Functional Analysis Sobolev Spaces And Partial Differential Equa....pdf

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Cover

Universitext

Functional Analysis, Sobolev Spaces and Partial Differential Equations

9780387709130

Preface

Contents

1. The Hahn–Banach Theorems. Introduction to the Theory of Conjugate Convex Functions

1.1 The Analytic Form of the Hahn–Banach Theorem: Extension of Linear Functionals

1.2 The Geometric Forms of the Hahn–Banach Theorem: Separation of Convex Sets

1.3 The Bidual Ε^{**}. Orthogonality Relations

1.4 A Quick Introduction to the Theory of Conjugate Convex Functions

Comments on Chapter 1

Exercises for Chapter 1

2. The Uniform Boundedness Principle and the Closed Graph Theorem

2.1 The Baire Category Theorem

2.2 The Uniform Boundedness Principle

2.3 The Open Mapping Theorem and the Closed Graph Theorem

2.4 Complementary Subspaces. Right and Left Invertibility of Linear Operators

2.5 Orthogonality Revisited

2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint

2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators

Comments on Chapter 2

Exercises for Chapter 2

3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity

3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous

3.2 Definition and Elementary Properties of the Weak Topology σ(E,E*)

3.3 Weak Topology, Convex Sets, and Linear Operators

3.4 The Weak* Topology σ(E*,E)

3.5 Reflexive Spaces

3.6 Separable Spaces

3.7 Uniformly Convex Spaces

Comments on Chapter 3

Exercises for Chapter 3

4. L^p Spaces

4.1 Some Results about Integration That Everyone Must Know

4.2 Definition and Elementary Properties of L^p Spaces

4.3 Reflexivity. Separability. Dual of L^p

4.4 Convolution and regularization

4.5 Criterion for Strong Compactness in L^p

Comments on Chapter 4

Exercises for Chapter 4

5. Hilbert Spaces

5.1 Definitions and Elementary Properties. Projection onto a Closed Convex Set

5.2 The Dual Space of a Hilbert Space

5.3 The Theorems of Stampacchia and Lax–Milgram

5.4 Hilbert Sums. Orthonormal Bases

Comments on Chapter 5

Exercises for Chapter 5

6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators

6.1 Definitions. Elementary Properties. Adjoint

6.2 The Riesz–Fredholm Theory

6.3 The Spectrum of a Compact Operator

6.4 Spectral Decomposition of Self-Adjoint Compact Operators

Comments on Chapter 6

Exercises for Chapter 6

7. The Hille–Yosida Theorem

7.1 Definition and Elementary Properties of Maximal Monotone Operators

7.2 Solution of the Evolution Problem du/dt + Au = 0 on [0,+∞), u(0) = u_0. Existence and uniqueness

7.3 Regularity

7.4 The Self-Adjoint Case

Comments on Chapter 7

8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension

8.1 Motivation

8.2 The Sobolev Space W^{1,p}(I)

8.3 The Space W^{1,p}_0

8.4 Some Examples of Boundary Value Problems

8.5 The Maximum Principle

8.6 Eigenfunctions and Spectral Decomposition

Comments on Chapter 8

Exercises for Chapter 8

9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions

9.1 Definition and Elementary Properties of the Sobolev Spaces

9.2 Extension Operators

9.3 Sobolev Inequalities

9.4 The Space W^{1,p}_0(Ω)

9.5 Variational Formulation of Some Boundary Value Problems

9.6 Regularity of Weak Solutions

9.7 The Maximum Principle

9.8 Eigenfunctions and Spectral Decomposition

Comments on Chapter 9

10. Evolution Problems: The Heat Equation and the Wave Equation

10.1 The Heat Equation: Existence, Uniqueness, and Regularity

10.2 The Maximum Principle

10.3 The Wave Equation

Comments on Chapter 10

11. Miscellaneous Complements

11.1 Finite-Dimensional and Finite-Codimensional Spaces

11.2 Quotient Spaces

11.3 Some Classical Spaces of Sequences

11.4 Banach Spaces over \mathbb{C}: What Is Similar and What Is Different?

Solutions of Some Exercises

Problems

Partial Solutions of the Problems

Notation

References

Index

Haim Brezis Functional Analysis, Sobolev Spaces and Partial Differential Equations 1 C

Haim Brezis Distinguished Professor Department of Mathematics Rutgers University Piscataway, NJ 08854 USA brezis@math.rutgers.edu and Professeur émérite, Université Pierre et Marie Curie (Paris 6) and Visiting Distinguished Professor at the Technion Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczyński, Case Western Reserve University ISBN 978-0-387-70913-0 e-ISBN 978-0-387-70914-7 DOI 10.1007/978-0-387-70914-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938382 Mathematics Subject Classification (2010): 35Rxx, 46Sxx, 47Sxx © Springer Science+Business Media, LLC 2011 All 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, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec- tion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Felix Browder, a mentor and close friend, who taught me to enjoy PDEs through the eyes of a functional analyst

Preface This book has its roots in a course I taught for many years at the University of Paris. It is intended for students who have a good background in real analysis (as expounded, for instance, in the textbooks of G. B. Folland [2], A. W. Knapp [1], and H. L. Royden [1]). I conceived a program mixing elements from two distinct “worlds”: functional analysis (FA) and partial differential equations (PDEs). The ﬁrst part deals with abstract results in FA and operator theory. The second part concerns the study of spaces of functions (of one or more real variables) having speciﬁc differentiability properties: the celebrated Sobolev spaces, which lie at the heart of the modern theory of PDEs. I show how the abstract results from FA can be applied to solve PDEs. The Sobolev spaces occur in a wide range of questions, in both pure and applied mathematics. They appear in linear and nonlinear PDEs that arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, and physics. They belong to the toolbox of any graduate student in analysis. Unfortunately, FA and PDEs are often taught in separate courses, even though they are intimately connected. Many questions tackled in FA originated in PDEs (for a historical perspective, see, e.g., J. Dieudonné [1] and H. Brezis–F. Browder [1]). There is an abundance of books (even voluminous treatises) devoted to FA. There are also numerous textbooks dealing with PDEs. However, a synthetic presentation intended for graduate students is rare. and I have tried to ﬁll this gap. Students who are often fascinated by the most abstract constructions in mathematics are usually attracted by the elegance of FA. On the other hand, they are repelled by the never- ending PDE formulas with their countless subscripts. I have attempted to present a “smooth” transition from FA to PDEs by analyzing ﬁrst the simple case of one- dimensional PDEs (i.e., ODEs—ordinary differential equations), which looks much more manageable to the beginner. In this approach, I expound techniques that are possibly too sophisticated for ODEs, but which later become the cornerstones of the PDE theory. This layout makes it much easier for students to tackle elaborate higher-dimensional PDEs afterward. A previous version of this book, originally published in 1983 in French and fol- lowed by numerous translations, became very popular worldwide, and was adopted as a textbook in many European universities. A deﬁciency of the French text was the vii

viii Preface lack of exercises. The present book contains a wealth of problems. I plan to add even more in future editions. I have also outlined some recent developments, especially in the direction of nonlinear PDEs. Brief user’s guide 1. Statements or paragraphs preceded by the bullet symbol • are extremely impor- tant, and it is essential to grasp them well in order to understand what comes afterward. 2. Results marked by the star symbol can be skipped by the beginner; they are of interest only to advanced readers. 3. In each chapter I have labeled propositions, theorems, and corollaries in a con- tinuous manner (e.g., Proposition 3.6 is followed by Theorem 3.7, Corollary 3.8, etc.). Only the remarks and the lemmas are numbered separately. 4. In order to simplify the presentation I assume that all vector spaces are over R. Most of the results remain valid for vector spaces over C. I have added in Chapter 11 a short section describing similarities and differences. 5. Many chapters are followed by numerous exercises. Partial solutions are pre- sented at the end of the book. More elaborate problems are proposed in a separate section called “Problems” followed by “Partial Solutions of the Problems.” The problems usually require knowledge of material coming from various chapters. I have indicated at the beginning of each problem which chapters are involved. Some exercises and problems expound results stated without details or without proofs in the body of the chapter. Acknowledgments During the preparation of this book I received much encouragement from two dear friends and former colleagues: Ph. Ciarlet and H. Berestycki. I am very grateful to G. Tronel, M. Comte, Th. Gallouet, S. Guerre-Delabrière, O. Kavian, S. Kichenas- samy, and the late Th. Lachand-Robert, who shared their “ﬁeld experience” in dealing with students. S. Antman, D. Kinderlehrer, andY. Li explained to me the background and “taste” of American students. C. Jones kindly communicated to me an English translation that he had prepared for his personal use of some chapters of the original French book. I owe thanks to A. Ponce, H.-M. Nguyen, H. Castro, and H. Wang, who checked carefully parts of the book. I was blessed with two extraordinary as- sistants who typed most of this book at Rutgers: Barbara Miller, who is retired, and now Barbara Mastrian. I do not have enough words of praise and gratitude for their constant dedication and their professional help. They always found attractive solu- tions to the challenging intricacies of PDE formulas. Without their enthusiasm and patience this book would never have been ﬁnished. It has been a great pleasure, as

Preface ix ever, to work with Ann Kostant at Springer on this project. I have had many oppor- tunities in the past to appreciate her long-standing commitment to the mathematical community. The author is partially supported by NSF Grant DMS-0802958. Haim Brezis Rutgers University March 2010

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