Sobolev Spaces on Domains

Victor I. Burenkov Sobolev Spaces on Domains
B. G. Teubner Gmbh | English | 1998-01 | ISBN: 3815420687 | 312 pages | File type: PDF | 5,4 mb

Contents
Notation and basic inequalities 11
1 Preliminaries 15
1.1 Mollifiers 15
1.2 Weak derivatives 18
1.3 Sobolev spaces (basic properties) 28
2 Approximation by infinitely differentiable functions 39
2.1 Approximation by C^\infty_0-functions on R^n 39
2.2 Nonlinear mollifiers with variable step 42
2.3 Approximation by C^\infty-functions on open sets 47
2.4 Approximation with preservation of boundary values 56
2.5 Linear mollifiers with variable step 60
2.6 The best possible approximation with preservation of boundary values 73
3 Sobolev's integral representation 81
3.1 The one-dimensional case 81
3.2 Star-shaped sets and sets satisfying the cone condition 92
3.3 Multidimensional Taylor's formula 100
3.4 Sobolev's integral representation 104
3.5 Corollaries 111
4 Embedding theorems 119
4.1 Embeddings and inequalities 119
4.2 The one-dimensional case 127
4.3 Open sets with quasi-resolvable, quasi-continuous, smooth and Lipschitz boundaries 148
4.4 Estimates for intermediate derivatives 160
4.5 Hardy-Littlewood-Sobolev inequality for integral of potential type 177
4.6 Embeddings into the space of continuous functions 181
4.7 Embeddings into the space L^q 186
5 Trace theorems 197
5.1 Notion of the trace of a function 197
5.2 Existence of the traces on subspaces 201
5.3 Nikol'skil-Besov spaces 202
5.4 Description of the traces on subspaces 214
5.5 Traces on smooth surfaces 238
6 Extension theorems 247
6.1 The one-dimensional case 247
6.2 Pasting local extensions 264
6.3 Extensions for sufficiently smooth boundaries 269
6.4 Extensions for Lipschitz boundaries 271
7 Comments 289
Bibliography 296
Index 311



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