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Lieberman G.M. Ч Second Order Parabolic Differential Equations
Lieberman G.M. Ч Second Order Parabolic Differential Equations

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Ќазвание: Second Order Parabolic Differential Equations

јвтор: Lieberman G.M.


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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2005

 оличество страниц: 447

ƒобавлена в каталог: 10.04.2007

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
$H_{\zeta}$ condition      236Ч238 253Ч254 318
$L^{p,q}$ spaces      121 129 130 154 229
$M^{p,\lambda}$ spaces      see УMorrey spacesФ
$W^{1,2}$ solution      30Ч41
Barrier      38
Barrier, global      38Ч40
Barrier, local      40
Barrier, local, from earlier time      92 231Ч257 397 408 413
Barrier, local, two-sided      40
Bellman equations      363 375 382
Boundary point lemma      12
Brouwer fixed point theorem      204
Calderon Ч Zygmund decomposition      159Ч161 189
Campanato space      49Ч51
Campanato space, equivalent to Holder space      50Ч51 61Ч62
Caristi fixed point theorem      208Ч210
Cauchy Ч Dirichlet problem for fully nonlinear equations      369Ч378 381 385Ч420
Cauchy Ч Dirichlet problem for linear equations      36Ч41 77Ч78 89Ч95
Cauchy Ч Dirichlet problem for quasilinear equations      219Ч320
Comparison principles      13 37 159 188 219Ч222 363Ч364
Conormal problems      137Ч140 179 324Ч341 346Ч347 355Ч356
Contraction mapping principle      29
Convex-increasing domain      238Ч241 317 413Ч414
Cube decomposition      see УCalderon Ч Zygmund decompositionФ
Curvature conditions      243Ч247
Cylinder condition      233Ч234
DeGiorgi class      143Ч150
DeGiorgi iteration      144Ч145
Dini continuity      83Ч84
Distance functions, regularized      71Ч72
Distance functions, spatial      241Ч243
Domains with $H_1$ boundary      75 105Ч108 241
Domains with $H_{\delta}$ boundary      75Ч79 87Ч89 95 97 243 306Ч307 309 314Ч318
Duality argument      172Ч173
Elliptic part of operator      205
Extension of a function      73Ч74 175 183Ч185 235Ч236 328Ч331 341Ч343
False mean curvature equation      265Ч266 287Ч289 316Ч318
Fully nonlinear equation      361Ч384
Fully nonlinear equation, boundary gradient estimate      389 413Ч414
Fully nonlinear equation, boundary regularity      375Ч378 389
Fully nonlinear equation, Cauchy Ч Dirichlet problem      385Ч420
Fully nonlinear equation, global bound      389 412Ч413
Fully nonlinear equation, global solvability      369Ч371 374Ч375 377 381
Fully nonlinear equation, gradient bound      408
Fully nonlinear equation, gradient estimate      372Ч373 378Ч379 389Ч390 414
Fully nonlinear equation, Holder gradient estimate      373Ч375 379
Fully nonlinear equation, Holder second derivative estimate      365Ч371 377Ч378 400
Fully nonlinear equation, maximum estimate      364 378
Fully nonlinear equation, oblique derivative problem      378Ч381
Fully nonlinear equation, second derivative bounds      391Ч400 409Ч412 414
Fully nonlinear equation, uniformly parabolic      361Ч363 365Ч384
Gateaux variation      211Ч212 370
Global solvability      36Ч41 92Ч93 95Ч97 104Ч108 138 141Ч142 182 185 207Ч208 213Ч215 313Ч318 351 355Ч356 400Ч401 407 412 414
Gradient estimates, boundary      15 62Ч63 170 231Ч257 326Ч346 371
Gradient estimates, global      32Ч33 259Ч266 270Ч271 275Ч276 281 286Ч291
Gradient estimates, Holder      55Ч56 301Ч313 346Ч355
Gradient estimates, local      62Ч63 266Ч269 277Ч289 291Ч294
Harnack inequality for strong solutions      192
Harnack inequality for weak solutions      129
Harnack inequality, weak      186Ч192
Hessian equations      385Ч420
Hessian quotient equations      406Ч407
Holder continuity      129Ч132 134Ч135 140
Holder, continuity      46Ч49
Holder, norm      46Ч47
Holder, norm, weighted      47
Holder, semi-norm      46
Integro-differential equations      19 85 98
Interpolation inequalities      47Ч49 73 174Ч175
Linear parabolic equations (operators), classical solutions, Cauchy Ч Dirichlet problem      77Ч78 87Ч95
Linear parabolic equations (operators), classical solutions, maximum principles      7Ч20
Linear parabolic equations (operators), classical solutions, oblique derivative problem      79 88Ч89 95Ч97
Linear parabolic equations (operators), divergence form      see УWeak solutionsФ
Linear parabolic equations (operators), first initial-boundary value problem      see УCauchy Ч Dirichlet problemФ
Linear parabolic equations (operators), strong solutions      155Ч201
Linear parabolic equations (operators), strong solutions, $L^p$ estimates      173Ч185
Linear parabolic equations (operators), strong solutions, boundary estimates      177Ч185
Linear parabolic equations (operators), strong solutions, boundary regularity      192Ч197
Linear parabolic equations (operators), strong solutions, Cauchy Ч Dirichlet problem      181Ч183
Linear parabolic equations (operators), strong solutions, global bound      155Ч159
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      185Ч186
Linear parabolic equations (operators), strong solutions, maximum principle      155Ч159
Linear parabolic equations (operators), strong solutions, oblique derivative problem      183Ч185
Linear parabolic equations (operators), weak solutions      21Ч43 55Ч56 62Ч67 101Ч154
Linear parabolic equations (operators), weak solutions, boundary regularity      132Ч135
Linear parabolic equations (operators), weak solutions, Cauchy Ч Dirichlet problem      36Ч41 77 104Ч108 141Ч142
Linear parabolic equations (operators), weak solutions, continuous      35Ч43 55Ч56
Linear parabolic equations (operators), weak solutions, global bound      116Ч121 138Ч139
Linear parabolic equations (operators), weak solutions, Holder continuity      129Ч132 134Ч135 140
Linear parabolic equations (operators), weak solutions, local bound      121Ч132
Linear parabolic equations (operators), weak solutions, mixed boundary value problem      65Ч68 140Ч141
Linear parabolic equations (operators), weak solutions, oblique derivative problem      79 142
Local solvability      30Ч36 89Ч92 206Ч207
Maclaurin inequalities      402 406
Marcinkiewicz interpolation theorem      161Ч163
Maximal function      163Ч172
Maximum principles for fully nonlinear equations      364
Maximum principles for linear equations, classical solutions      7Ч20
Maximum principles for linear equations, strong solutions      155Ч159
Maximum principles for linear equations, weak solutions      129 139Ч140
Maximum principles for quasilinear equations      220Ч226 322Ч326
Mean curvature      242Ч244
Mean curvature equation      231 244 265 277 286Ч287 291Ч293 315Ч316 318 339Ч340
Method of continuity      29Ч30 369Ч370
Mixed boundary value problems      140Ч141 179 215Ч216 351
Monge Ч Ampere equation      385 386 407Ч414
Morrey space      130
Morrey space, weighted      56Ч58
Moser iteration      119Ч120 122 143
Newton inequalitites      402Ч403
Newton Ч Maclaurin inequalities      402Ч403 406
Nonlinear boundary condition      see УOblique derivative problemФ
Oblique derivative problem for fully nonlinear equations      378Ч381
Oblique derivative problem for linear equations      8 13Ч14
Oblique derivative problem for quasilinear equations      211Ч215 321Ч360
Parabolic frustum      12 13 15 19 41 93 236 239
Paraboloid condition, interior      13 18
Perron process      36Ч41 89Ч93 95Ч96
PoincareТs inequality      28 114Ч116
Pucci operators      362
Quasilinear parabolic equations      203Ч360
Quasilinear parabolic equations, boundary regularity      231Ч257 305Ч309
Quasilinear parabolic equations, Cauchy Ч Dirichlet problem      206Ч208
Quasilinear parabolic equations, conormal problem      324Ч341 346Ч347 355Ч356
Quasilinear parabolic equations, global bounds      220Ч226 322Ч326
Quasilinear parabolic equations, gradient estimate      259Ч299
Quasilinear parabolic equations, gradient estimate, boundary      231Ч257 326Ч346
Quasilinear parabolic equations, Holder gradient estimate      301Ч313 346Ч355
Quasilinear parabolic equations, maximum principle      220Ч226
Quasilinear parabolic equations, oblique derivative problem      211Ч215 321Ч360
Quasilinear parabolic equations, weak solutions      301Ч302 324Ч341 346Ч347 355Ч356
Regular point      38Ч41
Regularized distance      see УDistance functions regularizedФ
Schauder estimates, boundary      65Ч67 69Ч71
Schauder estimates, global      75Ч79 141Ч142
Schauder estimates, interior      56Ч60
Schauder estimates, intermediate      74Ч75
Schauder fixed point theorem      205Ч206
Sobolev imbedding theorem      109Ч112
Sobolev inequalities      109Ч114 135Ч137 271Ч275
Sobolev inequalities, weighted      112Ч114 275 327Ч328
Steklov average      102 108 118 126
Strong maximum principle      129
Strong solutions, Cauchy Ч Dirichlet problem      181Ч183
Strong solutions, oblique derivative problem      183Ч185
Subsolution      92 117
Subsolution, strict      387Ч391 393 396Ч397 407 408 412
Supersolution      117
Symmetric function      386 387
Symmetric polynomials      401Ч407
Uniformly parabolic equations, quasilinear      204 253 264 270Ч271 290 292Ч294 313Ч315 318 340Ч346
Vitali Covering Lemma      162Ч163
Weak maximum principle      7Ч9
Weak solutions      101Ч154 221Ч226
Weak solutions, $L^p$ estimates      165Ч173 177Ч179
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