Degenerate Parabolic Equations

Cover......Page 1
Degenerate Parabolic Equations......Page 2
0387940200......Page 3
1. Elliptic equations: Harnack estimates and Holder continuity......Page 4
2. Parabolic equations: Harnack estimates and Holder continuity......Page 5
3. Parabolic equations and systems......Page 6
4. Main results......Page 7
Contents......Page 10
§1. Some notation......Page 16
§2. Basic facts about W^{1,p}(Ω) and W^{1,p}_0{Q)......Page 18
§3. Parabolic spaces and embeddings......Page 22
§4. Auxiliary lemmas......Page 27
§5. Bibliographical notes......Page 30
§1. Quasilinear degenerate or singular equations......Page 31
§2. Boundary value problems......Page 35
§3. Local integral inequalities......Page 37
§4. Energy estimates near the boundary......Page 46
§5. Restricted structures: the levels k and the constant γ......Page 53
§6. Bibliographical notes......Page 55
§1. The regularity theorem......Page 56
§2. Preliminaries......Page 58
§3. The main proposition......Page 59
§4. The first alternative......Page 64
§5. The first alternative continued......Page 67
§6. The first alternative concluded......Page 70
§7. The second alternative......Page 73
§8. The second alternative continued......Page 77
§9. The second alternative concluded......Page 79
§10. Proof of Proposition 3.1......Page 83
§11. Regularity up to t = 0......Page 84
§12. Regularity up to S_T. Dirichlet data......Page 87
§14. Remarks on stability......Page 89
§15. Bibliographical notes......Page 90
§1. Singular equations and the regularity theorems......Page 92
§2. The main proposition......Page 94
§3. Preliminaries......Page 96
§4. Rescaled iterations......Page 99
§5. The first alternative......Page 103
§6. Proof of Lemma 5.1. Integral inequalities......Page 107
§7. An auxiliary proposition......Page 110
§8. Proof of Proposition 7.1 when (7.6) holds......Page 112
§9. Removing the assumption (6.1)......Page 116
§10. The second alternative......Page 117
§11. The second alternative concluded......Page 121
§12. Proof of the main proposition......Page 124
§13. Boundary regularity......Page 125
§14. Miscellaneous remarks......Page 129
§15. Bibliographical notes......Page 131
V. Boundedness of weak solutions......Page 0
§1. Introduction......Page 132
§2. Quasilinear parabolic equations......Page 133
§3. Sup-bounds......Page 135
§4. Homogeneous structures. The degenerate case p > 2......Page 137
§5. Homogeneous structures. The singular case 1 < p < 2......Page 140
§6. Energy estimates......Page 143
§7. Local iterative inequalities......Page 146
§8. Local iterative inequalities (p > max {1; \\frac{2N}{N+2}})......Page 149
§9. Global iterative inequalities......Page 150
§10. Homogeneous structures and 1 < p \\leq max {1; \\frac{2N}{N+2}})......Page 152
§11. Proof of Theorems 3.1 and 3.2......Page 153
§12. Proof of Theorem 4.1......Page 155
§13. Proof of Theorem 4.2......Page 157
§14. Proof of Theorem 4.3......Page 158
§15. Proof of Theorem 4.5......Page 159
§16. Proof of Theorems 5.1 and 5.2......Page 162
§17. Natural growth conditions......Page 164
§18. Bibliographical notes......Page 170
§1. Introduction......Page 171
§2. The intrinsic Harnack inequality......Page 172
§3. Local comparison functions......Page 174
§4. Proof of Theorem 2.1......Page 178
§5. Proof of Theorem 2.2......Page 182
§6. Global versus local estimates......Page 184
§7. Global Harnack estimates......Page 186
§8. Compactly supported initial data......Page 187
§9. Proof of Proposition 8.1......Page 189
§10. Proof of Proposition 8.1 continued......Page 192
§11. Proof of Proposition 8.1 concluded......Page 194
§12. The Cauchy problem with compactly supported initial data......Page 195
§13. Bibliographical notes......Page 198
§1. The Harnack inequality......Page 199
§2. Extinction in finite time (bounded domains)......Page 203
§3. Extinction in finite time (in \\mathbb{R}^N)......Page 206
§4. An integral Harnack inequality for all 1 §5. Sup-estimates for \\frac{2N}{N+2} §6. Local subsolutions......Page 214
§7. Time expansion of positivity......Page 218
§8. Space-time configurations......Page 219
§9. Proof of the Harnack inequality......Page 221
§10. Proof of Theorem 1.2......Page 226
§11. Bibliographical notes......Page 229
§1. Introduction......Page 230
§2. Boundedness of weak solutions......Page 233
§3. Weak differentiability of |Du|^{\\frac{p-2}{2}} Du and energy estimates for |Du|......Page 238
§4. Boundedness of |Du|. Qualitative estimates......Page 246
§5. Quantitative sup-bounds of |Du|......Page 253
§6. General structures......Page 258
§7. Bibliographical notes......Page 259
§1. The main theorem......Page 260
§2. Estimating the oscillation of Du......Page 263
§3. Holder continuity of Du (the case p > 2)......Page 266
§4. Holder continuity of Du (the case 1§5. Some algebraic Lemmas......Page 273
§6. Linear parabolic systems with constant coefficients......Page 278
§7. The perturbation lemma......Page 283
§8. Proof of Proposition 1.1-(i)......Page 290
§9. Proof of Proposition 1.1-(ii)......Page 293
§10. Proof of Proposition 1.1-(iii)......Page 297
§11. Proof of Proposition 1.1 concluded......Page 299
§12. Proof of Proposition 1.2-(i)......Page 301
§13. Proof of Proposition 1.2 concluded......Page 303
§15. Bibliographical notes......Page 306
§1. Introduction......Page 307
§2. Flattening the boundary......Page 309
§3. An iteration lemma......Page 312
§4. Comparing w and v (thecase p > 2)......Page 314
§5. Estimating the local average of |Dw| (the case p > 2)......Page 319
§6. Estimating the local averages of w (the case p > 2)......Page 320
§7. Comparing w and v (the case max {1; \\frac{2N}{N+2}} §8. Estimating the local average of |Dw|......Page 328
§9. Bibliographical notes......Page 330
§1. Introduction......Page 331
§2. Behaviour of non-negative solutions as |x| -> \\infty and as t  earrow 0......Page 332
§3. Proof of (2.4)......Page 334
§4. Initial traces......Page 337
§5. Estimating |Du|^{p-1} in Σ_T......Page 338
§6. Uniqueness for data in L_{loc}^1(\\mathbb{R}^N)......Page 341
§7. Solving the Cauchy problem......Page 345
§8. Bibliographical notes......Page 348
§1. Introduction......Page 349
§2. Weak solutions......Page 352
§3. Estimating |Du|......Page 355
§4. The weak Harnack inequality and initial traces......Page 359
§5. The uniqueness theorem......Page 361
§6. An auxiliary proposition......Page 365
§8. Solving the Cauchy problem......Page 377
§9. Compactness in the space variables......Page 378
§10. Compactness in the t variable......Page 381
§11. More on the time—compactness......Page 385
§12. The limiting process......Page 386
§13. Bounded solutions. A counterexample......Page 391
§14. Bibliographical notes......Page 394
Bibliography......Page 396