The analysis of solutions of elliptic equations:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1997
|
| Series: | Mathematics and its applications
406 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus dem Russ. übers. |
| Physical Description: | XX, 479 S. |
| ISBN: | 0792345312 |
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MARC
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| 240 | 1 | 0 | |a Rjad lorana dlja rešenij ėlliptičeskich sistem |
| 245 | 1 | 0 | |a The analysis of solutions of elliptic equations |c by Nikolai N. Tarkhanov |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1997 | |
| 300 | |a XX, 479 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 406 | |
| 500 | |a Aus dem Russ. übers. | ||
| 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_ | 1804126207818596352 |
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| adam_text | Table of Contents
Preface to the English Translation xvii
Preface to the Russian Edition xix
List of Main Notations 1
1 Removable Singularities 5
1.1 Bochner s Theorems 5
1.1.1 Sheaf of solutions 5
1.1.2 Removable sets 6
1.1.3 Hausdorff measure 7
1.1.4 Capacity 8
1.1.5 The fundamental theorem of Bochner 12
1.1.6 The second theorem of Bochner 13
1.1.7 Examples 14
1.2 Sufficient Conditions for Removability in Terms of Hausdorff Measure 15
1.2.1 A fundamental lemma 15
1.2.2 Removability for W ^(X) 18
1.2.3 Removability for C?0C(X) 20
1.2.4 Removability for C (X) 21
1.2.5 Painleve s theorem 24
1.2.6 Some examples 28
1.2.7 Generalization to cohomology spaces 28
1.3 Removable Singularities on Hypersurfaces 29
1.3.1 Transversal order 30
1.3.2 Transversally non characteristic hypersurfaces 31
1.3.3 Affine singularities 35
1.3.4 Rado s theorem 39
1.4 Characterization of Removable Singularities in Terms of Capacity . . 43
1.4.1 Historical information 43
1.4.2 Solutions regular at infinity 43
1.4.3 The capacity Cap(a, L, P,(h)) 45
1.4.4 The main result 48
1.4.5 Some corollaries 50
1.4.6 Capacitary extremals 51
vii
viii
1.4.7 Examples 52
1.4.8 Further horizons 54
1.5 Metric Properties of the Capacity Associated to Holder Spaces .... 55
1.5.1 Some history 55
1.5.2 A metric lemma 55
1.5.3 Commensurability with Hausdorff content for compact sets . . 57
1.5.4 Conclusion of proof 65
1.5.5 Semiadditivity of the capacity 65
2 Laurent Series 67
2.1 Laurent Expansion for Differential Complexes 67
2.1.1 History of the subject 67
2.1.2 Cohomology of differential complexes with point support . . . . 68
2.1.3 Cohomology of Hilbert complexes with point support 69
2.1.4 Laurent expansions 71
2.1.5 Laurent series for elliptic complexes 72
2.2 Laurent Series for Solutions of Elliptic Systems 76
2.2.1 Admissible scalar products in the space of polynomials 76
2.2.2 Expansions in series of fundamental matrices 77
2.2.3 Properties of the matrices Ma{y) 80
2.2.4 Analog of the Cauchy estimates for the coefficients of Laurent
series 84
2.2.5 Laurent series for solutions of the system Pu = 0 85
2.2.6 Concluding remarks 87
2.3 Taylor Series for Solutions of Elliptic Systems 87
2.3.1 Dual situation 87
2.3.2 Analog of the Cauchy estimates for the coefficients of Taylor
series 89
2.3.3 Taylor series for solutions of the system Pu = 0 89
2.3.4 Analyticity of the cohomology of elliptic complexes 91
2.4 Examples 91
2.4.1 Spherical averaging 91
2.4.2 The scalar product, in connection with calculating the Cauchy
principal value 93
2.4.3 The Gauss representation for homogeneous polynomials .... 94
2.4.4 More on the Almanzi formula 96
2.4.5 Laurent expansion on the line 97
2.4.6 Remarks on Clifford analysis 98
2.4.7 Laurent series for matrix factorizations of the Laplace equation 98
2.4.8 Laurent series for holomorphic functions 100
2.4.9 Laurent series for harmonic functions 100
2.4.10 Laurent series for polyharmonic functions 101
2.4.11 A scalar product connected with approximations by solutions . 102
2.5 Local Properties of Solutions of Elliptic Equations 104
2.5.1 A digression 104
Table of Contents ix
2.5.2 Behavior of a solution near a finite singular point 104
2.5.3 Behavior of solutions near the point at infinity 106
2.5.4 Homogeneous solutions 106
2.5.5 Expanding a solution as a series of polynomial solutions . . . . 107
2.5.6 Expanding solutions via the Euler operator 107
2.5.7 The Cauchy principal value 110
2.5.8 A generalization for cohomology classes 113
3 Representation of Solutions with Non Discrete Singularities 115
3.1 The Topology in Spaces of Solutions of Elliptic Systems 115
3.1.1 Locally convex inductive limit topology 115
3.1.2 The Stieltjes Vitali Theorem 116
3.1.3 Solutions on Closed Sets 117
3.1.4 Regular compacta 118
3.1.5 The extension theorem 119
3.1.6 The Hilbert spaces l2(r)K 120
3.1.7 Another topology in the space of solutions on a compact set . . 121
3.1.8 The role of regularity 123
3.1.9 The equivalence of two topologies on Sol(/ Q 125
3.1.10 Notes 125
3.2 The Structure of Solutions with Compact Singularities 126
3.2.1 Golubev series 126
3.2.2 Sufficiency of the representation as a Golubev series 127
3.2.3 Releasing condition 2) on the coefficients 128
3.2.4 The general form of a continuous linear functional on Sol (K, P ) 129
3.2.5 Conclusion of the proof of Theorem 3.2.1 130
3.2.6 Golubev series for solutions of elliptic systems ivith real ana¬
lytic coefficients 131
3.2.7 Some consequences of Theorem 3.2.6 132
3.2.8 Golubev series with finitely many terms 132
3.2.9 Example 134
3.3 Duality in the Spaces of Solutions of Elliptic Systems 135
3.3.1 Preliminaries 135
3.3.2 Grothendieck duality 136
3.3.3 Green s function 138
3.3.4 Grothendieck duality for harmonic functions 140
3.3.5 A corollary 144
3.3.6 Miscellaneous 148
3.3.7 Duality for solutions of Pu = 0 149
3.3.8 Proof of the main theorem 151
3.3.9 The converse theorem 158
3.3.10 Duality in complex analysis 158
3.4 A Representation Theorem for Solutions off a Plane 159
3.4.1 Motivation 159
3.4.2 Statement of the main results 160
X
3.4.3 The converse theorem 161
3.4.4 The abstract framework 162
3.4.5 Duality in the space of sequences 164
3.4.6 Transpose 165
3.4.7 Conclusion of proof 167
3.4.8 An example for harmonic functions 169
3.4.9 Hyperfunctions 169
3.5 Spectral Decomposition of the Green Type Integral in a Ball 170
3.5.1 The Green type integral 170
3.5.2 A digression 171
3.5.3 Spectrum 172
3.5.4 Examples 178
3.5.5 Theorem on iterations 179
3.5.6 Solvability of the system Pu = f in a ball 184
3.5.7 Applications 189
4 Uniform Approximation 191
4.1 Runge Theorem 191
4.1.1 A tour of the theory of holomorphic approximation 191
4.1.2 Holomorphic approximations in several variables 193
4.1.3 From holomorphic functions to solutions of elliptic equations . 194
4.1.4 Approximation by solutions of overdetermined systems 197
4.1.5 Differentiate functions on a closed set 197
4.1.6 A theorem of Malgrange 198
4.1.7 Approximation of finitely smooth solutions by infinitely differ
entiable solutions 200
4.1.8 Local solvability of homogeneous elliptic systems 203
4.1.9 A Runge theorem 203
4.1.10 Miscellaneous 206
4.2 Walsh Type Theorems 207
4.2.1 Approximation by linear combinations of fundamental solutions207
4.2.2 Approximation on domains with strong cone property 211
4.2.3 Approximation on compact sets of zero measure 212
4.2.4 Approximation on nowhere dense compact sets 214
4.2.5 Other approximation theorems 215
4.2.6 A closer look at the case of differential operators with constant
coefficients 217
4.2.7 The additive Cousin problem 221
4.2.8 Domain of existence for solutions 223
4.2.9 Some remarks 224
4.3 The Notion of Capacity in Problems of Uniform Approximation . . . 225
4.3.1 Preliminary results 225
4.3.2 Further look at the capacity Ca.p{o ,C (X),P,(h)) 225
4.3.3 Other expressions for the capacity 228
4.3.4 General properties 229
Table of Contents xi
4.3.5 Comparison with Hausdorff measure 231
4.3.6 Behavior under affine transformations 232
4.3.7 The capacity of a ball 233
4.3.8 The capacity of a point ¦. 235
4.3.9 Capacity and removable singularities 236
4.3.10 More on the capacity Ca.p( r,C (X)) 237
4.4 Vitushkin s Constructive Technique 239
4.4.1 Statements of problems 239
4.4.2 Special partitions of unity inRn 240
4.4.3 The localization operator 241
4.4.4 Separation of singularities of functions being approximated . . 246
4.4.5 The basic technical lemma 246
4.4.6 Approximation of functions by parts 250
4.4.7 A preparatory theorem 251
4.4.8 Localized version of the approximation theorem 256
4.4.9 Approximation on compact sets with complement having the
cone property 257
4.4.10 Unsolved problems 257
4.5 Capacitary Criteria 259
4.5.1 A counterexample 259
4.5.2 An auxiliary result 261
4.5.3 Formulation of the fundamental result 262
4.5.4 Proof of the implication 1) =» 2) 263
4.5.5 Proof of the implication 3) = 1) 266
4.5.6 A closer look at the case of equations of order p n 268
4.5.7 Some problems 269
5 Mean Approximation 271
5.1 Reduction to the Spectral Synthesis in Sobolev Spaces 271
5.1.1 Preliminaries 271
5.1.2 Approximation in the mean by holomorphic functions 272
5.1.3 Sketch of the theory for solutions of elliptic equations 275
5.1.4 Sobolev functions on a closed set 280
5.1.5 Reduction 281
5.2 Approximation on Nowhere Dense Compact Sets 282
5.2.1 Approximation in W {K), for small q 282
5.2.2 Capacitary criteria 283
5.2.3 From nowhere dense compact sets to those having interior points284
5.2.4 Criteria of Wiener type 284
5.2.5 Bases with double orthogonality 285
5.2.6 A basis of holomorphic monomials 288
5.2.7 A basis of harmonic polynomials 289
5.2.8 Instability phenomena 290
5.2.9 Holomorphic moments 293
5.2.10 Harmonic moments 295
xii
5.2.11 Factorizations 296
5.3 The Notion of Capacity in Problems of Mean Approximation 298
5.3.1 Overview 298
5.3.2 L estimates for potentials 298
5.3.3 Decay estimates for potentials 301
5.3.4 The capacities Cap (a, W {X),P,{h)) 303
5.3.5 Pathologies of the capacities 305
5.4 Construction of Mean Approximations 306
5.4.1 A lemma of Lindberg 306
5.4.2 Estimates for the localization operator in Sobolev spaces .... 306
5.4.3 Separation of singularities 308
5.4.4 The basic technical lemma 309
5.4.5 Proof of the basic lemma under an additional hypothesis .... 309
5.4.6 Conclusion of the proof 311
5.4.7 Approximation of a function by parts 314
5.5 Capacitary Criteria for the Mean Approximation 315
5.5.1 A preparatory lemma 316
5.5.2 The fundamental result: formulation and commentary 317
5.5.3 Spectral synthesis in Sobolev spaces 318
6 BMO Approximation 319
6.1 BMO Functions on Compact Sets 319
6.1.1 The space BMO(Rn) 320
6.1.2 Multiplication by functions of compact support 322
6.1.3 Local BMO spaces 324
6.1.4 Higher order BMO spaces 324
6.1.5 VMO functions 325
6.1.6 Approximation o/BMO functions by C°° functions of compact
support 328
6.1.7 VMO functions on a closed set 329
6.2 Duality 330
6.2.1 Local Hardy spaces 330
6.2.2 The dual of the local Hardy space 331
6.2.3 The dual of VMO 333
6.3 Boundedness Theorems for Pseudodifferential Operators on BMO
spaces 335
6.3.1 Boundedness in local Hardy spaces 335
6.3.2 Boundedness in local BMO spaces 336
6.3.3 Fundamental theorem of calculus 336
6.3.4 Behavior in higher order Hardy spaces 338
6.3.5 Behavior in higher order BMO spaces 338
6.4 Higher Order Approximation 338
6.4.1 The annihilator of the subspace of solutions 339
6.4.2 Description of the closure of the subspace of solutions 340
6.4.3 Notes 341
Table of Contents xiii
6.5 Lower Order Approximation 341
6.5.1 A counterexample 341
6.5.2 Reduction to spectral synthesis in Hardy spaces 342
6.5.3 Approximation on nowhere dense compact sets 343
6.5.4 Capacitary criteria 344
7 Conditional Stability 345
7.1 Conditionally Stable Linear Problems 345
7.1.1 Ill posed problems 345
7.1.2 Conditionally stable problems 347
7.1.3 Example of a conditionally stable problem 348
7.1.4 The case of normed spaces 348
7.2 An Abstract Carleman Formula 349
7.2.1 A brief reference 349
7.2.2 An abstract formula for solutions of stable problems 349
7.2.3 A Carleman formula 350
7.2.4 Two norm spaces 351
7.2.5 Proof of necessity in Theorem 7.2.4 354
7.2.6 Proof of sufficiency in Theorem 7.2.4 355
7.2.7 Some remarks 355
7.3 A Carleman Formula for Solutions 356
7.3.1 Basic assumption 356
7.3.2 Continuation problems for solutions 356
7.3.3 Analog of the Stieltjes Vitali theorem 357
7.3.4 A Carleman formula 358
7.3.5 Analytic continuation 359
7.3.6 The Cauchy problem 360
7.3.7 Other examples 362
7.4 An Inversion Formula for Toeplitz Operators 363
7.4.1 Toeplitz operators on Hilbert spaces of solutions 363
7.4.2 Statement of the problem 364
7.4.3 A variant of the Carleman formula 364
7.4.4 Conditional stability of the problem 365
7.5 Carleman Function of the Cauchy Problem 366
7.5.1 The Cauchy problem 366
7.5.2 Carleman functions 557
7.5.3 Carleman functions and regularization 557
7.5.4 Carleman function and conditional stability 369
7.5.5 Existence 370
7.5.6 Further horizons 371
8 The Cauchy Problem 373
8.1 Traces of Holomorphic Functions 373
8.1.1 The abstract framework 575
8.1.2 Criteria for analytic continuation 575
xiv
8.1.3 A generalization of the Riesz theorem 379
8.2 Solvability of Systems with Surjective Symbol 381
8.2.1 P convex manifolds 382
8.2.2 Formulation of the result 382
8.2.3 Necessity 383
8.2.4 An excursion in the theory of Sobolev spaces 383
8.2.5 Proof of sufficiency 384
8.2.6 Left fundamental solutions 386
8.3 Hardy Spaces 387
8.3.1 Special exhausting functions for domains 387
8.3.2 Dirichlet systems 388
8.3.3 A Green formula 389
8.3.4 Generalized Hardy spaces 392
8.3.5 Boundary values of solutions of Hardy class 396
8.3.6 Open problems 397
8.4 Solvability of the Cauchy Problem with Data Given on the Whole
Boundary 398
8.4.1 A uniqueness theorem 398
8.4.2 Solvability in Hardy spaces 399
8.4.3 Local regularity 401
8.5 Solvability of the Cauchy Problem with Data Given on a Part of the
Boundary 405
8.5.1 Statement of results 405
8.5.2 Proof of necessity 406
8.5.3 Special approximations by solutions of the transposed system . 407
8.5.4 Completion of the proof 408
9 Quasiconformality 411
9.1 The Stability Concept 412
9.1.1 Basic classes of mappings 412
9.1.2 Examples 413
9.1.3 Closeness functionals 414
9.1.4 Stability 417
9.1.5 Problems of the theory of stability 418
9.1.6 Liouville s theorem 420
9.2 First Order Elliptic Systems 422
9.2.1 Cauchy s theorem 422
9.2.2 Morera s theorem 423
9.2.3 Cauchy s formula 424
9.2.4 Some singular integral operators 424
9.2.5 The ellipticity is a necessary condition for the stability .... 425
9.3 Beltrami Equation 426
9.3.1 A decomposition of the differential 426
9.3.2 The Beltrami equation 428
Table of Contents xv
9.3.3 Local closeness to the sheaf of solutions and the Beltrami equa¬
tion 428
9.3.4 Notes 431
9.4 Stability of the Sheaf of Solutions 432
9.4.1 Statement of the main theorems 432
9.4.2 Generalized Cauchy s formula 433
9.4.3 An estimate for the double layer potential 434
9.4.4 An estimate for the volume potential 435
9.4.5 Lq estimates of the derivatives of solutions to the Beltrami
equation 436
9.4.6 Global closeness to the sheaf of solutions and the Beltrami
equation 439
9.4.7 Order of closeness 442
9.5 Properties of Mappings Close to the Sheaf of Solutions 444
9.5.1 Proximity of the derivatives 444
9.5.2 Generalized Cauchy s theorem 446
9.5.3 Generalized Morera s theorem 447
9.5.4 Generalized Liouville s theorem 448
Bibliography 451
Name Index 472
Subject Index 475
Index of Notation 478
|
| any_adam_object | 1 |
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| id | DE-604.BV011672558 |
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| indexdate | 2024-07-09T18:13:47Z |
| institution | BVB |
| isbn | 0792345312 |
| language | English |
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| spelling | Tarchanov, Nikolaj Nikolaevič 1955-2020 Verfasser (DE-588)121160521 aut Rjad lorana dlja rešenij ėlliptičeskich sistem The analysis of solutions of elliptic equations by Nikolai N. Tarkhanov Dordrecht [u.a.] Kluwer 1997 XX, 479 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 406 Aus dem Russ. übers. Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s DE-604 Mathematics and its applications 406 (DE-604)BV008163334 406 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007868964&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tarchanov, Nikolaj Nikolaevič 1955-2020 The analysis of solutions of elliptic equations Mathematics and its applications Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4014485-9 |
| title | The analysis of solutions of elliptic equations |
| title_alt | Rjad lorana dlja rešenij ėlliptičeskich sistem |
| title_auth | The analysis of solutions of elliptic equations |
| title_exact_search | The analysis of solutions of elliptic equations |
| title_full | The analysis of solutions of elliptic equations by Nikolai N. Tarkhanov |
| title_fullStr | The analysis of solutions of elliptic equations by Nikolai N. Tarkhanov |
| title_full_unstemmed | The analysis of solutions of elliptic equations by Nikolai N. Tarkhanov |
| title_short | The analysis of solutions of elliptic equations |
| title_sort | the analysis of solutions of elliptic equations |
| topic | Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007868964&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT tarchanovnikolajnikolaevic rjadloranadljaresenijellipticeskichsistem AT tarchanovnikolajnikolaevic theanalysisofsolutionsofellipticequations |