An Introduction to Variational Inequalities and Their Applications (Classics in Applied Mathematics)

An Introduction to Variational Inequalities and Their Applications

SIAM's Classics in Applied Mathematics series consists of books that were previously allowed to go out of print. These books are republished by SIAM as a professional service because they continue to be important resources for mathematical scientists. Editor-in-Chief Robert E. O'Malley, Jr.,UniversityofWashington Editorial Board Richard A. Brualdi, University of Wisconsin-Madison Herbert B. Keller,California Institute of Technology Andrzej Z. Manitius, George Mason University Ingram Olkin, Stanford University Stanley Richardson, University of Edinburgh Ferdinand Verhulst, Mathematisch Instituut, University of Utrecht Classics in Applied Mathematics C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences Johan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential UnconstrainedMinimisation Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F. Carrier and Carl E. Pearson, Ordinary Differential Equations Leo Breiman, Probability R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding Abraham Berman and Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences Olvi L. Mangasarian, Nonlinear Programming *Carl Friedrich Gauss, Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement.Translated by G. W. Stewart Richard Bellman, Introduction to Matrix Analysis U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of InitialValue Problems in Differential-Algebraic Equations Charles L. Lawson and Richard J. Hanson, Solving Least Squares Problems J. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimisation and Nonlinear Equations Richard E. Barlow and Frank Proschan, Mathematical Theory of Reliability *First time in print.

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Richard Bellman, Introduction to MatrixAnalysis,Second Edition BeresfordN. Parlett, The Symmetric Eigenvalue Problem Richard Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic flow Peter W. M. John, Statistical Design and AnalysisofExperiments Tamer Basar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, Second Edition Emanuel Parzen, Stochastic Processes Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and Ordering Populations: A New Statistical Methodology James A. Murdock, Perturbations: Theory and Methods Ivar Ekeland and Roger Temam, Convex Analysis and VariationalProblems Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II J. M. Ortega and W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their Applications

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An Introduction to Variational Inequalities and Their Applications David Kinderlehrer Guido Stampacchia

Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2000 by the Society for Industrial and Applied Mathematics. This SIAM edition is an unabridged republication of the work first published by Academic Press, New York and London, 1980.

10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Kinderlehrer, David. An introduction to variational inequalities and their applications / David Kinderlehrer, Guido Stampacchia. p. cm. — (Classics in applied mathematics ; 31) Originally published : New York : Academic Press, 1980. Includes bibliographical references and index. ISBN 0-89871-466-4 (pbk.) 1. Variational inequalities (Mathematics) I. Stampacchia, Guido. II. Title. III. Series. QA316.K552000 515'.64—dc21 00-033845 is a registered trademark.

Questo volume e dedicate alia memoria del nostro diletto amico e collega Nestor Riviere

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Contents

Preface to the Classics Edition Preface Glossary of Notations

Introduction Chapter I 1. 2. 3. 4. 5.

xiii xvii xix

l

Variational Inequalities in R" Fixed Points The Characterization of the Projection onto a Convex Set A First Theorem about Variational Inequalities Variational Inequalities Some Problems Which Lead to Variational Inequalities Comments and Bibliographical Notes Exercises

7 8 11 13 15 18 18

Chapter II Variational Inequalities in Hilbert Space 1. Bilinear Forms 2. Existence of a Solution 3. Truncation 4. Sobolev Spaces and Boundary Value Problems 5. The Weak Maximum Principle 6. The Obstacle Problem: First Properties 7. The Obstacle Problem in the One Dimensional Case Appendix A. Sobolev Spaces Appendix B. Solutions to Equations with Bounded Measurable Coefficients

23 24 27 28 35 40 47 49 62

IX

X

CONTENTS Appendix C. Local Estimates of Solutions Appendix D. Holder Continuity of the Solutions Comments and Bibliographical Notes Exercises

66 72 76 77

Chapter HI Variational Inequalities for Monotone Operators 1. 2. 3. 4.

An Abstract Existence Theorem Noncoercive Operators Semilinear Equations Quasi-Linear Operators Comments and Bibliographical Notes Exercises

83 87 93 94 100 101

Chapter IV Problems of Regularity 1. Penalization 2. Dirichlet Integral 3. Coercive Vector Fields 4. Locally Coercive Vector Fields 5. Another Penalization 6. Limitation of Second Derivatives 7. Bounded Variation of Au 8. Lipschitz Obstacles 9. A Variational Inequality with Mixed Boundary Conditions Appendix A. Proof of Theorem 3.3 Comments and Bibliographical Notes Exercises

105 106 113 116 120 124 130 134 139 143 146 147

Chapter V Free Boundary Problems and the Coincidence Set of the Solution 1. 2. 3. 4. 5. 6.

Introduction The Hodograph and Legendre Transformations The Free Boundary in Two Dimensions A Remark about Singularities The Obstacle Problem for a Minimal Surface The Topology of the Coincidence Set When the Obstacle Is Concave 7. A Remark about the Coincidence Set in Higher Dimensions Comments and Bibliographical Notes Exercises

Chapter VI

149 153 155 166 167 173 178 181 182

Free Boundary Problems Governed by Elliptic Equations and Systems

1. Introduction 2. Hodograph and Legendre Transforms: The Theory of a Single Equation

184 185

CONTENTS 3. 4. 5. 6.

XI Elliptic Systems A Reflection Problem Elliptic Equations Sharing Cauchy Data A Problem of Two Membranes Comments and Bibliographical Notes Exercises

190 202 204 212 218 218

Chapter VII Applications of Variational Inequalities 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction A Problem in the Theory of Lubrication The Filtration of a Liquid through a Porous Medium The Resolution of the Filtration Problem by Variational Inequalities The Filtration of a Liquid through a Porous Medium with Variable Cross Section The Resolution of the Filtration Problem in Three Dimensions Flow past a Given Profile: The Problem in the Physical Plane Flow past a Given Profile: Resolution by Variational Inequalities The Deflection of a Simply Supported Beam Comments and Bibliographical Notes Exercises

222 223 227 235 242 249 257 260 270 273 274

Chapter VIII A One Phase Stefan Problem 1. 2. 3. 4.

Introduction Existence and Uniqueness of the Solution Smoothness Properties of the Solution The Legendre Transform Comments and Bibliographical Notes

278 281 289 297 299

Bibliography

300

Index

309

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Preface to the Classics Edition

With the enormous range of topics and vast literature now connected with variational inequalities, the subject has become part of the standard material taught in partial differential equations. With this objective in mind, I am extremely grateful to SIAM for the opportunity to maintain the availability of this volume, for the curious and aficionado alike. Current technology enables anyone to review the present status of the literature, so I would just like to briefly note a few of the exciting new areas where our subject has found a home. The references below are just brief indications. Finance The American put option, after a change of variables, becomes a one phase Stefan problem. More generally, any derivative security problem where there is an early exercise feature involves a free boundary problem [10]. Phase transformations The Allen-Cahn equation, one of the models of the kinetics of grain growth in polycrystals, has been treated as a parabolic variational inequality [2], For a treatment of superconducting vortices see [4], [13]. ems and contact problems Static contact problems, factional contact problems, and questions related to thermal expansion may be considered as variational inequalities, often for systems, or as generalizations of them [9], [11], [14]. A variational inequality, technically speaking, is a variational problem or an evolution problem with a convex constraint. Since the set where the constraint is active is not known beforehand, a free boundary problem often arises and, indeed, variational inequalities are inextricably bound to free

xiii

xiv

PREFACE TO THE CLASSICS EDITION

boundary problems. So the reader is encouraged to seek information about free boundary problems as part of his research activity, e.g., [3]. There are also a number of new books since this volume originally appeared, e.g., [5], [6], [7], [8], [12]. Guido Stampacchia was a legendary mentor and teacher. He was also an exquisite author and lecturer who succeeded in unifying elegance and clarity, aspects of his science seriously degraded by his collaboration with the undersigned. The Opere Scelte of Guido Stampacchia have been published in two volumes by Edizioni Cremonese (1996, 1997) with an introduction b Louis Nirenberg and reprints of the appreciations of Stampacchia's work by Jacques-Louis Lions and Enrico Magenes that originally appeared in the BolletinodellaUnioneMatematicaItaliana15 (1978), 715-756. I strongly recommend consulting the original papers. Steven Shreve and Vanessa Styles provided valuable assistance with the references. As always, I am indebted to Sara Stampacchia Naldini and to Mauro Stampacchia for their collaboration and to Ha'im Brezis for his encouragement and support. David Kinderlehrer REFERENCES [1] Baiocchi, C. and Capelo, A. Variational and Quasivariationai Inequalities,(Jayakar, L. trans.), John Wiley and Sons, New York, 1984J2] Blowey, J. F. and Elliott, C. M. Curvature dependent phase boundary motion and parabolic obstacle problems, Proc. IMA Workshop on Degenerate Diffusion, Vol. 47, W. Ni, L. Peletier, and J. L. Vasquez, eds., Springer-Verlag, Berlin, 1993, 19-60. [3] Caffarelli, L. A. The obstacle problem revisited, J. Fourier Anal. Appl4 (1998), 383-402. [4] Chapman, S. J. A mean-field model of superconducting vortices in three dimensions, SIAM J. Appl Math. 55 (1995), 1252-1274. [5] Chipot, M. Variational Inequalities and Flow through Porous Media, Appl. Math. Sci., Vol. 52, Springer-Verlag, New York, 198416] Chipot, M. On Some Issues of Nonlinear Analysis, Birkhauser, Basel, 2000. [7] Elliott, C. M. and Ockendon, J. R. Weak and Variational Methods for Moving Boundary Problems, Research Notes in Mathematics, Vol. 59, Pitman (Advanced Publishing Program), Boston, MA, 1982.

PREFACE TO THE CLASSICS EDITION

XV

[8] Friedman, A. Variational Principles and Free-Boundary Problems, Wiley-Interscience, New York, 1982. [9] Gwinner, J. and Stephan, E. P. A boundary element procedure for contact problems in plane linear elastostatics, RAIRO Model. Math. Anal. Numer. 27 (1993), 457-480. [10] Karatzas, I. and Shreve, S. E. Methods of Mathematical Finance, Springer-Verlag, New York, 1998. [11] Kinderlehrer, D. Remarks on Signorini's problem in elasticity, Annafidella S.N.S. Pisa 8 (1981), 605-645. [12] Rodrigues, J.-F. Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987. [13] Schatzle, R. and Styles, V. Analysis of a mean field model of superconducting vortices, European J. Appl. Math. 10 (1999), 319-352. [14] Shillor, M. (ed). Recent Advances in Contact Mechanics, Comput. Modelling, Vol. 28, Pergamon Press, Exeter, 1998.

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Preface

The rapid development of the theory of variational inequalities and the prolific growth of its applications made evident to us the need of an introduction to the field. This is our response; we hope that it will be found useful and enlightening. We drew an outline of our enterprise in July of 1976, confident that it would be completed by August or, perhaps, September. Our conception of the necessary labor was optimistic; nonetheless, the finished book follows the original plan. Many of the chapters have been adapted from courses we gave at the Scuola Normale Superiore, the University of Minnesota, the University of Paris, the College de France, the Mittag-Leffler Institut, and Northwestern University. A few suggestions about the use of this book in courses are discussed in the introduction. The life and work of the late Guido Stampacchia are discussed by Jacques Louis Lions and Enrico Magenes in the Bollettino della Unione Matematica Italiana15 (1978), 715-756. I wish to express my profound indebtedness to our many friends and collaborators who so generously assisted us during these years. Most important, without the nurturing and encouragement of Mrs. Sara Stampacchia, it would not have been possible for us to conceive this project nor for me to bring it to its completion. In addition, I would like to thank especially Haim Brezis for his many useful suggestions and his constant interest in our progress. Silvia Mazzone, xvn

xviii

PREFACE

Michael Crandall, and David Schaeffer offered useful contributions and assistance in various stages of the writing, all of which are sincerely appreciated. Finally, I would like to thank Matania Ben Artzi and Bevan Thompson for their careful reading of portions of the material. Ronchi and Minneapolis 1976-1979

Glossary of Notations

Common Notations Euclidean N-dimensional space, the product of N copies of the real line IR complex N-dimensional space, the product of N copies of the complex numbers C coordinates in scalar product

the op[en ball o readius r and cemtedsdkv

open, generally bounded and connected,subseto e boundary of heclosurefi interior ofU the support of the function u, which is the smallest closed set outside of which u = 0 ial derivative of u with respect to gers, a multi-index of lengt

Lebesgue measure on Laplace operator or Laplacian

xix

XX

GLOSSARY OF NOTATIONS

a pairing between a (real) Banach space V and its dual V'\ Chapter VI only) summation convention: repeated indices are summed over their ranges, which are usually 1 S is a contraction mapping if for some a, 0 < a < 1. When we allow a = 1, the mapping F is called nonexpansiue. We state the contraction mapping theorem. 7

8

IVARIATIONAL INEQUALITIES IN U

Theorem 1.2.Let S be a complete metric space and let F:S-»S be a contraction mapping. Then there exists a unique fixed point ofF. Note that the theorem is not generally true ifF is nonexpansive. For example a translation of a linear space into itself does not admit fixed points. Another fundamental theorem is that of Brouwer. There are many proofs of this theorem connected with many branches of mathematics and yet its statement is very simple.WedenotebyUNthe(real) Euclidean space of dimension N > 1. Theorem 1.3(Brouwer). Let F be a continuous mapping of a closed ball Z c (RNintoitself.Then F admits at least one fixed point. The proofs of these theorems may be found in many books (cf. Massey [1 ]). We point out that in Brouwer's theorem, the ball Z may be replaced by a compactconvexsubsetofUN.One object of the next section is to present this extension.

2. The Characterization of the Projection onto a Convex Set In this section we consider the projection onto a convex set in a real Hilbert space H, since it is in this context that it will be useful to us in the sequel. The demonstrations are identical in the case H is finite dimensional. ThereadermaychoosetoreadIRNorH. Lemma 2.1.LetIKbeaclosedconvexsubsetofaHilbertspaceH.AThen foreachxeHtheresauniquey e IKsuchthat

Remark2.2.The point y satisfying (2.1) is called the projection of x on IK and we write

NoticethatPrKx=x,or all x £ IK. Proof of the lemma. Let r\keIK be a minimizing sequence, namely,

2

PROJECTION ONTO A CONVEX SET

9

From the parallelogram law, a consequence of the existence of an inner product on H, we compute that Now IK is convex, so %j]k+^)e IK and Therefore and we conclude from (2.2) that

Hence, since H is complete, there is an element yeh. such that

Moreover,

To see thaty is unique, merely observe that any two elementsy,y' e K which satisfy (2.1) may be inserted in (2.3) in place ofr\

or y = y'. Q.E.D. We proceed to characterize the projection. Theorem 2.3. Let (K be a closed convex, set of a Hilbert space H. Then y =PrKx,heprojectionofxonIK,if and only if Proof.

Let x e H and y =Pr K xeIK.Since IK is convex

and hence, by (2.1), the function

10

I

VARIATIONAL INEQUALITIES IN UN

attains its minimum at t = 0. So '(0) > 0, namely, or

On the other hand, if then Therefore so, finally,

Corollary 2.4.Let Kbea dosed convex set of a Hilbert space H. Then the operatorPr is nonexpansive, that is, Proof.Given x, x' 6H, lety= Pr x andy' =PrKx'.Then

We choose 77 = y' in the first inequality andr\= y in the second. Adding we obtain

or

Notice that the proof of the uniqueness of the projection follows again. We conclude this section with the proof of Brouwer's theorem for a compact convex set. Theorem 2.5 (Brouwer).LetKcUNbecompacandconvexandlet F:K-+Kbe continuous. Then F admits a fixed point.

3

A FIRST THEOREM ABOUT VARIATIONAL INEQUALITIES

11

Proof.LetIbeaclosedballinRNsuchthatIKcI.From Corollary 2.4,PrKiscontinuous;hence the mapping is a continuous mapping of £ into itself. It admits a fixed point x by Theorem 1.3, namely, In particular,Pr Kx— .v so(x)=x.Q.E.D.

3.

A First Theorem about Variational Inequalities

In the study of variational inequalities we are frequently concerned with mapping F from a linear space X, or a convex subset IK cr X, into its dual X'. This will be particularly evident in Chapters II and HI. Recall that the dual (UNyof UNisthepaceoflllinearforms definedonUN.Indeed the bilinear mapping is referred to as a pairing. On the other hand we may always identify (RN)' with (RN,for example, we may identify a e (IRN)'withtheelementrcaeUN such that =(na, x).Neither the identification nor the realization of the pairing it determines is unique, but we always assume that where n: (RN)-> UNistheidentificationand-,•)is the scalar product on UN.Finally, a function is continuous if each of the functions Fj(x), . . . ,FN(x)determined by the relation

is continuous. The reader may verify that this is equivalent to the "natural" definition of continuity.

12

I

VAR1ATIONAL INEQUALITIES IN RN

Theorem 3.1.LetKcUNbecompactandconvexandlet

be continuous. Then there is an xeK such that

Proof.Proving the theorem is equivalent to showing there exists Now the mapping

where 7x = x is continuous; hence, by Theorem 2.5, it admits a fixed point x e IK, namely,

Consequently by Theorem 2.3, which is the characterization of the projection,

o

Corollary 3.2.Let x be a solution to (3.1) and suppose that x e IK, the interior ofK. Then F(x) — 0. Proof.If x e fl 0 and y € IK such that £ = e(y — x). Consequently from which it follows that F(x) = 0. Q.E.D. Definition 3.3.LetIKbeconvexsetofUNandx63IK.A hyperplane

is said a hyperplane of support, or a supporting hyperplane, of IK if

Corollary 3.4.Let x be a solution of (3.1)andsupposethatx€ dIK.Then F(x) determines a hyperplane of support for (K, provided F(x) ^ 0. Namely, the affine function /(y) = is nonnegative for y e IK.

4

4.

VAR1ATIONAL INEQUALITIES

13

Variational Inequalities We consider Problem 4.1.GivenKclosedandconvexinUNand

continuous, find

If IK is bounded, we have proved the existence of a solution to Problem 4.1 in the previous section. On the other hand, notice that the problem does not always admit a solution. For example, if IK = R, has no solution for/(x) =ex.The following theorem gives a necessary and sufficient condition for the existence of solutions. Given a convex set IK, we set KJJ =IKnZjjwhereZistheclosedballofradiusRandcenter0eUN. Returning to ourF: IK ->(UN)'we notice that there exists at least one

wheneverKR^0 by the previous theorem. Theorem 4.2.Let K c:MNbeclosedandconvexand

be continuous. A necessary and sufficient condition that there exist a solution to Problem 4.1 is that there exist an R > 0 such that a solution X satisfies

Proof.It is clear that if there exists a solution to Problem 4.1, then x is a solution to (4.1) whenever \x\ < R. SupposenowthatXeKRatisfies(4.2).ThenXRisalsoasolutionto Problem 4.1. Indeed, since \XR\ 0 sufficiently small. Consequently whichmeansthatXRisasolutiontoProblem4.1.Q.E.D.

14

I

VARIATIONAL INEQUALITIES IN UN

From this theorem we may deduce many sufficient conditions for existence. We state one which is both useful and introduces the notion of coerciveness. Corollary 4.3.LetF:IK -> (UN)'satisfy

for some x0 e IK. Then there exists a solution to Problem 4.1. Proof.

Choose H > \F(x0)|and R > \x0\such that

Then

NowletXReKRbethesolutionof(4.1). Then

so, in view of (4.4), \XR\^R. In other words, \XR\ (UN)' monotone if It is calledstrictlymonotone if equality holds only whenx =x',that is, when condition (4.5) is valid. As a related application of strictly monotone mappings we state Proposition 4.6.Let F:K be a continuous strictly monotone mappingoftheclosedconvexKcrUN.LetIK 2=KI be closed and convex. Suppose there exist solutions of the problems (i) //F(x2)=0,thenXj =x2. (ii) //F(x2)7^0andx t^x2,thenhehyperplane=0 separates Xj /ramIK The proof is left as an exercise.

5. Some Problems Which Lead to Variational Inequalities In this section we lightly touch on some elementary problems that are associated to variational inequalities. In particular, we discuss the connection between convex functions and monotone operators. Let/e C^tK), IK cUN,be a closed convex set, and set F(x) = grad/(x). AtthispointwedonotdistinguishbetweenUNand(UN)'. Proposition 5.1.Suppose there exists an x e IK such that

Then x is a solution of the variational inequality (SeeExample2oftheIntroduction.)

16

I

VARIATIONAL INEQUALITIES IN UN

Proof.Ify e IK, then z = x +t(y -x) e IK for 0 ^N.Findx0€UN+suchthatF(x0) +and (F(x0), x 0 )=0. Theorem5.5.Thepointx0eIR+is a solution to complementarity Problem 5.4 if and only if

Proof.First note thatifx0sasolutionoComplementarityProblem 5.4, (F(x0),y) >0foranyy€RN+,so On the other hand suppose that x0 6 1R+ is a solution to the variational inequality. Then

is an element of R +, so

or F(x0)eMN+.Hence, since y =0eRN+,

18

I

VARIATIONAL INEQUALITIES IN UN

N

0,F(x0)€R

impliesthat(F(x0), x0)>0, so

COMMENTS AND BIBLIOGRAPHICAL NOTES The characterization of the projection onto a convex set by a variational inequality is contained in Section 3. It is also used in Lions and Stampacchia [1]. Theorem 3.1 was proved in Hartman and Stampacchia [1]. Another proof, assuming F to be monotone, is due to Browder [1]. The proof given here employs a simplification due to H. Brezis. Theorem 4.2 may be found in Hartman and Stampacchia [1] or Stampacchia [4]. The condition of Corollary 4.3 has been used frequently beginning with Browder [1, 2] and Minty [1, 2]. The relations between convex functions and variational inequalities are considered in Rockafellar [1] and Moreau [1]. The concept of subdifferential (subgradient) was developed by the latter author. The fact that the complementarity problem may be reduced to a variational inequality was noticed by Karamardian. For the connections between mathematicalprogrammingandvariationalinequalitiesinUNseeMancino and Stampacchia [1].

EXERCISES 1. Prove Proposition 4.6. 2.AmappingFfromUNintotRN)'is calledcyclicallymonotone if one has

for any set of points (x0,x . . . ,xn}(n arbitrary). Show that F(x) = grad /(x) is cyclically monotone if/(x)saC1convex function. 3. (i) Deduce Brouwer's fixed point from Theorem 3.1. (ii) Let F be a continuous mapping of a closed ball "L 0 such that The bilinear form a(u, v) is coercive if and only if the mapping A defined by (1.1) is coercive in the sense of Chapter I, Definition 4.4. Evidently, a coercive symmetric bilinear forma(u, v)defines a norm(a(v,))}/2onHequivalentto ||y||. We consider now: Problem 1.2.Let IK c= H be closed and convex andfe H'. To find:

2. Existence of a Solution The purpose of this section is to solve Problem 1.2 and to prove Theorem 2.1.Let a(u, v) be a coercive bilinear form on H, K c H close and convex andfe H'.ThenhereexistsauniquesolutiontoProblem1.2.I addition,themapping/-»u is Lipschitz, that is,iful, u2aresolutionsto Problem 1.2 corresponding toj\,f2eH',then

Observe that the mapping/ -> u is linear if (K is a subspace of H. Proof. We begin with the demonstration of (2.1). Suppose there exist M,,M 2eHsolutionsofthevariationalinequalities

2

EXISTENCE OF A SOLUTION

25

Setting v =u2inthevariationalinequalityforMIandv=ulinhatfor u 2e obtain, upon adding, Hence by the coerciveness of a, and therefore (2.1) holds. It remains to show the existence of u, which we present in several steps. First suppose that a(u, v) is symmetric, and define the functional Let d = infiK I(u). Since

we see that Letunbeaminimizingsequenceof/in IK such that Applying the parallelogram law, and keeping in mind that IK is convex, w see that

We have used Hence the sequence {«„} is Cauchy and the closed set IK contains an element usuchthatu n-u in H and /(«„) -*• /(u). So /(M) = d. Now for any u e IK, u + e(t; — M) e IK, 0 < e < 1, and I(u + e(v — u)) > /(M). Then (d/de)I(u + E(V - v))\e =0>0. In other words, or

a(«, v — u) > — %ea(v — u,v — u)

for any

Setting e = 0 we see that M is a solution to Problem 1.2.

e, 0 < e < 1.

26

II

VARIATIONAL INEQUALITIES IN HILBERT SPACE

We treat now the general case as a perturbation of the symmetric one. Introduce the coercive bilinear form where and

are the symmetric and antisymmetric parts of a.Observethata t (u, v) = a(u, v)andthatat(u, v) is coercive with the same constant a. Lemma2.2.If Problem 1.2 is solvabl fora t (w, v) and allfe H', then it is solvable for a,(w, v) and a///e H', where r < t < r +t0,t0eL2(£). We set evidently a closed convex set. Let us fix

thescalarproductonL2(£), which is a coercive bilinear form. For a given /eL2(£), there exists

by Theorem 2.1. We claim that

In fact, defining u by (3.2), we compute

sincev —\(p> 0 for anyv e IK. Consequently,

sou is the solution to the variational inequality.

28

II

VARIATIONAL INEQUALITIES IN HILBERT SPACE

The interested reader will realize at this point that it is possible to raise questions of regularity of the solution to Problem 1.2. So, one result we mention in this connection is Theorem 3.1.LetQcRNbepenandlet