Lieberman G.M. — Second Order Parabolic Differential Equations :: Электронная библиотека попечительского совета мехмата МГУ

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Lieberman G.M. — Second Order Parabolic Differential Equations

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Название: Second Order Parabolic Differential Equations

Автор: Lieberman G.M.

Аннотация:

This book is an introduction to the general theory of second order parabolic differential equations, which model many important, time-dependent physical systems. It studies the existence, uniqueness, and regularity of solutions to a variety of problems with Dirichlet boundary conditions and general linear and nonlinear boundary conditions by means of a priori estimates. The first seven chapters give a description of the linear theory and are suitable for a graduate course on partial differential equations. The last eight chapters cover the nonlinear theory for smooth solutions. They include much of the author's research and are aimed at researchers in the field. A unique feature is the emphasis on time-varying domains.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 447

Добавлена в каталог: 10.04.2007

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Предметный указатель

$H_{\zeta}$ condition      236—238 253—254 318
$L^{p,q}$ spaces      121 129 130 154 229
$M^{p,\lambda}$ spaces      see “Morrey spaces”
$W^{1,2}$ solution      30—41
Barrier      38
Barrier, global      38—40
Barrier, local      40
Barrier, local, from earlier time      92 231—257 397 408 413
Barrier, local, two-sided      40
Bellman equations      363 375 382
Boundary point lemma      12
Brouwer fixed point theorem      204
Calderon — Zygmund decomposition      159—161 189
Campanato space      49—51
Campanato space, equivalent to Holder space      50—51 61—62
Caristi fixed point theorem      208—210
Cauchy — Dirichlet problem for fully nonlinear equations      369—378 381 385—420
Cauchy — Dirichlet problem for linear equations      36—41 77—78 89—95
Cauchy — Dirichlet problem for quasilinear equations      219—320
Comparison principles      13 37 159 188 219—222 363—364
Conormal problems      137—140 179 324—341 346—347 355—356
Contraction mapping principle      29
Convex-increasing domain      238—241 317 413—414
Cube decomposition      see “Calderon — Zygmund decomposition”
Curvature conditions      243—247
Cylinder condition      233—234
DeGiorgi class      143—150
DeGiorgi iteration      144—145
Dini continuity      83—84
Distance functions, regularized      71—72
Distance functions, spatial      241—243
Domains with boundary      75 105—108 241
Domains with $H_{\delta}$ boundary      75—79 87—89 95 97 243 306—307 309 314—318
Duality argument      172—173
Elliptic part of operator      205
Extension of a function      73—74 175 183—185 235—236 328—331 341—343
False mean curvature equation      265—266 287—289 316—318
Fully nonlinear equation      361—384
Fully nonlinear equation, boundary gradient estimate      389 413—414
Fully nonlinear equation, boundary regularity      375—378 389
Fully nonlinear equation, Cauchy — Dirichlet problem      385—420
Fully nonlinear equation, global bound      389 412—413
Fully nonlinear equation, global solvability      369—371 374—375 377 381
Fully nonlinear equation, gradient bound      408
Fully nonlinear equation, gradient estimate      372—373 378—379 389—390 414
Fully nonlinear equation, Holder gradient estimate      373—375 379
Fully nonlinear equation, Holder second derivative estimate      365—371 377—378 400
Fully nonlinear equation, maximum estimate      364 378
Fully nonlinear equation, oblique derivative problem      378—381
Fully nonlinear equation, second derivative bounds      391—400 409—412 414
Fully nonlinear equation, uniformly parabolic      361—363 365—384
Gateaux variation      211—212 370
Global solvability      36—41 92—93 95—97 104—108 138 141—142 182 185 207—208 213—215 313—318 351 355—356 400—401 407 412 414
Gradient estimates, boundary      15 62—63 170 231—257 326—346 371
Gradient estimates, global      32—33 259—266 270—271 275—276 281 286—291
Gradient estimates, Holder      55—56 301—313 346—355
Gradient estimates, local      62—63 266—269 277—289 291—294
Harnack inequality for strong solutions      192
Harnack inequality for weak solutions      129
Harnack inequality, weak      186—192
Hessian equations      385—420
Hessian quotient equations      406—407
Holder continuity      129—132 134—135 140
Holder, continuity      46—49
Holder, norm      46—47
Holder, norm, weighted      47
Holder, semi-norm      46
Integro-differential equations      19 85 98
Interpolation inequalities      47—49 73 174—175
Linear parabolic equations (operators), classical solutions, Cauchy — Dirichlet problem      77—78 87—95
Linear parabolic equations (operators), classical solutions, maximum principles      7—20
Linear parabolic equations (operators), classical solutions, oblique derivative problem      79 88—89 95—97
Linear parabolic equations (operators), divergence form      see “Weak solutions”
Linear parabolic equations (operators), first initial-boundary value problem      see “Cauchy — Dirichlet problem”
Linear parabolic equations (operators), strong solutions      155—201
Linear parabolic equations (operators), strong solutions, estimates      173—185
Linear parabolic equations (operators), strong solutions, boundary estimates      177—185
Linear parabolic equations (operators), strong solutions, boundary regularity      192—197

Linear parabolic equations (operators), strong solutions, Cauchy — Dirichlet problem      181—183
Linear parabolic equations (operators), strong solutions, global bound      155—159
Linear parabolic equations (operators), strong solutions, Holder estimate      192
Linear parabolic equations (operators), strong solutions, Holder estimates      197
Linear parabolic equations (operators), strong solutions, local bound      185—186
Linear parabolic equations (operators), strong solutions, maximum principle      155—159
Linear parabolic equations (operators), strong solutions, oblique derivative problem      183—185
Linear parabolic equations (operators), weak solutions      21—43 55—56 62—67 101—154
Linear parabolic equations (operators), weak solutions, boundary regularity      132—135
Linear parabolic equations (operators), weak solutions, Cauchy — Dirichlet problem      36—41 77 104—108 141—142
Linear parabolic equations (operators), weak solutions, continuous      35—43 55—56
Linear parabolic equations (operators), weak solutions, global bound      116—121 138—139
Linear parabolic equations (operators), weak solutions, Holder continuity      129—132 134—135 140
Linear parabolic equations (operators), weak solutions, local bound      121—132
Linear parabolic equations (operators), weak solutions, mixed boundary value problem      65—68 140—141
Linear parabolic equations (operators), weak solutions, oblique derivative problem      79 142
Local solvability      30—36 89—92 206—207
Maclaurin inequalities      402 406
Marcinkiewicz interpolation theorem      161—163
Maximal function      163—172
Maximum principles for fully nonlinear equations      364
Maximum principles for linear equations, classical solutions      7—20
Maximum principles for linear equations, strong solutions      155—159
Maximum principles for linear equations, weak solutions      129 139—140
Maximum principles for quasilinear equations      220—226 322—326
Mean curvature      242—244
Mean curvature equation      231 244 265 277 286—287 291—293 315—316 318 339—340
Method of continuity      29—30 369—370
Mixed boundary value problems      140—141 179 215—216 351
Monge — Ampere equation      385 386 407—414
Morrey space      130
Morrey space, weighted      56—58
Moser iteration      119—120 122 143
Newton inequalitites      402—403
Newton — Maclaurin inequalities      402—403 406
Nonlinear boundary condition      see “Oblique derivative problem”
Oblique derivative problem for fully nonlinear equations      378—381
Oblique derivative problem for linear equations      8 13—14
Oblique derivative problem for quasilinear equations      211—215 321—360
Parabolic frustum      12 13 15 19 41 93 236 239
Paraboloid condition, interior      13 18
Perron process      36—41 89—93 95—96
Poincare’s inequality      28 114—116
Pucci operators      362
Quasilinear parabolic equations      203—360
Quasilinear parabolic equations, boundary regularity      231—257 305—309
Quasilinear parabolic equations, Cauchy — Dirichlet problem      206—208
Quasilinear parabolic equations, conormal problem      324—341 346—347 355—356
Quasilinear parabolic equations, global bounds      220—226 322—326
Quasilinear parabolic equations, gradient estimate      259—299
Quasilinear parabolic equations, gradient estimate, boundary      231—257 326—346
Quasilinear parabolic equations, Holder gradient estimate      301—313 346—355
Quasilinear parabolic equations, maximum principle      220—226
Quasilinear parabolic equations, oblique derivative problem      211—215 321—360
Quasilinear parabolic equations, weak solutions      301—302 324—341 346—347 355—356
Regular point      38—41
Regularized distance      see “Distance functions regularized”
Schauder estimates, boundary      65—67 69—71
Schauder estimates, global      75—79 141—142
Schauder estimates, interior      56—60
Schauder estimates, intermediate      74—75
Schauder fixed point theorem      205—206
Sobolev imbedding theorem      109—112
Sobolev inequalities      109—114 135—137 271—275
Sobolev inequalities, weighted      112—114 275 327—328
Steklov average      102 108 118 126
Strong maximum principle      129
Strong solutions, Cauchy — Dirichlet problem      181—183
Strong solutions, oblique derivative problem      183—185
Subsolution      92 117
Subsolution, strict      387—391 393 396—397 407 408 412
Supersolution      117
Symmetric function      386 387
Symmetric polynomials      401—407
Uniformly parabolic equations, quasilinear      204 253 264 270—271 290 292—294 313—315 318 340—346
Vitali Covering Lemma      162—163
Weak maximum principle      7—9
Weak solutions      101—154 221—226
Weak solutions, estimates      165—173 177—179

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