The Cauchy problem for solutions of elliptic equations

목차 (Table of Contents)

• 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