Adams / Fournier Sobolev Spaces

2. Auflage 2003
ISBN: 978-0-08-054129-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 320 Seiten

Sobolev Spaces presents an introduction to the theory of Sobolev Spaces and other related spaces of function, also to the imbedding characteristics of these spaces. This theory is widely used in pure and Applied Mathematics and in the Physical Sciences. This second edition of Adam's 'classic' reference text contains many additions and much modernizing and refining of material. The basic premise of the book remains unchanged: Sobolev Spaces is intended to provide a solid foundation in these spaces for graduate students and researchers alike.Self-contained and accessible for readers in other disciplinesWritten at elementary level making it accessible to graduate students

Autoren/Hrsg.

Adams, Robert A.

Fournier, John J. F.

Inhaltsverzeichnis

1;Front Cover;1
2;SOBOLEV SPACES;4
3;Copyright Page;5
4;CONTENTS;6
5;Preface;10
6;List of Spaces and Norms;13
7;CHAPTER 1. PRELIMINARIES;16
7.1;Notation;16
7.2;Topological Vector Spaces;18
7.3;Normed Spaces;19
7.4;Spaces of Continuous Functions;25
7.5;The Lebesgue Measure in Rn;28
7.6;The Lebesgue Integral;31
7.7;Distributions and Weak Derivatives;34
8;CHAPTER 2. THE LEBESGUE SPACES Lp(.);38
8.1;Definition and Basic Properties;38
8.2;Completeness of LP (.);44
8.3;Approximation by Continuous Functions;46
8.4;Convolutions and Young's Theorem;47
8.5;Mollifiers and Approximation by Smooth Functions;51
8.6;Precompact Sets in LP (.);53
8.7;Uniform Convexity;56
8.8;The Normed Dual of LP (.);60
8.9;Mixed-Norm LP Spaces;64
8.10;The Marcinkiewicz Interpolation Theorem;67
9;CHAPTER 3. THE SOBOLEV SPACES Wm,,P (.);74
9.1;Definitions and Basic Properties;74
9.2;Duality and the Spaces W -m,p' (.);77
9.3;Approximation by Smooth Functions on .;80
9.4;Approximation by Smooth Functions on Rn;82
9.5;Approximation by Functions in C08 (.);85
9.6;Coordinate Transformations;92
10;CHAPTER 4. THE SOBOLEV IMBEDDING THEOREM;94
10.1;Geometric Properties of Domains;96
10.2;Imbeddings by Potential Arguments;102
10.3;Imbeddings by Averaging;108
10.4;Imbeddings into Lipschitz Spaces;114
10.5;Sobolev's Inequality;116
10.6;Variations of Sobolev's Inequality;119
10.7;W m,p (.) as a Banach Algebra;121
10.8;Optimality of the Imbedding Theorem;123
10.9;Nonimbedding Theorems for Irregular Domains;126
10.10;Imbedding Theorems for Domains with Cusps;130
10.11;Imbedding Inequalities Involving Weighted Norms;134
10.12;Proofs of Theorems 4.51–4.53;146
11;CHAPTER 5. INTERPOLATION, EXTENSION, AND APPROXIMATION THEOREMS;150
11.1;Interpolation on Order of Smoothness;150
11.2;Interpolation on Degree of Sumability;154
11.3;Interpolation Involving Compact Subdomains;158
11.4;Extension Theorems;161
11.5;An Approximation Theorem;174
11.6;Boundary Traces;178
12;CHAPTER 6. COMPACT IMBEDDINGS OF SOBOLEV SPACES;182
12.1;The Rellich-Kondrachov Theorem;182
12.2;Two Counterexamples;188
12.3;Unbounded Domains — Compact Imbeddings of Wom'p (.);190
12.4;An Equivalent Norm for Wom'p (.);198
12.5;Unbounded Domains m Decay at Infinity;201
12.6;Unbounded Domains — Compact Imbeddings of W m,p (.);210
12.7;Hilbert-Schmidt Imbeddings;215
13;CHAPTER 7. FRACTIONAL ORDER SPACES;220
13.1;Introduction;220
13.2;The Bochner Integral;221
13.3;Intermediate Spaces and Interpolation—The Real Method;223
13.4;The Lorentz Spaces;236
13.5;Besov Spaces;243
13.6;Generalized Spaces of Hölder Continuous Functions;247
13.7;Characterization of Traces;249
13.8;Direct Characterizations of Besov Spaces;256
13.9;Other Scales of Intermediate Spaces;262
13.10;Wavelet Characterizations;271
14;CHAPTER 8. ORLICZ SPACES AND ORLICZ-SOBOLEV SPACES;276
14.1;Introduction;276
14.2;N-Functions;277
14.3;Orlicz Spaces;281
14.4;Duality in Orlicz Spaces;287
14.5;Separability and Compactness Theorems;289
14.6;A Limiting Case of the Sobolev Imbedding Theorem;292
14.7;Orlicz-Sobolev Spaces;296
14.8;Imbedding Theorems for Orlicz-Sobolev Spaces;297
15;References;310
16;Index;316