Book:Haïm Brezis/Functional Analysis, Sobolev Spaces and Partial Differential Equations

Subject Matter

 * Functional Analysis

Contents
Preface


 * 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 $E^{**}$. 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 Complimentary Subspaces. Right and Left Invertability 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 $\map \sigma {E,E^*}$


 * 3.3 Weak Topology, Convex Sets and Linear Operators


 * 3.4 The Weak* Topology $\map \sigma {E^*,E}$


 * 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 Elemenary 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 Problem


 * 7.1 Definition and Elementary Properties of Maximal Monotone Operators


 * 7.2 Solution of the Evolution Problem $\dfrac {\d u} {\d t} + A u = 0$ on $\hointr 0 {+ \infty}$, $\map u 0 = u_0$. 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 $\map {W^{1,p} } {I}$


 * 8.3 The Space $W_0^{1,p}$


 * 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 $\map {W^{1,p}} {\Omega}$


 * 9.2 Extension Operators


 * 9.3 Sobolev Inequalities


 * 9.4 The Space $\map {W_0^{1,p} } \Omega$


 * 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 Quotien Spaces


 * 11.3 Some Classical Spaces and Sequences


 * 11.4 Banach Space over $\C$: What Is Similar and What Is Different

Solutions of Some Exercises

Problems

Partial Solutions

Notation

References

Index