Table Of Content

Progress in Nonlinear Differential Equations and Their Applications Volume 37 Editor Haim Brezis Universite Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Nonnale Superiore, Pisa A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainennan, Princeton University Robert Kohn, New York University P. L. Lions, University of Paris IX Jean Mawhin, Universite Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath Bemard Dacorogna Paolo Marcellini Implicit Partial Differential Equations Springer Science+Business Media, LLC Bemard Dacorogna Pao10 Marcellini Department of Mathematics Dipartimento di Matematica "u. Dini" Ecole Polytechnique Federale de Lausanne Universita di Firenze 1015 Lausanne, Switzerland 50134 Firenze,ltaly Library of Congress Cataloging-in-Publication Data Dacorogna, Bemard, 1953- Implicit partial differential equations I Bemard Dacorogna, Paol0 Marcellini. p. cm. - (Progress in nonlinear partial differential equations ; v. 37) Inc\udes bibliographical references and index. ISBN 978-1-4612-7193-2 ISBN 978-1-4612-1562-2 (eBook) DOI 10.1007/978-1-4612-1562-2 1. Differential equations, Nonlinear. I. MarcelJini, Paolo. II. Title. III. Series. QA377.D331999 515' .323-dc21 99-38323 CIP AMS Subject Classifications: 34A60, 34B99, 35G30, 35R70, 49J45, 49K15, 19K20, 49L25, 73C50 Printed on acid-free paper. © 1999 Springer Science+Business Media New York Originally published by Birkhiiuser Boston in 1999 Softcover reprint of the hardcover 1s t edition 1999 AU rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dis similar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-1-4612-7193-2 SPIN 19901625 Typeset by the authors in It\TEX. 987 6 543 2 1 Contents Preface ix Acknowledgments xi 1 Introduction 1 1.1 The first order case ........ 1 1.1.1 Statement of the problem . 1 1.1.2 The scalar case ...... 2 1.1.3 Some examples in the vectorial case 4 1.1.4 Convexity conditions in the vectorial case 8 1.1.5 Some typical existence theorems in the vectorial case . 9 1.2 Second and higher order cases ............ 10 1.2.1 Dirichlet-Neumann boundary value problem 10 1.2.2 Fully nonlinear partial differential equations . 12 1.2.3 Singular values . 13 1.2.4 Some extensions 14 1.3 Different methods . . . . . 15 1.3.1 Viscosity solutions 15 1.3.2 Convex integration 17 1.3.3 The Baire category method . 18 1.4 Applications to the calculus of variations . 20 1.4.1 Some bibliographical notes . 21 1.4.2 The variational problem 22 1.4.3 The scalar case ....... 23 vi Contents 1.4.4 Application to optimal design in the vector-valued case. 24 1.5 Some unsolved problems . . . . . 26 1.5.1 Selection criterion .... 26 1.5.2 Measurable Hamiltonians 26 1.5.3 Lipschitz boundary data . 27 1.5.4 Approximation of Lipschitz functions by smooth functions 27 1.5.5 Extension of Lipschitz functions and compatibility conditions . . . . . . . . 27 1.5.6 Existence under quasiconvexity assumption 28 1.5.7 Problems with constraints 28 1.5.8 Potential wells .... 29 1.5.9 Calculus of variations .. 30 I First and Second Order PDE's 31 2 First Order Equations 33 2.1 Introduction.... 33 2.2 The convex case . . 34 2.2.1 The main theorem 34 2.2.2 An approximation lemma 36 2.2.3 The case independent of (x, u) . 40 2.2.4 Proof of the main theorem . 43 2.3 The nonconvex case . . . . . . . . . 47 2.3.1 The pyramidal construction 47 2.3.2 The general case . . 52 2.4 The compatibility condition . 56 2.5 An attainment result . 60 3 Second Order Equations 69 3.1 Introduction..................... 69 3.2 The convex case . . . . . . . . . . . . . . . . . . . 70 3.2.1 Statement of the result and some examples 70 3.2.2 The approximation lemma . . . . . . . . . 72 3.2.3 The case independent of lower order terms 73 3.2.4 Proof of the main theorem . 77 3.3 Some extensions .................. 81 3.3.1 Systems of convex functions . . . . . . . . 81 3.3.2 A problem with constraint on the determinant . 82 3.3.3 Application to optimal design ........ . 90 4 Comparison with Viscosity Solutions 95 4.1 Introduction....... 95 4.2 Definition and examples 97 4.3 Geometric restrictions. . . 100 Contents vii 4.3.1 Main results. . . . . . .. . ..... . · .100 4.3.2 Proof of the main results. . . . . . . . · . 103 4.4 Appendix . . . . . . . . . . . . . . ......... . · .113 4.4.1 Subgradient and differentiability of convex functions . · . 113 4.4.2 Gauges and their polars. . . . . . . . . . . . . . . · . 113 4.4.3 Extension of Lipschitz functions . . . . . . . · . 115 4.4.4 A property of the sub and super differentials. · .117 II Systems of Partial Differential Equations 119 5 Some Preliminary Results 121 5.1 Introduction ....................... . · . 121 5.2 Different notions of convexity ............. . · . 121 5.2.1 Definitions and basic properties (first order case) · . 121 5.2.2 Definitions and basic properties (higher order case) . . · . 124 5.2.3 Different envelopes. . .... · . 126 5.3 Weak lower semicontinuity .. · . 127 5.3.1 The first order case .. · . 127 5.3.2 The higher order case. · . 129 5.4 Different notions of convexity for sets · .130 5.4.1 Definitions ............ . · .130 5.4.2 The different convex hulls .... . · . 131 5.4.3 Further properties of rank one convex hulls · . 135 5.4.4 Extreme points . . . . . . . . . . . . . . .. · .138 6 Existence Theorems for Systems 141 6.1 Introduction.............. · 141 6.2 An abstract result . . . . . . .142 6.2.1 The relaxation property .. .142 6.2.2 Weakly extreme sets . . . . . . . . · . 147 6.3 The key approximation lemma . . . . . . . · . 148 6.4 Sufficient conditions for the relaxation property · . 152 6.4.1 One quasiconvex equation ...... . · . 152 6.4.2 The approximation property . . . . . . · . 153 6.4.3 Relaxation property for general sets .154 6.5 The main theorems . . . . . . . . . . . . . . . . · .157 III Applications 167 7 The Singular Values Case 169 7.1 Introduction ................... . · 169 7.2 Singular values and functions of singular values . · 171 7.2.1 Singular values ............. . · . 171 viii Contents 7.2.2 Functions depending on singular values .... 174 7.2.3 Rank one convexity in dimension two .. . ... 181 7.3 Convex and rank one convex hulls . . . . . . . · .185 7.3.1 The case of equality ofthe Qi •••••• · . 186 7.3.2 The main theorem for general matrices · . 187 7.3.3 The diagonal case in dimension two .. . ...... 193 7.3.4 The symmetric case in dimension two .. .195 7.4 Existence of solutions (the first order case). . . . .199 7.5 Existence of solutions (the second order case) .200 8 The Case of Potential Wells 205 .......... 8.1 Introduction . . . . . . .205 8.2 The rank one convex hull .206 .......... 8.3 Existence of solutions . . .215 9 The Complex Eikonal Equation 217 9.1 Introduction . . . . . . . . . . . . . . · .... .217 9.2 The convex and rank one convex hulls · .... .218 · .... 9.3 Existence of solutions . . . . . . . . . .222 IV Appendix 223 10 Appendix: Piecewise Approximations 225 10.1 Vitali covering theorems and applications .225 10.1.1 Vitali covering theorems . . . . . . . . . .225 10.1.2 Piecewise affine approximation ..... .232 10.2 Piecewise polynomial approximation . . . . . . . · .241 10.2.1 Approximation of functions of class eN . .242 10.2.2 Approximation of functions of class WN•oo . .245 References 249 Index 271 Preface Nonlinear partial differential equations has become one of the main tools of mod ern mathematical analysis; in spite of seemingly contradictory terminology, the subject of nonlinear differential equations finds its origins in the theory of linear differential equations, and a large part of functional analysis derived its inspiration from the study of linear pdes. In recent years, several mathematicians have investigated nonlinear equations, particularly those of the second order, both linear and nonlinear and either in divergence or nondivergence form. Quasilinear and fully nonlinear differential equations are relevant classes of such equations and have been widely examined in the mathematical literature. In this work we present a new family of differential equations called "implicit partial differential equations", described in detail in the introduction (c.f. Chapter 1). It is a class of nonlinear equations that does not include the family of fully nonlinear elliptic pdes. We present a new functional analytic method based on the Baire category theorem for handling the existence of almost everywhere solutions of these implicit equations. The results have been obtained for the most part in recent years and have important applications to the calculus of variations, nonlin ear elasticity, problems of phase transitions and optimal design; some results have not been published elsewhere. The book is essentially self-contained, and includes some background mate rial on viscosity solutions, different notions of convexity involved in the vectorial calculus of variations, singular values, Vitali type covering theorems, and the ap proximation of Sobolev functions by piecewise affine functions. Also, a compari son is made with other methods - notably the method of viscosity solutions and x Preface briefly that of convex integration. Many mathematical examples stemming from applications to the material sciences are thoroughly discussed. The book is divided into four parts. In Part 1 we consider the scalar case for first (Chapter 2) and second (Chapter 3) order equations. We also compare (Chap ter 4) our approach for obtaining existence results with the celebrated viscosity method. While most of our existence results obtained in this part of the book are consequences of vectorial results considered in the second part, we have avoided (except for very briefly in Section 3.3) vectorial machinery in order to make the material more readable. In Part 2 we first (Chapter 5) recall basic results on generalized notions of con vexity, such as quasiconvexity, as well as on some important lower semicontinuity theorems of the calculus of variations. Central existence results of Part 2 are in Chapter 6, where Nth order vectorial problems are discussed. In Part 3 we study in great detail applications of vectorial existence results to important problems originating, for example, from geometry or from the mate rial sciences. These applications concern singular values, potential wells and the complex eikonal equation. Finally, in Part 4 we gather some nonclassical Vitali type covering theorems, as well as several fine results on the approximation of Sobolev functions by piece wise affine or polynomial functions. These last results may be relevant in other contexts, such as numerical analysis.