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.
Author(s): Robert A. Adams, John J. F. Fournier
Series: Academic Press: 140
Edition: 2
Publisher: Academic Press
Year: 2003
Language: English
Pages: 321
Front Cover......Page 1
SOBOLEV SPACES......Page 4
Copyright Page......Page 5
CONTENTS......Page 6
Preface......Page 10
List of Spaces and Norms......Page 13
Notation......Page 16
Topological Vector Spaces......Page 18
Normed Spaces......Page 19
Spaces of Continuous Functions......Page 25
The Lebesgue Measure in Rn......Page 28
The Lebesgue Integral......Page 31
Distributions and Weak Derivatives......Page 34
Definition and Basic Properties......Page 38
Completeness of LP (Ώ)......Page 44
Approximation by Continuous Functions......Page 46
Convolutions and Young's Theorem......Page 47
Mollifiers and Approximation by Smooth Functions......Page 51
Precompact Sets in LP (Ω)......Page 53
Uniform Convexity......Page 56
The Normed Dual of LP (Ω)......Page 60
Mixed-Norm LP Spaces......Page 64
The Marcinkiewicz Interpolation Theorem......Page 67
Definitions and Basic Properties......Page 74
Duality and the Spaces W -m,p' (Ω)......Page 77
Approximation by Smooth Functions on Ω......Page 80
Approximation by Smooth Functions on Rn......Page 82
Approximation by Functions in C0∞ (Ω)......Page 85
Coordinate Transformations......Page 92
CHAPTER 4. THE SOBOLEV IMBEDDING THEOREM......Page 94
Geometric Properties of Domains......Page 96
Imbeddings by Potential Arguments......Page 102
Imbeddings by Averaging......Page 108
Imbeddings into Lipschitz Spaces......Page 114
Sobolev's Inequality......Page 116
Variations of Sobolev's Inequality......Page 119
W m,p (Ω) as a Banach Algebra......Page 121
Optimality of the Imbedding Theorem......Page 123
Nonimbedding Theorems for Irregular Domains......Page 126
Imbedding Theorems for Domains with Cusps......Page 130
Imbedding Inequalities Involving Weighted Norms......Page 134
Proofs of Theorems 4.51–4.53......Page 146
Interpolation on Order of Smoothness......Page 150
Interpolation on Degree of Sumability......Page 154
Interpolation Involving Compact Subdomains......Page 158
Extension Theorems......Page 161
An Approximation Theorem......Page 174
Boundary Traces......Page 178
The Rellich-Kondrachov Theorem......Page 182
Two Counterexamples......Page 188
Unbounded Domains — Compact Imbeddings of Wom'p (Ω)......Page 190
An Equivalent Norm for Wom'p (Ω)......Page 198
Unbounded Domains m Decay at Infinity......Page 201
Unbounded Domains — Compact Imbeddings of W m,p (Ω)......Page 210
Hilbert-Schmidt Imbeddings......Page 215
Introduction......Page 220
The Bochner Integral......Page 221
Intermediate Spaces and Interpolation—The Real Method......Page 223
The Lorentz Spaces......Page 236
Besov Spaces......Page 243
Generalized Spaces of Hölder Continuous Functions......Page 247
Characterization of Traces......Page 249
Direct Characterizations of Besov Spaces......Page 256
Other Scales of Intermediate Spaces......Page 262
Wavelet Characterizations......Page 271
Introduction......Page 276
N-Functions......Page 277
Orlicz Spaces......Page 281
Duality in Orlicz Spaces......Page 287
Separability and Compactness Theorems......Page 289
A Limiting Case of the Sobolev Imbedding Theorem......Page 292
Orlicz-Sobolev Spaces......Page 296
Imbedding Theorems for Orlicz-Sobolev Spaces......Page 297
References......Page 310
Index......Page 316
