Functional Analysis, Sobolev Spaces and Partial Differential Equations

ISBN-10: 0387709134
ISBN-13: 9780387709130
Edition: 2011
Authors: Haim Br�zis
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Description: 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  More...

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Book details

Copyright year: 2011
Publisher: Springer London, Limited
Publication date: 11/10/2010
Binding: Paperback
Pages: 600
Size: 6.25" wide x 9.50" long x 1.50" tall
Weight: 1.892
Language: English

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.

Preface
The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions
The Analytic Form of the Hahn-Banach Theorem: Extension of Linear Functional
The Geometric Forms of the Hahn-Banach Theorem: Separation of Convex Sets
The Bidual E<sup>**</sup>. Orthogonality Relations
A Quick Introduction to the Theory of Conjugate Convex Functions
The Uniform Boundedness Principle and the Closed Graph Theorem
The Baire Category Theorem
The Uniform Boundedness Principle
The Open Mapping Theorem and the Closed Graph Theorem
Complementary Subspaces. Right and Left Invertibility of Linear Operators
Orthogonality Revisited
An Introduction to Unbounded Linear Operators. Definition of the Adjoint
A Characterization of Operators with Closed Range. A Characterization of Surjective Operators
Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity
The Coarsest Topology for Which a Collection of Maps Becomes Continuous
Definition and Elementary Properties of the Weak Topology �(E, E<sup>*</sup>)
Weak Topology, Convex Sets, and Linear Operators
The Weak<sup>*</sup> Topology �(E<sup>*</sup> E)
Reflexive Spaces
Separable Spaces
Uniformly Convex Spaces
L<sup>p</sup> Spaces
Some Results about Integration That Everyone Must Know
Definition and Elementary Properties of L<sup>p</sup> Spaces
Reflexivity. Separability. Dual of L<sup>p</sup>
Convolution and regularization
Criterion for Strong Compactness in L<sup>p</sup>
Hilbert Spaces
Definitions and Elementary Properties. Projection onto a Closed Convex Set
The Dual Space of a Hilbert Space
The Theorems of Stampacchia and Lax-Milgram
Hilbert Sums. Orthonormal Bases
Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators
Definitions. Elementary Properties. Adjoint
The Riesz-Fredholm Theory
The Spectrum of a Compact Operator
Spectral Decomposition of Self-Adjoint Compact Operators
The Hille-Yosida Theorem
Definition and Elementary Properties of Maximal Monotone Operators
Solution of the Evolution Problem du/dt + Au = 0 on (0, +&#8734;), u(0) = u<sub>0</sub>. Existence and uniqueness
Regularity
The Self-Adjoint Case
Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension
Motivation
The Sobolev Space W<sup>1.p</sup> (I)
The Space &$$$;
Some Examples of Boundary Value Problems
The Maximum Principle
Eigenfunctions and Spectral Decomposition
Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions
Definition and Elementary Properties of the Sobolev Spaces W<sup>1.p</sup> (�)
Extension Operators
Sobolev Inequalities
The Space &$$$; (�)
Variational Formulation of Some Boundary Value Problems
Regularity of Weak Solutions
The Maximum Principle
Eigenfunctions and Spectral Decomposition
Evolution Problems: The Heat Equation and the Wave Equation
The Heat Equation: Existence, Uniqueness, and Regularity
The Maximum Principle
The Wave Equation
Miscellaneous Complements
Finite-Dimensional and Finite-Codimensional Spaces
Quotient Spaces
Some Classical Spaces of Sequences
Banach Spaces over C: What Is Similar and What Is Different?
Solutions of Some Exercises
Problems
Partial Solutions of the Problems
Notation
References
Index

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