Elliptic and parabolic equations

Contents......Page 10
Preface......Page 6
1.1.1 Some frequently applied inequalities......Page 17
1.1.2 Spaces Ck(O) and Ck0(O)......Page 18
1.1.3 Smoothing operators......Page 19
1.1.4 Cut-off functions......Page 21
1.1.6 Local flatting of the boundary......Page 22
1.2.1 Spaces Ck a(O) and Ck a(O)......Page 23
1.2.2 Interpolation inequalities......Page 24
1.2.3 Spaces C2k+a k+a/2(QT)......Page 29
1.3.1 Weak derivatives......Page 30
1.3.2 Sobolev spaces Wk p(O) and Wk p0(O)......Page 31
1.3.4 Interpolation inequality......Page 33
1.3.5 Embedding theorem......Page 35
1.3.6 Poincare's inequality......Page 37
1.4.1 Spaces W2k kp(QT) W2k kp(QT) W2k kp(QT) V2(QT) and V(QT)......Page 40
1.4.2 Embedding theorem......Page 42
1.4.3 Poincare's inequality......Page 44
1.5.1 Some propositions on functions in H1(Q+)......Page 45
1.5.2 Trace of functions in H1(O)......Page 49
1.5.3 Trace of functions in H1(QT) = W1 12(QT)......Page 51
2.1 Weak Solutions of Poisson's Equation......Page 55
2.1.1 Definition of weak solutions......Page 56
2.1.2 Riesz's representation theorem and its application......Page 57
2.1.3 Transformation of the problem......Page 59
2.1.4 Existence of minimizers of the corresponding functional......Page 60
2.2.1 Difference operators......Page 63
2.2.2 Interior regularity......Page 66
2.2.3 Regularity near the boundary......Page 69
2.2.4 Global regularity......Page 72
2.2.5 Study of regularity by means of smoothing operators......Page 74
2.3.1 Weak solutions......Page 76
2.3.2 Riesz's representation theorem and its application......Page 77
2.3.3 Variational method......Page 78
2.3.4 Lax-Milgram's theorem and its application......Page 80
2.3.5 Fredholm's alternative theorem and its application......Page 83
3.1 Energy Method......Page 87
3.1.1 Definition of weak solutions......Page 88
3.1.2 A modified Lax-Milgram's theorem......Page 89
3.1.3 Existence and uniqueness of the weak solution......Page 91
3.2 Rothe's Method......Page 95
3.3 Galerkin's Method......Page 101
3.4 Regularity of Weak Solutions......Page 105
3.5.1 Energy method......Page 110
3.5.2 Rothe's method......Page 112
3.5.3 Galerkin's method......Page 113
4.1.1 Weak maximum principle for solutions of Laplace's equation......Page 121
4.1.2 Weak maximum principle for solutions of Poisson's equation......Page 123
4.2.1 Weak maximum principle for solutions of the homogeneous heat equation......Page 127
4.2.2 Weak maximum principle for solutions of the nonhomogeneous heat equation......Page 128
4.3.1 Weak subsolutions (supersolutions)......Page 132
4.3.2 Local boundedness estimate for weak solutions of Laplace's equation......Page 134
4.3.3 Local boundedness estimate for solutions of Poisson's equation......Page 136
4.3.4 Estimate near the boundary for weak solutions of Poisson's equation......Page 138
4.4.2 Local boundedness estimate for weak solutions of the homogeneous heat equation......Page 139
4.4.3 Local boundedness estimate for weak solutions of the nonhomogeneous heat equation......Page 142
5.1.1 Mean value formula......Page 147
5.1.3 Estimate of sup u......Page 149
5.1.4 Estimate of inf u......Page 151
5.1.5 Harnack's inequality......Page 157
5.1.6 Holder's estimate......Page 159
5.2 Harnack's Inequalities for Solutions of the Homogeneous Heat Equation......Page 161
5.2.1 Weak Harnack's inequality......Page 162
5.2.2 Holder's estimate......Page 171
5.2.3 Harnack's inequality......Page 172
6.1 Campanato Spaces......Page 175
6.2.1 Estimates to be established......Page 181
6.2.2 Caccioppoli's inequalities......Page 184
6.2.3 Interior estimate for Laplace's equation......Page 189
6.2.4 Near boundary estimate for Laplace's equation......Page 191
6.2.5 Iteration lemma......Page 193
6.2.6 Interior estimate for Poisson's equation......Page 194
6.2.7 Near boundary estimate for Poisson's equation......Page 197
6.3 Schauder's Estimates for General Linear Elliptic Equations......Page 203
6.3.2 Interior estimate......Page 204
6.3.3 Near boundary estimate......Page 207
6.3.4 Global estimate......Page 209
7.1 t-Anisotropic Campanato Spaces......Page 213
7.2.1 Estimates to be established......Page 215
7.2.2 Interior estimate......Page 216
7.2.3 Near bottom estimate......Page 224
7.2.4 Near lateral estimate......Page 230
7.2.5 Near lateral-bottom estimate......Page 243
7.2.6 Schauder's estimates for general linear parabolic equations......Page 247
8.1.1 The case of elliptic equations......Page 249
8.1.2 The case of parabolic equations......Page 252
8.2.1 Existence and uniqueness of the classical solution for Poisson's equation......Page 256
8.2.2 The method of continuity......Page 262
8.2.3 Existence and uniqueness of classical solutions for general linear elliptic equations......Page 264
8.3 Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations......Page 265
8.3.1 Existence and uniqueness of the classical solution for the heat equation......Page 266
8.3.2 Existence and uniqueness of classical solutions for general linear parabolic equations......Page 267
9.1.1 Lp estimates for Poisson's equation in cubes......Page 271
9.1.2 Lp estimates for general linear elliptic equations......Page 276
9.1.3 Existence and uniqueness of strong solutions for linear elliptic equations......Page 280
9.2.1 Lp estimates for the heat equation in cubes......Page 282
9.2.2 Lp estimates for general linear parabolic equations......Page 287
9.2.3 Existence and uniqueness of strong solutions for linear parabolic equations......Page 288
10.1.2 Solvability of quasilinear elliptic equations......Page 293
10.1.3 Solvability of quasilinear parabolic equations......Page 296
10.2 Maximum Estimate......Page 298
10.3 Interior Holder's Estimate......Page 300
10.4 Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation......Page 303
10.5 Boundary Holder's Estimate and Boundary Gradient Estimate......Page 305
10.6 Global Gradient Estimate......Page 312
10.7.1 An iteration lemma......Page 317
10.7.2 Morrey's theorem......Page 318
10.7.3 Holder's estimate......Page 319
10.8.1 Interior Holder's estimate for gradients of solutions......Page 323
10.8.2 Boundary Holder's estimate for gradients of solutions......Page 324
10.9.1 Solvability of more general quasilinear elliptic equations......Page 326
10.9.2 Solvability of more general quasilinear parabolic equations......Page 327
11.1.1 Brouwer degree......Page 329
11.1.2 Leray-Schauder degree......Page 331
11.2 Existence of a Heat Equation with Strong Nonlinear Source......Page 333
12.1 Monotone Method for Parabolic Problems......Page 339
12.1.2 Iteration and monotone property......Page 340
12.1.3 Existence results......Page 343
12.1.4 Application to more general parabolic equations......Page 346
12.1.5 Nonuniqueness of solutions......Page 348
12.2 Monotone Method for Coupled Parabolic Systems......Page 352
12.2.2 Definition of supersolutions and subsolutions......Page 353
12.2.3 Monotone sequences......Page 355
12.2.4 Existence results......Page 366
12.2.5 Extension......Page 369
13.1 Linear Equations......Page 371
13.1.1 Formulation of the first boundary value problem......Page 372
13.1.2 Solvability of the problem in a space similar to H1......Page 377
13.1.3 Solvability of the problem in Lp(O)......Page 378
13.1.4 Method of elliptic regularization......Page 381
13.1.5 Uniqueness of weak solutions in Lp(O) and regularity......Page 382
13.2 A Class of Special Quasilinear Degenerate Parabolic Equations - Filtration Equations......Page 384
13.2.1 Definition of weak solutions......Page 385
13.2.2 Uniqueness of weak solutions for one dimensional equations......Page 387
13.2.3 Existence of weak solutions for one dimensional equations......Page 389
13.2.4 Uniqueness of weak solutions for higher dimensional equations......Page 394
13.2.5 Existence of weak solutions for higher dimensional equations......Page 397
13.3 General Quasilinear Degenerate Parabolic Equations......Page 400
13.3.1 Uniqueness of weak solutions for weakly degenerate equations......Page 401
13.3.2 Existence of weak solutions for weakly degenerate equations......Page 409
13.3.3 A remark on quasilinear parabolic equations with strong degeneracy......Page 415
Bibliography......Page 419
Index......Page 421