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BK

New York ; Dordrecht ; Heidelberg ; London : Springer, [2011]

xiii, 599 stran ; 24 cm

ISBN 978-0-387-70913-0 (brožováno)

Universitext

Obsahuje bibliografii na stranách 585-594 a rejstřík

001463120

Universitext // Haim Brezis // Functional Analysis, Sobolev Spaces and Partial Differential Equations // Uniquely, this book presents a coherent, concise and unified way of combining elements from two distinct “worlds,” functional analysis (FA) and partial differential equations (PDFs), and is intended for students who have a good background in real analysis. This text presents a smooth transition from FA to PDFs by analyzing in great detail the simple case of one-dimensional PDFs (i.e., ODEs), a more manageable approach for the beginner. Although there are many books on functional analysis and many on PDFs, this is the first to cover both of these closely connected topics. Moreover, the wealth of exercises and additional material presented, leads the reader to the frontier of research. // This book has its roots in a celebrated course taught by the author for many years and is a completely revised, updated, and expanded English edition of the important Analyse Fonctionnelle (1983). Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English version is a welcome addition to this list. // The first part of the text deals with abstract results in FA and operator theory. The second part is concerned with the study of spaces of functions (of one or more real variables) having specific differentiability properties, e.g., the celebrated Sobolev spaces, which lie at the heart of the modern theory

of PDFs. The Sobolev spaces occur in a wide range of questions, both in pure and applied mathematics, appearing in linear and nonlinear PDFs which arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, physics etc. and belong in the toolbox of any graduate student studying analysis. // Mathematics // ISBN 978-0-387-70913-0 // 9 // ? springer.com // Preface... vn // 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 Functionals... 1 // 1.2 The Geometric Forms of the Hahn-Banach Theorem: Separation // of Convex Sets... 4 // 1.3 The Bidual Ł**. 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 // Contents // xii // 3.2 Definition and Elementary Properties of the Weak Topology // v(E,E*) ... 57 // 3.3 Weak Topology, Convex Sets, and Linear Operators... 60 // 3.4 The Weak* Topology o(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 ... 95 // 4.4 Convolution and regularization...104 // 4.5 Criterion for Strong Compactness in ZT...Ill // 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...I44 // Exercises for Chapter 5 ...146 // 6 Compact Operators. Spectral Decomposition of Self-Adjoint // Compact Operators...I57 // 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  7 + Am = 0 on [0, 4-oo), // m(0) = mq. Existence and uniqueness...184 // 7.3 Regularity...191 // 7.4 The Self-Adjoint Case ...193 // Comments on Chapter 7...I97 // Contents // xiii // 8 Sobolev Spaces and the Variational Formulation of Boundary Value // Problems in One Dimension...201 // 8.1 Motivation...201 // 8.2 The Sobolev Space WX p{I) ...202 // 8.3 The Space WqP...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 // Wl-P(Q)...263 // 9.2 Extension Operators...272 // 9.3 Sobolev Inequalities...278 // 9.4 The Space Vr(J,/7(Ł2) ...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