on the given energy surface, i.e., H(z(t)) = c. The reason why we can set up the type 2 problem is that Hamiltonian systems are conservative: d H(z(t)) = (H (z(t)), z(t)) ˙ dt = (H (z(t)), JH (z(t))) = 0 ,

(5.24)

i.e., along a trajectory t → z(t), the Hamiltonian function is a constant. Note: the periodic trajectory z on Σ = H −1 (c) is virtually independent of the behavior of the function H outside Σ , that is, we have: Lemma 5.3.1 Let H, H1 ∈ C 1 (R2n , R1 ), c ∈ R1 . Assume that Σ = H −1 (c) is a compact manifold, If H|Σ = H1 |Σ , then z˙ = JH (z) and z˙ = JH1 (z) has the same periodic trajectories on Σ.

372

5 Topological and Variational Methods

Proof. By the hypotheses, ∃ µ ∈ C(Σ, R1 ) satisfying H (z) = µ(z)H1 (z),

∀z ∈ Σ .

Since µ(z) = 0 and Σ is compact, there are constants m, M > 0, m µ(z) M,

∀z ∈ Σ .

(5.25)

Let z be an integral curve of H on Σ, deﬁne z1 (t) = z ◦ s(t) , where s : R1 → R1 will be determined later. First we solve the equation: 1 σ(t) ˙ = (µ◦z)(σ(t)) (5.26) σ(0) = 0 . Since the functions µ, z are all bounded and continuous, (5.26) has a solution on every open interval in R1 . Let T be the period of z, since σ(t) → +∞ as t → +∞, there exists τ > 0 such that σ(τ ) = T . Let t t s(t) = T +σ t− τ , τ τ where [t] denotes the largest integer less than t, then s ∈ C 1 (R1 , R1 ), s(0) = 0. It remains to verify the diﬀerentiability at t = jτ, j = 0, ±1, ±2, . . . . In fact, for j = 1, 1 1 1 = = (µ ◦ z)(σ(0)) (µ ◦ z)(0) (µ ◦ z)(T ) = σ(τ ˙ − 0) = s (τ − 0)

˙ = s (τ + 0) = σ(0)

Similarly, we prove this for j = 1. Hence, s : R1 → R1 is a diﬀeomorphism and satisﬁes 1 s(t) ˙ = (µ◦z)(s(t)) , s(0) = 0 . It follows that

˙ z(t) ˙ ◦ s(t) = JH1 (z1 (t)) , z˙1 (t) = s(t)

and z1 (0) = z(0) ∈ Σ, hence z1 is an integral curve of H1 on Σ. The converse can be proved in the same way. Thus a given hypersurface Σ determines the orbits of every Hamiltonian vector ﬁeld J grad H with Σ as regular energy surface H −1 (c) for some constant c. In this sense, one can ask the question: Which hypersurfaces carry a periodic orbit?

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

373

5.3.1 Hamiltonian Operator Let S 1 be the unit circle {eit t ∈ [ 0, 2π ]} and V = Rn × Rn . Deﬁne the real Hilbert space L2 (S 1 , V ) as follows: z ∈ L2 (S 1 , V ) means that z is deﬁned on S 1 with range in V and |z|2 is integrable, the inner product is deﬁned by z, w =

1 2π

2n

S1

∀z, w ∈ L2 (S 1 , V ) .

zj (t)wj (t)dt,

j=1

Similarly deﬁne the Sobolev space H 1 (S 1 , V ) with the inner product 1 z, w = 2π

2n

S 1 j=1

[zj (t)wj (t) + z˙j (t)w˙ j (t)] dt .

Let J=

0 In

−In 0

,

where In is the n × n identity matrix. Consider the linear operator A : z → d z in L2 (S 1 , V ) with domain D(A) = H 1 (S 1 , V ). −J dt The real space V is isomorphic to the complex linear space Cn = Rn +iRn . The isomorphism is deﬁned as follows: Let e1 , . . . , e2n be an orthonormal basis in V , and let φj = ej + iej+n , j = 1, 2, . . . , n . 2n

ˆ = They form a basis in Cn . Let z = j=1 zj ej be corresponding to z n j=1 (zj − izj+n )φj . Deﬁne the inner product [ zˆ, w ˆ ] = Re

n

(zj − izj+n )(wj − iwj+n )

j=1

=

n 2n (zj wj + zj+n wj+n ) = zj wj . j=1

j=1

Then the correspondence z → zˆ is linear and inner product preserving, and then {φj j = 1, . . . , n} is an orthogonal basis in Cn . The real Hilbert space L2 (S 1 , V ) is isomorphic and homeomorphic to the complex Hilbert space L2 (S 1 , Cn ) with inner product deﬁned by

1 [ zˆ(t), w(t) ˆ ]dt = z, w , 2π S 1 then z → zˆ is an inner product preserving linear isomorphism of L2 (S 1 , V ) → L2 (S 1 , Cn ). The space L2 (S 1 , Cn ) has an orthonormal basis: {e−imt φj j = 1, 2, . . . , n; m = 0, ±1, ±2, . . . } .

374

5 Topological and Variational Methods

Every zˆ ∈ L2 (S 1 , Cn ) has the Fourier expansion +∞ n −imt zˆ = cjm e φj . m=−∞

j=1

One has

+∞ n

|cjm |2 = z2 =

j=1 m=−∞

1 2π

[ zˆ, zˆ ]dt . S1

Hence zˆ ↔ c = {cjm } is an isometric isomorphism of L2 (S 1 , Cn ) → (l2 )n =

n :

l2 .

j=1 n

According to the Fourier expansion, zˆ ∈ H 1 (S 1 , C ) if and only if +∞ n

(1 + |m|)2 |cjm |2 < +∞ .

j=1 m=−∞

Note that

d −imt (e φj ) = me−imt φj ; dt it is easy to verify that A is a self-adjoint operator with domain D(A) = H 1 (S 1 , Cn ), and then {e−imt φj } is just the diagonalized orthonormal basis. Let M (m) = span{e−imt φ1 , . . . , e−imt φn }, m ∈ Z , −J

then L2 (S 1 , Cn ) = ⊕m∈Z M (m) ; this is the spectral decomposition or A. 5.3.2 Periodic Solutions Theorem 5.3.2 Suppose that H ∈ C 2 (Rn × Rn , R1 ) satisﬁes: (1) H(θ) = 0, H (θ) = θ, H (θ) = 0, H(z) 0. (2) H(z) = γ2 |z|2 , for |z| R, R > 0, γ ∈ (1, 2). Then the Hamiltonian system (5.22) possesses a nontrivial 2π-periodic solution. d Proof. Deﬁne G(z) = H(z) − γ2 |z|2 . Set A = −J dt ,

H(z(t))dt , ψ(z) = S1

and

G(z(t))dt ;

g(z) = S1 n

then ψ ∈ C 2 (H, R1 ), (H = L2 (S 1 , C )) possesses the following properties:

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

(1) ψ(θ) = 0, ψ (θ) = θ, ψ (θ) = 0, ψ(x) 0 (2) ∃ b, M > 0 such that ∀x ∈ H

375

∀x ∈ H.

g (x)L(H) M , γ ψ(x) x2H − b , 2 ψ (x) = γx + g (x) . 1

1

Denote the Sobolev space H 2 (S 1 , Cn ) = D(|A| 2 ) by E, and let E ∗ denote the subspace associated with the ∗ eigenvalues of E, where ∗ = +, 0, −. Let P ∗ be the orthogonal projections onto E ∗ , ∗ = +, 0, −. We deﬁne a bilinear form on E: 1 1 ((z, w)) = (|A| 2 z, |A| 2 w) , where ( , ) is the inner product on H. The system (5.22) is rewritten as Az = ψ (z), and equivalently,

z ∈ D(A) ,

(5.27)

(A − γ)z = g (z) .

We introduce a functional on E: Φ(x) =

1 (((P + − P − )x, x)) − ψ(x) . 2

Obviously, if z ∈ D(A) is a critical point of Φ, then z solves (5.27). We choose an arbitrary element e in M (1) with norm one, and set Q = ((E − ⊕ E 0 ) ∩ BR (θ)) × [0, R]e S = E + ∩ ∂Bε (θ)

for R > 0 large, and

for ε > 0 small .

Since ψ(x) = o(x2H ) as xH → 0, we have β > 0 such that inf Φ S

ε2 + o(ε2 ) β . 2

We turn out to estimate the values of Φ on ∂Q: Φ(x) −ψ(x) 0 ∀x = x− + x0 ∈ E − ⊕ E 0 , 1 1 γ Φ(x− + x0 + se) − x− 2 + (1 − γ)s2 + b − x0 2 0 , 2 2 2 as x− + x0 + se ∈ ∂Q, s > 0 and R large enough. In summary Φ|∂Q 0. Since both S and Q are inﬁnite dimensional, and the intersection of ∂Q and S has not been discussed previously, we shall use the Galerkin method instead. Let Ek = ⊕|j |k M (j), and Φk = Φ|Ek . Then Sk and ∂Qk link, where Sk = S ∩ Ek , Qk = Q ∩ Ek , ∀k. We verify the (P S) condition for Φk . Denote by Πk the orthogonal projection onto Ek .

376

5 Topological and Variational Methods

Let {xj } ⊂ Ek be a sequence along which Φk (xj ) → θ. Then (A − γ)xj = Πk g (xj ) + o(1) , i.e.,

xj = (A − γ)−1 (Πk g (xj ) + o(1)) .

Since g (xj ) is bounded, {xj } is bounded, and then {xj } is subconvergent. Applying the minimax theorem for linkings, we have a critical point zk ∈ Ek satisfying (A − γ)zj = Πk g (zk ) , Φ(zk ) β > 0 . It remains to prove that {zk } is subconvergent in E. Since we have zk = (A − γ)−1 Πk g (zk ) , and that the linear map (A − γ)−1 : L2 → H 1 → E is compact, {zk } has a convergent subsequence with limit zˆ. Thus Φ(ˆ z ) β > 0 and zˆ = (A − z ), which implies zˆ ∈ D(A) and that zˆ is a nontrivial solution of γ)−1 g (ˆ (5.27). The proof is complete. 5.3.3 Weinstein Conjecture As to the given energy question, the breakthrough is due to Rabinowitz and Weinstein. In 1978, Rabinowitz [Ra 3] proved that a strongly star-shaped hypersurface (Weinstein [Wei 1] proved for a convex hypersurface) carries at least one periodic orbit. Nevertheless, the star-shapedness, and then the convexity, is not invariant under canonical diﬀeomorphism. Weinstein [Wei 2] called for a symplectic invariant version. He made the following conjecture: Every compact smooth hypersurface of contact type carries at least a periodic orbit. A hypersurface Σ is said to be of contact type if there exists a vector ﬁeld X in a neighborhood of Σ, such that (1) X Σ. (2) The ﬂow φt , generated by X (i.e., (dφt )J(dφt )T = et J, ∀t.

d t dt φ

= X ◦ φt and φ0 = id) satisﬁes

A matrix A satisfying AJAT = µJ, where µ is a nonzero constant, is called a canonical matrix with multiple µ. A diﬀeomorphism φ : R2n → R2n satisfying (dφ)J(dφ)T = µJ, is called a canonical diﬀeomorphism with multiple µ. Remark 5.3.3 In the deﬁnition of contact type, the condition (2) is equivalent to LX ω = ω, where ω is the sympectic form ω(u, v) = Ju, v , ∀u, v ∈ R2n , · is the inner product in R2n , and LX is the Lie-derivative of the 2-form.

5.3 Periodic Orbits for Hamiltonian System and Weinstein Conjecture

377

Lemma 5.3.4 Suppose that φ is a canonical diﬀeomorphism with multiple µ, and that z4 is a solution of the Hamiltonian system z˙ = JH (z) . 4 Let H(u) = µH(z), where u = φ(z). Then u ˆ = φ(ˆ z ) is a solution of 4 (u) . u˙ = J H

Proof. By changing variables: u˙ = dφ(z) · z˙ = (dφ(z)) J(∇z H) 1 T 4 = (dφ(z)) J (dφ(z)) ∇u H µ 7 6 1 T 4 = (dφ(z)) · J · (dφ(z)) · ∇u H µ 4 (u) . = JH Example 5.3.5 A hypersurface Σ is called strongly star-shaped, if there exists a point x0 ∈ R2n , such that x − x0 , n(x) > 0

∀x ∈ Σ ,

where n(x) is the unit outward normal vector on Σ. A strongly star-shaped hypersurface is of contact type. Indeed, without loss of generality, we may assume x0 = θ. Let the vector ﬁeld 1 X(x) = x . 2 The strong star-shapedness implies that X Σ; the ﬂow φt generated by X is deﬁned to be φt (x) = et/2 x , i.e., φt (x) = et/2 id. Therefore, (dφt )J(dφt )T = et J . Now we turn to the Weinstein conjecture. By deﬁnition, for a contact-type hypersurface Σ, we have@a+family Σε = φε (Σ), ε ∈ (−1, 1). & % of hypersurfaces 2n Σε |ε| < δ has two components: The interior Let 0 < δ < 1, then R B, which with no with loss of generality contains & exterior + %the origin, and the ε ∈ (−1, −δ) ⊂ B. A, which is unbounded. Also one may assume Σ ε + Let d = diam {Σε | ε ∈ (−1, 1)}. We deﬁne d < r < 2d, γ2 r2 < b < r2 , γ = 3 ∞ ∞ 2 , and functions f ∈ C (−1, 1), g ∈ C (0, ∞) satisfying

378

5 Topological and Variational Methods

0, s ∈ ( −1, −δ ] , f (s) = b, s ∈ [ δ, 1) , b, s ≤ r, g(s) = γ 2 2 s , s ≥ R, where R > r , g(s) ≥ f (s) > 0

γ 2 s for s > r , 2 0 < g (s) < γs

for s ∈ (−δ, δ)

sr.

Then we deﬁne a Hamiltonian function: ⎧ 0 z∈B, ⎪ ⎪ ⎪ ⎨f (ε) z ∈ Σε |ε| δ , H(z) = ⎪ b z ∈ A and |z| r , ⎪ ⎪ ⎩ g(|z|) |z| > r . The function H satisﬁes assumptions (1) and (2) of Theorem 5.3.2. According to Theorem 5.3.2, there is a nontrivial 2π-periodic solution zˆ for this Hamiltonian system, with

d 1 H(ˆ z (t))dt β for some β > 0 . −J zˆ, zˆ − Φ(ˆ z) = 2 dt S1 Lemma 5.3.6 The solution zˆ lies on Σε for some ε ∈ (−δ, δ). Proof. 1. If |ˆ z (0)| > r, then |ˆ z (t)| = |ˆ z (0)|, because the energy surface of H at zˆ(0) is a sphere. Thus

2π 1 g (|ˆ z (0)|) |ˆ z (0)|2 · 2π − g(|ˆ z (t)|)dt 2 0 = γπ|ˆ z (0)|2 − γπ|ˆ z (0)|2 = 0 .

Φ(ˆ z) =

This is impossible. + Σε , then H(ˆ z (0)) = b or 0. In either case, 2. If |ˆ z (0)| r but zˆ(0) ∈ / |ε| 0 is a necessary condition for the existence of a solution. However, in this section we always assume that K(x) > 0 ∀x ∈ S 2 . We introduce

1 [|∇u|2 + 2u] , S(u) = 4π S 2 and J(u) = S(u) − log

1 4π

Ke2u . S2

The Euler–Lagrange equation for J reads as 4πKe2u on S 2 . u = 1 − 2u Ke 2 S

(5.29)

point of J if and only Noticing that J(u+c) = J(u) ∀c ∈ R1 , u0 is a critical 1 2u0 Ke . Once we obtain if u solves (5.28), where u = u0 −c, and c = 12 log 4π S2 a critical point of J, after adjusting a constant, we solve equation (5.29). In order to avoid the indetermination caused by the translation invariance of J, we add a constraint. Let

e2u = 4π , X = u ∈ H 1 (S 2 )| S2

and consider the functional J on X. The Euler–Lagrange equation is again (5.29). 5.4.1 The Conformal Group and the Best Constant In this section, we ﬁrst analyze the special case K ≡ 1, and ﬁnd out all the minimizers. Now, we introduce the following stereographic projection: ˆ = C ∪ {∞} π : S2 → C x1 + ix2 π(x1 , x2 , x3 ) = z = ∀(x1 , x2 , x3 ) ∈ S 2 ⊂ R3 . 1 − x3 This is a conformal diﬀeomorphism with (π −1 )∗ g0 =

4 dzdz . (1 + |z|2 )2

5.4 Prescribing Gaussian Curvature Problem on S 2

381

Let Conf(S 2 ) be the conformal group consisting of all conformal diﬀeoobius group morphisms from S 2 into itself. Conf(S 2 ) is isomorphic to the M¨ SL(2, C). A subgroup D of Conf(S 2 ) is deﬁned as follows: for any Q ∈ S 2 , we choose an orthonormal frame {e1 , e2 , e3 }, such that Q is the north pole, i.e., Q = e3 , by which we deﬁne a stereographic projection πQ , ∀t ∈ [1, ∞), let ˆ φQ,t = π −1 ◦ τt ◦ πQ , and let τt z = tz ∀z ∈ C, Q D = {φQ,t | (Q, t) ∈ S 2 × [1, ∞)} . Since φQ,1 = id, and φ−Q,t = φQ,t−1 , D is a submanifold and is parameterized to be B 3 (S 2 × [1, ∞))/(S 2 × {1}) under the correspondence: φQ,t → t−1 t Q. For any Riemannian metric g on S 2 , ∀φ ∈ Conf(S 2 ), the function ψ=

1 log det(dφ) 2

(5.30)

satisﬁes φ∗ g0 = e2ψ g0 . The conformal group action on the function v reads as: (5.31) φ∗ v := vφ = v ◦ φ + ψ . We deﬁne the conformal Laplacian as follows: Lg ϕ = −g ϕ + Kg

∀ϕ ∈ C ∞ (S 2 ) ,

(5.32)

where g is the Laplacian–Beltrami operator and Kg is the Gaussian curvature with respect to g, respectively. Thus, for g = e2u g0 , we have Lg ϕ = −g ϕ + Kg = e2ϕ Ke2ϕ g −2u

=e

−2u

=e

·e

2(u+ϕ)

(5.33) Ke2(u+ϕ) g0

Lg0 (u + ϕ) .

Lemma 5.4.1 Lg0 vφ = det(dφ)(Lg0 v) ◦ φ

(5.34)

Proof. (Lg0 v) ◦ φ = Le2ψ g0 (v ◦ φ) = e−2ψ Lg0 (v ◦ φ + ψ) = det(dφ)−1 Lg0 vφ ,

provided by (5.33) and (5.31).

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5 Topological and Variational Methods

Lemma 5.4.2 All solutions of the equation: −ψ + 1 = e2ψ on S 2

(5.35)

are of the form (5.30). Proof. Since Lg0 ψ = −ψ + 1, plugging ψ = oφ into (5.34), (5.35) follows from Lemma (5.4.1). Conversely, if ψ is a solution of (5.35), then Kg = 1 for g = e2ψ g0 . According to the isometric theorem in Riemannian geometry, there exists an isometry φ : (S 2 , g) → (S 2 , g0 ) satisfying φ∗ g0 = g, i.e., φ ∈ Conf(S 2 ). By (5.30), ψ has the form 12 log det(dφ). 1 Lemma 5.4.3 Both the functionals S and T (u) = 4π e2u are conformal S2 invariant. Furthermore, S(ψ) = 0 ∀ψ of form (5.30). Proof. 1. Obviously, ∀φ ∈ Conf(S 2 ), ∀v ∈ H 1 (S 2 ),

1 1 2(v◦φ) T (vφ ) = e det(dφ) = e2v = T (v) . 4π S 2 4π S 2 2. We prove that S(ψ) = 0 ∀ψ of the form (5.30). For any φ0 ∈ Conf(S 2 ), we introduce a diﬀerentiable path φt ∈ Conf(S 2 ), d ψt |t=0 . Then by step 1, with φt |t=0 = φ0 . Let ψt = 12 log det(dφt ), and w = dt

d 1 0 = T (oφt )|t=0 = e2ψ0 w , (5.36) dt 2π S 2 and

d 1 S(ψt )|t=0 = [∇ψ0 · ∇w + w] dt 2π S 2

1 = [−ψ0 + 1]w 2π S 2

1 e2ψ0 w = 0 . = 2π S 2

Since the path φt is arbitrary, we have S(ψ) = const. In particular, let φ0 = id, then ψ0 = 0 and S(ψ) = 0. 3. Finally, we turn to proving the conformal invariant of S. ∀φ ∈ Conf(S 2 ),

1 |∇(v ◦ φ)|2 + 2 [∇(v ◦ φ)∇ψ + v ◦ φ] + S(ψ) S(vφ ) = 4π 2 S2 S

1 |∇v|2 + 2 (Lg0 ψ)v ◦ φ = 4π 2 S2 S

1 = |∇v|2 + 2 v ◦ φe2ψ 4π S2 S2 = S(v) .

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In the sequel of this subsection we are going to prove that S(u) ≥ 0 ∀u ∈ X, and then, all the minimizers of S on X are of the form (5.30). Firstly, we give a necessary condition for the solvability of (5.28). Namely: Theorem 5.4.4 (Kazdan–Warner) If u is solution of (5.28) then

∇K, ∇xi e2u = 0 i = 1, 2, 3 .

(5.37)

S2

Proof. By Lemma 5.4.3, ∀φQ,t ∈ D, J(uφQ,t ) = S(u) − log

1 4π

S2

2u K ◦ φ−1 . Q,t e

(5.38)

If u is a solution of (5.28), then u is a critical point of J, therefore ; < d d J(uφQ,t )|t=1 = J (u), uφQ,t |t=1 = 0 . dt dt By (5.38), d 1 J(uφQ,t )|t=1 = 2u dt K ◦ φ−1 Q,t e S2

S2

∇K, ∇(x, Q) e2u .

We choose Q = e1 , e2 , and e3 , respectively; (5.37) follows.

For any u ∈ H 1 (S 2 ), we deﬁne the mass center of u to be

−1

e2u · x1 e2u , x2 e2u , x3 e2u . P (u) = S2

S2

S2

S2

Obviously, P (u) ∈ B 3 . We want to show that ∀u ∈ H 1 (S 2 ), after a conformal transform, the mass center can be moved to the origin, i.e., there exists (Q, t) ∈ S 2 × [1, ∞) such that P (u ◦ φQ,t ) = θ, or equivalently, the nonlinear system:

xi e2uφθ,t = 0, i = 1, 2, 3 , (5.39) S2

is solvable. We solve the nonlinear system by IFT. Let F : H 1 (S 2 ) × B 3 → B 3 be the map

x ◦ φ−Q,t e2u , (5.40) F (u, ξ) = S2

where ξ = s(t)Q, and s(t) =

1 − t−1 1 − t−2 ln t

is a diﬀeomorphism: [1, ∞) → [0, 1). We need some estimate for ∂ξ F .

t ∈ [1, 2] , for t large ,

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5 Topological and Variational Methods

Lemma 5.4.5 Assume {Q(t)} ⊂ S 2 , Q(t) → Q0 as t → +∞. Then

x ◦ φQ(t),t e2u → Q0 e2u as t → +∞ S2

(5.41)

S2

uniformly in {u ∈ H 1 (S 2 )| u ≤ c} ∀c > 0. Moreover, e2ψQ,t → 4πδ(−Q) as t → +∞ .

(5.42)

Proof. Since the mapping H 1 (S 2 ) → L1 (S 2 ) : w → e2w is compact, it is suﬃcient to prove (5.41) for ﬁxed u. Noticing that x ◦ φQ,t = D−1 {2t(x − (Q · x)Q) + [t2 (1 + Q · x) − (1 − Q · x)]Q} , where D = t2 (1 + Q · x) + (1 − Q · x) , we have x ◦ φQ(t),t → Q0 except x = −Q0 . Our conclusion follows from the Lebesgue dominance theorem.

Lemma 5.4.6 For any compact subset C ⊂ H 1 (S 2 ) and any constants b > 1, the matrix ∂ξ F (u, ξ) has a uniformly bounded inverse ∀u ∈ C, t ≤ b, where ξ = s(t)Q solves the equation F (u, ξ) = θ. Proof. We write x ! = x ◦ φQ,t , and compute 1 (Q − (! x · Q)! x), t (t2 − 1)2 t2 − 1 ∂e x (e − (! x · e)! x) + [(! x · e)Q − (! x · Q)e] , != 2t 2t

∂t x !=

∀e ∈ TQ (S 2 ). One chooses an orthonormal frame {e1 , e2 } in TQ (S 2 ); the 2 2 −1 −1 matrix ∂ξ F (u, ξ) is then an array of the three vectors ( t 2t g1 , t 2t g2 , ts1(t) g3 ), where

1 (ej − (ˆ x · ej )ˆ x)e2u j = 1, 2, 3 , gi = 4π S 2 x ˆ = x ◦ φ−Q,t and e3 = Q, because ξ is a solution. Let A be the matrix consisting of (g1 , g2 , g3 ). It is symmetric and positive 3 deﬁnite. Indeed, ∀λ = i=1 λi ei with |λ| = 1, 3 λi gi · λ (Aλ) · λ = i=1

= 1−

1 4π

S2

(ˆ x · λ)2 e2u .

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Fixing u, ∃δ = δ(u) > 0 such that

1 (ˆ x · λ)2 e2u ≤ 1 − δ 4π S 2 uniformly in (Q, t) ∈ S 2 × [1, b]. Since C is compact, ∃δ0 > 0 such that

1 (ˆ x · λ)2 e2u ≤ 1 − δ0 , ∀(u, ξ) ∈ C × (S 2 × [1, b]) . 4π S 2 i.e., (Aλ) · λ ≥ δ0 . Thus ∃δ1 > 0 such that | det (A)| ≥ δ1 , and then | det (∂ξ F (u, ξ))| ≥

(t2 − 1)2 δ1 . 4t3 s (t)

The lemma is proved.

We are ready to solve (5.40) by the continuity method. According to Lemma 5.4.6, equation (5.39) is locally solvable for any solution (u0 , ξ0 ) of (5.39) from the implicit function theorem. One chooses a starting point (u0 , ξ0 ), satisfying F (u0 , ξ0 ) = θ. For any u ∈ H 1 (S 2 ), we consider a path u(τ ) = 12 log(τ e2u + (1 − τ )e2u0 ) connecting u0 and u, and study the equation F (u(τ ), ξ(τ )) = θ for τ ∈ [0, 1]. Let [0, τ0 ) be the maximal existence interval. If τ0 = 1, then (5.39) is solvable for such a u. Otherwise, there must be a sequence τj → τ0 such that tj = t(τj ) → +∞. Because, if not, t(τ ) is bounded on [0, τ0 ), and then by Lemma 5.4.6, (∂ξ F (u(τ ), ξ(τ )))−1 is uniformly bounded. The solution is extendible beyond τ0 . This contradicts the maximality of τ0 . After subtracting a subsequence from {τj }, again denoted by {τj }, we have Qj = Q(τj ) → Q0 ∈ S 2 . By Lemma 5.4.5, θ = F (u(τj ), ξ(τj ))

2u(τj ) x ◦ φ−Qj ,tj e → −Q0 = S2

e2u(τ0 ) .

S2

But this is impossible. Thus we have completed the proof. Theorem 5.4.7 ∀u ∈ H 1 (S 2 ), there exists (Q, t) ∈ S 2 × [1, ∞) satisfying (5.39). Now let us deﬁne

1 e2u . I(u) = S(u) − log 4π S 2 Obviously, I(u) = S(u) ∀u ∈ X. We shall prove:

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5 Topological and Variational Methods

Theorem 5.4.8 I(u) ≥ 0 ∀u ∈ H 1 (S 2 ), and the equality holds if and only if u = 12 log det(dφ) + c, ∀φ ∈ Conf(S 2 ) ∀c ∈ R1 . Thus all the minimizers of S on X are of the form (5.30). Proof. First we prove the existence of a minimizer of I on Y = {u ∈ H 1 (S 2 )| P (u) = θ, S 2 u = 0}. According to Aubin’s inequality Sect. 4.2, ∀u ∈ Y .

1 1 2 [|∇u| + 2u] − log e2u I(u) = 4π S 2 4π S 2 1 1 − ≥ u 2 − log C , 4π 8π − i.e., I is coercive on Y . Since Y is weakly closed in H 1 (S 2 ), we conclude that there exists u0 ∈ Y satisfying I(u0 ) = min{I(u)| u ∈ Y }. The Euler–Lagrange equation reads as −1

1 2u0 e e2u0 = µ · xe2u0 , for some µ ∈ R3 . −u0 + 1 − 4π S 2 2u 1 Let c = 12 log 4π e 0 and v = u0 − c, then −v + 1 = (1 + µ · xe2c )e2v . Applying the necessary condition in Theorem 5.4.4, we have

µe2u0 = θ , S2

i.e., µ = θ. Hence v is a solution of (5.35), it must be of form (5.30). However, P (v) = θ; it follows that v = θ, or u0 = const. Therefore I(u0 ) = 0 and then I(u) ≥ 0 ∀u ∈ Y . Since I is translation invariant: I(u+c) = I(u) ∀c ∈ R1 , and also conformal invariant: I(uφ ) = I(u) ∀φ ∈ Conf(S 2 ), our conclusion follows from Theorem 5.4.7. Returning to equation (5.28), in which K ≡ 1, all solutions are also the minimizers of the functional I. We may even write down these solutions explicitly: 1 − α2 1 (5.43) ψ = log 2 (1 + αQ · x)2 2

2 2 where α = 1−t 1+t2 , for (Q, t) ∈ S × [1, ∞). Although Conf(S ) is a group with 1 six parameters, the representation ψQ,t = 2 log det(dφQ,t ) only has three, i.e., it is totally determined by the subgroup D.

Corollary 5.4.9 (Onofri) ∀u ∈ H 1 (S 2 ), we have

1 1 2u 2 e ≤ exp (|∇u| + 2u) . 4π S 2 4π S 2 see also Hong [Hon 1].

5.4 Prescribing Gaussian Curvature Problem on S 2

387

5.4.2 The Palais–Smale Sequence From Theorem 5.4.8, we see that J is bounded below. We claim that if K = const, then the inﬁmum of J is not achieved. Indeed, let P0 ∈ S 2 be the maximum of K. On one hand, we have J(u) ≥ − log K(P0 ), provided by Corollary 5.4.9. On the other hand,

1 1 2ψ−P0 ,t Ke = − log Ke2ψ−P0 ,t . J(ψ−P0 ,t ) = S(ψ−P0 ,t ) − log 4π S 2 4π S 2 Applying (5.42), we obtain J(ψ−P0 ,t ) → − log K(P0 ), i.e., inf J = − log K(P0 ). If u0 were a minimum of J, then we would have J(u0 ) = − log K(P0 ). Deﬁne

1 ˆ J(u) = S(u) − log K(P0 )e2u , 4π S 2 then min Jˆ = − log K(P0 ), which is achieved by u = ψQ,t + c ∀c ∈ R1 , ∀(Q, t) ∈ S 2 × [1, ∞). Then ˆ 0 ) ≤ J(u0 ) = − log K(P0 ) , − log K(P0 ) ≤ J(u ˆ Therefore, u0 = ψQ,t + c for some which implies that u0 is a minimum of J. (Q, t) ∈ S 2 × [1, ∞) and c ∈ R1 . But this is impossible. The above argument proved indirectly that the PS condition does not hold for J (Ekeland variational principle). Now, we shall analyze the (PS) sequence carefully, and then reveal the reason why the (PS) condition breaks down. Firstly, we continue the proof in Theorem 5.4.7, and obtain: Theorem 5.4.10 Let P (u) be the mass center of u, and let X0 = {u ∈ X| P (u) = θ} , then

◦

X∼ = X0 × B 3 , where ∼ = means diﬀeomorphism. Proof. After Theorem 5.4.7, it remains to prove that the solution of equation (5.39) is unique. Suppose not, then we may assume that ξi = s(ti )Qi satisﬁes (5.39), i = 0, 1. Deﬁne wi = uφQi ,ti , i = 0, 1; we pick a path (Q(τ ), t(τ )) connecting (Q0 , t0 ) and (Q1 , t1 ), and let w(τ ) = 12 log(τ e2w1 + (1 − τ )e2w0 ) ∀ τ ∈ [0, 1]. Set , u(τ ) = w(τ )φ−1 Q(τ ),t(τ )

then F (u(τ ), ξ(τ )) =

S2

x ◦ φ−Q(t),t(τ ) e2u(τ ) = θ .

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5 Topological and Variational Methods

We introduce a new parameter σ, and deﬁne for (σ, τ ) ∈ [0, 1]2 , u(σ, τ ) =

1 log[(1 − σ)e2u(τ ) + σe2u ] 2

and we solve the equation: F (u(σ, τ ), ξ(σ, τ )) = θ . By the same argument as used in Theorem 5.4.8, we have solutions (Q(σ, τ ), t(σ, τ )) depending on σ, τ continuously, and satisfying Q(0, τ ) = Q(τ ), t(0, τ ) = t(τ ) , Q(σ, i) = Qi , t(σ, i) = ti , i = 0, 1, and F (u, ξ(1, τ )) = θ . This contradicts the local uniqueness of the solution of (5.39), which follows from the implicit function theorem. The smoothness of (Q, t) depending on u follows directly. Theorem 5.4.10 provides a parameterization of X : X → X0 × (S 2 × [1, ∞)/S 2 × {1}), u → (w, Q, t). Theorem 5.4.11 Suppose that {uj = (wj , Qj , tj )} is a Palais–Smale sequence of J on X, then either it is subconvergent or tj → +∞. Proof. Since

J(uj ) = S(wj ) − log

K ◦ φQj ,tj e2wj → c ,

and 4πK(! xj )e2wj + λj e2wj + (µj · x)e2wj + ◦(1) , −wj + 1 = K(! xj )e2wj

(5.44)

where (λj , µj ) ∈ R1 × R3 , if {tj } is bounded, then after a subsequence, x !j = ! = x ◦ φQ,t for some (Q, t) ∈ S 2 × [1, ∞). x ◦ φQj ,tj → x It follows that S(wj ) and λj are both bounded. Since − K(! xj )xe2wj + ◦(1) , ∧(wj )µj = K(! xj )e2wj

where ∧(wj ) =

1 4π

S2

xi xk e2wj

, 3×3

and ∧(wj ) is uniformly positive deﬁnite, {µj } is bounded. Moreover, the embedding w → e2w from H 1 to Lp , p > 1, is compact; we may use (5.44) to conclude that {wj } is subconvergent.

5.4 Prescribing Gaussian Curvature Problem on S 2

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5.4.3 Morse Theory for the Prescribing Gaussian Curvature Equation on S2 From Theorem 5.4.11, we know that the Palais–Smale Condition for J breaks down at t = +∞. We are inspired with courage to compactify the open ball ◦

B 3 . By simple estimates, we obtain the limit of the integral:

1 K ◦ φQ,t e2w = K(Q) + ◦(1) as t → ∞ . 4π S 2 Let us extend J to be J! deﬁned on the manifold with boundary M = X0 × B where ∂M = X0 × S 2 : J(u) u∈X ! J(u) = S(w) − log K(Q) (w, Q) ∈ X0 × S 2 .

3

If J! were C 1 on M , then Remark 5.1.31 would be applicable directly to this problem. Unfortunately, this is not true. We are forced to ﬁgure out the asymptotic expansions for the partial derivatives for J. Namely we have v ∈ Tw (X0 ) such that Lemma 5.4.12 ∃b > 0, cb > 0 and ∃! 1

S (w), v! ≥ cb S 2 (w) v! , if S(w) ≤ b. Furthermore, ∀ µ, δ > 0, ∃N = N (δ, b, µ), ∃T = T (δ, b, µ) such that ∂e K(Q) | ≤ N t−1+µ log t ∀e ∈ TQ (S 2 ) , K(Q) 1 1 2K(Q) | ≤ N (|∇K(Q)|t log− 2 tS 2 (w) |∂s J − K(Q) |∂e J +

+ (log t)−1 + t2 (log t)−1 S 2 (w)) . if K(Q) ≥ δ and t ≥ T . The proof is referred to Chang and Liu [CL 2]. Combining Lemma 5.4.12 with the partition of unity, we obtain: Theorem 5.4.13 Assume K > 0. If K = 0 whenever ∇K(x) = θ, then 3 there exists a function f deﬁned on X0 × B possessing the same critical set as J, and ∃ a neighborhood U of X0 × S 2 , such that on U f (w, sQ) = S(w) − log K(Q) −

2K(Q) (1 − s) , K(Q)

where s : [1, ∞) → [0, 1) is a diﬀeomorphism with s(t) = 1 − t−2 log t for t large.

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5 Topological and Variational Methods

Furthermore, f satisﬁes the (PS) condition. Now we are ready to apply the Morse theory under general boundary 3 conditions to f on X0 × B . For given a < b such that e−a , e−b are regular values of K, we assume: (I) K has only a ﬁnite number of critical points with values in the interval [e−b , e−a ]. Set Ω = {x ∈ S 2 | K(x) < 0}, CR0 (a, b) = {x ∈ Ω| K(x) ∈ (e−b , e−a ), x is a local maximum of K}, CR1 (a, b) = {x ∈ Ω| K(x) ∈ (e−b , e−a ), x is a saddle point of K} . Theorem 5.4.14 Under assumption (I), assume (II) K(x) = 0 whenever ∇K(x) = θ . If J has only isolated critical points, then the following Morse inequalities hold: (mq + µq − βq )tq = (1 + t)Q(t) , where Q is a formal series with nonnegative coeﬃcients, and mq = qth Morse type number of J in J −1 [a, b] , βq = rank Hq (Jb , Ja ) , 0 q≥2 µg = CRq (a, b) q = 0, 1 .

Proof. It is equivalent to study the Morse inequalities for f .We notice that f is of separate variables, and that S(w) has no critical point except the minimum θ on X0 , from Lemma 5.4.12. Moreover, f possesses the same critical set as J and equals J in a neighborhood of the critical set. Now the restriction fˆ on the boundary X0 × S 2 reads as S(w) − log K(Q), which has only isolated critical points. Since f (w, sQ) = S(w) − log K(Q) − 2

K(Q) (1 − s) K(Q)

near the boundary, and the normal n(w, Q) = (θ, Q) ∈ Tw (X0 ) × R3 , we see that Σ− = {(w, Q) ∈ X0 × S 2 | f (u), n(u) ≤ 0} = {(w, Q) ∈ X0 × S 2 | Q ∈ Ω ∩ K −1 [e−b , e−a ]} .

5.4 Prescribing Gaussian Curvature Problem on S 2

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It remains to compute the critical groups of the critical points of − log K on Ω. Now ∀Q ∈ CR0 (a, b), rankCq (− log K, Q) = δq0 , and ∀Q ∈ CR1 (a, b) rankCq (− log K, Q) = δq1 , from (II). We claim that rankC2 (− log K, Q) = 0 ∀Q ∈ Ω. If not, Q must be a local minimum of K, and then K(Q) ≥ 0. This is a contradiction. Obviously we have rankCq (− log K, Q) = 0 ∀q > 2 . On the other hand, θ is the minimum of S on X0 ; it follows that rankCq (S, θ) = δq0 ∀q . Thus the Morse type numbers of fˆ read as 0 ∀q ≥ 2 µq = CRq (a, b) q = 0, 1 .

The proof is complete. Corollary 5.4.15 Under assumptions (I) and (II). Let p = local maxima of K in Ω , q = saddle points of K in Ω . If p = q + 1, then (5.28) admits a solution.

¯ 3 is conProof. By taking e−b < min K and e−a > max K. Since X0 × B tractible, the conclusion follows from the Morse inequalities. Even assumption (I) can be dropped. Namely, Theorem 5.4.16 Assume (II) and if deg(Ω, ∇K, θ) = 1, then (5.28) admits a solution. ! satisfying: Proof. First, we perturb K to K ! > 0 possesses only nondegenerate critical points. (1) K ! (2) For a given neighborhood U of the critical set V of K, K(x) = K(x) ∀x ∈ U. ! C 2 is small. (3) K − K

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5 Topological and Variational Methods

! is constructed as follows: Find a cover ∪l B (xi ) ⊂ U of V, ∀i, we pull K i=1 back the linear function on B (xi ), and obtain a function φa,i (x) such that ∇φa,i (x) = a ∀x ∈ B (xi ), where a is any given vector in R2 . Let {ρi }l1 be the partition of unity of V with suppρi ⊂ B (xi ), and let ! K(x) = K(x) − φa,i (x)ρi (x) . By Sard’s theorem, a can be chosen as small as we wish such that (1) and (3) hold. Second, as in Theorem 5.4.13, we can construct a functional f! similar to f satisfying: ! and (1) In a neighborhood O of X0 × S 2 , f! is similar to f with respect to K, there are no critical points there. (2) Outside O, f! = f = J. ! Now, let p, q be the numbers of local maxima and saddle points of K 2 ! ! in Ω = {x ∈ S | K(x) < 0}, respectively. By homotopy invariance of the Brouwer degree: ! ∇K, ! θ) = p − q . 1 = det(Ω, ∇K, θ) = deg(Ω, This proves the existence of a critical point of f! so does for J.

Remark 5.4.17 Other suﬃcient conditions were obtained by Chen and Ding [CD]. Chang and Yang [CY 1, 2], Han [Han], and Hong [Hon 2]. All these results are contained in Theorem 5.4.14, see [CL 2]. Remark 5.4.18 The prescribing scalar curvature problem on S n is a similar problem. For n = 3, it was due to Bahri and Coron [BC] and later, Schoen and Zhang [SZ]. For general n, see Li [Li], and Chen and Lin [ChL].

5.5 Conley Index Theory The Conley index is another topological tool in nonlinear analysis. It can be seen as an extension of both the Leray–Schauder degree and the critical groups. In fact, the Conley index theory is an extension of the Morse theory. It is based on ﬂows without a variational structure. In contrast with the degree theory and the Morse theory, Conley’s theory studies the invariant sets of a ﬂow rather than either the ﬁxed points for a compact vector ﬁeld or the critical points for diﬀerentiable functions. Let (X, d) be a metric space, and let ϕ : R1 × X → X be a ﬂow, i.e., it is continuous and satisﬁes: (1) ϕ(0, x) = x ∀x ∈ X, (2) ϕ(t, ϕ(s, x)) = ϕ(t + s, x)

x ∈ X, ∀t, s ∈ R1 .

5.5 Conley Index Theory

393

Deﬁnition 5.5.1 A set S ⊂ X is called invariant with respect to the ﬂow ϕ, if ϕ(R1 , S) = S, i.e., ∀x ∈ S, ∀t ∈ R1 , ϕ(t, x) ∈ S. Example 1. Let X be a Banach space, and let K : X → X be compact. Let ϕ be the ﬂow derived by the compact vector ﬁeld f = id − K: ϕ(t, ˙ x) = f (ϕ(t, x)) ∀(t, x) ∈ R1 × X , ϕ(0, x) = x , then any subset of the ﬁxed-point set of K is an invariant set for the ﬂow ϕ. Example 2. Let M be a smooth Banach–Finsler manifold and let f ∈ C 1 (M, R1 ) satisfy the (PS) condition. If V is a pseudo-gradient vector ﬁeld of f and ϕ is the associated ﬂow: ϕ(t, ˙ x) = −g(ϕ(t, x))

V (ϕ(t, x)) with ϕ(0, x) = x , V (ϕ(t, x))

(5.45)

where g(x) = Min{d(x, K), 1}, and K is the critical set of f . Then any subset of the critical set of f is an invariant set of the ﬂow ϕ. Deﬁnition 5.5.2 We call the triple (M, f, ϕ) a pseudo-gradient ﬂow if M is a smooth Banach–Finsler manifold, f ∈ C 1 (M, R1 ) satisﬁes the (PS) condition, and ϕ is deﬁned by equation (5.45.) 5.5.1 Isolated Invariant Set Invariant sets can be extremely complicated. It is known from dynamical systems that there are chaotic dynamics and fractal structure. As we have learned in Chap. 3 and in Sects. 5.1 and 5.2, we single out isolated invariant sets as the object of our study. Before going on, we introduce some notions from dynamical systems. 1 ∪ {+∞}, for any subset A of X, we write ∀T ∈ R+ GT (A) = ∩ ϕ(t, A), and ΓT (A) = {x ∈ GT (A)| ϕ([0, T ], x) ∩ ∂A = ∅} . |t|≤T

In particular, we write I(A) = G∞ (A). It is the maximal invariant subset of A. By deﬁnition, we have the following properties: (1) GT (A) = GT (A), GT1 (A) ⊂ GT2 (A), if T1 ≥ T2 , A1 ⊂ A2 ⇒ GT (A1 ) ⊂ GT (A2 ), GT (A) is closed, GT1 +T2 (A) = GT2 (GT1 (A)). (2) If GT (A) ⊂ int(A), then G2T (A) ⊂ int(GT (A)).

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5 Topological and Variational Methods

Proof. Suppose not, then ∃y ∈ G2T (A) ∩ ∂GT (A), i.e., ∃yn → y with ϕ([−T, T ], yn ) ⊂ A, or ∃tn ∈ [−T, T ], ϕ(tn , yn ) ∈ A. There exists a subsequence {tn } of {tn } such that tn → t ∈ [−T, T ] and ϕ(t, y) ∈ ∂A. But y ∈ G2T (A), i.e., ϕ([−2T, 2T ], y) ∈ A, which implies that ϕ(t, y) ∈ GT (A) = GT (A) ⊂ int(A), a contradiction. Now, we introduce a family of closed sets: Σ = Σ(ϕ) = {A ⊂ X| A is closed, and ∃ T > 0 such that GT (A) ⊂ int(A)} . Deﬁnition 5.5.3 A neighborhood U of an invariant set S for the ﬂow ϕ is called isolating if U ∈ Σ and I(U ) = S. An invariant set S is called an isolated invariant set if there exists an isolating neighborhood. Remark 5.5.4 If U is compact and if I(U ) = S ⊂ int(U ), then U ∈ Σ, i.e., ∃T > 0 such that GT (U ) ⊂ int(U ). In fact, if not, i.e., Gn (U ) ⊂ int(U ) ∀n ∈ N, we have yn ∈ Gn (U )\int(U ), i.e., ϕ([−n, n], yn ) ⊂ U but yn ∈ int(U ). Since U is compact, there is a subsequence yni → y0 ∈ I(U ), but y0 ∈ int(U ). A contradiction. Moreover, we have: (3) If U ∈ Σ, then ∃T1 > 0 such that GT1 (U ) ∈ Σ, and ∀t ∈ R1 , ϕ(t, U ) ∈ Σ. Proof. It is known that both GT1 (U ) and ϕ(t, U ) are closed. Since ∃T > 0 such that GT (U ) ⊂ int(U ). Taking T1 = T , from (1) and (2), we have GT (GT1 (U )) = G2T (U ) ⊂ int(GT (U )) = int(GT1 (U )). Also GT (ϕ(t, U )) = ∩ ϕ(t+s, U ) = ϕ(t, GT (U )) ⊂ ϕ(t, int(U )) = int(ϕ(t, U )) . |s|≤T

(4) If U ∈ Σ, then ΓT (U ) ⊂ ∂GT (U ) is closed. Proof. If {xn } ∈ ΓT (U ) and xn → x, then ∃tn ∈ [0, T ] such that ϕ(tn , xn ) ∈ ∂U , after a subsequence, we have tn → t ∈ [0, T ] and ϕ(t, x) ∈ ∂U . Thus x ∈ ΓT (U ). This shows that ΓT (U ) is closed. Next, ∀x ∈ ΓT (U ), we verify x ∈ ∂GT (U ). By deﬁnition, ∃t¯ ∈ [0, T ], ∃yn ∈ U such that yn → ϕ(t, x). Let xn = ϕ(−t, yn ), we have xn → x and / int(GT (U)), therefore xn ∈ GT (U ). Since GT (U ) is closed, we have x ∈ x ∈ ∂GT (U ). The following terminologies are taken from dynamical systems: Deﬁnition 5.5.5 ∀x ∈ X, the set ω(x) = ∩ ϕ([t, ∞), x) t>0

is called the ω-limit set of x, and the set

5.5 Conley Index Theory

395

ω ∗ (x) = ∩ ϕ((−∞, −t], x) t>0

is called the ω ∗ -limit set of x. Given a subset S ⊂ X, the set [S] = {x ∈ X| ω(x) ∪ ω ∗ (x) ⊂ S} is called the invariant hull of S. By deﬁnition, ω(x) = ω(ϕ(t, x)), ω ∗ = ω ∗ (ϕ(t, x)) ∀t ∈ R1 , and then [S] is invariant. Moreover, if S is invariant, then S ⊂ [S]. Lemma 5.5.6 Let (M, f, ϕ) be a pseudo-gradient ﬂow. Then for any x ∈ M the set ω(x) is compact and is a subset of Kc for some critical value c, where Kc = K ∩ f −1 (c). The same holds true for the set ω ∗ (x). Proof. Since the function is nonincreasing along the pseudo-gradient ﬂow, ﬁrst we show that ω(x) is on one level, say ω(x) ⊂ f −1 (c) for some c ∈ R1 . Indeed, if not, then there exist tn , tn ↑ ∞ such that ϕ(tn , x) → y and ϕ(tn , x) → y with f (y) < f (y ). We may always assume that tn > tn , which means that f (y ) = limf (ϕ(tn , x)) ≤ limf (ϕ(tn , x)) = f (y) . n

n

This is a contradiction. Next, we prove that ω(x) ⊂ K. Indeed, if ∃y ∈ ω(x)\K, then we choose regular values a < b such that both f (x) and f (y) are in (a, b). Since Kab = K ∩ f −1 [a, b] is compact, there exists r > 0 such that Br (y) ∩ (Kab )r = ∅. By deﬁnition, there exists tn → +∞ such that xn = ϕ(tn , x) → y. From the (PS) condition, there exists δ > 0 such that f (x) ≥ δ for all x ∈ f −1 [a, b]\(Kab )r , We claim that there exists tn → +∞ such that xn = ϕ(tn , x) ∈ ∂(Kab )r . If not, ∃T > 0 such that ϕ([T, ∞), x) ∩ (Kab )r = ∅, then we would have f (y) = limf (xn ) = lim inf f (ϕ(t, x)) ≤ a , n

t→+∞

This is impossible. Now we choose tn → +∞ with tn < tn such that xn = ϕ(tn , x) ∈ Br (y), ϕ([tn , tn ], x) ∩ (Kab )r = ∅ . It follows that f (xn ) − f (xn ) ≥ δxn − xn ≥ δdist(Br (y), (Kab )r ) . Again, this is impossible. The lemma is proved.

Deﬁnition 5.5.7 A subset W of X is said to have a mean value property (MVP) with respect to the ﬂow ϕ, if ∀x ∈ X, ∀t1 < t2 , ϕ(ti , x) ∈ W, i = 1, 2 implies that ϕ([t1 , t2 ], x) ⊂ W . We now introduce the following key concept:

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Deﬁnition 5.5.8 (Dynamically isolated critical set) Let (M, f, ϕ) be a pseudo-gradient ﬂow. A subset S of the critical set K is said to be a dynamically isolated set if there exist a closed neighborhood O of S and regular values α < β of f such that O ⊂ f −1 [α, β] , and

! ∩ K ∩ f −1 [α, β] = S , cl(O)

! = ∪ ϕ(t, O). We shall then say that (O, α, β) is an isolating triplet where O 1 t∈R

for S. Lemma 5.5.9 Let (M, f, ϕ) be a pseudo-gradient ﬂow and K be the critical set of f . If U is a closed (MVP) neighborhood of S satisfying U ∩ K = S, then [S] = I(U ). If further, there exist real numbers α < β and a (MVP) closed set W , satisfying U ⊂ W ⊂ f −1 [α, β] and W ∩ K = S, then ∃T > 0 such that GT (W ) ⊂ int(U). Proof. 1. [S] = I(U ) “⊂” ∀x ∈ [S], by deﬁnition ω(x) ∪ ω ∗ (x) ⊂ S, then ∃t± n → ±∞ such that ± + ϕ(tn , x) ∈ U , and then ϕ([t− n , tn ], x) ⊂ U . Since n is arbitrary, it follows that ϕ(t, x) ∈ U ∀t ∈ R1 , i.e., x ∈ I(U ). “⊃” ∀x ∈ I(U ), ϕ(t, x) ∈ U ∀t ∈ R1 . Since U is closed, ω(x) ∪ ω ∗ (x) ⊂ U . From Lemma 5.5.6, ω(x) ∪ ω ∗ (x) ⊂ K, therefore ω(x) ∪ ω ∗ (x) ⊂ U ∩ K = S, i.e., x ∈ [S]. 2. GT (W ) ⊂ int(U ). From the (PS) condition, ∃δ ∈ (0, 1) such that dist(x, K) ≥ δ and f (x) ≥ δ ∀x ∈ W \int(U ). Set T > δ −2 (β − α). We shall prove that ∀x ∈ int(U ) ∃t ∈ [−T, T ] such that ϕ(t, x) ∈ W . It is divided into three cases: (a) x ∈ W . By taking t = 0, it is done. ). If ϕ([−T, T ], x) ⊂ W , then (b) x ∈ W \int(U

f (ϕ(−T, x)) − f (ϕ(T, x)) =

T

−T

f (ϕ(s, x)), ϕ(s, ˙ x) ds

≥ 2T δ 2 > 2(β − α) . The contradiction shows ϕ([−T, T ], x) ⊂ W . )\int(U )) ∩ W = int(U ) ∩ W \int(U ). Since either x ∈ (c) x ∈ (int(U ∪ ϕ(t, int(U )) or x ∈ ∪ ϕ(t, int(U )). t>0

t 0 small. Again, we would have β − α ≥ δ 2 (t2 − t1 ) so that t2 < T . Therefore ϕ([−T, T ], x) ⊂ W . Similarly for the second case. Theorem 5.5.10 Let (M, f, ϕ) be a pseudo-gradient ﬂow. If (O, α, β) is an isolating triplet for a dynamically isolated critical set S for f , then [S] is an isolated invariant set. ! ∩ Moreover, any closed MVP neighborhood U of [S], satisfying U ⊂ cl(O) f −1 [α, β], is an isolating neighborhood for [S], and U ∈ Σ. ! ∩ f −1 [α, β] ∈ Σ, is an isolated Proof. 1. Applying Lemma 5.5.9, W = cl(O) invariant neighborhood of [S] . 2. To prove that U ∈ Σ is an isolating neighborhood for [S], it is suﬃcient to show that [S] = I(U ) ⊂ GT (U ) ⊂ int(U ) for some T > 0. Since [S] = I([S]) ⊂ I(U ) ⊂ I(W ) = [S] , ! ∩ f −1 [α, β], we obtain [S] = I(U ). where W = cl(O) Again by Lemma 5.5.9, we have GT (U ) ⊂ GT (W ) ⊂ int(U ).

Example 1. If c is an isolated critical value, i.e., Kc = K ∩ f −1 (c) = ∅, and there is no critical point on the levels in [c − , c + ]\{c} for some > 0, then the set Kc is a dynamically isolated critical set. Example 2. If x0 is an isolated critical point of f , then S = {x0 } is a dynamically isolated critical set. 5.5.2 Index Pair and Conley Index As we have seen in the Morse theory, isolated neighborhoods only are not enough in characterizing the isolated critical points, the local dynamic behavior provides the necessary information. This leads us to: Deﬁnition 5.5.11 Let (N, L) be a pair of subspaces of X. A subset L of N is called positively invariant in N with respect to the ﬂow ϕ, if x ∈ L and ϕ([0, t], x) ⊂ N imply ϕ([0, t], x) ⊂ L . It is called an exit of N , if ∀x ∈ N, ∃t1 > 0 such that ϕ(t1 , x) ∈ N , implies ∃t0 ∈ [0, t1 ) such that ϕ([0, t0 ], x) ⊂ N and ϕ(t0 , x) ∈ L. Example. Let (M, f, φ) be a pseudo-gradient ﬂow, and let α < β < γ. Let N = f −1 [α, γ] and L = f −1 [α, β]. L is positively invariant in N , and also an exit set of N . To an isolated invariant set S, we introduce: Deﬁnition 5.5.12 For U ∈ Σ, let (N, L) be a pair of closed subsets of U with L ⊂ N . It is called an index pair relative to U if: (1) N \L ∈ Σ,

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(2) L is positively invariant in N , (3) L is an exit set of N , (4) N \L ⊂ U and ∃T > 0 such that GT (U ) ⊂ N \L. According to the deﬁnition S := I(U ) = I(N \L) is an isolated invariant set, and both U and N \L are isolating neighborhoods of S. Example. Let (M, f, ϕ) be a pseudo-gradient ﬂow. If (O, α, β) and (O , α , β ) are two isolating triplets for a dynamically isolated critical set S for f , with O ⊂ O, [α , β ] ⊂ [α, β], then ∃T > 0 such that N = GT (W ), L = ϕ(−T, W− ) ! ∩ 1 ) ∩ f −1 [α , β ], where W = cl(O) is an index pair relative to U = cl(O −1 −1 ! f [α, β] and W− = cl(O) ∩ f (α). Moreover, both U and N are isolating neighborhoods of [S]. Note that N \L = N and int(N \L) = int(N ). In order to verify the conclusion, we need: 1. ∀T > 0, N = GT (W ) is a closed MVP neighborhood of [S]. It is suﬃcient to verify that [S] ⊂ int(N ). From Lemma 5.5.9, we have [S] ⊂ N . If the conclusion is not true, then ∃x ∈ [S] ∩ ∂N , i.e., ∃xn ∈ GT (W ) with xn → x. This means that there are tn ∈ [−T, T ], such that ϕ(tn , xn ) ∈ W . After a subsequence, we have tn → t ∈ [−T, T ] and then ϕ(t, x) ∈ int(W ). But x ∈ [S] implies that ϕ(t, x) ∈ [S] ⊂ int(W ), provided by Lemma 5.5.9. This is a contradiction. Now, we are going to verify conditions (1)–(4). 2. Applying Lemma 5.5.9 to U = N , ∃T0 > 0 such that GT +T0 (W ) ⊂ GT0 (W ) ⊂ int(N ) . Since GT +T0 (W ) = GT0 (N ). This shows that N \L ∈ Σ, (1) is veriﬁed. 3. Again applying Lemma 5.5.9, ∃T > 0 such that N = GT (W ) ⊂ int(U ) ⊂ U . Moreover, GT (U ) ⊂ GT (W ) = N = N \L . Poperty (4) is veriﬁed. 4. Since W− is an exit set of W , L is an exit set of N . Obviously, L is positively invariant in N . This completes the veriﬁcation. From Lemma 5.5.9, both U and N are isolating neighborhoods of [S]. For a system without variational structure, does there exist an index pair relative to any set U ∈ Σ? We have: Theorem 5.5.13 (Existence of an index pair) Let ϕ be a ﬂow on a metric space X. ∀U ∈ Σ, (GT (U ), ΓT (U )) is an index pair relative to U , where T > 0 is assumed such that GT (U ) ⊂ int(U ). Proof. From the properties (1) and (4), both GT (U ) and ΓT (U ) are closed. We shall verify the four conditions in Deﬁnition 5.5.11 successively.

5.5 Conley Index Theory

399

(1) By property (4), int(GT (U )\ΓT (U )) = int(GT (U )). Applying property (2), GT (GT (U )\ΓT (U )) ⊂ G2T (U ) ⊂ int(GT (U )) . Thus GT (U )\ΓT (U ) ∈ Σ. (2) ΓT (U ) is positively invariant in GT (U ), i.e., if x ∈ ΓT (U ) and ϕ([0, T1 ], x) ∈ GT (U ), then ϕ([0, T1 ], x) ⊂ ΓT (U ). Suppose not, then ∃t ∈ [0, T1 ] such that ϕ(t, x) ∈ ΓT (U ). Let t∗ = inf{s ∈ [0, T1 ]| ϕ(s, x) ∈ ΓT (U )}. Since ΓT (U ) is closed, y = ϕ(t∗ , x) ∈ ΓT (U ) and ∃n → +0 such that ϕ(t∗ + n , x) ∈ ΓT (U ). Thus ϕ([0, T ], y) ∩ ∂U = ∅ and ϕ([n , T ], y) ∩ ∂U = ∅; it follows that y ∈ ∂U . But y ∈ GT (U ) ⊂ int(U ). This is a contradiction. (3) ΓT (U ) is an exit set of GT (U ), i.e., if x ∈ GT (U ) and if ∃t1 > 0 such that ϕ(t1 , x) ∈ GT (U ), then ∃t0 ∈ [0, t1 ) such that ϕ([0, t0 ], x) ⊂ GT (U ) and y = ϕ(t0 , x) ∈ ΓT (U ). Let us deﬁne t0 = inf{s > 0| ϕ([s − T, s + T ], x) ⊂ U } = inf{s > 0| ϕ(s, x) ∈ GT (U )} ; we have t0 ∈ [0, t1 ]. Since GT (U ) is closed, ϕ([0, t0 ], x) ⊂ GT (U ), therefore t0 < t1 . Deﬁning y = ϕ(t0 , x), we have ϕ(T, y) = ϕ(t0 + T, x) ∈ ∂U , therefore y ∈ ΓT (U ). (4) From property (4), GT (U )\ΓT (U ) = GT (U ) ⊂ U . For T1 > T , we obtain GT1 (U ) ⊂ GT (U ) = GT (U )\ΓT (u) . A topological invariant is introduced to describe the index pair (N, L) relative to an isolating neighborhood U . Conley called the homotopy type h(U ) = [N/L] the invariant. In comparing with that in the Morse theory, we prefer to replace it by the relative homology groups. However, in order to match Conley’s deﬁnition, Alexander–Spanier cohomology is more suitable, because it possesses a special excision property not shared by singular cohomology theory. ∗ For a topological pair (X, A) and a coeﬁcient ﬁeld F , H (X, A; F ) stands for Alexander–Spanier cohomology. The following excision property holds. Suppose that X and Y are paracompact Hausdorﬀ spaces, and that A and B are closed in X and Y respectively. If X\A is homeomorphic to Y \B. Then ∗ ∗ H (X, A; F ) ∼ = H (Y, B; F ). Deﬁnition 5.5.14 (Conley index) For a given ﬂow ϕ and a given U ∈ Σ, let (N, L) be an index pair with respect to (U, ϕ). We call

400

5 Topological and Variational Methods ∗

h(U ) = h(U, ϕ) = H (N, L; F ) the Conley index for U (or for the isolated invariant set S = I(U )) with respect to the ﬂow ϕ. In contrast with equation (5.3), we also write h(t, U ) =

∞

¯ q (N, L; F ) . tq H

q=0

In the following we omit ϕ in the notation h, if there is no ambiguity. In order to show that the Conley index is well-deﬁned, one has to verify that if (N1 , L1 ) and (N2 , L2 ) are two index pairs relative to U , then ∗ ∗ H (N1 , L1 ; F ) ∼ = H (N2 , L2 ; F ). In fact by deﬁnition, ∃T > 0 such that GT (N1 \L1 ) ⊂ GT (U ) ⊂ N2 \L2 and GT (N2 \L2 ) ⊂ GT (U ) ⊂ N1 \L1 . Because Ni \Li ∈ Σ, i = 1, 2, for suﬃcient large T > 0, we have GT (N1 \L1 ) ⊂ int(N2 \L2 ) and GT (N2 \L2 ) ⊂ int(N1 \L1 ) .

(5.46)

Lemma 5.5.15 If (N1 , L1 ) and (N2 , L2 ) are two index pairs relative to U ∈ Σ, and if T > 0 such that (5.46) holds, then the map f : [T, ∞) × N1 /L1 → N2 /L2 deﬁned by ϕ(3t, x) if ϕ([0, 2t], x) ⊂ N1 \L1 and ϕ([t, 3t], x) ⊂ N2 \L2 , f (t, [x]) = otherwise [L2 ] is continuous. Proof. There are three cases in logic: (1) ϕ([t, 3t], x) ⊂ N2 \L2 , (2) ϕ([0, 2t], x) ⊂ N1 \L1 , (3) ϕ([t, 3t], x) ⊂ N2 \L2 and ϕ([0, 2t], x) ⊂ N1 \L1 . In case (1), by the continuity of the ﬂow, there exists a neighborhood W of (t, x) such that ϕ([s, 3s], y) ⊂ N2 \L2 ∀(s, y) ∈ W then f (s, [y]) = [L2 ] = f (t, [x]), i.e., f is continuous at (t, [x]). Similarly for case (2). In case (3), we consider two possibilities: (a) ϕ([t, 3t], x) ∩ L2 = ∅. Since ϕ([t, 3t], x) ⊂ N2 \L2 , it follows that ϕ(2t, x) ∈ Gt (N2 \L2 ) ⊂ int(N1 \L1 ) ⊂ N1 \L1 . Again from ϕ([0, 2t], x) ⊂ N1 \L1 and the positive invariance of L1 in N1 , it follows that ϕ([0, 2t], x) ⊂ N1 \L1 . Therefore f (t, [x]) = ϕ(3t, x) ∈ N2 \L2 . For any open neighborhood U of ϕ(3t, x) in N2 \L2 , by the continuity of the ﬂow ϕ, ∃ a neighborhood W in N1 /L1 of (t, x) such that ϕ([0, 2s], y) ⊂ N1 \L1 , ϕ([s, 3s], y) ⊂ N2 \L2 and ϕ(3s, y) ∈ U ∀(s, y) ∈ W . Thus f (s, [y]) = ϕ(3s, y) ∈ U as (s, y) ∈ W with y ∈ N1 , i.e., f is continuous at this point (t, x).

5.5 Conley Index Theory

401

(b) ϕ([t, 3t], x) ∩ L2 = ∅. Since L2 is an exit set of N2 , ϕ(3t, x) ∈ L2 . Let [U ] be any neighborhood of [L2 ] in N2 \L2 , deﬁne V = {x ∈ N2 \L2 | x ∈ [U ]} ∪ (X\N2 ) ∪ L2 , then V is a neighborhood of L2 in X, and [U ] = (V ∩ (N2 \L2 )) ∪ [L2 ]. By the continuity of the ﬂow ϕ, ∃ a neighborhood W of (t, x) such that ϕ(3s, y) ∈ V ∀(s, y) ∈ W . Thus f (s, [y]) ∈ {[ϕ(3s, y)], [L2 ]} ⊂ (V ∩ (N2 \L2 )) ∪ [L2 ] ⊂ [U ] , ∀(s, y) ∈ W with y ∈ N1 . Again f is continuous at this point. The proof is complete. Theorem 5.5.16 If (N1 , L1 ) and (N2 , L2 ) are two index pairs relative to ∗ ∗ U ∈ Σ, then H (N1 , L1 ; F ) ∼ = H (N2 , L2 ; F ). Proof. According to Lemma 5.5.15, there are continuous functions: f : [T, ∞)× N1 /L1 → N2 /L2 and g : [T, ∞) × N2 /L2 → N1 /L1 deﬁned by ϕ(3t, x) if ϕ([0, 2t], x) ⊂ N1 \L1 and ϕ([t, 3t], x) ⊂ N2 \L2 , f (t, [x]) = otherwise , [L2 ] and

ϕ(3t, x) g(t, [x]) = [L1 ]

if ϕ([0, 2t], x) ⊂ N2 \L2 and ϕ([t, 3t], x) ⊂ N1 \L1 , otherwise ,

respectively. One deﬁnes ηi : [0, T ] × Ni /Li → Ni /Li ϕ(6t, x) if ϕ([0, 6t], x) ⊂ Ni \Li , ηi (t, [x]) = otherwise , [Li ] i = 1, 2. Again they are continuous and satisfy η1 (T, [x]) = g(T, f (T, [x])), η2 (T, [x]) = f (T, g(T, [x])) and ηi (0, [x]) = idNi /Li i = 1, 2. This shows that N1 /L1 and N2 /L2 have the same homotopy type. It follows that ∗

∗

H (N1 /L1 , [L1 ]; F ) ∼ = H (N2 /L2 , [L2 ]; F ) . By the special excision property, we have ∗ ∗ H (Ni , Li ; F ) ∼ = H (Ni /Li , [Li ]; F ), i = 1, 2 .

Combining them together, we obtain ∗ ∗ H (N1 , L1 ; F ) ∼ = H (N2 , L2 ; F ) .

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5 Topological and Variational Methods

Example 1. (Hyperbolic ﬁxed point) Recall the hyperbolic system and the Hartman–Grobman theorem in Chap. 1. Let f : Rn → Rn be C 1 . Assume that θ is a hyperbolic ﬁxed point with k-dimensional unstable manifold, i.e., A = eL with L = f (θ) satisfying σ(A) ∩ S 1 = ∅. Then we have the decomposition Rn = E u ⊕ E s where E u and E s are invariant subspaces of A, on which the eigenvalues of Au = A|E u lie outside the unit circle, and those of As = A|E s lie inside the unit circle with dim E u = k. Thus L = Lu ⊕ Ls , where Lu = L|E u and Ls = L|E s . The real parts of all eigenvalue of Lu are greater than zero, while those of Ls are less than zero. According to the Hartman–Grobman theorem the ﬂow in a neighborhood of θ for the ODE x˙ = f (x) is topologically equivalent to the ﬂow in a neighborhood of x˙ = Lx, i.e., ϕ(t, x) = eLt x. An isolating neighborhood of the origin is given by B k × B n−k and the exit set by S k−1 × B n−k . Let U = N = B k × B n−k and L = S k−1 × B n−k . It follows that h(t, U ) = tk . Example 2. An invariant manifold S is an invariant set with manifold structure. It is called normally hyperbolic, if the tangent bundle over Rn restricted on S has the bundle decomposition: (TRn )S = E u ⊕ E s with u s Eϕ(t,x) = T ϕ(t, ·)Exu , Eϕ(t,x) = T ϕ(t, ·)Exs

∀x ∈ S, ∀t ∈ R1 .

According to Thom’s isomorphism theorem, see [BT],[Mi 3],[Sp], we have the following result: Let S be a normally hyperbolic invariant manifold. Let E u be the vector bundle over S deﬁned by the local unstable manifold of S. If E u is a rank k orientable bundle, then h(U ) = H

∗+k

(S; F ) ,

for U ∈ Σ with I(U ) = S. Thus, if S is a hyperbolic periodic orbit with an oriented unstable manifold of dimension k + 1, then h(t, U ) = tk + tk+1 . Example 3. (Critical groups) Let (M, f, φ) be a pseudo-gradient ﬂow, and let {O, α, β} be an isolating triplet for a dynamically isolated critical set S of f . Then ∗ ! ∩ fβ , O ! ∩ fα ; F ) . h(O) = H (O In particular, if S is an isolated critical point p, then h(O) = C∗ (f, p) , where C∗ (f, p) is the critical groups of f at S. The Conley index possesses many important properties: Wazewski’s principle, the continuation and the Morse–Smale decomposition, which are the

5.5 Conley Index Theory

403

counterparts of the Kronecker existence, the homotopy invariance and the (sub) additivity of the degree theory. First, we introduce a type of compactness condition on U ∈ Σ. Deﬁnition 5.5.17 (Condition(B)) U ∈ Σ is said to satisfy Condition (B) if ∀ closed neighborhoods W of I(U ) there exists T > 0 such that GT (U ) ⊂ W . In particular, if U ∈ Σ satisﬁes Condition (B) and I(U ) = ∅, then ∃T > 0 such that GT (U ) = ∅. It is known that if U ∈ Σ is compact, then U satisﬁes Condition (B) (see Remark 5.5.4). We investigate other suﬃcient conditions of Condition (B): Rubakowski [Ry] introduced: Condition (A) U ∈ Σ is said to satisfy Condition (A) if ∀xn ∈ U, ∀tn → +∞, ϕ([0, tn ], xn ) ⊂ U implies that ϕ(tn , xn ) is subconvergent. Condition (A) ⇒ Condition (B). Indeed, if not, ∃W , a closed neighborhood of I(U ) such that Gn (U ) ⊂ W ∀n, i.e., ∃yn ∈ Gn (U )\W . Let xn = ϕ(−n, yn ), then ϕ([0, n], xn ) ⊂ U . By Condition (A), yn = ϕ(n, xn ) subconvergent to y0 ∈ U , and then y0 ∈ I(U ). But W is closed, therefore y0 ∈ int(W ). This is a contradiction. The isolating neighborhoods of the examples of dynamically isolated critical sets after Theorem 5.5.10 satisfy Condition (B). More precisely, let (M, f, ϕ) be a pseudo-gradient ﬂow, and let c be an isolated critical value. If Kc = K ∩ f −1 (c) is isolated (or, p ∈ Kc is isolated), then the set ˜ ∩ f −1 [c − , c + ] is an isolating neighborhood of Kc (or p resp.) W = cl(O) satisfying Condition (B), if > 0 is small so that f −1 [c − , c + ] ∩ K = Kc , and O is a neighborhood of Kc (or p resp.) containing only critical points in Kc (or p resp.). From Lemma 5.5.9, it is suﬃcient to show that any neighborhood V of p, containing p as the only critical point, contains a (MVP) neighborhood U of p. In fact, let BR (p) ⊂ V for some R > 0, we choose r = R2 , O = Br (p), ˜ ∩ f −1 (c − δ, c + δ) for small δ > 0. By standard estimates, and U = cl(O) U ⊂ BR (p) ⊂ V . The following Wazewski principle holds: Theorem 5.5.18 Let U ∈ Σ satisfy Condition (B). If h(U ) = 0 then I(U ) = ∅. ∗

Proof. We prove that I(U ) = ∅ implies H (N, L; F ) = 0 for all index pairs (N, L) relative to U . According to Theorem 5.5.13, it is suﬃcient to verify that for a special index pair (N, L) = (GT (U ), ΓT (U )). Condition (B) yields the existence of large T > 0 such that GT (U ) = ∅. The conclusion follows trivially.

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Next, we turn to the continuation. A special form of the homotopy invariance of the topological degree asserts that the degree deg(f, Ω, p) is continuous with respect to the vector ﬁeld f . In Conley index theory, the vector ﬁeld f is replaced by the ﬂow ϕ, so we should study: Under what condition on U ∈ Σ, and in which sense of the variance between ﬂows ϕ and ψ, does one have h(U, ϕ) = h(U, ψ)? First we notice: Lemma 5.5.19 Assume that U ∈ Σ(ϕ) satisﬁes Condition (B) and that I(U ) is compact. Then U satisﬁes the condition: Σ0 (ϕ) : ∃T, δ > 0 such that Nδ (GT (U )) ⊂ int(U ) , where Nδ (A) is the δ-neighborhood of the set A. Moreover, ∃T, δ > 0 such that Nδ (GT (U )) ⊂ GT /2 (U ), Nδ (GT /2 (U )) ⊂ int(U ) .

(5.47)

Proof. 1. Let = d(∂U, I(U )) where d(·, ·) is the distance between two sets induced by the metric d. Since I(U ) is compact, > 0. Let V = N/2 (I(U )); by Condition (B), one has T > 0 such that GT (U ) ⊂ V . Let δ = /2; the ﬁrst conclusion follows. 2. To prove the second conclusion, we claim that ∂GT (U ) ∩ I(U ) = ∅. Indeed, if not, ∃x0 ∈ ∂GT (U ) ∩ I(U ). From x0 ∈ ∂GT (U ), it implies that ∃t0 ∈ [−T, T ] such that ϕ(t0 , x0 ) ∈ ∂U . But from x0 ∈ I(U ), we have ϕ([t0 − T, t0 + T ], x0 ) ⊂ U , then ϕ(t0 , x0 ) ∈ GT (U ) ⊂ int(U). This is a contradiction. Then 1 := d(∂GT (U ), I(U )) > 0; we repeat the proof in step 1, and obtain T1 > 0 such that GT1 (U ) ⊂ V := N 21 (I(U )). Thus, N 21 (GT1 ) ⊂ GT (U ). One deﬁnes the Hausdorﬀ metric between two sets U and V as follows: dH (U, V ) = sup d(x, V ) + sup d(U, x) . x∈U

x∈V

Lemma 5.5.20 For U ∈ Σ0 (ϕ), there exist T, > 0 such that if a ﬂow ψ satisﬁes (5.48) d(ϕ(t, x), ψ(t, x)) ≤ ∀(t, x) ∈ [−T, T ] × Nδ (U ) and a closed set V satisﬁes dH (U, V ) ≤ , then U, V ∈ Σ0 (ψ) and V ∈ Σ0 (ϕ).

(5.49)

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Proof. By deﬁnition, one has T, δ > 0 such that Nδ (GTϕ (U )) ⊂ int(U ) . Letting ≤ δ/4, we have N (GTψ (U )) ⊂ N2 (GTϕ (U )) ⊂ int(U ) , i.e., U ∈ Σ0 (ψ), and N2 (GTψ (V )) ⊂ N4 (GTϕ (U )) ⊂ int(U ) . Thus dH (N (GTψ (V )), ∂V ) ≥ ; it follows that N (GTψ (V )) ⊂ int(V ) i.e., V ∈ Σ0 (ψ). Similarly, we prove that V ∈ Σ0 (ϕ). Theorem 5.5.21 (Continuation) For U ∈ Σ0 (ϕ), ∃T, > 0 such that h(U, ϕ) = h(V, ψ) , whenever (ψ, V ) satisﬁes (5.48) and (5.49), and that both ϕ and ψ are uniformly continuous on [−T, T ] × N2δ (U ), namely, d(φ(t, x), φ(t, y))

0, and show (N, L) = (GT , ΓT ), (N ! T /Γ !T , that g2 ◦ g1 and g1 ◦ g2 are homotopic to identity maps on GT /ΓT and G respectively. Let us deﬁne for λ ∈ [0, 1] the map [ϕ(2T, ξ−λ (T, x))] ϕ([0, 2T ], ξ−λ (T, x)) ⊂ GT H(λ, [x]) = [ΓT ] otherwise . Note: It is equivalent to replace ϕ([0, 2T ], ξ−λ (T, x)) ⊂ GT by ϕ(2T, ξ−λ (T, x)) ∈ GT on the right-hand side of the deﬁnition. Since H(1, [x]) = g2 ◦ g1 and [ϕ(2T, x)] ϕ([0, 2T ], x) ⊂ GT H(0, [x]) = [ΓT ] otherwise , the latter is homotopic to id|GT /ΓT as shown in Theorem 5.5.16. It remains to verify that the deﬁnition is well-deﬁned and that H(λ, [x]) is continuous. We shall show that x ∈ ΓT ⇒ H(λ, x) = [ΓT ] i.e., ϕ([0, 2T ], ξ−λ (T, x)) ⊂ T G . From x ∈ ΓT , by step 1, ξ−λ (T, x) ∈ N (ΓT ), i.e., ∃x ∈ ΓT such that d(ξ−λ (T, x), x) < . Following the argument in 3, ϕ([ 12 T, 32 T ], x) ∩ ∂U = ∅, we have ϕ([0, 2T ], ξ−λ (T, x)) ⊂ GT . As to the continuity, we deﬁne h2 : GT /ΓT → GT /ΓT by [ϕ(T, x)] ϕ(T, x) ∈ GT h2 ([x]) = [ΓT ] otherwise . Since ϕ(T, x) ∈ GT can be replaced by ϕ([0, T ], x) ⊂ GT , the continuity of h2 has been proved in Theorem 5.5.16, and from ϕ([0, T ], x) ⊂ GT ⊂ int(U ), it implies x ∈ ΓT , h2 is well deﬁned. By deﬁnition H(λ, [x]) = h2 ◦ h1 ◦ ξ−λ (T, x) , therefore H is continuous on [0, T ] × GT /ΓT . 6. Finally, we consider the case U = V , but with ϕ ! = ϕ, and write GT (U ) = T T T T Gϕ (U ), G (V ) = Gϕ (V ). Since Nδ (G (U )) ⊂ int(U ), Nδ (GT (V )) ⊂ int(V ), if one takes 0 < < δ/3 and dH (U, V ) < , we have N (GT (U )) ⊂ V and N (GT (V )) ⊂ U .

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! , L) ! be index pairs related to U and V resp., then Let (N, L) and (N ! \L. ! ∃T1 , T2 > 0 such that GT1 (N \L) ⊂ GT1 (U ) ⊂ V , and GT2 (V ) ⊂ N

! \L. ! Similarly one has GT1 +T2 (N ! \L) ! ⊂ N \L. By Thus GT1 +T2 (N \L) ⊂ N ∗ ∗ ! , L; ! F ), i.e., H (N the proof of Theorem 5.5.16, we obtain H (N, L; F ) ∼ = h(U, ϕ) = h(V, ϕ). Combining steps 5 and 6, we have h(U, ϕ) = h(V, ϕ) = h(V, ψ) .

The proof is complete. 5.5.3 Morse Decomposition on Compact Invariant Sets and Its Extension

In the study of the dynamics on invariant sets the Morse–Smale decomposition is an extension of the Morse relation. Before going on, let us introduce the notion of attractor–repeller pairs. On a metric space (X, d) with ﬂow ϕ, ∀Y ⊂ X, deﬁne ω(Y ) = ∩ ϕ([t, ∞), Y ) and ω ∗ (Y ) = ∩ ϕ((−∞, −t], Y ) . t>0

t≥0

By deﬁnition, ω(Y ) is an invariant closed set. Deﬁnition 5.5.22 Let S be a compact invariant set for ϕ; a subset A ⊂ S is called an attractor in S, if there exists a neighborhood U of A such that ω(U ∩ S) = A. The dual repeller of A in S is deﬁned by A∗ = {x ∈ S| ω(x) ∩ A = ∅} . The pair (A, A∗ ) is called an attractor–repeller pair. The set C(A∗ , A, S) = {x ∈ S| ω(x) ⊂ A, ω ∗ (x) ⊂ A∗ } is called the set of connecting orbits from A∗ to A in S. The following properties hold. (1) A and A∗ are disjoint compact invariant sets. Proof. In the following, we use the notation U as in Deﬁnition 5.5.22. In fact, A = ω(U ∩ S) is invariant. Both ω(x) and A are invariant, so is A∗ . If ∃x ∈ A ∩ A∗ , then ω(x) ⊂ A. But by deﬁnition of A∗ , ω(x) ∩ A = ∅. Since S is compact, ω(x) = ∅. This is a contradiction. Therefore A ∩ A∗ = ∅. By deﬁnition, A is closed. It remains to verify the closedness of A∗ . If {xn } ⊂ A∗ with xn → x ∈ S, and if ω(x) ∩ A = ∅, then ϕ(t, x) ∈ U for some t > 0. Consequently ϕ(t, xn ) ∈ U for n large. Therefore ω(xn ) ⊂ ω(U ∩ S) = A. This is impossible. Therefore ω(x) ∩ A = ∅, i.e., x ∈ A∗ .

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(2) For any open neighborhood V of A, ∃T = T (V ) such that ∪ ϕ(t, U ∩ t≥T

S) ⊂ V . Proof. If not, ∃V ⊃ A open, ∃xn ∈ U ∩ S ∃tn → +∞ such that ϕ(tn , xn ) ∈ V . Since S is a compact invariant set, ϕ(tn , xn ) → y ∈ S\V . But y ∈ ω(U ∩ S) = A. This is a contradiction. (3) If B is a closed set disjoint from A, then ∀ > 0 ∃T = T > 0 such that d(x, A∗ ) < , whenever x ∈ S and t ≥ T such that ϕ(t, x) ∈ B. Proof. If not, ∃ > 0 ∃tn → ∞ ∃xn ∈ S such that ϕ(tn , xn ) ∈ B but d(xn , A∗ ) ≥ . Since S is compact and invariant, ∃x ∈ S such that xn → x. Then d(x, A∗ ) ≥ , which implies that x ∈ A∗ , i.e., ω(x) ∩ A = ∅. Thus for the neighborhood U of A; we have t1 ∈ R1 such that ϕ(t1 , x) ∈ U ∩ S and then ϕ(t1 , xn ) ∈ U ∩ S for n large. Let V = X\B, which is an open neighborhood of A, from (2), ∃T > 0 such that ϕ(t, xn ) ∈ V ∀t ≥ T . This contradicts ϕ(tn , xn ) ∈ B ∀n. For any x ∈ X, we denote the orbit passing through x by o(x) = {ϕ(t, x)| t ∈ R1 }. (4) ω(y) ∩ A∗ = ∅ ⇒ o(y) ⊂ A∗ . Proof. Let B be a closed neighborhood of A∗ such that A ∩ B = ∅. Since ω(y) ∩ A∗ = ∅, ∃tn → +∞ such that ϕ(tn , y) ∈ B. ∀t ∈ R1 , let z = ϕ(t, y); we have ϕ(tn − t, z) = ϕ(tn , y) ∈ B, thus from (3), ∀ > 0, we have d(z, A∗ ) < . Since > 0 is arbitrarily small, o(y) ⊂ A∗ . (5) ω ∗ (y) ∩ A = ∅ ⇒ o(y) ⊂ A. Proof. By the assumption that ∃tn → +∞ such that ϕ(−tn , y) ∈ U ∩ S, then ∀t ∈ R1 , t + tn ≥ 0 for n large, from ϕ(t, y) = ϕ(tn + t, ϕ(−tn , y)) we have ϕ(t, y) ∈ ω(U ∩ S), i.e., o(y) ⊂ A. Lemma 5.5.23 Let (A, A∗ ) be an attractor–repeller pair of a compact invariant set S, then S = A ∪ A∗ ∪ C(A∗ , A, S) . Proof. It is suﬃcient to prove that ∀x ∈ S\(A ∪ A∗ ), ω(x) ⊂ A, and ω ∗ (x) ⊂ A∗ . 1. ω(x) ⊂ A : ∀y ∈ ω(x), i.e., ∃tn → +∞ such that ϕ(tn , x) → y. Let B = S\U , where U is as in Deﬁnition 5.5.22, then either ∃n0 ∈ N such that z = ϕ(tn0 , x) ∈ U or ϕ(tn , x) ∈ B, ∀n. But the latter case is impossible, because from (3) we would have x ∈ A∗ . In the former case, y ∈ ω(S ∩U ) = A. 2. ω ∗ (x) ⊂ A∗ : ∀y ∈ ω ∗ (x), ∃tn → +∞ such that zn = ϕ(−tn , x) → y. Let B = {x}. Since {x} ∩ A = ∅, we have d(zn , A∗ ) → 0, provided by (3). Thus y ∈ A∗ .

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Combining the conclusions of Lemma 5.5.23 and properties (4) and (5), we see that ∀x ∈ C(A∗ , A, S), ω(x) ⊂ A and ω ∗ (x) ⊂ A∗ , and there are connecting orbits from A to A∗ but none from A∗ to A. Let us deﬁne the Morse decomposition for a compact invariant set. Deﬁnition 5.5.24 Let S be a compact invariant set of X with respect to the ﬂow ϕ. An ordered collection (M1 , . . . , Mn ) of invariant subsets Mj ⊂ S is called a Morse decomposition of S, if there exists an increasing sequence of attractors in S: ∅ = A0 ⊂ A1 ⊂ · · · ⊂ An = S such that Mj = Aj ∩ A∗j−1 , 1 ≤ j ≤ n. Example 1. An attractor–repeller pair (A, A∗ ) of a compact invariant set S is a Morse decomposition, where A0 = ∅, A1 = A, A2 = S. Example 2. Suppose f ∈ C 1 (M, R1 ), where M is a compact manifold. Assume that f −1 [a, b] ∩ K = {p1 , . . . , pn }, where a, b are regular values of f with f (pi ) ≤ f (pi+1 ), i = 1, . . . , n − 1. Then ({p1 }, . . . , {pn }) is a Morse decomposition of S = I(f −1 ([a, b])). In fact, by setting A0 = ∅, and Ai = {x ∈ S| ω ∗ (x) ⊂ {p1 , . . . , pi }}, i = 1, 2, . . . , n. We shall verify that this is an increasing sequence of attractors in S with A∗i = {x ∈ S| ω(x) ⊂ {pi+1 , . . . , pn }}, and then Ai ∩ A∗i−1 = {pi }. It is proved by induction. Let Sk = I(f −1 [a, ak ]), where ak ∈ (f (pk ), f (pk+1 )), k = 1, . . . , n − 1, and an = b. Thus Sn = S. We verify that Ai = {x ∈ Sk | ω ∗ (x) ⊂ {p1 , p2 , . . . , pi }}, i = 1, 2, . . . , k, is an increasing sequence of attractors in Sk , and A∗i = {x ∈ Sk | ω(x) ⊂ {pi+1 , . . . , pk }}. For n = 1. Obviously, A1 = {p1 } is an attractor in S1 = {p1 }, with A∗1 = ∅ and A∗0 = S1 . Thus A1 ∩ A∗0 = {p1 }. If the conclusion holds for n = k, i.e., Ai = {x ∈ Sk | ω ∗ (x) ⊂ {p1 , . . . , pi }}, i = 1, . . . , k, is an increasing sequence of attractors in Sk . For n = k + 1. It is easily seen that Sk is an attractor in Sk+1 . Indeed, let Uk = f −1 (a, ak ), then ω(Sk+1 ∩ Uk ) = Sk with Sk∗ = {pk+1 }. Then Ak = {x ∈ Sk+1 | ω ∗ (x) ⊂ {p1 , . . . , pk }} = Sk is an attractor in Sk+1 with A∗k = Sk∗ = {pk+1 }. In this case ∅ = A0 ⊂ A1 ⊂ . . . ⊂ Ak = Sk ⊂ Ak+1 = Sk+1 is an increasing sequence of attractors in S = Sk+1 . By easy computation, A∗i = {x ∈ Sk+1 | ω(x) ⊂ {pi+1 , . . . , pk+1 }} and then Ai ∩ A∗i−1 = {pi } . Our conclusion follows from the mathematical induction. By the deﬁnition of the Morse decomposition (M1 , . . . , Mn ) of S, we have the following conclusions: (1) {Mi | 1 ≤ i ≤ n} are pairwise disjoint.

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Indeed if i < j, then Mi ∩ Mj = Ai ∩ A∗i−1 ∩ Aj ∩ A∗j−1 ⊂ Ai ∩ A∗j−1 ⊂ Aj−1 ∩ A∗j−1 = ∅, from property (1). (2) ∀x ∈ S\( ∪ Mi ), ∃i < j +1 such that ω(x) ⊂ Mi and ω ∗ (x) ⊂ Mj+1 . 1≤i≤n

Indeed, set i = min{k ∈ N| ω(x) ⊂ Ak } and j = max{k ∈ N| ω ∗ (x) ⊂ A∗k } . From A0 = ∅ and An = S, we see 0 < i and j < n. Since ω(x) ⊂ Ai−1 , provided by the decomposition lemma, x ∈ A∗i−1 , and then by property (4), o(x) ⊂ A∗i−1 , in particular, ω(x) ⊂ A∗i−1 . By the same reason, ω ∗ (x) ⊂ Aj+1 , we have o(x) ⊂ Aj+1 and ω ∗ (x) ⊂ Aj+1 . We claim that i ≤ j + 1. For otherwise, j + 1 ≤ i − 1, then o(x) ⊂ Aj+1 ∩ A∗i−1 ⊂ Ai−1 ∩ A∗i−1 = ∅, a contradiction. For i = j + 1, we have / Mi . Therefore the only possibility o(x) ⊂ Ai ∩ A∗i−1 = Mi , it contradicts x ∈ is i < j + 1. In this case we have ω(x) ⊂ Ai ∩ A∗i−1 = Mi and ω ∗ (x) ∈ Aj+1 ∩ A∗j = Mj+1 . Comparing with Example 2, the following conclusion is easily veriﬁed: (3) Aj = {x ∈ S| ω(x) ∪ ω ∗ (x) ⊂ M1 ∪ . . . ∪ Mj } and A∗j = {x ∈ S| ω(x) ∪ ω ∗ (x) ⊂ Mj+1 ∪ . . . ∪ Mn }, j = 1, 2, . . . , n. (4) If (M1 , . . . , Mn ) is a Morse decomposition of S, and if ∅ = A0 ⊂ A1 ⊂ . . . ⊂ An = S is the associated increasing sequence of attractors, then for 0 < i ≤ j < n, (M1 , . . . , Mi−1 , Mji , Mj+1 , . . . , Mn ) is again a Morse decomposition of S, where Mji = Aj ∩ A∗i−1 = {x ∈ S| ω(x) ∪ ω ∗ (x) ⊂ Mi ∪ . . . ∪ Mj }. (5) If S is an isolating invariant compact set, and if {M1 , . . . , Mn } is a Morse decomposition of S, then Mi , 1 ≤ i ≤ n are isolating. Proof. We assume that there exists a compact neighborhood U of S such that I(U ) = S ⊂ int(U ). According to (1), Mi and Mj are disjoint, ∀i = j, we have a neighborhood Ui ⊂ U of Mi such that Ui ∩ Mj = ∅ ∀j = i. We claim that I(Ui ) = Mi . “⊂” ∀x ∈ I(Ui ), o(x) ⊂ I(Ui ) ⊂ Ui ⊂ U . Therefore ω(x) ∪ ω ∗ (x) ⊂ Ui ∩ S. According to (2), ω(x) ∪ ω ∗ (x) ⊂ Mi and then x ∈ Mi . “⊃” Since Mi is an invariant set in Ui , Mi ⊂ I(Ui ). For a compact isolating invariant set S, let U ∈ Σ be its compact isolated neighborhood, i.e., I(U ) = S ⊂ int(U ). We deﬁne 1 , x) ⊂ U, ω(x) ⊂ Mj ∪ . . . ∪ Mn } and Ij+ = {x ∈ U | ϕ(R+ 1 , x) ⊂ U, ω ∗ (x) ⊂ M1 ∪ . . . ∪ Mj }, 1 ≤ j ≤ n . Ij− = {x ∈ U | ϕ(R−

(6) 1 I1+ ⊃ I2+ ⊃ . . . ⊃ In+ , ϕ(R+ , Ij+ ) = Ij+ , 1 ≤ j ≤ n , 1 I1− ⊃ I2− ⊃ . . . ⊃ In− , ϕ(R− , Ij− ) = Ij− , 1 ≤ j ≤ n .

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Aj ⊂ Ij− , A∗j−1 ⊂ Ij+ , and then

Mji = Aj ∩ A∗i−1 = Ii+ ∩ Ij− ,

for i ≤ j. (7) Ij± , 1 ≤ j ≤ n are compact. Proof. It is suﬃcient to show that they are closed. For I1+ and In− it follows from the continuity of the ﬂow and the closedness of U . In the case where n = 2, it suﬃces to verify the closedness of I2+ and I1− . Let (M1 , M2 ) be the Morse decomposition, and let {xn } ⊂ I2+ and xn → x; we want to show that x ∈ I2+ . Since x ∈ I1+ , we have ω(x) ⊂ M1 ∪ M2 . It is suﬃcient to show that ω(x) ⊂ M1 . Suppose not, i.e., ω(x) ⊂ M1 . We choose open neighborhoods Vi of Mi , i = 1, 2 such that V 1 ∩ V 2 = ∅. Since ω(xn ) ⊂ M2 and xn → x, we have tn ≥ 0 and tn ≥ 0 such that ϕ([tn , ∞), xn ) ⊂ V2 and ϕ(tn , xn ) ∈ V1 . Thus ∃tn ∈ [tn , tn ] such that ϕ([tn , ∞), xn ) ⊂ U \V1 and ϕ(tn , xn ) ∈ U \(V1 ∪ V2 ). From the compactness of U , after a subsequence ϕ(tn , xn ) → y ∈ M1 ∪ M2 and ϕ([0, +∞), y) ⊂ U \V1 . From xn ∈ I1+ , we have ϕ(tn , xn ) ∈ I1+ ; it follows that y ∈ I1+ , and then ω(y) ⊂ M2 . If {tn } is bounded, then after a subsequence, tn → s; we have ω(ϕ(−s, y)) = ω(y) ⊂ M2 . But ϕ(−s, y) = x. This contradicts ω(x) ⊂ M1 . If {tn } is unbounded, without loss of generality may assume, tn → +∞. ∀t > 0 ∃n large such that ϕ([−t, 0], ϕ(tn , xn )) = ϕ([tn −t, tn ], xn ) ⊂ ϕ([0, ∞), x) ⊂ U ; it follows that ϕ([−t, 0], y) ⊂ U . Since t > 0 is arbitrary, 1 1 , y) ⊂ U . Combining with ϕ(R+ , y) ⊂ U \V1 , it follows that we have ϕ(R− o(y) ⊂ U , i.e., y ∈ I(U ) = S. Since (M1 , M2 ) is a Morse decomposition of S, and ω(y) ⊂ M2 , it follows that y ∈ M2 , again a contradiction. Similarly, I1− is closed. For n > 2, it can be reduced to the case where n = 2 by deﬁning M1 = Mj−1,1 and M2 = Mn,j . Thus (M1 , M2 ) is again a Morse decomposition of S from (4), since Ij+ = I2+ , where I2+ is that in (M1 , M2 ). Thus Ij+ is closed.

Similarly Ij− = Ij− , where M1 = Mj1 and M2 = Mn,j+1 . The proof is complete. (8) Let U ∈ Σ be compact. If Z ⊂ U is closed, and if ∀x ∈ Z ∃t > 0 such that ϕ(t, x) ∈ U , then the minimal positively invariant set in U which contains Z, deﬁned by P (Z, U ) = {x ∈ U | ∃t ≥ 0 such that ϕ([−t, 0], x) ⊂ U and ϕ(−t, x) ∈ Z} is closed. It follows directly from the continuity of the ﬂow and the closedness of U . Lemma 5.5.25 Suppose that U ∈ Σ is a compact isolating neighborhood with S = I(U ), and that (M1 , . . . , Mn ) is a Morse decomposition of S. Then for every open neighborhood V of Ij− , there exists a compact neighborhood Nj such that Nj ⊂ V and Nj is positively invariant in U .

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413

+ Proof. Since Ij+1 ∩ Ij− = ∅, we choose a compact neighborhood W ⊂ U of + − Ij+1 ∩ In such that W ∩ Ij− = ∅. 1. ∃t∗ > 0 such that ϕ([−t∗ , 0], x) ⊂ U \W ⇒ x ∈ V ∩ (U \W ). If not, 1 with tn → +∞ such that ϕ([−tn , 0], xn ) ⊂ U \W , ∃{xn } ⊂ U and {tn } ⊂ R+ but xn ∈ V ∩ (U \W ). Let x be a limit point of xn , then x ∈ V ∩ (U \W ) 1 , x) ⊂ U \W . Thus ω ∗ (x) ⊂ M1 ∪ . . . ∪ Mj . Therefore x ∈ Ij− ⊂ and ϕ(R− V ∩ (U \W ). A contradiction. 2. Deﬁne two disjoint subsets of Ij−

A = {x ∈ Ij− | ϕ([0, t∗ ], x) ⊂ U } and B = {x ∈ Ij− | ϕ([0, t∗ ], x) ⊂ U } , then ∀x ∈ A ∃δ = δ(x) > 0 such that ϕ([0, t∗ ], Bδ (x)) ⊂ V ∩(X\W ), from step 1. ∀x ∈ B ∃t = t(x) > 0 such that ϕ([0, t], x) ⊂ V ∩ (X\W ) and ϕ(t, x) ∈ U . This enables us to choose δ = δ(x) > 0 such that ϕ([0, t(x)], Bδ (x)) ⊂ V ∩ (X\W ) and ϕ(t(x), Bδ(x) (x)) ∩ U = ∅. Since Ij− is compact, ∃x1 , . . . , xk ∈ Ij− k

such that Ij− ⊂ ∪ Bδ(xi ) (xi ), and then we choose a compact neighborhood of i=1

k

Ij− , Z ⊂ ∪ Bδ(xi ) (xi ). i=1

3. We claim that P (Z, U ) ⊂ V ∩ (U \W ). ∀x ∈ P (Z, U ), by deﬁnition ∃t ≥ 0 such that ϕ([−t, 0], x) ⊂ U , and ϕ(−t, x) ∈ Z. Then by the deﬁnition of Z, ∃i ∈ [1, k], such that ϕ(−t, x) ∈ Bδ(xi ) (xi ). If x ∈ V ∩ (U \W ), there are two cases: Either xi ∈ A, then ϕ([−t, t∗ − t], x) ⊂ V ∩ (X\W ) which implies t > ∗ t , and then ∃t1 ∈ [0, t∗ − t] such that ϕ([−t, −t1 ], x) ⊂ V ∩ (X\W ) and y = ϕ(−t1 , x) ∈ V ∩ (X\W ). It implies that ϕ([−t∗ , 0], y) ⊂ U \W , but y ∈ V ∩ (X\W ). This contradicts step 1. Or xi ∈ B, then ϕ([−t, t(xi ) − t], x) ⊂ V ∩ (X\W ) and ϕ(t(xi ) − t, x) ∈ U . From ϕ([−t, 0], x) ⊂ U , we see that t(xi ) > t. But if x ∈ V ∩ (X\W ), we would have t(xi ) < t. Again this is a contradiction. We have proved that x ∈ V ∩ (X\W ) ∩ U = V ∩ (U \W ). 4. P (Z, U ) is compact. In fact, if {xn } ⊂ P (Z, U ) with xn → x, i.e., ∃tn ≥ 0 such that ϕ([−tn , 0], xn ) ⊂ U and ϕ(−tn , xn ) ∈ Z, then ϕ([−tn , 0], xn ) ⊂ P (Z, U ) ⊂ 1 , x) ⊂ V ∩ (U \W ). It follows V ∩ (U \W ). If {tn } is unbounded, then ϕ(R− ∗ that ω (x) ∈ Mi for some i ≤ j, and then x ∈ Ij− ⊂ P (Z, U ). Otherwise, {tn } is bounded; after a subsequence, tn → s ≥ 0, then ϕ([−s, 0], x) ⊂ U and ϕ(−s, x) ∈ Z; again x ∈ P (Z, U ). Now we set Nj = P (Z, U ). Combining steps 1 and 4 and (8), the conclusion follows. Inductively applying Lemma 5.5.25, we have the following: Theorem 5.5.26 Let S be an isolating invariant set with a compact isolated neighborhood U0 . If (M1 , . . . , Mn ) is a Morse decomposition of S, then there exists an increasing sequence of compact sets: N0 ⊂ N1 ⊂ . . . ⊂ Nn such that

414

5 Topological and Variational Methods

(Nj , Ni−1 ) is an index pair for Mji ∀i ≤ j. In particular, (Nn , N0 ) is an index pair for S and (Nj , Nj−1 ) is an index pair for Mj , ∀j. Proof. According to Theorem 5.5.13, in combination with Remark 5.5.4, there exists an index pair (Nn , N0 ) for S. Setting U = Nn \N0 , we deﬁne compact sets Ij± j = 1, 2, . . . , n see (7), and Un = U ∩ Nn . Applying Lemma 5.5.25, we − have a compact neighborhood Un−1 ⊂ U of In−1 , which is positively invariant + in Nn , and Un−1 ∩ In = ∅. Deﬁne Nn−1 = Un−1 ∪ N0 . Inductively, one deﬁnes a compact neighborhood Uj ⊂ U of Ij− , which is positively invariant in Nj+1 + and Uj ∩ Ij+1 = ∅, and Nj = Uj ∪ N0 , j = n − 1, . . . , 1. Recalling (5.2) and (5.3), as a consequence of Theorem 5.5.26, we have the Morse relations for the Morse decomposition (M1 , . . . , Mn ), i.e., n

P (t; Nj , Nj−1 ) = P (t; Nn , N0 ) + (1 + t)Q(t) ,

j=1

where Q is a formal series with nonnegative coeﬃcients, and P (t; Nj , Nj−1 ) =

∞

tq

q

rank H (Nj , Nj−1 ; F ) ,

q=0

or, in the spirit of Theorem 5.1.29, n

h(t; Mj ) = h(t; U0 ) + (1 + t)Q(t) .

j=1

Let us return to Example 2, and assume that f is a Morse function, i.e., ∀p ∈ K, f (p) is invertible, or in other words, the gradient system x˙ = −f (x) is hyperbolic. ∀p ∈ K, let W s (p) = {x ∈ M | ω(x) = p} and W u (p) = {x ∈ M | ω ∗ (x) = p} be the stable and unstable manifolds at p, respectively. One has dim W s (p) = ind(f, p) = codimW u (p) . According to the Sard–Smale theorem, one may choose a generic Riemannian metric on M such that W s (p)W u (q) ∀p, q ∈ K. Such Morse functions are said to be of Morse–Smale type. Thus, if ind(f, q) − ind(f, p) = k, and let M (q, p) = W u (q) ∩ W s (p), dim M (q, p) = k . We have seen that if K = {p1 , . . . , pn }, then ({p1 }, . . . , {pn }) is a Morse decomposition of M , where p1 , . . . , pn are ordered by their values {f (pi )| i ≤ i ≤

5.5 Conley Index Theory

415

n}. However, there is another way to make a Morse decomposition according to their Morse indices. Setting Sk = {p ∈ K| ind(f, p) = k}, 0 ≤ k ≤ m = dim M , and letting Aj = {x ∈ M | ω(x) ∪ ω ∗ (x) ⊂ S0 ∪ . . . ∪ Sj } 0 ≤ j ≤ m , then (S0 , S1 , . . . , Sm ) is a Morse decomposition of M . According to Theorem 5.5.26, there exists an increasing sequence of compact sets: ∅ = N−1 ⊂ N0 ⊂ . . . ⊂ Nm such that (Nj , Nj−1 ) is an index pair of Sj , 0 ≤ j ≤ m. Noticing the following exact sequences, we have: (1) j is injective, ◦ j = 0, and (3) ∂k = j ◦ i. Thus ∂k−1 ◦ ∂k = 0. (2) ∂k−1

0 = Hk (Nk−1 , Nk−2 )

Hk (Nk+1 , Nk−2 )

ξ

Hk (Nk , Nk−2 ) i

...

Hk+1 (Nk+1 , Nk )

∂k

j

Hk (Nk , Nk−1 )

Hk (Nk+1 , Nk−1 )

...

∂k−1

Hk−1 (Nk−1 , Nk−2 )

Fig. 5.6.

With suitable geometric representation of {Hk (Nk , Nk−1 ; G)| k = 0, 1, . . . , m}, Floer used ∂ to establish the Floer homology. In order to extend the notion of Morse decomposition to noncompact isolated invariant sets, let us keep in mind the Morse relations for (PS) functions on Banach–Finsler manifolds, and introduce the following:

416

5 Topological and Variational Methods

Deﬁnition 5.5.27 Let U1 , U2 ∈ Σ with int(U1 ) ∩ int(U2 ) = ∅. One deﬁnes an order U2 > U1 , if ∃T > 0 such that U1 ∩ GT (U1 ∪ U2 ) is positively invariant with respect to GT (U1 ∪ U2 ). If U2 > U1 and U1 > U2 , then we say that U1 and U2 are ϕ-disconnected. Otherwise, we say that they are ϕ-connected. Lemma 5.5.28 Let U1 , U2 ∈ Σ with U2 > U1 , and let U = U1 ∪ U2 . Then there exists closed subsets N0 ⊂ N1 ⊂ N2 such that (N2 , N0 ), (N2 , N1 ) and (N1 , N0 ) are index pairs for U, U2 and U1 , respectively. Proof. Taking T > 0 large enough such that (GT (U ), ΓT (U )) is an index pair for U , GT (U1 ) ⊂ int(U1 ) and U1 ∩GT (U ) is positively invariant with respect to GT (U ). Setting N0 = ΓT (U ), N1 = (U1 ∩ GT (U )) ∪ ΓT (U ) and N2 = GT (U ). Obviously (N2 , N0 ) is an index pair for U (Theorem 5.5.13). (1) In order to verify that (N1 , N0 ) is an index pair for U1 , it is suﬃcient to verify the four conditions in Deﬁnition 5.5.12 as follows: 1. GT (N1 \N0 ) = GT (U1 ∩ GT (U )) ⊂ GT (U1 ) ∩ G2T (U ) ⊂ int(U1 ∩ GT (U )) = int(N1 \N0 ) . 2. Since ΓT (U ) is positively invariant in GT (U ), and U1 ∩ GT (U ) is positively invariant in GT (U ), ΓT (U ) is positively invariant in N1 . 3. By the same reasoning, ΓT (U ) is the exit set of N1 . 4. N1 \N0 = (U1 ∩ GT (U )) ⊂ U1 , GT (U1 ) ⊂ U1 ∩ GT (U ) = N1 \N0 ⊂ N1 \N0 . (2) Similarly, we verify that (N2 , N1 ) is an index pair for U2 . Since N2 \N1 = GT (U )\U1 = GT (U ) ∩ U2 , the veriﬁcation of GT (N2 \N1 ) ⊂ int(N2 \N1 ) follows similarly to that of (N1 , N0 ). Moreover GT (U2 ) ⊂ GT (U ) ∩ U2 = N2 \N1 .

Other veriﬁcations are omitted.

Deﬁnition 5.5.29 Let U ∈ Σ, a family of subsets of U , {U1 , . . . , Un } is called a Morse decomposition of U , if (1) Ui ∈ Σ, i = 1, . . . , n, (2) U = ∪ Ui , 1≤i≤n

(3) int(Ui ) ∩ int(Uj ) = ∅ as i = j, (4) Ui+1 > ∪ Uj , for i = 1, 2, . . . , n − 1. 1≤j≤i

By observing the exactness of sequences q

q

q

→ H (N2 , N1 ; G) → H (N2 , N0 ; G) → H (N1 , N0 ; G) → . . . . and the proof of (5.3), we arrive at:

5.5 Conley Index Theory

417

Theorem 5.5.30 If {U1 , . . . , Un } is a Morse decomposition of U ∈ Σ, then there is a formal series Q with nonnegative coeﬃcients such that n j=1

h(t; Uj ) = h(t; U ) + (1 + t)Q(t) .

Notes

Chapter 1 Section 1.1 contains the basic diﬀerential calculus on Banach spaces; the material can be found in any nonlinear functional analysis book, for instance, Schwartz [Scw], Nirenberg [Ni 1], Deimling [De], Zeidler [Zei], Berger [Ber 1] etc. The discussion on Nemytscki operator can be found in Vainberg [Va], but the proof is much simpler than that in the reference. The presentations of the implicit function theorem and the inverse function theorem are standard. Interesting applications are scattered in the literature, for instance, Nirenberg [Ni 4], Kazdan [Ka], Chow, Hale [CH], Mawhin [Maw 3] etc. The continuity method is extensively used in the existence proof of diﬀerential equations. The global implicit function theorem is due to Hadamard [Ha], Caccioppoli [Cac 1]; extensions can be found in Browder [Bd 1] and Plastock [Pl]. However, the proof presented here is very diﬀerent. The application of the continuity method to an a priori bound as a simpler proof for semi-linear elliptic equations with quadratic growth was given by Amann and Crandall [AC]; the result for quasi-linear elliptic equations can be found in Ladyzenskaya and Ural’zeva [LU]. The Lyapunov–Schmidt reduction is extensively used in nonlinear problems. The application to the study of bifurcation problems introduced here is due to Crandall and Rabinowitz [CR 1], [CR 2]. The references on bifurcation theory are recommended to Chow and Hale [CH]. Sections 1.3.3–1.3.4 provide examples on gluing, which is an important technique in symplectic geometry. The material here is taken from Floer and Weinstein [FW] and Oh [Oh]. Further references are Floer [Fl 1] [Fl 2], Hofer and Zehnder [HZ 2]. Parallel to the implicit function theorem method, a gluing technique via the variational method was developed by Sere [Se]; it has been applied to the homoclinic orbits in Hamiltonian systems, multi-bump solutions, and multi-peak solutions for elliptic diﬀerential equations, see also Coti Zelati and Rabinowitz [CR], Li [Li 1], Gui [Gu] etc. Section 1.3.5 is another important technique in applying the implicit function theorem. The

420

Notes

notion of transversality is taken from diﬀerential geometry. Combining with the Sard theorem, it provides a method for proving various generic type results. The ﬁnite-dimensional form of the transversality theorem can be found in Guillemin and Podollak [GP]. The Sard–Smale theorem is taken from Smale [Sm 3]. The simplicity of eigenvalues of the Laplacian on generic domains is due to Uhlenbeck [Uh]. Section 1.4 is on the Nash–Moser technique. See Nash [Na 2] and Moser [Mos 1] [Mos 2]. KAM theory is due to Kolmogorov [Ko], Arnold [Ar 2] and Moser [Mo 3]. The presentation here can be found in Hormander [Hor 2]. Applications in diﬀerential geometry can be found in Hamilton [Ham]; a version setting up on Frechet spaces is given therein. Other versions can be found in Nirenberg [Ni 1], see also Zehnder [Ze]. For further reading on recent developments of KAM theory to partial diﬀerential equations, Bourgain [Bou], Kuksin [Kuk], Wayne [Way], and Poschel [Po] are recommended. Chapter 2 The order method is very diﬀerent from other methods in this book. In concrete problems, once the assumptions are met, the method is simple and powerful. Our discussions start with the Bourbaki–Kneser principle [Bo], [Kn], see also Tarski [Ta]. The Amann theorem [Am 1] is a version that is easy to apply. The sub- and super-solutions method is extensively applied in ODE and PDE whenever the maximum principle is applicable. However, the constructions of sub- and super-solutions require special knowledge and techniques. We are satisﬁed introducing the method by an example. The Caristi ﬁxed-point theorem [Ca] is among a few ﬁxed-point theorems without assuming the continuity of the nonlinear mappings, applications can be found in [Lie]. Also, an equivalent version of this theorem is the very important Ekeland variational principle [Ek 1]; various applications in diﬀerent branches of analysis can be found in [Ek 1] and de Figueiredo [dF]. There are lots of books on convex analysis; in Sect. 2.2, we only present very brieﬂy the necessary material for the sequel discussions. References can be found in Ekeland and Temam [ET], Aubin and Ekeland [AE] etc. Fixed points for nonexpansive mappings have been studied by Browder [Bd 2], Goebel [GK], etc. The importance of this class of mappings is that one of the ﬁxed points can be ﬁgured out by iteration methods. Many algorithms for ﬁnding feasible solutions for convex programming have been studied in recent years, see [B], [BB]. There are many ways to introduce the Schauder ﬁxed-point theorem and related topics. We use the KKM theorem [KKM] and Ky Fan’s inequality [FK 3] as the starting point. The Nash equilibrium [Na 3], the Von Neumann– Sion minimax theorem [VN] [Si], the Schauder ﬁxed-point theorem [Sc], the Schauder–Techonoﬀ theorem, the Ky Fan–Glicksberg theorem [FK 1], [FK 2], and the existence result of Hartman and Stampacchia on variational inequalitiy [HS] are direct consequences. The variational inequality [LS], [Bd 3], [Stm]

Notes

421

is another direction in convex analysis with many applications in free boundary problems from mathematical physics, see Duvaut and Lions [DL], Kinderlehrer and Stampacchia [KS], Friedman [Fr] etc. However, the approach based on the ﬁxed points of set-valued mappings is due to the author, see [Ch 1], [Ch 2]. The theory of monotone operators and pseudo-monotone operators attracted much attention in the 1960s and 70s. The works of Minty [Min], Browder [Bd 4], Hartman and Stampacchia [HS], H. Brezis [Br 1], etc. constitute the basic content of the theory. Again we use the version of the Ky Fan inequality due to Brezis, Nirenberg and Stampacchia [BNS] to derive the most important results on this topics. For applications to quasi-linear elliptic equations see Leray and Lions [LL], to boundary value problems in nonlinear partial diﬀerential equations see Lions [LJ 1], and to nonlinear semigroups of operators see Brezis [Br 1], Crandall and Ligget [CLi]. Chapter 3 The Brouwer degree is a topological invariant; roughly speaking, there are two approaches: algebraic (see for instance Spanier [Sp], Greenberg [GH]) and diﬀerential (Milnor [Mi 2]). The Leray–Schauder degree is its extension to compact vector ﬁelds [LS] on Banach space. The analytic presentation can be found in many books and lecture notes, for instance: J. Schwartz [Scw], Nirenberg [Ni 1], Rabinowitz [Ra 2], Guillemin and Podollak [GP], Zeidler [Zei] etc. The materials of Sects. 3.1–3.4 are taken from these references. An application of the transversality theorem to the proof for the Borsuk–Ulam theorem and to the computation of S 1 -invariant degree are due to Nirenberg [Ni 3]. Sections 3.5 and 3.6.3 are adapted from Rabinowitz [Ra 1], [Ra 2]. Section 3.6.2 is based on Dancer [Dan 1]. The material of Sects. 3.6.4 and 3.6.5 is taken from Amann [Am 2], Krasnosel’ski [Kr 2], de Figueiredo, Lions and Nussbaum [FLN], Chang [Ch 3], and Dancer [Dan 2]. The blowing up method in a priori estimates is a useful technique, see for instance, Gidas and Spruck [GS]. Section 7 contains various extensions of the Leray–Schauder degree; for α-set contraction mappings, see Darbo [Dar], Stuart and Toland [ST]; for condensing mappings, see Nussbaum [Nu], Sadovskii [Sa], set-valued mappings see Browder [Bd 5], Cellina [Ce], Ma [Ma], Chang [Ch 2]. There are other directions: Fredholm operators, see Elworthy and Tromba [ElT 1] [ElT 2], Nirenberg [Ni 4], Llyord [Ll], Fitzpatrick, Pejsachowicz and Rabier [FP], [FPR], [PR 1, PR 2]; coincidence degree theory, Mawhin [Maw 1], etc. Chapter 4 Section 4.1 is an introduction to the calculus of variations. The derivations of the Euler–Lagrange equation, the Legendre–Hadamard condition and the Ljusternik theorem on constraint variational problems can be found in any

422

Notes

standard textbook. For the dual variational principle, i.e., the Legendre– Fenchel transformation, see for instance, Arnold [Ar 1], Ekeland and Temam [ET] or Aubin and Ekeland [AE]. For the Hamiltonian systems, the second version of the dual variational principle, i.e., the Legendre transform to the Hamiltonian function of all variables if the latter is strictly convex, see Clarke and Ekeland [CE]. In Sect. 4.2, the direct method is a general principle in the calculus of variations. We introduce a few interesting examples showing how the principle works. Harmonic maps were introduced by Eells and Sampson [ES] in representing homotopy classes of mappings between two manifolds. Here we only touch on the existence of a weak solution. For m = 2 the harmonic map with minimal energy is smooth, see Morrey [Mo 2]; for m > 2, Schoen and Uhlenbeck [ScU 1], [ScU 2] proved that the singularity has at most a codimension 3 ﬁnite Hausforﬀ measure. A nonsmooth minimal energy harmonic map was given by Lin [Lin]. As to nonminimal energy harmonic maps, the smoothness for m = 2 was proved by Helein [Hel 1], and the existence of a nowhere continuous harmonic for m = 3 constructed by Riviere [Ri]. The example on constant mean curvature surface is taken from Hildebrandt [Hi 1]; a systematic introduction can be found in Struwe [St 2]. The prescribing scalar curvature problem can be found in Kazdan [Ka], Kazdan and Warner [KW 1], [KW 2]. The result for χ(M ) = 0 is due to Berger [Ber 2], that for real projective space P 2 is due to Moser [Mos 4]; however, we present an easy proof with the aid of Aubin’s inequality [Au 3]. Section 4.3 is from Morrey [Mo 1] [Mo 2], Dacorogna [Dac], Acerbi and Fusco [AF], Marcellini [Mar], Muller [Mul 1] [Mul 2] Sverak [Sv 1] [Sv 2], Zhang [Zh 1] and Ball [Bal 1]. The relaxation method is from Buttazzo [Bu] and Dacorogna [Dac]. For the two-well problem, Zhang [Zh 2] has obtained an explicit expression of the quasi-convex envelope for the square distance function. The Young measure was introduced by Young [Yo]; the presentation here is due to Ball [Bal 2], see also [Mul 1], Kinderlehrer and Pedregal [KP 1, KP 2]. In Chipot [Chi], there are examples on the computation of Young measures. There are few books in dealing with the BV space, see for instance, Giusti [Gi], Evans and Gariepy [EG], Zimer [Zi], Ambrosio [Amb] etc. Section 4.5.1 is adapted from these books. The Hardy space and BMO space are important parts in harmonic analysis, see Stein [Ste 1, Ste 2] and Stein–Weiss [SW]. They are applied to PDEs in compensated compactness (as seen in Sect. 4.5.2) and in regularity (for harmonic maps see Helein [Hel 2] Evans [Ev 2]. For the background material we recommend Brezis and Nirenberg [BN 3], and Semmes [Se] as references. The connection between Hardy space and the compensated compactness is revealed by Coifman, Lions, Meyer and Semmes [CLMS]. In the applications to the calculus of variations, the bitting theorem due to Chacon is required, see Brooks and Chacon [BC]. Compensated compactness is also an important

Notes

423

tool in nonlinear analysis; we refer the reader to Tartar [Tar 1, Tar 2], DiPerna [Di] and Evans [Ev 2]. We brieﬂy introduce the Γ-convergence in Sect. 4.6 by a result of Modica and Mortola [MM], see also Alberti [Al]. Readers who want to know more can read Dal Maso [D]. The Munford–Shah model [MS] in the segmentation of the image processing is an interesting subject; the existence proof was given by Ambrisio [Amb], De Giorgi and [DG 2], and De Giorgi and Ambrosio [GA]. More details can be found in Ambrosio Fusco and Pallara [AFP], Dal Maso, Morel and Solimini [DMS] and Morel and Solimini [MS]. As to the regularity of the edge curve, see Morel [Mor]. However, all of these results require deep knowledge on BV functions. A much simpliﬁed model presented here is from Nordstr¨ om [No]. Concentration compactness [LP] in combination with blowing up analysis [SU] is one of the most important techniques in nonlinear problems without compactness. Since interesting concrete problems require more background knowledge, they are out of the scope of this book. We present in Sect. 4.7 just an introduction of the idea: The concentration phenomenon and the role of the bubble (or best constant). Two examples on semilinear elliptic equations are studied: The subcritical exponent on Rn , see Coti Zelati [CZ], and the critical exponent on bounded domain, see Brezis and Nirenberg [BN 1]. Section 4.8 is the preliminary of the minimax method. The Palais–Smale condition [PS] and the mountain pass lemma due to Ambrosetti and Rabinowitz [AR] are introduced in accordance with the Ekeland minimizing principle. The proof was given by Shi [Shi] and Aubin and Ekeland [AE]. Examples are taken from Mawhin [Maw 2] and Ambrosetti and Rabinowitz [AR]. Chapter 5 Section 5.1: The most recommended book in introducing Morse theory on compact manifolds is Milnor [Mi 1]. The classics are Morse [Mo] and Morse and Cairns [MC]. The extension to inﬁnite-dimensional space can be found in Chang [Ch 4], [Ch 5]. See also Mawhin and Willem [MW]. The background material in this section can be found in Palais [Pa 1], Palais and Smale [PS], Rothe [Ro 1], [Ro 2], [Ro 3], Marino and Prodi [MP 1], [MP 2], Chang [Ch 7], [Ch 8], Wang [Wa 1], [Wa 2], Castro and Lazer [CaL]. The critical point theory has been extended to non-diﬀerentiable functionals, see Chang [Ch 6] for locally Lipschitzian functionals on Banach space, and Kartiel [Ka], Ioﬀe, Schwartzmann [IS], and Corvellec, De Giovanni, Marzocchi [CGM] for continuous functions on metric spaces. Section 5.2: There are several books on minimax methods, see for instance Rabinowitz [Ra 4], Struwe [St 1], Ghoussoub [GH], and Willem [Wi] etc. The background material can be found in Rabinowitz [Ra 4], [Ra 5], Palais [Pa 2], [Pa 3], Ni [Ni], Chang [Ch 5], Tian [Ti], Liu [Liu 1], [Liu 2], Viterbo [Vi 2], Solimini [So], Chang, Long and Zehnder [CYZ], Fournier and Willem [FoW].

424

Notes

The notion of category was discovered by Ljusternik and Schnirelmenn [LjS 1]. The genus was introduced by Krasnosel’ski in [Kr 1], while for general index theory see Fadell and Rabinowitz [Fr]. The geometric index for S 1 is taken from Benci [Be 1] and Nirenberg [Ni 3]; other extensions of the index theory can be found in Benci and Rabinowitz [BR 2]. Section 5.3: There are many papers on periodic solutions for Hamiltonian systems via variational methods. The results related to this section are Rabinowitz [Ra 3], Weinstein [We 1], [We 2], Viterbo [Vi 1], Hofer and Zehnder [HZ 2], Struwe [St 1]. Further development on the Weinstein conjecture can be found in Hofer and Viterbo [HV], Hofer [Ho 2] and Liu and Tian [LT 1]. The following books are recommended: Hofer and Zehnder [HZ 1], Ekeland [Ek 2], Mawhin and Willem [MW]. Other important topics on the periodic solutions include: • For Arnold conjecture on the symplectic ﬁxed points and Lagrangian intersections, see Conley and Zehnder [CZ 1], Floer [Fl 1], [Fl 2], [Fl 3], Liu and Tian, [LT 2], Fukaya Ono [FO]. • The number of the periodic orbits on compact convex hypersurface has been estimated by Long and Zhu [LZ]. • For the N-body problem, see Bahri and Rabinowitz [Br]. Section 5.4: The main result of this section is taken from Chang and Yang [CY 1], [CY 2], and Chang and Liu [CL 2]. See also Han [Han], Chen and Li [ChL 1], [ChL 2]. The related prescribing scalar curvature problem on S n has received extensive attention. For n = 3 see Bahri and Coron [BC], and for high dimension, see Schoen and Zhang [SZ], Bahri [Bar 1], [Bar 2], Li [Li 1], and Chen and Lin [ChL]. Section 5.5: Conley’s index theory was introduced by Conley [Co], in which isolating neighborhoods for isolated invariant sets are compact. The theory is completed by Conley and Zehnder [CZ 2], Salamon [Sal], Salamon and Zehnder [SZ]. There are many ways to extend the theory to inﬁnite-dimensional spaces, see Rybakowski [Ry], Rybakowski and Zehnder [RZ], Benci [Be 2] etc. The relationship between the Conley theory and the Morse theory can be found in Chang and Ghoussoub [CG]. For further reading see Michaikov [Mic] and Smoller [Smo].

References

[AF] [Ad] [Al]

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[AE] [Au 1] [Au 2]

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