ISBN-10:
0387709134
ISBN-13:
9780387709130
Pub. Date:
11/10/2010
Publisher:
Springer New York
Functional Analysis, Sobolev Spaces and Partial Differential Equations / Edition 1

Functional Analysis, Sobolev Spaces and Partial Differential Equations / Edition 1

by Haim Brezis

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Overview

This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.

Product Details

ISBN-13: 9780387709130
Publisher: Springer New York
Publication date: 11/10/2010
Series: Universitext
Edition description: 2011
Pages: 600
Sales rank: 921,378
Product dimensions: 6.20(w) x 9.20(h) x 1.50(d)

Table of Contents

Preface vii

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

1.1 The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functional 1

1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets 4

1.3 The Bidual E**. Orthogonality Relations 8

1.4 A Quick Introduction to the Theory of Conjugate Convex Functions 10

Comments on Chapter 1 17

Exercises for Chapter 1 19

2 The Uniform Boundedness Principle and the Closed Graph Theorem 31

2.1 The Baire Category Theorem 31

2.2 The Uniform Boundedness Principle 32

2.3 The Open Mapping Theorem and the Closed Graph Theorem 34

2.4 Complementary Subspaces. Right and Left Invertibility of Linear Operators 37

2.5 Orthogonality Revisited 40

2.6 An Introduction to Unbounded Linear Operators. Definition of the Adjoint 43

2.7 A Characterization of Operators with Closed Range. A Characterization of Surjective Operators 46

Comments on Chapter 2 48

Exercises for Chapter 2 49

3 Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity 55

3.1 The Coarsest Topology for Which a Collection of Maps Becomes Continuous 55

3.2 Definition and Elementary Properties of the Weak Topology σ(E, E*) 57

3.3 Weak Topology, Convex Sets, and Linear Operators 60

3.4 The Weak* Topology σ(E* E) 62

3.5 Reflexive Spaces 67

3.6 Separable Spaces 72

3.7 Uniformly Convex Spaces 76

Comments on Chapter 3 78

Exercises for Chapter 3 79

4 Lp Spaces 89

4.1 Some Results about Integration That Everyone Must Know 90

4.2 Definition and Elementary Properties of Lp Spaces 91

4.3 Reflexivity. Separability. Dual of Lp 95

4.4 Convolution and regularization 104

4.5 Criterion for Strong Compactness in Lp 111

Comments on Chapter 4 114

Exercises for Chapter 4 118

5 Hilbert Spaces 131

5.1 Definitions and Elementary Properties. Projection onto a Closed Convex Set 131

5.2 The Dual Space of a Hilbert Space 135

5.3 The Theorems of Stampacchia and Lax-Milgram 138

5.4 Hilbert Sums. Orthonormal Bases 141

Comments on Chapter 5 144

Exercises for Chapter 5 146

6 Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators 157

6.1 Definitions. Elementary Properties. Adjoint 157

6.2 The Riesz-Fredholm Theory 159

6.3 The Spectrum of a Compact Operator 162

6.4 Spectral Decomposition of Self-Adjoint Compact Operators 165

Comments on Chapter 6 168

Exercises for Chapter 6 170

7 The Hille-Yosida Theorem 181

7.1 Definition and Elementary Properties of Maximal Monotone Operators 181

7.2 Solution of the Evolution Problem du/dt + Au = 0 on (0, +∞), u(0) = u0. Existence and uniqueness 184

7.3 Regularity 191

7.4 The Self-Adjoint Case 193

Comments on Chapter 7 197

8 Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension 201

8.1 Motivation 201

8.2 The Sobolev Space W1.p (I) 202

8.3 The Space &$$$; 217

8.4 Some Examples of Boundary Value Problems 220

8.5 The Maximum Principle 229

8.6 Eigenfunctions and Spectral Decomposition 231

Comments on Chapter 8 233

Exercises for Chapter 8 235

9 Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions 263

9.1 Definition and Elementary Properties of the Sobolev Spaces W1.p (Ω) 263

9.2 Extension Operators 272

9.3 Sobolev Inequalities 278

9.4 The Space &$$$; (Ω) 287

9.5 Variational Formulation of Some Boundary Value Problems 291

9.6 Regularity of Weak Solutions 298

9.7 The Maximum Principle 307

9.8 Eigenfunctions and Spectral Decomposition 311

Comments on Chapter 9 312

10 Evolution Problems: The Heat Equation and the Wave Equation 325

10.1 The Heat Equation: Existence, Uniqueness, and Regularity 325

10.2 The Maximum Principle 333

10.3 The Wave Equation 335

Comments on Chapter 10 340

11 Miscellaneous Complements 349

11.1 Finite-Dimensional and Finite-Codimensional Spaces 349

11.2 Quotient Spaces 353

11.3 Some Classical Spaces of Sequences 357

11.4 Banach Spaces over C: What Is Similar and What Is Different? 361

Solutions of Some Exercises 371

Problems 435

Partial Solutions of the Problems 521

Notation 583

References 585

Index 595

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