Analytic semigroups and semilinear initial boundary value problems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2016
|
| Edition: | Second edition |
| Series: | London Mathematical Society lecture note series
434 |
| Subjects: | |
| Online Access: | BSB01 FHN01 UEI01 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (xvi, 331 Seiten) Diagramme |
| ISBN: | 9781316729755 9781316758434 |
| DOI: | 10.1017/CBO9781316729755 |
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Record in the Search Index
| _version_ | 1804176364824166400 |
|---|---|
| adam_text | Titel: Analytic semigroups and semilinear initial boundary value problems
Autor: Taira, Kazuaki
Jahr: 2016
Contents
Preface to the Second Edition page xi
Preface to the First Edition xv
1 Introduction and Main Results 1
1.1 Formulation of the elliptic boundary value problem (*) 2
1.2 Existence and uniqueness theorem for problem (*) 4
1.3 Generation theorem for analytic semigroups of
problem (*) 5
1.4 The semilinear initial boundary value problem for
problem (*) 7
1.5 The global existence and uniqueness theorems 8
1.6 Summary of the Contents 10
2 Preliminaries from Functional Analysis 22
2.1 Linear Operators and functionals 23
2.2 Topological spaces 24
2.3 Quasinormed linear spaces 25
2.3.1 Bounded sets 27
2.3.2 Continuity of linear Operators 27
2.3.3 Topologies of linear Operators 27
2.3.4 The Banach-Steinhaus theorem 29
2.4 Normed linear spaces 29
2.4.1 Finite dimensional spaces 34
2.4.2 The Hahn-Banach extension theorem 34
2.4.3 Dual spaces 35
2.4.4 Annihilators 35
2.4.5 Dual spaces of normed factor spaces 36
2.4.6 Bidual spaces 36
2.4.7 Transpose Operators 37
vii
viii
Contents
2.5 Closed Operators 38
2.6 Complemented subspaces 40
2.7 The Riesz-Schauder theory for compact Operators 40
2.7.1 Compact Operators 41
2.7.2 Spectral analysis of compact Operators 41
2.8 Fredholm Operators 43
2.9 Hilbert Spaces 48
2.9.1 Orthogonality 49
2.9.2 The closest-point theorem and applications 50
2.9.3 Orthonormal sets 53
2.9.4 Adjoint Operators 54
2.10 The Hilbert-Schmidt theory 55
3 Theory of Analytic Semigroups 58
3.1 Generation theorem for analytic semigroups 60
3.2 Fractional powers 71
3.3 The linear Cauchy problem 84
3.3.1 The homogeneous case 84
3.3.2 The non-homogeneous case 85
3.4 The semilinear Cauchy problem 91
3.4.1 The space Ea of fractional powers 92
3.4.2 Local existence and uniqueness theorems for
the semilinear Cauchy problem 93
3.4.3 Global existence and uniqueness theorem for
the semilinear Cauchy problem 101
4 Sobolev Imbedding Theorems 108
4.1 Holder spaces and Sobolev spaces 110
4.2 Interpolation theorems 112
4.2.1 Proof of Theorem 4.1, Part (i) 113
4.2.2 Proof of Theorem 4.1, Part (ii) 123
4.2.3 Proof of Theorem 4.1, Part (iii) 144
4.3 Imbeddings of the spaces Wm p(Rn) 148
4.3.1 Imbeddings of Wm,p{Rn) into JBa(Rn) 153
4.3.2 Imbeddings of Wm p(Rn) into H^ r(Rn) 159
4.4 Imbeddings of the spaces Wm p(fl) 161
4.5 Trace theorems 168
4.6 Jump formulas 171
4.7 Regulär distributions with respect to one variable 172
5 Lp Theory of Pseudo-Differential Operators 176
5.1 Generalized Sobolev spaces and Besov spaces 177
Contents
ix
5.1.1 Fourier transforms and Bessel potentials 178
5.1.2 Definitions and basic properties of Sobolev
and Besov Spaces 181
5.2 Fourier integral Operators 184
5.2.1 Symbol classes 184
5.2.2 Phase functions 186
5.2.3 Oscillatory Integrals 188
5.2.4 Definitions and basic properties of Fourier
integral Operators 191
5.3 Pseudo-differential Operators 195
5.3.1 Definitions and basic properties of pseudo-
differential Operators 195
5.3.2 Pseudo-differential Operators on a manifold 206
5.3.3 Hypoelliptic pseudo-differential Operators 208
5.3.4 Subelliptic pseudo-differential Operators 209
5.3.5 The sharp Gärding inequality 210
6 Lp Approach to Elliptic Boundary Value Problems 215
6.1 The Dirichlet problem 217
6.1.1 Volume and surface potentials 219
6.1.2 Uniqueness theorem for the Dirichlet problem 224
6.1.3 The Poisson Operator P 225
6.2 Degenerate elliptic boundary value problems 227
6.2.1 The space Br1-1/p,p(r) 227
6.2.2 Formulation of the boundary value problem (*)232
6.3 A Special reduction to the boundary 232
6.4 The Dirichlet-Neumann Operator II 239
6.5 Proof of Theorem 6.16 244
6.6 Elliptic boundary value problems 250
6.6.1 The Lopatinski Shapiro ellipticity condition 251
6.6.2 The fundamental theorem for elliptic bound¬
ary value problems 255
7 Proof of Theorem 1.1 257
7.1 Regularity Theorem for problem (*) 258
7.2 Uniqueness Theorem for problem (*) 265
7.3 Existence Theorem for problem (*) 266
7.3.1 Proof of Theorem 7.7 267
7.3.2 Proof of Proposition 7.9 275
7.4 Melin s inequality revisited 277
X
Contents
8 Proof of Theorem 1.2 282
8.1 A priori estimates for problem (*) 283
8.2 Generation theorem for analytic semigroups of 21p 291
8.2.1 Proof of Theorem 1.2, Part (i) 291
8.2.2 Proof of Theorem 1.2, Part (ii) 296
8.3 Proof of Theorem 8.7 297
9 Proof of Theorems 1.3 and 1.4 303
9.1 Fractional powers and imbedding theorems 304
9.1.1 The space Xa of fractional powers 304
9.1.2 Proof of Theorem 9.1 305
9.2 The semilinear Cauchy problem for problem (*) 308
9.2.1 Proof of Theorem 1.3 309
9.2.2 Proof of Theorem 1.4 309
9.2.3 The global version of Theorem 1.3 312
9.2.4 The global version of Theorem 1.4 313
Appendix A The Laplace Transform 315
Appendix B The Maximum Principle 318
Appendix C Vector Bundles 320
References 322
Index 327
|
| any_adam_object | 1 |
| author | Taira, Kazuaki 1946- |
| author_GND | (DE-588)120483858 |
| author_facet | Taira, Kazuaki 1946- |
| author_role | aut |
| author_sort | Taira, Kazuaki 1946- |
| author_variant | k t kt |
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| discipline | Mathematik |
| doi_str_mv | 10.1017/CBO9781316729755 |
| edition | Second edition |
| format | Electronic eBook |
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| spelling | Taira, Kazuaki 1946- Verfasser (DE-588)120483858 aut Analytic semigroups and semilinear initial boundary value problems Kazuaki Taira Second edition Cambridge Cambridge University Press 2016 1 Online-Ressource (xvi, 331 Seiten) Diagramme txt rdacontent c rdamedia cr rdacarrier London Mathematical Society lecture note series 434 Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Analytische Halbgruppe (DE-588)4376792-8 gnd rswk-swf Anfangswertproblem (DE-588)4001991-3 gnd rswk-swf Anfangsrandwertproblem (DE-588)4001990-1 gnd rswk-swf Analytische Halbgruppe (DE-588)4376792-8 s Anfangsrandwertproblem (DE-588)4001990-1 s DE-604 Anfangswertproblem (DE-588)4001991-3 s Elliptischer Differentialoperator (DE-588)4140057-4 s 1\p DE-604 Erscheint auch als Druckausgabe 978-1-316-62086-1 London Mathematical Society lecture note series 434 (DE-604)BV000000130 434 https://doi.org/10.1017/CBO9781316729755 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029045124&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Taira, Kazuaki 1946- Analytic semigroups and semilinear initial boundary value problems London Mathematical Society lecture note series Elliptischer Differentialoperator (DE-588)4140057-4 gnd Analytische Halbgruppe (DE-588)4376792-8 gnd Anfangswertproblem (DE-588)4001991-3 gnd Anfangsrandwertproblem (DE-588)4001990-1 gnd |
| subject_GND | (DE-588)4140057-4 (DE-588)4376792-8 (DE-588)4001991-3 (DE-588)4001990-1 |
| title | Analytic semigroups and semilinear initial boundary value problems |
| title_auth | Analytic semigroups and semilinear initial boundary value problems |
| title_exact_search | Analytic semigroups and semilinear initial boundary value problems |
| title_full | Analytic semigroups and semilinear initial boundary value problems Kazuaki Taira |
| title_fullStr | Analytic semigroups and semilinear initial boundary value problems Kazuaki Taira |
| title_full_unstemmed | Analytic semigroups and semilinear initial boundary value problems Kazuaki Taira |
| title_short | Analytic semigroups and semilinear initial boundary value problems |
| title_sort | analytic semigroups and semilinear initial boundary value problems |
| topic | Elliptischer Differentialoperator (DE-588)4140057-4 gnd Analytische Halbgruppe (DE-588)4376792-8 gnd Anfangswertproblem (DE-588)4001991-3 gnd Anfangsrandwertproblem (DE-588)4001990-1 gnd |
| topic_facet | Elliptischer Differentialoperator Analytische Halbgruppe Anfangswertproblem Anfangsrandwertproblem |
| url | https://doi.org/10.1017/CBO9781316729755 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029045124&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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