The Cauchy problem for solutions of elliptic equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin
Akad.-Verl.
1995
|
| Edition: | 1. ed. |
| Series: | Mathematical topics
7 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 457 - 472 |
| Physical Description: | 478 S. |
| ISBN: | 3055016637 |
Staff View
MARC
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| 100 | 1 | |a Tarchanov, Nikolaj Nikolaevič |d 1955-2020 |e Verfasser |0 (DE-588)121160521 |4 aut | |
| 245 | 1 | 0 | |a The Cauchy problem for solutions of elliptic equations |c Nikolai N. Tarkhanov |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Berlin |b Akad.-Verl. |c 1995 | |
| 300 | |a 478 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical topics |v 7 | |
| 500 | |a Literaturverz. S. 457 - 472 | ||
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Record in the Search Index
| _version_ | 1804124499362185216 |
|---|---|
| adam_text | Contents
Introduction 17
List of main notations 27
1 Function spaces 29
1.1 An abstract theory 29
1.1.1 Semilocal spaces 29
1.1.2 Functions of positive smoothness 31
1.1.3 Function spaces on closed sets 32
1.1.4 Dual spaces 34
1.1.5 Functions of negative smoothness 36
1.2 Spaces of smooth functions 38
1.2.1 Spaces of continuous functions 38
1.2.2 Spaces of functions of finite smoothness 39
1.2.3 The space of infinitely differentiable functions 39
1.2.4 Standard regularization 40
1.2.5 Approximation by C°° functions 41
1.2.6 Spectral synthesis in spaces of smooth functions 43
1.2.7 Smooth functions on closed sets 44
1.2.8 Dual spaces 45
1.2.9 Functions of negative smoothness 46
1.3 Holder spaces 47
1.3.1 Spaces of Holder functions 47
1.3.2 Holder functions of finite smoothness 48
1.3.3 Holder continuous functions 50
1.3.4 Standard regularization of Holder functions 51
1.3.5 Approximation by C°° functions 53
1.3.6 Spectral synthesis in spaces of Holder functions 54
1.3.7 Holder functions on closed sets 56
1.3.8 Dual spaces 58
1.3.9 The negative Holder spaces 58
1.4 Sobolev spaces 60
1.4.1 Lebesgue spaces 60
1.4.2 The spaces W$(X) 60
8 Contents
1.4.3 Standard regularization of Sobolev functions 61
1.4.4 Approximation by C°° functions 62
1.4.5 Geometrical properties of domains 63
1.4.6 Embedding theorems 65
1.4.7 Spectral synthesis in Sobolev spaces 66
1.4.8 Sobolev functions on closed sets 68
1.4.9 The negative Sobolev spaces 69
1.4.10 Duality 70
1.4.11 Fractional order Sobolev spaces 72
1.4.12 Besov spaces 74
1.4.13 Traces of Sobolev functions 75
2 Pseudodifferential operators in the spaces of distributions on closed
sets 78
2.1 Calderon Zygmund operators 78
2.1.1 The Kernel Theorem of L. Schwartz 78
2.1.2 Calderon Zygmund kernels 80
2.1.3 Singular integrals 81
2.1.4 Extension to I«(Kn) 82
2.1.5 Maximal operator 83
2.1.6 Classical examples 84
2.1.7 Calderon Zygmund operators 85
2.1.8 The maximal function 86
2.1.9 Bounded mean oscillation 87
2.1.10 The Calderon Zygmund theory 88
2.2 Pseudodifferential operators 89
2.2.1 The Fourier integral representation of Calderon Zygmund opera¬
tors 89
2.2.2 The definition 90
2.2.3 Symbols 92
2.2.4 Schwartz kernels of pseudodifferential operators 93
2.2.5 C* algebra of pseudodifferential operators 95
2.2.6 Pseudohomogeneous kernels 97
2.2.7 Seeley s theorem 100
2.2.8 Operators on manifolds 103
2.2.9 Elliptic operators and parametrices 104
2.2.10 Symbols with limited smoothness 106
2.3 Boundedness theorems for pseudodifferential operators in local spaces . 107
2.3.1 Fundamental theorem of calculus 107
2.3.2 Behavior in local Holder spaces 109
2.3.3 Behavior in local Zygmund spaces 110
2.3.4 Behavior in local spaces of Holder continuous functions 110
2.3.5 Behavior in local Sobolev spaces Ill
Contents 9
2.3.6 Behavior in local Besov spaces 112
2.3.7 Behavior in local BMO spaces 112
2.3.8 Potential spaces 113
2.4 Boundedness theorems for pseudodifferential operators in non local spa¬
ces 115
2.4.1 Surface layer potentials 115
2.4.2 Surface values of layer potentials 116
2.4.3 Symbols with the transmission property 117
2.4.4 Operators with the transmission property 119
2.4.5 Pseudodifferential operators on manifolds with boundary .... 120
2.4.6 Potential operators 121
2.4.7 Continuity in Holder spaces 122
2.4.8 Continuity in Sobolev spaces 123
3 Capacity 124
3.1 Generalized form of capacity associated with a seminormed space . . . 124
3.1.1 More on the traces of distributions 124
3.1.2 Removable singularities 126
3.1.3 Solutions regular at infinity 128
3.1.4 The equivalence of two forms of capacity 129
3.1.5 Capacitary extremals 131
3.1.6 Approximation on nowhere dense compact sets 132
3.1.7 The unified capacity 133
3.2 Capacity in spaces of smooth functions 136
3.2.1 Fundamental solutions of homogeneous elliptic equations .... 136
3.2.2 Orthogonal decomposition in the space of polynomials 138
3.2.3 A Laurent expansion at infinity 139
3.2.4 Higher order capacities 143
3.2.5 Examples 145
3.2.6 Other expressions for the capacity 148
3.2.7 Behavior under affine transformations 149
3.2.8 The capacity of a point 151
3.2.9 More on outer capacity 152
3.2.10 Comparison with Hausdorff measure 153
3.3 Capacity in Holder spaces 153
3.3.1 A definition 153
3.3.2 Behavior under affine transformations 154
3.3.3 A nondegeneraey property 154
3.3.4 A further look at outer capacity 155
3.3.5 Hausdorff measure 155
3.3.6 Commensurability with Hausdorff content 156
3.3.7 Semiadditivity of the capacity 159
3.4 Capacity in Sobolev spaces 160
10 Contents
3.4.1 Bessel capacity 160
3.4.2 Metric properties of Bessel capacity 163
3.4.3 Quasicontinuous representatives of Sobolev functions 163
3.4.4 An application to spectral synthesis in Sobolev spaces 165
3.4.5 A brief review of higher order capacities 165
3.4.6 Comparison with Bessel capacity 166
3.4.7 Nguyen s theorem 168
4 Systems of differential equations with injective (surjective) symbols 172
4.1 Elliptic complexes 172
4.1.1 (Over ) underdetermined systems 172
4.1.2 Complexes of differential operators 173
4.1.3 Resolutions of overdetermined systems 175
4.1.4 Laplacians 176
4.1.5 Parametrices of elliptic complexes 179
4.2 A solvability criterion for a system with surjective symbol in terms of
convexity of supports 180
4.2.1 P convex sets 181
4.2.2 Statement of the theorem 181
4.2.3 Proof of the necessity 182
4.2.4 Proof of the sufficiency 183
4.2.5 Solvability in the space of distributions 184
4.3 Uniqueness condition for the Cauchy problem in the small 184
4.3.1 The sheaf of solutions 184
4.3.2 The uniqueness condition 185
4.3.3 Topological conditions for solvability 186
4.4 Left (right) fundamental solutions for a system with injective (surjective)
symbol 187
4.4.1 Fundamental solutions to differential complexes 187
4.4.2 An existence theorem 188
4.4.3 Some examples 189
5 Coarse results on approximation on compact sets by solutions of a
system with surjective symbol 190
5.1 Runge theorem for solutions of a system with surjective symbol .... 190
5.1.1 Problem of approximation 191
5.1.2 Some examples 192
5.1.3 A brief survey 193
5.1.4 The annihilator of sol{K) 194
5.1.5 P convex hull 194
5.1.6 Runge theorem 196
5.2 Approximation of finitely smooth solutions by infinitely differentiable
solutions 197
5.2.1 More on the hypoellipticity of elliptic complexes 197
Contents i j
5.2.2 An auxiliary result 198
5.2.3 Proof of the theorem 198
5.2.4 A generalization of the Stone Weierstrass Theorem 199
5.3 Approximation by potentials 201
5.3.1 A digression 201
5.3.2 Analogy with rational approximation 201
5.3.3 Examples 202
5.4 Localization property under approximation on compact a by solutions of
a system with surjective symbol 203
5.4.1 The validity range 203
5.4.2 Localization property 203
5.4.3 The necessity of condition (U)s 204
6 Approximation in spaces of smooth functions 206
6.1 Approximation of high order 206
6.1.1 Further look at the approximation problem 206
6.1.2 The main theorem 207
6.1.3 Notes 208
6.2 Approximation on the closure of a domain with the strong cone proper¬
ty 208
6.2.1 Approximation of lower order 208
6.2.2 The role of the connectedness of the complement 209
6.2.3 Walsh theorem 210
6.2.4 Bernstein theorems for elliptic equations 211
6.3 Approximation on nowhere dense compact sets 213
6.3.1 Hartogs Rosenthal theorem for systems with surjective symbol . 213
6.3.2 A generalization of the Lavrent ev Theorem 213
6.3.3 Further remarks on the Hartogs Rosenthal theorem 214
6.3.4 Systems elliptic in the sense of Douglis Nirenberg 215
6.3.5 The case of totally disconnected compact sets 217
6.3.6 More on the Weierstrass Theorem 219
6.3.7 The general case 223
6.3.8 Overdetermined systems of canonical type 226
6.3.9 Approximation by harmonic vector fields 228
6.4 Capacitary criteria of Vitushkin type for approximation in spaces of
smooth functions 229
6.4.1 A capacitary criterion 229
6.4.2 Discussion of the theorem 230
6.4.3 Approximation on compacta whose complements have the cone
property 231
12 Contents
7 Approximation in Holder spaces 233
7.1 Approximation of high order in Holder spaces 233
7.1.1 Description of the annihilated of the subspace of solutions .... 233
7.1.2 The range s p 234
7.2 Approximation of lower order in Holder spaces 235
7.2.1 A counterexample 235
7.2.2 A brief review 236
7.2.3 Reduction 236
7.3 Approximation criteria in terms of Hausdorff content 237
7.3.1 Approximation on compacta of measure zero 237
7.3.2 Approximation on nowhere dense compacta 238
7.3.3 Further results 238
7.4 Capacitary criteria of Vitushkin type for approximation in spaces of
Holder functions 239
7.4.1 A capacitary criterion 239
7.4.2 Discussion of the theorem 240
7.4.3 Approximation on compacta whose complements have the cone
property 241
8 Approximation in Sobolev spaces 242
8.1 Approximation of high order in Sobolev spaces 243
8.1.1 The annihilated of sol(K) in W^iK)1* 243
8.1.2 The range s p 244
8.2 Approximation of lower order in Sobolev spaces 245
8.2.1 Reducing approximation of lower order to a problem of spectral
synthesis 245
8.2.2 Approximation in Sobolev spaces on compact sets by potentials
with densities supported on the boundary 247
8.2.3 Degenerate cases of approximation in Sobolev spaces on compact
sets with empty interior 249
8.2.4 Degenerate cases of approximation in Sobolev spaces on arbitrary
compact sets 251
8.2.5 Uniform approximation on compact sets by potentials with den¬
sities supported on the boundary 254
8.2.6 Degenerate cases of uniform approximation on nowhere dense
compact sets 255
8.2.7 Distinguished case of uniform approximation on nowhere dense
compact sets 257
8.2.8 Absence of degenerate cases of uniform approximation on com¬
pact sets with nonempty interior 259
8.3 Approximation criteria in terms of Bessel capacity 262
8.3.1 The case of nowhere dense compact sets 262
8.3.2 The problem for arbitrary compact sets 263
Contents j;{
8.3.3 Approximation criteria in terms of sporial rapacities 264
8.3.4 Bounded point evaluations 2CG
8.4 Capacitary criteria of Vitushkin type for approximation in spaces of
Sobolev functions 208
8.4.1 Statement of the theorem 2G8
8.4.2 Comments 269
8.4.3 Proof of the direct part, 1) = • 2) 270
8.4.4 Proof of the converse part. 3) = 1) 273
9 Generalized boundary values of solutions of a system with injective
symbol 277
9.1 Golubev series for solutions of elliptic equations 278
9.1.1 Statement of the main results 278
9.1.2 The converse theorem 280
9.1.3 A basic special case 281
9.1.4 Inductive limit topology in the space of solutions on a compact
set 282
9.1.5 Banach spaces I (r)K 283
9.1.6 Inductive limit of the spaces lq (r)K 285
9.1.7 Another topology in the space of solutions on a compact set . . 286
9.1.8 The role of local connectedness 287
9.1.9 Equivalence of two topologies on Sol{K. P ) 290
9.1.10 Conclusion of proof 291
9.1.11 A variant of Laurent series expansion 293
9.1.12 Separation of singularities into atomic singularities 293
9.1.13 Representation of solutions by boundary integrals 294
9.1.14 Solutions with poles 294
9.1.15 An example for harmonic functions 296
9.1.16 Further results 296
9.1.17 Hyperfunctions 297
9.2 The Dirichlet problem for the generalized Laplacian by means of gener¬
alized functions 298
9.2.1 Green operators 298
9.2.2 Dirichlet systems 298
9.2.3 Green s formula for the generalized Laplacian 300
9.2.4 The Dirichlet problem 302
9.2.5 Function spaces 304
9.2.6 The operator related to the Dirichlet problem in the complete
scale of Sobolev spaces 306
9.2.7 Fredholm operators 307
9.2.8 Theorem on a Complete Set of Isomorphisms 309
9 3 Traces on the boundary of generalized solutions of the Dirichlet equa¬
tion 312
14 Contents
9.3.1 Weak solutions of the Dirichlet problem 313
9.3.2 Traces on the boundary of weak solutions to the Dirichlet equa¬
tion 314
9.3.3 Traces on the boundary of solutions in the domain 317
9.3.4 Remarks 319
9.3.5 Traces of generalized solutions on parallel hypersurfaces 320
9.3.6 Solutions of finite order of growth near the boundary 322
9.3.7 Local increase of smoothness 323
9.3.8 Green s function 324
9.3.9 Problems with power singularities 328
9.4 Weak limit values on the boundary of solutions of a system with injective
symbol 328
9.4.1 Green s formula for solutions of finite order of growth 328
9.4.2 Weak limit values 329
9.4.3 Equivalence of strong and weak limit values 331
9.4.4 A characterization 333
9.4.5 Miscellaneous 334
10 The Cauchy problem for a system with injective symbol 335
10.1 Green type integral 336
10.1.1 Definition and simple properties 336
10.1.2 The Sokhotskii Plemelj formulas 338
10.1.3 An application to the Cauchy problem 340
10.2 Iterations of the Green type integral 342
10.2.1 Prologue 342
10.2.2 A theorem on iterations 344
10.2.3 The inner product h( , •) 346
10.2.4 Solvability conditions for Pu = f 350
10.2.5 A remark about the 9 problem 353
10.2.6 An application to the Dirichlet problem 355
10.3 Solvability of the Cauchy problem in the class of distributions of finite
order 355
10.3.1 Further look at the Cauchy problem 355
10.3.2 Tangential equation 357
10.3.3 Reduction to the Cauchy problem for the generalized Laplacian 358
10.3.4 Solvability of the Cauchy problem with data on the whole boun¬
dary 362
10.3.5 Criterion of solvability of the Cauchy problem with data on a
boundary subset 363
10.3.6 A concluding remark 365
10.4 Carleman function 366
10.4.1 Definition 366
10.4.2 Existence 367
Contents 15
10.4.3 Carleman formula 3C8
10.4.4 Conditional stability of the Cauchy problem 369
10.4.5 The system of elasticity theory 370
11 Method of Fischer Riesz equations in the Cauchy problem for a system
with injective symbol 374
11.1 Operator theoretic foundations of the method of Fischer R.iesz equa¬
tions 375
11.1.1 Abstract problem in Hilbert spaces 375
11.1.2 Special bases 375
11.1.3 Solvability 377
11.1.4 Approximate solution 379
11.2 Hardy spaces 380
11.2.1 A further look at generalized boundary values 380
11.2.2 Generalized Hardy spaces 384
11.2.3 Boundary kernel function 386
11.2.4 Bergman formula 388
11.2.5 Relation with Green s function 389
11.3 Analysis of the Cauchy problem 389
11.3.1 Special bases for the Cauchy problem 389
11.3.2 Examples of special bases 391
11.3.3 Solvability of the Cauchy problem 393
11.3.4 Approximate solutions of the Cauchy problem 396
11.3.5 Zin s theorems 398
11.3.6 Traces of holomorphic functions on subsets of Shilov s boundary 400
11.3.7 Another approach 402
11.4 Analysis of the Dirichlet problem 402
11.4.1 Basic assumptions 403
11.4.2 Special bases in the Dirichlet problem 405
11.4.3 Examples of special bases 406
11.4.4 A criterion of solvability of the Dirichlet problem 408
11.4.5 Regularization of solutions of the Dirichlet problem 410
11.4.6 Some calculations for the classical Dirichlet problem 412
12 Bases with double orthogonality in the Cauchy problem for a system
with injective symbol 415
12.1 An operator theoretic approach 417
12.1.1 The abstract framework 417
12.1.2 Abstract Bergman Theory 420
12.1.3 Further horizons 423
12.1.4 An alternative method 425
12.1.5 Extremal property 427
12.2 Analysis of the Cauchy problem in terms of surface bases with double
orthogonality 129
16 Contents
12.2.1 The main step 429
12.2.2 Surface bases 430
12.2.3 Analysis of the Cauchy problem 431
12.2.4 Notes 432
12.3 Analysis of the Cauchy problem in terms of solid bases with double or¬
thogonality 433
12.3.1 Formulation of the problem 433
12.3.2 Green type integral 434
12.3.3 Main lemma 435
12.3.4 The Cartan Kahler Theorem 437
12.3.5 Extension problem 438
12.3.6 Solid bases 439
12.3.7 Solvability of the Cauchy problem 441
12.3.8 Approximate solution 443
12.3.9 Example for harmonic functions 446
12.3.10 A stability set 446
12.4 Applications to matrix factorizations of the Laplace equation 448
12.4.1 The Cauchy problem 448
12.4.2 Green type integral 449
12.4.3 A solid basis of harmonic polynomials 451
12.4.4 An expansion of the fundamental solution 452
12.4.5 A solvability criterion 453
12.4.6 Regularization 454
Bibliography 457
Index of names 473
Subject index 476
Index of notation 479
|
| any_adam_object | 1 |
| author | Tarchanov, Nikolaj Nikolaevič 1955-2020 |
| author_GND | (DE-588)121160521 |
| author_facet | Tarchanov, Nikolaj Nikolaevič 1955-2020 |
| author_role | aut |
| author_sort | Tarchanov, Nikolaj Nikolaevič 1955-2020 |
| author_variant | n n t nn nnt |
| building | Verbundindex |
| bvnumber | BV010107504 |
| ctrlnum | (OCoLC)246878584 (DE-599)BVBBV010107504 |
| dewey-full | 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. ed. |
| format | Book |
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| id | DE-604.BV010107504 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:46:38Z |
| institution | BVB |
| isbn | 3055016637 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006711116 |
| oclc_num | 246878584 |
| open_access_boolean | |
| owner | DE-12 DE-19 DE-BY-UBM DE-634 DE-11 DE-188 |
| owner_facet | DE-12 DE-19 DE-BY-UBM DE-634 DE-11 DE-188 |
| physical | 478 S. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Akad.-Verl. |
| record_format | marc |
| series | Mathematical topics |
| series2 | Mathematical topics |
| spelling | Tarchanov, Nikolaj Nikolaevič 1955-2020 Verfasser (DE-588)121160521 aut The Cauchy problem for solutions of elliptic equations Nikolai N. Tarkhanov 1. ed. Berlin Akad.-Verl. 1995 478 S. txt rdacontent n rdamedia nc rdacarrier Mathematical topics 7 Literaturverz. S. 457 - 472 Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Cauchy-Anfangswertproblem (DE-588)4147404-1 s DE-604 Mathematical topics 7 (DE-604)BV008671507 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006711116&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tarchanov, Nikolaj Nikolaevič 1955-2020 The Cauchy problem for solutions of elliptic equations Mathematical topics Elliptische Differentialgleichung (DE-588)4014485-9 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4147404-1 |
| title | The Cauchy problem for solutions of elliptic equations |
| title_auth | The Cauchy problem for solutions of elliptic equations |
| title_exact_search | The Cauchy problem for solutions of elliptic equations |
| title_full | The Cauchy problem for solutions of elliptic equations Nikolai N. Tarkhanov |
| title_fullStr | The Cauchy problem for solutions of elliptic equations Nikolai N. Tarkhanov |
| title_full_unstemmed | The Cauchy problem for solutions of elliptic equations Nikolai N. Tarkhanov |
| title_short | The Cauchy problem for solutions of elliptic equations |
| title_sort | the cauchy problem for solutions of elliptic equations |
| topic | Elliptische Differentialgleichung (DE-588)4014485-9 gnd Cauchy-Anfangswertproblem (DE-588)4147404-1 gnd |
| topic_facet | Elliptische Differentialgleichung Cauchy-Anfangswertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006711116&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008671507 |
| work_keys_str_mv | AT tarchanovnikolajnikolaevic thecauchyproblemforsolutionsofellipticequations |