The Cauchy problem for hyperbolic operators: multiple characteristics ; micro-local approach
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin
Akad.-Verl.
1997
|
| Ausgabe: | 1. ed. |
| Schriftenreihe: | Mathematical topics
12 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 379 - 393 |
| Beschreibung: | 397 S. |
| ISBN: | 3055017390 |
Internformat
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| 100 | 1 | |a Yagdjian, Karen |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The Cauchy problem for hyperbolic operators |b multiple characteristics ; micro-local approach |c Karen Yagdjian |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Berlin |b Akad.-Verl. |c 1997 | |
| 300 | |a 397 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Mathematical topics |v 12 | |
| 500 | |a Literaturverz. S. 379 - 393 | ||
| 650 | 4 | |a Caractéristique multiple | |
| 650 | 7 | |a Cauchy, Problème de |2 ram | |
| 650 | 4 | |a Equation différentielle avec point-centre | |
| 650 | 7 | |a Fourier, Opérateurs intégraux de |2 ram | |
| 650 | 4 | |a Inégalité asymptotique Gärding | |
| 650 | 7 | |a Liapounov, Fonctions de |2 ram | |
| 650 | 4 | |a Microlocalisation | |
| 650 | 4 | |a Opérateur hyperbolique | |
| 650 | 7 | |a Opérateurs différentiels partiels |2 ram | |
| 650 | 4 | |a Théorème stabilité Liapounov | |
| 650 | 7 | |a Équations différentielles hyperboliques |2 ram | |
| 650 | 4 | |a Cauchy problem | |
| 650 | 4 | |a Differential equations, Hyperbolic | |
| 650 | 4 | |a Partial differential operators | |
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| 650 | 0 | 7 | |a Cauchy-Anfangswertproblem |0 (DE-588)4147404-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Mikrolokale Analysis |0 (DE-588)4169832-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hyperbolischer Differentialoperator |0 (DE-588)4140064-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Introduction n
Notations 15
1 Fourier integral operators 16
1.1 Definition of Fourier integral operators 16
1.1.1 Oscillatory integrals 16
1.1.2 Singular support of oscillatory integrals 20
1.1.3 The Kernel Theorem 22
1.1.4 Definition of Fourier integral operators 27
1.2 The method of stationary phase 29
1.2.1 The Morse lemma 30
1.2.2 Fourier transforms of Gaussian functions 32
1.3 Background on differential and symplectic geometry 35
1.3.1 Smooth manifolds of dimension n 35
1.3.2 The tangent space 37
1.3.3 Canonical 1 and 2 forms 42
1.4 The strictly hyperbolic Cauchy problem. Construction of a parametrix 45
1.4.1 Outline of construction 45
1.4.2 Hamilton s canonical field 53
1.4.3 Construction of phase function 57
1.4.4 Solution of the transport equations 60
1.5 Fourier integral operators with inhomogeneous phase functions 63
1.5.1 Definition of Fourier integral operators with inhomogeneous phase
functions 63
1.5.2 Product of the Fourier integral operators Pi^/^* 67
1.5.3 Operators with double symbols 70
1.5.4 Product of the Fourier integral operators Pi 0.P2,* 71
1.5.5 Product of Fourier integral operator and pseudo differential op¬
erator: Pi^Pi and QiQij 72
1.5.6 Product of Fourier integral operator and pseudo differential op¬
erator: P Pi and 5i,0 (?2 74
1.5.7 Continuity in Sobolev spaces 75
8 Contents
1.5.8 Fundamental solution of the Cauchy problem for first order hy¬
perbolic operators 76
1.5.9 Egorov s theorem 81
1.5.10 Propagation of singularities 86
2 Ordinary differential equations with turning point 89
2.1 Representation theorem for the solutions of equations with turning point 89
2.1.1 Introduction 89
2.1.2 Model second order equations with real characteristics. The in¬
voking of confluent hypergeometric functions 91
2.1.3 Expansion of implicit functions in power series 108
2.1.4 Expansion of implicit functions in power series (multidimensional
case)* 112
2.1.5 Zones. On the zeros of the complete symbol 115
2.1.6 Classes of symbols 123
2.1.7 Reduction to a first order diagonal system 127
2.1.8 The representation formula 134
2.2 The Gevrey asymptotic representation of the solutions to the equations
with turning point 142
2.2.1 Introduction 142
2.2.2 On the zeros of the complete symbol 145
2.2.3 Classes of symbols 149
2.2.4 Reduction to a first order diagonal system 153
2.2.5 The representation formula 157
2.3 WKB solutions of the equations with infinite order turning point .... 161
2.3.1 Introduction 162
2.3.2 On the zeros of the complete symbol 166
2.3.3 Classes of symbols 169
2.3.4 Reduction to a first order diagonal system in the exterior zone 171
2.3.5 Construction of exact solutions in the exterior zone 173
2.3.6 Construction in the inner zone 176
2.4 Applications 179
2.4.1 The Cauchy problem. Propagation of singularities 179
2.4.2 The Cauchy problem in Gevrey classes 187
2.4.3 Tunneling phenomena 193
2.4.4 Problems of hypoellipticity and local solvability 197
3 Fundamental solution of the Cauchy problem for operators with mul¬
tiple characteristics. Degeneration with respect to time 214
3.1 Introduction 214
3.2 Hyperbolicity 217
3.2.1 Hadamard s and Garding s definitions of hyperbolic operators
with constant coefficients 217
3.2.2 Petrovskij s definition of hyperbolic operators 219
Contents 9
3.2.3 Zones. On the zeros of the complete symbol. Hyperbolicity con¬
dition 219
3.3 Classes of symbols 225
3.4 Reduction to a Cauchy problem for a first ordersystem 233
3.5 Diagonalization in the hyperbolic zone 237
3.6 Hamiltonian vector fields 240
3.7 The construction of phase functions 245
3.8 The fundamental solution of the Cauchy problem for elementary operators247
3.8.1 Construction of the fundamental solution 247
3.8.2 Egorov s theorem 253
3.8.3 Ideals of the Fourier integral operators with phase functions gen¬
erated by X(t, x,£) 256
3.9 Exponential functions of certain pseudo differential operators 258
3.10 Completion of the construction of the fundamental solution 264
3.11 Well posedness of the Cauchy problem 270
3.11.1 On equations with fast oscillating coefficients 273
3.11.2 The energy method for weakly hyperbolic equations 276
4 Fundamental solution of the Cauchy problem for hyperbolic operators
with multiple characteristics. Degeneration with respect to the spatial
variables 283
4.1 Introduction 283
4.2 Hyperbolicity 287
4.3 Classes of symbols 291
4.4 Reduction to a first order system 299
4.5 Diagonalization in the hyperbolic zone 301
4.6 Hamiltonian vector fields. Construction of the phase function 305
4.7 Fundamental solution of the Cauchy problem for an elementary operator 308
4.8 Ideals of Fourier integral operators with phase functions generated by
X(t,x,0 311
4.9 Completion of the construction of the fundamental solution 315
4.10 Well posedness of the Cauchy problem 317
4.11 Other constructions a brief survey 323
5 Necessary correctness conditions for the Cauchy problem for operators
with multiple characteristics 326
5.1 Introduction 326
5.2 Hadamard s example in the light of the Lyapunov stability theory of
ordinary differential equations 333
5.3 A priori estimates which follow from the well posedness of the Cauchy
problem 336
5.4 Proof of Theorem 5.1.3 338
5.4.1 Partial hyperbolicity 338
5.4.2 Construction of an unstable solution: the case s — 0 343
10 Contents
5.4.3 Garding s asymptotic inequality 347
5.4.4 The Lyapunov function of third kind. The lower bound for the
energy of the unstable solution 349
5.4.5 Completion of the proof ( s = T ) 353
5.5 Proof of Theorem 5.1.4 355
5.6 Proof of Theorem 5.1.5 362
5.6.1 Reduction to an almost diagonal system 362
5.6.2 The construction of the asymptotic solution 366
5.6.3 Garding s asymptotic inequality 369
5.6.4 Lower bound for the energy of the asymptotic solution. Comple¬
tion of the proof 371
5.7 Necessary conditions obtained by other approaches 374
5.8 Examples 375
Bibliography 379
Author index 394
Subject index 396
Notation index 398
|
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| author | Yagdjian, Karen |
| author_facet | Yagdjian, Karen |
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| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV011329756 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:07:56Z |
| institution | BVB |
| isbn | 3055017390 |
| language | German |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007612841 |
| oclc_num | 37865712 |
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| owner_facet | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-706 DE-634 DE-11 DE-188 |
| physical | 397 S. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Akad.-Verl. |
| record_format | marc |
| series | Mathematical topics |
| series2 | Mathematical topics |
| spelling | Yagdjian, Karen Verfasser aut The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach Karen Yagdjian 1. ed. Berlin Akad.-Verl. 1997 397 S. txt rdacontent n rdamedia nc rdacarrier Mathematical topics 12 Literaturverz. S. 379 - 393 Caractéristique multiple Cauchy, Problème de ram Equation différentielle avec point-centre Fourier, Opérateurs intégraux de ram Inégalité asymptotique Gärding Liapounov, Fonctions de ram Microlocalisation Opérateur hyperbolique Opérateurs différentiels partiels ram Théorème stabilité Liapounov Équations différentielles hyperboliques ram Cauchy problem Differential equations, Hyperbolic Partial differential operators Operatortheorie (DE-588)4075665-8 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Fourier-Integraloperator (DE-588)4155104-7 gnd rswk-swf Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd rswk-swf Mikrolokale Analysis (DE-588)4169832-0 gnd rswk-swf Hyperbolischer Differentialoperator (DE-588)4140064-1 gnd rswk-swf Hyperbolischer Differentialoperator (DE-588)4140064-1 s Cauchy-Anfangswertproblem (DE-588)4147404-1 s Mikrolokale Analysis (DE-588)4169832-0 s Fourier-Integraloperator (DE-588)4155104-7 s DE-604 Hyperbolische Differentialgleichung (DE-588)4131213-2 s Operatortheorie (DE-588)4075665-8 s Mathematical topics 12 (DE-604)BV008671507 12 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007612841&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Yagdjian, Karen The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach Mathematical topics Caractéristique multiple Cauchy, Problème de ram Equation différentielle avec point-centre Fourier, Opérateurs intégraux de ram Inégalité asymptotique Gärding Liapounov, Fonctions de ram Microlocalisation Opérateur hyperbolique Opérateurs différentiels partiels ram Théorème stabilité Liapounov Équations différentielles hyperboliques ram Cauchy problem Differential equations, Hyperbolic Partial differential operators Operatortheorie (DE-588)4075665-8 gnd Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Fourier-Integraloperator (DE-588)4155104-7 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd Mikrolokale Analysis (DE-588)4169832-0 gnd Hyperbolischer Differentialoperator (DE-588)4140064-1 gnd |
| subject_GND | (DE-588)4075665-8 (DE-588)4131213-2 (DE-588)4155104-7 (DE-588)4147404-1 (DE-588)4169832-0 (DE-588)4140064-1 |
| title | The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach |
| title_auth | The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach |
| title_exact_search | The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach |
| title_full | The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach Karen Yagdjian |
| title_fullStr | The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach Karen Yagdjian |
| title_full_unstemmed | The Cauchy problem for hyperbolic operators multiple characteristics ; micro-local approach Karen Yagdjian |
| title_short | The Cauchy problem for hyperbolic operators |
| title_sort | the cauchy problem for hyperbolic operators multiple characteristics micro local approach |
| title_sub | multiple characteristics ; micro-local approach |
| topic | Caractéristique multiple Cauchy, Problème de ram Equation différentielle avec point-centre Fourier, Opérateurs intégraux de ram Inégalité asymptotique Gärding Liapounov, Fonctions de ram Microlocalisation Opérateur hyperbolique Opérateurs différentiels partiels ram Théorème stabilité Liapounov Équations différentielles hyperboliques ram Cauchy problem Differential equations, Hyperbolic Partial differential operators Operatortheorie (DE-588)4075665-8 gnd Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Fourier-Integraloperator (DE-588)4155104-7 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd Mikrolokale Analysis (DE-588)4169832-0 gnd Hyperbolischer Differentialoperator (DE-588)4140064-1 gnd |
| topic_facet | Caractéristique multiple Cauchy, Problème de Equation différentielle avec point-centre Fourier, Opérateurs intégraux de Inégalité asymptotique Gärding Liapounov, Fonctions de Microlocalisation Opérateur hyperbolique Opérateurs différentiels partiels Théorème stabilité Liapounov Équations différentielles hyperboliques Cauchy problem Differential equations, Hyperbolic Partial differential operators Operatortheorie Hyperbolische Differentialgleichung Fourier-Integraloperator Cauchy-Anfangswertproblem Mikrolokale Analysis Hyperbolischer Differentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007612841&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008671507 |
| work_keys_str_mv | AT yagdjiankaren thecauchyproblemforhyperbolicoperatorsmultiplecharacteristicsmicrolocalapproach |