Pseudo-differential operators, singularities, applications:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
1997
|
| Series: | Operator theory
93 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 341 - 346 |
| Physical Description: | XIII, 349 S. |
| ISBN: | 3764354844 0817654844 |
Staff View
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| 245 | 1 | 0 | |a Pseudo-differential operators, singularities, applications |c Yuri V. Egorov ; Bert-Wolfgang Schulze |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 1997 | |
| 300 | |a XIII, 349 S. | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface XI
1 Sobolev spaces
1.1 Fourier transform 1
1.1.1 Definition 1
1.1.2 The Fourier transform in the Schwartz spaces 2
1.2 The first definition of the Sobolev space 4
1.2.1 The classical definition 4
1.2.2 The completeness of the classical Sobolev space 4
1.3 General definition of Sobolev spaces in R 5
1.3.1 General definition 5
1.3.2 Some properties of Sobolev spaces in Mn 6
1.4 Representation of a linear functional over H 7
1.5 Embedding theorems 9
1.5.1 Sobolev s theorem 9
1.5.2 Distributions with compact supports 10
1.5.3 Traces on the boundary 11
1.6 Sobolev spaces in a domain 12
1.6.1 Definition 12
1.6.2 The invariance under diffeomorphisms 14
1.6.3 The compactness of embeddings 15
2 Pseudo differential Operators
2.1 The algebra of differential operators 17
2.1.1 Differential operators in Rn 17
2.1.2 Differential operators on a manifold 19
2.1.3 The cotangent space and the characteristic form 21
2.1.4 Fundamental solutions of differential operators with
constant coefficients 22
2.1.5 Examples of fundamental solutions 24
2.1.6 Hypoelliptic operators 27
V
VI CONTENTS
2.2 Basic properties of pseudo differential operators 27
2.2.1 Definition and basic properties 27
2.2.2 Pseudo differential operators as integral operators 29
2.2.3 Continuity in the Sobolev spaces 30
2.3 Calculus of pseudo differential operators 32
2.3.1 A technical Lemma 32
2.3.2 The composition of pseudo differential operators 36
2.3.3 A more general definition 37
2.3.4 Formally adjoint operators 38
2.4 Pseudo differential operators on closed manifolds 38
2.4.1 Transformation of operators under a change of variables . . 38
2.4.2 Pseudo differential operators on a manifold 39
2.5 Garding inequality 40
2.5.1 Garding inequality for elliptic differential operators 40
2.5.2 Sharp Garding inequality for pseudo differential
operators 43
2.5.3 Some generalizations 48
3 Elliptic pseudo differential operators
3.1 Parametrices of the elliptic operators 51
3.1.1 Definitions and a technical lemma 51
3.1.2 The construction of a parametrix 53
3.2 Elliptic operators on a manifold 55
3.2.1 Definitions 55
3.2.2 The parametrix construction 55
3.2.3 A priori estimates and regularity of solutions 56
3.2.4 The Fredholm property 58
3.2.5 Vanishing of the index 61
4 Elliptic boundary value problems
4.1 Model elliptic boundary value problems 63
4.1.1 Statement of the problem and the condition of ellipticity . . 63
4.1.2 Construction of a parametrix 64
4.2 Elliptic boundary value problems in a domain 67
4.2.1 Ellipticity condition 67
4.2.2 Examples 68
4.2.3 Construction of a parametrix 68
4.2.4 Continuity of the parametrices 71
4.2.5 Fredholm property 72
4.2.6 Necessity of the ellipticity condition 73
CONTENTS VII
5 Kondratiev s theory
5.1 A model problem 75
5.2 The general problem 77
5.2.1 The conditions on a domain and differential operators ... 77
5.2.2 Functional spaces 77
5.2.3 The statement of the general boundary value problem
and main results 78
5.3 The boundary value problem in an infinite cone for operators
with constant coefficients 79
5.3.1 The statement of the problem and its transformations ... 79
5.3.2 The resolution of the model boundary value problem .... 81
5.3.3 The asymptotics of the solution 82
5.3.4 Lemmas 83
5.3.5 The proof of Theorem 3 84
5.4 Equations with variable coefficients in an infinite cone 87
5.4.1 Conditions on the coefficients 87
5.4.2 Lemmas 87
5.4.3 The existence of the solution 89
5.4.4 The smoothness of the solution 89
5.5 The boundary value problem in a bounded domain 90
5.5.1 Lemmas 90
5.5.2 The construction of the parametrix 93
5.5.3 Proof of Theorem 1 95
5.5.4 Smoothness of solutions 95
5.5.5 The solution of the boundary value problem
in usual Sobolev spaces 97
6 Non elliptic operators; propagation of singularities
6.1 Canonical transformations and Fourier integral operators 99
6.1.1 Definitions 99
6.1.2 Examples 100
6.1.3 Fourier integral operators and canonical transformations . . 102
6.1.4 Canonical transformations and quadratic forms 106
6.1.5 Reduction of operators of principal type to
canonical forms 108
6.2 Wave fronts of distributions Ill
6.2.1 Definitions Ill
6.2.2 Examples 113
6.2.3 Properties of wave fronts 114
6.2.4 Wave fronts under push forwards and pull backs 117
6.2.5 Wave fronts and traces of distributions on a manifold
of a lower dimension 119
6.2.6 Products of distributions 120
VIII CONTENTS
6.3 Wave fronts and Fourier integral operators 120
6.3.1 Wave fronts and integral operators 120
6.3.2 Wave fronts and pseudo differential operators 122
6.4 Propagation of singularities 125
6.4.1 Propagation of singularities for operators of
real principal type 125
6.4.2 Propagation of singularities in the Sobolev spaces 125
6.4.3 Solvability of real principal type 126
6.5 The Cauchy problem for a strongly hyperbolic equation 128
6.5.1 The Cauchy problem for the wave equation 128
6.5.2 The Cauchy problem for a hyperbolic equation 129
6.5.3 The construction of the phase function and the symbol . . . 131
6.5.4 The Cauchy problem for a hyperbolic system of
first order 132
7 Pseudo differential operators on manifolds with conical and
edge singularities; motivation and technical preparations
7.1 The general background 135
7.1.1 The program of the analysis of manifolds
with singularities 135
7.1.2 Typical differential operators on manifolds
with conical singularities 142
7.1.3 The typical differential operators on manifolds
with edges and corners 159
7.2 Parameter dependent pseudo differential operators and
operator valued Mellin symbols 169
7.2.1 Additional material on pseudo differential operators
on closed compact C°° manifolds 169
7.2.2 The parameter dependent calculus; reductions of orders . . 178
7.2.3 Mellin pseudo differential operators with
operator valued symbols 186
7.2.4 Kernel cut off and operator valued holomorphic
Mellin symbols 194
7.2.5 Meromorphic Fredholm families 201
8 Pseudo differential operators on manifolds with conical singularities
8.1 The cone algebra with asymptotics 209
8.1.1 Weighted Sobolev spaces with asymptotics
and Green operators 209
8.1.2 Smoothing Mellin operators 215
8.1.3 A Mellin operator convention 224
8.1.4 The cone algebra 227
8.1.5 Ellipticity and regularity with asymptotics 235
CONTENTS IX
8.2 The algebra on the infinite cone 238
8.2.1 Symbols in Mn with exit behaviour 238
8.2.2 Classical symbols 241
8.2.3 Pseudo differential operators in Rn with exit behaviour . . 246
8.2.4 The calculus on the infinite cylinder 254
8.2.5 The cone algebra onIA 258
9 Pseudo differential operators on manifolds with edges
9.1 Pseudo differential operators with operator valued symbols .... 263
9.1.1 The operator valued symbol spaces 263
9.1.2 Pseudo differential operators 269
9.1.3 Abstract wedge Sobolev spaces 272
9.2 The edge symbolic calculus 275
9.2.1 Green symbols 275
9.2.2 Smoothing Mellin symbols 280
9.2.3 Complete edge symbols 282
9.3 Edge pseudo differential operators 297
9.3.1 Edge Sobolev spaces with discrete asymptotics 297
9.3.2 The algebra of edge pseudo differential operators 303
9.3.3 Ellipicity and regularity with discrete edge asymptotics . . 309
9.3.4 Global constructions and Fredholm property 319
9.4 Applications, examples and remarks 330
9.4.1 Boundary value problems as particular edge problems . . . 330
9.4.2 The nature of asymptotics in singular configurations .... 332
9.4.3 Remarks on the role of the edge trace and
potential conditions 336
Bibliography 340
Index 347
|
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| author | Egorov, Jurij V. 1938- Schulze, Bert-Wolfgang 1944- |
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| author_sort | Egorov, Jurij V. 1938- |
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| building | Verbundindex |
| bvnumber | BV011196351 |
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| ctrlnum | (OCoLC)246606367 (DE-599)BVBBV011196351 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011196351 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:05:37Z |
| institution | BVB |
| isbn | 3764354844 0817654844 |
| language | English |
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| physical | XIII, 349 S. |
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| publisher | Birkhäuser |
| record_format | marc |
| series | Operator theory |
| series2 | Operator theory |
| spelling | Egorov, Jurij V. 1938- Verfasser (DE-588)121177181 aut Pseudo-differential operators, singularities, applications Yuri V. Egorov ; Bert-Wolfgang Schulze Basel [u.a.] Birkhäuser 1997 XIII, 349 S. txt rdacontent n rdamedia nc rdacarrier Operator theory 93 Literaturverz. S. 341 - 346 Pseudodifferentialoperator - Singularität <Mathematik> Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Pseudodifferentialoperator (DE-588)4047640-6 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 s Singularität Mathematik (DE-588)4077459-4 s Pseudodifferentialoperator (DE-588)4047640-6 s DE-604 Schulze, Bert-Wolfgang 1944- Verfasser (DE-588)120484579 aut Operator theory 93 (DE-604)BV000000970 93 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007510013&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Egorov, Jurij V. 1938- Schulze, Bert-Wolfgang 1944- Pseudo-differential operators, singularities, applications Operator theory Pseudodifferentialoperator - Singularität <Mathematik> Mannigfaltigkeit (DE-588)4037379-4 gnd Pseudodifferentialoperator (DE-588)4047640-6 gnd Singularität Mathematik (DE-588)4077459-4 gnd |
| subject_GND | (DE-588)4037379-4 (DE-588)4047640-6 (DE-588)4077459-4 |
| title | Pseudo-differential operators, singularities, applications |
| title_auth | Pseudo-differential operators, singularities, applications |
| title_exact_search | Pseudo-differential operators, singularities, applications |
| title_full | Pseudo-differential operators, singularities, applications Yuri V. Egorov ; Bert-Wolfgang Schulze |
| title_fullStr | Pseudo-differential operators, singularities, applications Yuri V. Egorov ; Bert-Wolfgang Schulze |
| title_full_unstemmed | Pseudo-differential operators, singularities, applications Yuri V. Egorov ; Bert-Wolfgang Schulze |
| title_short | Pseudo-differential operators, singularities, applications |
| title_sort | pseudo differential operators singularities applications |
| topic | Pseudodifferentialoperator - Singularität <Mathematik> Mannigfaltigkeit (DE-588)4037379-4 gnd Pseudodifferentialoperator (DE-588)4047640-6 gnd Singularität Mathematik (DE-588)4077459-4 gnd |
| topic_facet | Pseudodifferentialoperator - Singularität <Mathematik> Mannigfaltigkeit Pseudodifferentialoperator Singularität Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007510013&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000970 |
| work_keys_str_mv | AT egorovjurijv pseudodifferentialoperatorssingularitiesapplications AT schulzebertwolfgang pseudodifferentialoperatorssingularitiesapplications |