Degenerate elliptic equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht u.a.
Kluwer Acad. Publ.
1993
|
| Series: | Mathematics and its applications
258 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 431 S. |
| ISBN: | 079232305X |
Staff View
MARC
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| 100 | 1 | |a Levendorskij, Sergej Z. |d 1951- |e Verfasser |0 (DE-588)13349876X |4 aut | |
| 245 | 1 | 0 | |a Degenerate elliptic equations |c by Serge Levendorskii |
| 264 | 1 | |a Dordrecht u.a. |b Kluwer Acad. Publ. |c 1993 | |
| 300 | |a XI, 431 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 258 | |
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| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1804122549415575552 |
|---|---|
| adam_text | Contents
0 Introduction 1
1 General Calculus of Pseudodifferential Operators 9
1.1 Weyl Hormander Calculus 9
1.1.1 Main Notation 9
1.1.2 Metrics, Weight Functions, and Symbol Classes . 12
1.1.3 Classes of Operators. Main Theorem of Calculus . 19
1.1.4 Weighted Sobolev Spaces 24
1.1.5 Global Calculus 33
1.1.6 Formally Hypoelliptic Operators as Unbounded
Operators 37
1.1.7 The Garding Inequality 40
1.1.8 An Example: Operators with Polynomial Symbols 43
1.2 The Calculus of Pseudodifferential Operators with
Double Symbols 45
1.2.1 Metrics, Weight Functions and Symbols on fi x t 45
1.2.2 Double Symbols and Operators with Double
Symbols 50
1.2.3 Main Theorems of Calculus 56
1.2.4 Weighted Sobolev Spaces 59
1.2.5 Global Calculus 63
1.2.6 Formally Hypoelliptic Operators as Unbounded
Operators 68
1.2.7 The Garding Inequality 69
1.2.8 Operators and Weighted Sobolev Spaces Depend¬
ing on Parameter 70
v
vi
2 Model Classes of Degenerate Elliptic Differential
Operators 75
2.1 Classes of Operators and Weighted Sobolev Spaces ... 75
2.1.1 Main Definitions 75
2.1.2 Model Classes 78
2.1.3 On Methods of Investigation, Symbols and
Boundary Value Problems for Operators of Dif¬
ferent Types 79
2.2 Operators of Type 1 (Strong Degeneration Case) 81
2.2.1 Symbols 81
2.2.2 Main Theorems 81
2.2.3 Operators of Type 1 as Unbounded Operators . . 85
2.3 Operators of Type 2 (Ellipticity along Boundary and
Strong Degeneration in Normal Direction) 87
2.3.1 Symbols 87
2.3.2 Main Theorem 89
2.4 Operators of Type 3 (Ellipticity along Boundary and
Euler Operators in Normal Direction) 95
2.4.1 Symbols 95
2.4.2 Main Theorem 95
2.5 Operators of Type 4 (Equations which Require Bound¬
ary and Coboundary Conditions) 98
2.5.1 Investigation of the Auxiliary family of Operators
on R+ 98
2.5.2 Boundary Value Problems 103
2.5.2.1 Anisotropic Sobolev spaces and
Weighted Sobolev Spaces of Distributions
on Rn and R^ 103
2.5.2.2 Operators onR!f 105
2.5.2.3 Operators on a Domain 108
2.5.3 Conditions for an Operator to be Fredholm . . . .112
2.5.4 Dependence of Smoothness of Data 123
vii
3 General Classes of Degenerate Elliptic Differential
Operators 129
3.1 Definition of Types of Operators and their Symbols . . .129
3.1.1 Characterization of Types and Methods of Inves¬
tigation 129
3.1.2 Definition of Symbols 131
3.2 Operators of Type 1 135
3.2.1 Main Theorem 135
3.2.2 Estimates for Symbols onR^xl 137
3.2.3 End of the Proof of the Main Theorem 143
3.3 Operators of Type 4 145
3.3.1 Investigations of the Auxiliary Family on 1R+ . . . 145
3.3.2 Main Theorem 146
3.3.3 Remarks on Smoothness Conditions 156
3.4 Operators of Types 2, 3 158
3.4.1 Operators of Type 2 158
3.4.2 Operators of Type 3 162
4 Degenerate Elliptic Operators in Non Power Like
Degeneration Case 163
4.1 Operators of Type 1 3 163
4.1.1 Operators of Type 1 163
4.1.2 Operators of Type 2 164
4.1.3 Operators of Type 3 164
4.2 Operators of Type 4 165
4.2.1 Definitions 165
4.2.2 Main Theorem 166
5 Lp Theory for Degenerate Elliptic Operators 171
5.1 Lp Theory for PseudodifFerential Operators with
Double Symbols 171
5.1.1 Lp Boundedness 171
viii
5.1.2 Weighted Sobolev Spaces 174
5.1.3 Global Calculus 176
5.2 Operators of Type 1 180
5.3 Operators of Type 4 182
6 Coersiveness of Degenerate Quadratic Forms 187
6.1 Types of Degenerate Quadratic Forms and their Symbols 187
6.2 Forms of Type 1 190
6.3 Forms of Type 4 193
6.3.1 Main Theorem 193
6.3.2 Proof of Theorem 6.3.1.1 193
6.4 Forms of Types 2, 3 199
6.5 Forms of Type 3 202
7 Some Classes of Hypoelliptic Pseudodifferential Opera¬
tors on Closed Manifold 203
7.1 Operators of Slowly Varying Order 203
7.1.1 Formally Hypoelliptic Operators onR 203
7.1.2 Operators on Closed Manifold 207
7.1.3 Action in Sobolev Spaces 210
7.2 Hypoelliptic Operators with Multiple Characteristics . . 213
7.2.1 Strong Degeneration Case 213
7.2.2 Hypoelliptic Operators which do not belong to
HL% m , 0 8 p 1 216
7.3 Weighted Sobolev Spaces and Hypoelliptic Operators
with Multiple Characteristics as Fredholm Operators . . 225
7.3.1 Spaces of Distributions on Rn 225
7.3.2 Spaces and Operators on Closed Manifold .... 226
7.3.3 Simplest Spectral Properties of Hypoelliptic Op¬
erators with Multiple Characteristics 231
7.3.4 Analogues of Melin Inequality 232
7.4 Interior Boundary Value Problem 239
ix
7.4.1 Investigation of the Operator Valued Symbol . . 239
7.4.2 Interior Boundary Value Problem 240
8 Algebra of Boundary Value Problems for Class of Pseu
dodifferential Operators which Change Order on the
Boundary 245
8.1 Symbols on R; x I 245
8.1.1 Classical Symbols with Transmission Property . . 245
8.1.2 Symbols of Varying Order with Transmission
Property 247
8.1.3 Potential Symbols, Trace Symbols and Green
Symbols 249
8.2 Classes of operators on Half Space 251
8.2.1 Classical Operators 251
8.2.2 Operators of Varying Order 255
8.3 Weighted Sobolev Spaces 262
8.3.1 Definitions and Examples 262
8.3.2 Boundedness Theorems 265
8.4 Operators on Closed Manifolds withBoundaries 270
8.4.1 Classes of Operators 270
8.4.2 Action in Weighted Sobolev Spaces 272
8.5 An Index Theorem 275
9 General Schemes of Investigation of Spectral Asymp
totics for Degenerate Elliptic Equations 279
9.1 General Theorems on Spectral Asymptotics 279
9.1.1 Basic Variational Theorem 279
9.1.2 General Theorems Concerning the Approximate
Spectral Projection Method 285
9.2 General Schemes of Investigation of Spectral Asymp¬
totics and Generalizations of the Weyl Formula for De¬
generate Elliptic Operators 291
X
9.2.1 A General Scheme of Computation of Spectral
Asymptotics 291
9.2.2 Generalizations of the Classical Weyl Formula . . 296
10 Spectral Asymptotics of Degenerate Elliptic Operators301
10.1 Formal Computations of Spectral Asymptotics for Op¬
erators of All Types 301
10.1.1 Auxiliary Propositions 301
10.1.2 Formal computation of Spectral Asymptotics in
the Weak and Strong Degeneration Cases 304
10.1.3 Formal Computations of Spectral Asymptotics in
the Intermediate Degenerate Case 311
10.2 Proof of the Asymptotic Formulae 317
10.2.1 Operators of Type 1 317
10.2.2 Operators of Type 2 320
10.2.3 Operators of Type 3 329
10.2.4 Operators of Type 4 330
11 Spectral Asymptotics of Hypoelliptic Operators with
Multiple Characteristics 335
11.1 Formal Computations of SpectralAsymptotics 335
11.1.1 General Remarks on Weyl like Formulae for
Hypoelliptic Pseudodifferential Operators 335
11.1.2 Formal Computations of Spectral Asymptotics in
the Case of Weak and Strong Degeneration .... 338
11.1.3 Formal Computation of Spectral Asymptotics in
the Intermediate Degeneration Case 344
11.2 Proofs of the Asymptotic Formulae 355
11.2.1 Main Theorem and Reduction to Problems
in Domains 355
11.2.2 Proof in the Case 70 2 359
11.2.3 Proof in the Case 70 = 2 and £ Involutory .... 363
xi
11.2.4 Proof of the Lower Bound in the Case 70 = 2,
£ in General Position, Degeneration either Weak
or Intermediate 363
11.2.5 Proof of the Upper Bound in the Case 70 = 2,
S in General Position, Degeneration either Weak
or Intermediate 366
11.2.6 Proof in the Strong Degeneration Case 377
A Brief Review of the Bibligraphy 389
Bibliography 399
Index of Notation 423
|
| any_adam_object | 1 |
| author | Levendorskij, Sergej Z. 1951- |
| author_GND | (DE-588)13349876X |
| author_facet | Levendorskij, Sergej Z. 1951- |
| author_role | aut |
| author_sort | Levendorskij, Sergej Z. 1951- |
| author_variant | s z l sz szl |
| building | Verbundindex |
| bvnumber | BV008166366 |
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| id | DE-604.BV008166366 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:15:38Z |
| institution | BVB |
| isbn | 079232305X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005388444 |
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| physical | XI, 431 S. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Kluwer Acad. Publ. |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Levendorskij, Sergej Z. 1951- Verfasser (DE-588)13349876X aut Degenerate elliptic equations by Serge Levendorskii Dordrecht u.a. Kluwer Acad. Publ. 1993 XI, 431 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 258 Elliptisch entartete Differentialgleichung (DE-588)4152026-9 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptisch entartete Differentialgleichung (DE-588)4152026-9 s DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s Elliptische Differentialgleichung (DE-588)4014485-9 s Mathematics and its applications 258 (DE-604)BV008163334 258 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005388444&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Levendorskij, Sergej Z. 1951- Degenerate elliptic equations Mathematics and its applications Elliptisch entartete Differentialgleichung (DE-588)4152026-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4152026-9 (DE-588)4044779-0 (DE-588)4014485-9 |
| title | Degenerate elliptic equations |
| title_auth | Degenerate elliptic equations |
| title_exact_search | Degenerate elliptic equations |
| title_full | Degenerate elliptic equations by Serge Levendorskii |
| title_fullStr | Degenerate elliptic equations by Serge Levendorskii |
| title_full_unstemmed | Degenerate elliptic equations by Serge Levendorskii |
| title_short | Degenerate elliptic equations |
| title_sort | degenerate elliptic equations |
| topic | Elliptisch entartete Differentialgleichung (DE-588)4152026-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Elliptisch entartete Differentialgleichung Partielle Differentialgleichung Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005388444&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT levendorskijsergejz degenerateellipticequations |