Sobolev spaces:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Amsterdam [u.a.]
Acad. Press
2007
|
| Ausgabe: | 2. ed., reprint. |
| Schriftenreihe: | Pure and applied mathematics
140 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 305 S. graph. Darst. |
| ISBN: | 0120441438 9780120441433 |
Internformat
MARC
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| 100 | 1 | |a Adams, Robert A. |d 1940- |e Verfasser |0 (DE-588)141723874 |4 aut | |
| 245 | 1 | 0 | |a Sobolev spaces |c Robert A. Adams and John J. F. Fournier |
| 250 | |a 2. ed., reprint. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Acad. Press |c 2007 | |
| 300 | |a XIII, 305 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Pure and applied mathematics |v 140 | |
| 650 | 0 | 7 | |a Sobolev-Raum |0 (DE-588)4055345-0 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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| adam_text | CONTENTS
Preface ix
List of Spaces and Norms xii
1. PRELIMINARIES 1
Notation I
Topological Vector Spaces 3
Normed Spaces 4
Spaces of Continuous Functions 10
The Lebesgue Measure in W 13
The Lebesgue Integral I g
Distributions and Weak Derivatives 19
2. THE LEBESGUE SPACES LP(f2) 23
Definition and Basic Properties 23
Completeness of LP(Q) 29
Approximation by Continuous Functions 31
Convolutions and Young s Theorem 32
Mollifiers and Approximation by Smooth Functions 36
Precompact Sets in L (Q) 38
Uniform Convexity 41
The Normed Dual of L (Q) 45
Mixed Norm Lp Spaces 49
The Marcinkiewicz Interpolation Theorem 52
vi Contents
3. THE SOBOLEV SPACES Wm^(n) 59
Definitions and Basic Properties 59
Duality and the Spaces W~m P (£2) 62
Approximation by Smooth Functions on £2 65
Approximation by Smooth Functions on W 67
Approximation by Functions in C^°(£2) 70
Coordinate Transformations 77
4. THE SOBOLEV IMBEDDING THEOREM 79
Geometric Properties of Domains 81
Imbeddings by Potential Arguments 87
Imbeddings by Averaging 93
Imbeddings into Lipschitz Spaces 99
Sobolev s Inequality 101
Variations of Sobolev s Inequality 104
W p (£2) as a Banach Algebra 106
Optimality of the Imbedding Theorem 108
Nonimbedding Theorems for Irregular Domains 111
Imbedding Theorems for Domains with Cusps 115
Imbedding Inequalities Involving Weighted Norms 119
Proofs of Theorems 4.51 4.53 131
5. INTERPOLATION, EXTENSION, AND APPROXIMATION 135
THEOREMS
Interpolation on Order of Smoothness 135
Interpolation on Degree of Sumability 139
Interpolation Involving Compact Subdomains 143
Extension Theorems 146
An Approximation Theorem 159
Boundary Traces 163
6. COMPACT IMBEDDINGS OF SOBOLEV SPACES 167
The Rellich Kondrachov Theorem 167
Two Counterexamples 173
Unbounded Domains — Compact Imbeddings of W™; (£2) 175
An Equivalent Norm for W™P(S2) 183
Unbounded Domains — Decay at Infinity 186
Unbounded Domains — Compact Imbeddings of Wm p(Q) 195
Hilbert Schmidt Imbeddings 200
vi
Contents vii
7. FRACTIONAL ORDER SPACES 205
Introduction 205
The Bochner Integral 206
Intermediate Spaces and Interpolation — The Real Method 208
The Lorentz Spaces 221
Besov Spaces 228
Generalized Spaces of Holder Continuous Functions 232
Characterization of Traces 234
Direct Characterizations of Besov Spaces 241
Other Scales of Intermediate Spaces 247
Wavelet Characterizations 256
8. ORLICZ SPACES AND ORLICZ SOBOLEV SPACES 261
Introduction 261
N Functions 262
Orlicz Spaces 266
Duality in Orlicz Spaces 272
Separability and Compactness Theorems 274
A Limiting Case of the Sobolev Imbedding Theorem 277
Orlicz Sobolev Spaces 281
Imbedding Theorems for Orlicz Sobolev Spaces 282
References 295
Index 301
vii
|
| adam_txt |
CONTENTS
Preface ix
List of Spaces and Norms xii
1. PRELIMINARIES 1
Notation I
Topological Vector Spaces 3
Normed Spaces 4
Spaces of Continuous Functions 10
The Lebesgue Measure in W 13
The Lebesgue Integral I g
Distributions and Weak Derivatives 19
2. THE LEBESGUE SPACES LP(f2) 23
Definition and Basic Properties 23
Completeness of LP(Q) 29
Approximation by Continuous Functions 31
Convolutions and Young's Theorem 32
Mollifiers and Approximation by Smooth Functions 36
Precompact Sets in L '(Q) 38
Uniform Convexity 41
The Normed Dual of L '(Q) 45
Mixed Norm Lp Spaces 49
The Marcinkiewicz Interpolation Theorem 52
vi Contents
3. THE SOBOLEV SPACES Wm^(n) 59
Definitions and Basic Properties 59
Duality and the Spaces W~m P' (£2) 62
Approximation by Smooth Functions on £2 65
Approximation by Smooth Functions on W 67
Approximation by Functions in C^°(£2) 70
Coordinate Transformations 77
4. THE SOBOLEV IMBEDDING THEOREM 79
Geometric Properties of Domains 81
Imbeddings by Potential Arguments 87
Imbeddings by Averaging 93
Imbeddings into Lipschitz Spaces 99
Sobolev's Inequality 101
Variations of Sobolev's Inequality 104
W'" p (£2) as a Banach Algebra 106
Optimality of the Imbedding Theorem 108
Nonimbedding Theorems for Irregular Domains 111
Imbedding Theorems for Domains with Cusps 115
Imbedding Inequalities Involving Weighted Norms 119
Proofs of Theorems 4.51 4.53 131
5. INTERPOLATION, EXTENSION, AND APPROXIMATION 135
THEOREMS
Interpolation on Order of Smoothness 135
Interpolation on Degree of Sumability 139
Interpolation Involving Compact Subdomains 143
Extension Theorems 146
An Approximation Theorem 159
Boundary Traces 163
6. COMPACT IMBEDDINGS OF SOBOLEV SPACES 167
The Rellich Kondrachov Theorem 167
Two Counterexamples 173
Unbounded Domains — Compact Imbeddings of W™;' (£2) 175
An Equivalent Norm for W™P(S2) 183
Unbounded Domains — Decay at Infinity 186
Unbounded Domains — Compact Imbeddings of Wm p(Q) 195
Hilbert Schmidt Imbeddings 200
vi
Contents vii
7. FRACTIONAL ORDER SPACES 205
Introduction 205
The Bochner Integral 206
Intermediate Spaces and Interpolation — The Real Method 208
The Lorentz Spaces 221
Besov Spaces 228
Generalized Spaces of Holder Continuous Functions 232
Characterization of Traces 234
Direct Characterizations of Besov Spaces 241
Other Scales of Intermediate Spaces 247
Wavelet Characterizations 256
8. ORLICZ SPACES AND ORLICZ SOBOLEV SPACES 261
Introduction 261
N Functions 262
Orlicz Spaces 266
Duality in Orlicz Spaces 272
Separability and Compactness Theorems 274
A Limiting Case of the Sobolev Imbedding Theorem 277
Orlicz Sobolev Spaces 281
Imbedding Theorems for Orlicz Sobolev Spaces 282
References 295
Index 301
vii |
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| index_date | 2024-07-02T19:12:32Z |
| indexdate | 2024-07-09T21:09:07Z |
| institution | BVB |
| isbn | 0120441438 9780120441433 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016223690 |
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| physical | XIII, 305 S. graph. Darst. |
| publishDate | 2007 |
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| publisher | Acad. Press |
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| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Adams, Robert A. 1940- Verfasser (DE-588)141723874 aut Sobolev spaces Robert A. Adams and John J. F. Fournier 2. ed., reprint. Amsterdam [u.a.] Acad. Press 2007 XIII, 305 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 140 Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 s DE-604 Fournier, John J. F. Verfasser (DE-588)141723939 aut Pure and applied mathematics 140 (DE-604)BV010177228 140 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016223690&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Adams, Robert A. 1940- Fournier, John J. F. Sobolev spaces Pure and applied mathematics Sobolev-Raum (DE-588)4055345-0 gnd |
| subject_GND | (DE-588)4055345-0 |
| title | Sobolev spaces |
| title_auth | Sobolev spaces |
| title_exact_search | Sobolev spaces |
| title_exact_search_txtP | Sobolev spaces |
| title_full | Sobolev spaces Robert A. Adams and John J. F. Fournier |
| title_fullStr | Sobolev spaces Robert A. Adams and John J. F. Fournier |
| title_full_unstemmed | Sobolev spaces Robert A. Adams and John J. F. Fournier |
| title_short | Sobolev spaces |
| title_sort | sobolev spaces |
| topic | Sobolev-Raum (DE-588)4055345-0 gnd |
| topic_facet | Sobolev-Raum |
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