Sobolev Spaces

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 disciplines

- Written at elementary level making it accessible to graduate students

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 4.514.53;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;Th
e 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 InterpolationThe 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