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      The Cauchy problem for solutions of elliptic equations

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      https://www.riss.kr/link?id=M265954

      • 저자
      • 발행사항

        Berlin : Akademie Verlag ; New York : VCH Publishers, c1995

      • 발행연도

        1995

      • 작성언어

        영어

      • 주제어
      • DDC

        515/.353 판사항(20)

      • ISBN

        3055016637 (non-acid)

      • 자료형태

        단행본(다권본)

      • 발행국(도시)

        Germany

      • 서명/저자사항

        The Cauchy problem for solutions of elliptic equations / Nikolai N. Tarkhanov.

      • 판사항

        1st ed

      • 형태사항

        478 p. ; 25 cm.

      • 총서사항

        Mathematical topics, 0946-3844 ; v. 7 Mathematical topics (Berlin, Germany) ; v. 7.

      • 일반주기명

        Includes bibliographical references (p. 457-472) and indexes.

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      목차 (Table of Contents)

      • CONTENTS
      • Introduction = 17
      • List of main notations = 27
      • 1 Function spaces = 29
      • 1.1 An abstract theory = 29
      • 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 $$C^\infty$$ 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 H$$\ddot o$$lder spaces = 47
      • 1.3.1 Spaces of H$$\ddot o$$lder functions = 47
      • 1.3.2 H$$\ddot o$$lder functions of finite smoothness = 48
      • 1.3.3 H$$\ddot o$$lder continuous functions = 50
      • 1.3.4 Standard regularization of H$$\ddot o$$lder functions = 51
      • 1.3.5 Approximation by $$C^\infty$$ functions = 53
      • 1.3.6 Spectral synthesis in spaces of H$$\ddot o$$lder functions = 54
      • 1.3.7 H$$\ddot o$$lder functions on closed sets = 56
      • 1.3.8 Dual spaces = 58
      • 1.3.9 The negative H$$\ddot o$$lder spaces = 58
      • 1.4 Sobolev spaces = 60
      • 1.4.1 Lebesgue spaces = 60
      • 1.4.2 The spaces $$W^{s,q}_{opt}(X)$$ = 60
      • 1.4.3 Standard regularization of Sobolev functions = 61
      • 1.4.4 Approximation by $$C^\infty$$ 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 $$L^q(R^n)$$ = 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 Pseudo differential operators = 89
      • 2.2.1 The Fourier integral representation of Calderon-Zygmund operators = 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 H$$\ddot o$$lder spaces = 109
      • 2.3.3 Behavior in local Zygmund spaces = 110
      • 2.3.4 Behavior in local spaces of H$$\ddot o$$lder continuous functions = 110
      • 2.3.5 Behavior in local Sobolev spaces = 111
      • 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 spaces = 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 H$$\ddot o$$lder 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 H$$\ddot o$$lder spaces = 153
      • 3.3.1 A definition = 153
      • 3.3.2 Behavior under affine transformations = 154
      • 3.3.3 A nondegencracy 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
      • 3.4.1 Bcssel capacity = 160
      • 3.4.2 Metric properties of Bcssel capacity = 163
      • 3.4.3 Quasicontinuous representatives of Sobolev functions = 163
      • 3.4.4 An application to spectral synthesis in Sobolev spaces = 105
      • 3.4.5 A brief review of higher order capacities = 165
      • 3.4.6 Comparison with Bcssel 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
      • 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 compacta 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 Tlie 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 property = 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 surjectivc 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 Overdefcermined 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
      • 7 Approximation in H$$\ddot o$$lder spaces = 233
      • 7.1 Approximation of high order in H$$\ddot o$$lder spaces = 233
      • 7.1.1 Description of the annihilator of the subspace of solutions = 233
      • 7.1.2 The range s ≥ p = 234
      • 7.2 Approximation of lower order in H$$\ddot o$$lder 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 H$$\ddot o$$lder 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 annihilatator of sol(K) in $$W^{s,q}(K)^k$$ = 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 Soboicv spaces on arbitrary compact sets = 251
      • 8.2.5 Uniform approximation on compact sots 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 compact 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
      • 8.3.3 Approximation criteria in terms of special capacities = 264
      • 8.3.4 Bounded point evaluations = 266
      • 8.4 Capacitary criteria of Vituslikin type for approximation in spaces of Sobolev functions = 268
      • 8.4.1 Statement of the theorem = 268
      • 8.4.2 Comments = 269
      • 8.4.3 Proof of the direct part, 1) $$\Rightarrow$$ 2) = 270
      • 8.4.4 Proof of the converse part, 3) $$\Rightarrow$$ 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 $$l^{q^\prime}(r)^K$$ = 283
      • 9.1.6 Inductive limit of the spaces $$l^{q^\prime}(r)^K$$ = 285
      • 9.1.7 Another topology in the apace 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 generalized 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 equation = 312
      • 9.3.1 Weak solutions of the Dirichlet problem = 313
      • 9.3.2 Traces on the boundary of weak solutions to the Dirichlct equation = 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 hypersurfaccs = 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 injcctive symbol = 328
      • 9.4.1 Green's formula for solutions of finite order of growth = 328
      • 9.4.2 Weak limit value = 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($$\cdot,\cdot$$ = 346
      • 10.2.4 Solvability conditions for Pu = f = 350
      • 10.2.5 A remark about the $$\bar{\partial$$-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 boundary = 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
      • 10.4.3 Carleman formula = 368
      • 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-Riesz equations = 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 Shiiov'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 Rcgularization 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 = 429
      • 12.2.1 The main step = 429
      • 12.2.2 Surface bases = 430
      • 12.2.3 Analysis of the Canchy problem = 431
      • 12.2.4 Notes = 432
      • 12.3 Analysis of the Cauchy problem m terms of solid bases with double orthogonality = 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
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