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Bergman S., Schiffer M. — Kernel Functions and Elliptic Differential Equations in Mathematical Physics
Bergman S., Schiffer M. — Kernel Functions and Elliptic Differential Equations in Mathematical Physics



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Название: Kernel Functions and Elliptic Differential Equations in Mathematical Physics

Авторы: Bergman S., Schiffer M.

Язык: en

Рубрика: Математика/Анализ/Дифференциальные уравнения/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1953

Количество страниц: 432

Добавлена в каталог: 06.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$E_\lambda$-metric      273 344
$M^{(n)}$-kernel      313
$S_u$ transformation      337
$S_u$ transformation, scalar product      339
$T_u$ transformation      316
$T_u$ transformation, complete continuity      324
$T_u$ transformation, properties      338—41
$\gamma_u$ transformations      341
$\gamma_u$ transformations, eigenfunctions      346
$\gamma_u$ transformations, involutory character      342
$\gamma_u$ transformations, saltus condition      343
$\mathfrak G$-kernel      109 369
$\mathfrak T$-transform      345—346
$\Sigma$ linear space      260
$\Sigma$-monogenic functions      134
Adiabatic equation of state      135
Adiabatic gas, theory of      139
Airy stress function      247
Almansi, E.      232
Ampere's law      165—166
Analytic functions      72
Analytic functions, generalized      134
Associated partial differential equations      125—128
Axially symmetric flow      119
Bateman, H.      3
Bending of thin plates      232
Bergman, S.      116 119 132 141 147 206 229 236 246 297 355 402
Bergman, S., and Schiffer, M.      246 275 280 309 315 346 351 371 384 388
Bernoulli's equation      29 30 72 128
Bers, L., and Gelbart, A.      133 134 141
Betti's formulas      219 222
Bieberbach, L.      93 105 283
Biharmonic equation      229 232 234 235 247
Biharmonic equation, boundary value problems      236—237 242
Biharmonic equation, complex kernel      243
Biharmonic equation, complex solutions      243
Biharmonic equation, fundamental singularity      237
Biharmonic equation, fundamental solutions      238
Biharmonic equation, Green's function      238—239 244
Biharmonic functions      229
Biharmonic functions, representation of      229 238—240 241 243
Blasius formula, compressible flows      153—154
Blasius formula, force      88
Blasius formula, moment      90
Bochner, S.      119
Bolza, O.      105 108
Boundary value problems, elasticity      218 223—224
Boundary value problems, elasticity, plane strain      249 254
Boundary value problems, elasticity, thin plates      236
Boundary value problems, fluid flow, axially symmetric      122—123
Boundary value problems, fluid flow, compressible      141—145
Boundary value problems, fluid flow, incompressible      30 37
Boundary value problems, fluid flow, plane flows      77 82
Boundary value problems, heat conduction      4—7 12—16 22
Boundary value problems, Laplace's equation      9 19 30 31 77 79 82 110 168 199
Boundary value problems, of the first kind      50 77
Boundary value problems, of the second kind      57 77
Boundary value problems, of the third kind      267
Boundary value problems, statements and solutions for, $\Delta u-qu=0$      263—238 271—272
Boundary value problems, statements and solutions for, $\Delta u-qu=j(P)$      263—238
Boundary value problems, statements and solutions for, associated partial differential equations      125
Boundary value problems, statements and solutions for, biharmonic equation      236—237 242—243
Boundary value problems, where the domain is, a circle      82
Boundary value problems, where the domain is, infinite      38
Boundary value problems, where the domain is, non-schlicht      149
Boussinesq, J.      232
Branch points      76 136
Built-in edge      236
Capacity      see Electrostatic capacity
Caratheodory, C.      108
Cauchy — Riemann equations      72 73 77
Cauchy — Riemann equations, generalization      124
Chaplygin, S.A.      135 136
Charge coefficient      187
Charge density      155
Circulation      76
Circulation constant      89
Circulation constant, determination of      91
Cisotti, L      28 71
Coefficient of heat conductivity      1
Coefficient of heat conductivity, discontinuity      15
Coefficients, of mapping function, variational formulas      101—102
Complete continuity      323
Complete systems      116 202 226 241 243 249
Complete systems, construction of      282—5 286—297
Complete systems, for $\Sigma$      280
Complete systems, of eigenfunctions      324 347—348 352
complex derivative      72
Complex differential operators      111
Complex kernel function      112—118 243—246
Complex potential      74
Compressible fluid flow      128—154
Compressible fluid flow, approximate solution      144
Compressible fluid flow, boundary value problems      141—145
Compressible fluid flow, Chaplygin's solutions      135
Compressible fluid flow, construction of solutions in hodograph plane      133—141
Compressible fluid flow, non-linear differential equation      129
Conducting surface, polarization      191
Conductor potential, of a surface, by itself      174
Conductor potential, of a surface, with respect to a domain      158 195
Conductor potential, relation to capacity      185
Conductor potential, relation to Green's function      185 187
Conductor system, forces and moments      197
Conformal mapping, Riemann's theorem      80 84
Conjugate functions      73
Conservation of energy      155 162
Conservation of mass      26 30 32 33
Conservative forces      28
Continuity equation      26 72 128
Continuous source distribution      9 33
Courant, R.      20 119 261 315 371;
Courant, R., and Hilbert, D.      7 20 71 178 260 261 265 269 273 292 311 326 331 332 385
D'Alembert's paradox      88
Davis, P.      355
Dead water regions      65 103
Dead water regions, existence proof      104
Deformation tensor      208
Diaz, J.B., and Weinstein, A.      56
Dielectric coefficient      155
Differential equations, elastic state      215
Differential equations, elastic state, plane strain      247
Differential equations, elastic state, thin plates      234
Differential equations, electrostatic field      155
Differential equations, fluid flow      25—28
Differential equations, fluid flow, axially symmetric      119—120 124
Differential equations, fluid flow, compressible      128—129 132
Differential equations, fluid flow, incompressible      30
Differential equations, fluid flow, irrotational      29 30
Differential equations, fluid flow, stationary      29 30
Differential equations, fluid flow, two-dimensional      72 73
Differential equations, heat transfer      2
Differential equations, heat transfer, endothermal reaction      21
Differential equations, heat transfer, homogeneous body      3 4
Differential equations, heat transfer, steady flow      3 4
Differential equations, magnetostatic field      166
Differential operators, complex      111
Diffusion      3
Dilatation, axes      208
Dilatation, principal      208
Dipole      8 33 75
Dipole flow, complex potential      85
Dipole, at infinity      35 41 75
Dipole, axially symmetric case      122
Dipole, two-dimensional complex      75
Dirichlet identities      46 297
Dirichlet product      47 259 297
Dirichlet's integral      17 20 21 23 31 47 161 216 259 273
Dirichlet's integral, definition      20
Dirichlet's integral, monotonic property      20
Dirichlet's principle      19 372
Discontinuity surfaces      15 64 70
Displacement vector field      206
Dominants, method of      138 289
Doubly-orthogonal systems      352
Doubly-orthogonal systems, application of continuation problems      355
Doubly-orthogonal systems, completeness      354
E-metric      259
Eigenfunctions      see Complete systems
Eigenvalues, estimates      391
Eigenvalues, stability      388
Elastic body, differential equation      215
Elastic potential      212
Electrostatic capacity      185
Electrostatic capacity, expressed by Green's function      186
Electrostatic capacity, lower bounds      188 204
Electrostatic capacity, relation to conductor potential      185 186 203
Electrostatic capacity, relation to harmonic kernel      203—204
Electrostatic field, energy      161
Electrostatic field, energy, general equations      155
Electrostatic field, energy, minimum principle      162
Electrostatic polarization      54 191
Electrostatic polarization potential      189
Electrostatic polarization tensor      53 191
Electrostatic polarization, inequalities for      54 193
Endothermal processes      21
Energy integral      258—259
Energy, elastic body      215
Energy, electrostatic field      161
Energy, fluid flow      31 44 86
Energy, plane strain      247—248
Energy, polarization      192
Energy, reaction field      190—192
Energy, thin plates      234—235
Equation of continuity      26 72 128
Equation of state      25 29
Equation of state, adiabatic      135
Equilibrium conditions, elastic bodies      210
Equilibrium conditions, thin plates      234
Error term, for kernel function      355—357
Existence proof, for boundary value problem      371—81
Extremal functions      19 23 31 32 58 147 162 188 277 301
Extremum problems, lift      98
Extremum problems, virtual mass      56 66—69
Farrell, O.      285 286
Flow patterns      33 35 74 76 136
Fluid dynamics, fundamental equations      25—28
Fluid flow, axially symmetric      119
Fluid flow, compressible two-dimensional      128
Fluid flow, incompressible      29 30 71
Fluid flow, irrotational      28 30 72
Fluid flow, stationary      28 30
Fluid flow, two-dimensional      71
Fluid flow, walled-in region      32
Fluids, non-viscous      25
Flux integral      262
Forces, mass      209
Forces, surface      209
Frank, P., and Mises, R. v.      3 7 28 71 157 178 236 246 273
Frechet, M.      261
Fredholm, J.      226 313 333 334
Free boundaries      66
Free boundaries, as extremal problem      66—70
Free boundaries, characterizations      65 67 68
Free boundaries, existence proofs      68
Free boundaries, plane flow      103
Free edge      236
Friedrichs, K.      246; see also Mises and Friedrichs
Functional      18
Functional derivative      18 147
Fundamental functions      272
Fundamental singularity, axially symmetric flow      121
Fundamental singularity, biharmonic equation      237
Fundamental singularity, complex formula      74
Fundamental singularity, construction      392
Fundamental singularity, definition      10 269
Fundamental singularity, Laplace's equation in three variables      9
Fundamental singularity, Laplace's equation in two variables      74
Fundamental singularity, pseudo-logarithmic plane      149
Fundamental singularity, used to construct a complete orthonormal system      282—285
Fundamental solutions      10 35 272
Fundamental solutions, biharmonic equation      238
Fundamental solutions, plane flows      76—81
Fundamental solutions, stability      384
Fundamental tensor fields      222
Fundamental tensor fields, construction      228—229
Galerkin, B.G.      232
Garabedian, P.R., and Schiffer, M.      108 119 381
Garabedian, P.R., and Spencer, D.C.      1U8
Garabedian, P.R., Lewy, H., and Schiffer, M.      108
Gelbart, A.      154; see also Bers and Gelbart
Geometric quantity      49 298
Goursat, E.      15 265 268 273 334
Green's formula, for functions of $\Sigma$      270
Green's formula, three dimensions      2 11
Green's formula, two dimensions      260
Green's function, definition      12 78 263 270
Green's function, for $\Delta u-qu$      270
Green's function, for $\mathrm{div} (\kappa\, \mathrm{grad}\, u) = 0$      12
Green's function, for biharmonic equation      238—239 244
Green's function, for Laplace's equation, in three dimensions      50 179
Green's function, for Laplace's equation, in two dimensions      78 263
Green's function, in the case of a circle      81 304
Green's function, in the case of infinite domains      50
Green's function, in the case of one argument at infinity      95 187
Green's function, inequalities for      22 52 179 360 366—368 383 386 393
Green's function, physical interpretation      12 179
Green's function, properties, conformal invariance      304
Green's function, properties, dependence on differential equation      386
Green's function, properties, dependence on domain      367
Green's function, properties, monotonicity      367 386
Green's function, properties, non-negativity      13 50 366
Green's function, properties, superadditivity      368
Green's function, properties, symmetry      50 78 263 271
Green's function, properties, unic multiplier      273
Green's function, relation to capacity      186
Green's function, relation to complex kernel function      112 245
Green's function, relation to conductor potential      185 187
Green's function, relation to mapping function      80
Green's function, series development      393
Green's function, used in deriving Poisson's formula      82
Green's function, used in solving boundary value problems      12 79 82 126 157 199 239 242 271
Green's function, variational and comparison formulas      95 194 364 366
Green's tensor field      222
Gunther, N.M.      15 261 268 273
Hadamard, J.      64 95 98 111 198 246 273 365 371
Hamel, G.      268
Harmonic functions, behavior at infinity      34—35
Harmonic functions, inequalities      40
Harmonic functions, representation by, $\mathfrak G$-kernel      110
Harmonic functions, representation by, Green's function      79 199
Harmonic functions, representation by, Poisson's formula      82
Harmonic functions, singular points      33—35
Harmonic kernel function, three-dimensional case      198—206
Heat conduction      1
Heat equation      2
Heat equation, for a homogeneous body      3 4
Heat equation, for steady flow      3 4
Heat equation, modified      21
Hellinger, E., and Toeplitz, O.      268 326
Helmholtz, H.      64 65
Hilbert space      116 240 322
Hilbert space, fundamental theorem of linear transformations      324
Hilbert, D.      268 273 333 334 355 402;
Hodograph method      141
Hodograph plane      130
Hodograph plane, transition from      130—132 143—144
Hooke's law      213
Hopf, E.      262
Hurwitz, A., and Courant, R.      76
Hypergeometric series      140
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