Degenerate Parabolic Equations

Inhaltsverzeichnis

• I. Notation and function spaces.
• §1. Some notation.
• §2. Basic facts aboutW1, p(?) andWo1, p(?).
• §3. Parabolic spaces and embeddings.
• §4. Auxiliary lemmas.
• §5. Bibliographical notes.
• II. Weak solutions and local energy estimates.
• §1. Quasilinear degenerate or singular equations.
• §2. Boundary value problems.
• §3. Local integral inequalities.
• §4. Energy estimates near the boundary.
• §5. Restricted structures: the levelskand the constant ?.
• §6. Bibliographical notes.
• III. Hölder continuity of solutions of degenerate parabolic equations.
• §1. The regularity theorem.
• §2. Preliminaries.
• §3. The main proposition.
• §4. The first alternative.
• §5. The first alternative continued.
• §6. The first alternative concluded.
• §7. The second alternative.
• §8. The second alternative continued.
• §9. The second alternative concluded.
• §10. Proof of Proposition 3.1.
• §11. Regularity up tot= 0.
• §12. Regularity up toST. Dirichlet data.
• §13. Regularity atST. Variational data.
• §14. Remarks on stability.
• §15. Bibliographical notes.
• IV. Hölder continuity of solutions of singular parabolic equations.
• §1. Singular equations and the regularity theorems.
• §2. The main proposition.
• §3. Preliminaries.
• §4. Rescaled iterations.
• §5. The first alternative.
• §6. Proof of Lemma 5.1. Integral inequalities.
• §7. An auxiliary proposition.
• §8. Proof of Proposition 7.1 when (7.6) holds.
• §9. Removing the assumption (6.1).
• §10. The second alternative.
• §11. The second alternative concluded.
• §12. Proof of the main proposition.
• §13. Boundary regularity.
• §14. Miscellaneous remarks.
• V. Boundedness of weak solutions.
• §1. Introduction.
• §2. Quasilinear parabolic equations.
• §3. Sup-bounds.
• §4. Homogeneous structures. 2.
• §5. Homogeneous structures. The singular case 1 < 2.
• §6. Energy estimates.
• §7. Local iterative inequalities.
• §8. Local iterative inequalities $$
  \left( {p > max\left\{ {1;\frac{{2N}}
  {{N + 2}}} \right\}} \right)
  $$.
• §9. Global iterative inequalities.
• §10. Homogeneous structures and $$
  1 < p \leqslant max\left\{ {1;\frac{{2N}}
  {{N + 2}}} \right\}
  $$.
• §11. Proof of Theorems 3.1 and 3.2.
• §12. Proof of Theorem 4.1.
• §13. Proof of Theorem 4.2.
• §14. Proof of Theorem 4.3.
• §15. Proof of Theorem 4.5.
• §16. Proof of Theorems 5.1 and 5.2.
• §17. Natural growth conditions.
• §18. Bibliographical notes.
• VI. Harnack estimates: the casep>2.
• §2. The intrinsic Harnack inequality.
• §3. Local comparison functions.
• §4. Proof of Theorem 2.1.
• §5. Proof of Theorem 2.2.
• §6. Global versus local estimates.
• §7. Global Harnack estimates.
• §8. Compactly supported initial data.
• §9. Proof of Proposition 8.1.
• §10. Proof of Proposition 8.1 continued.
• §11. Proof of Proposition 8.1 concluded.
• §12. The Cauchy problem with compactly supported initial data.
• §13. Bibliographical notes.
• VII. Harnack estimates and extinction profile for singular equations.
• §1. The Harnack inequality.
• §2. Extinction in finite time (bounded domains).
• §3. Extinction in finite time (in RN).
• §4. An integral Harnack inequality for all 1   2).
• §4. Hölder continuity ofDu (the case 1 < 2).
• §5. Some algebraic Lemmas.
• §6. Linear parabolic systems with constant coefficients.
• §7. The perturbation lemma.
• §8. Proof of Proposition 1.1-(i).
• §9. Proof of Proposition 1.1-(ii).
• §10. Proof of Proposition 1.1-(iii).
• §11. Proof of Proposition 1.1 concluded.
• §12. Proof of Proposition 1.2-(i).
• §13. Proof of Proposition 1.2 concluded.
• §14. General structures.
• X. Parabolicp-systems: boundary regularity.
• §2. Flattening the boundary.
• §3. An iteration lemma.
• §4. Comparing w and y (the casep> 2).
• §5. Estimating the local average of |Dw| (the casep> 2).
• §6. Estimating the local averages of w (the casep> 2).
• §7. Comparing w and y (the case max $$
  \left\{ {1;\tfrac{{2N}}
  {{N + 2}}} \right\} < p< 2
  $$).
• §8. Estimating the local average of |Dw|.
• §9. Bibliographical notes.
• XI. Non-negative solutions in ? T. The casep>2.
• §2. Behaviour of non-negative solutions as |x| ? ? and as t ? 0.
• §3. Proof of (2.4).
• §4. Initial traces.
• §5. Estimating |Du|p?1 in ? T.
• §6. Uniqueness for data inLloc1(RN).
• §7. Solving the Cauchy problem.
• §8. Bibliographical notes.
• XII. Non-negative solutions in ? T. The case 1 The uniqueness theorem.
• §6. An auxiliary proposition.
• §7. Proof of the uniqueness theorem.
• §8. Solving the Cauchy problem.
• §9. Compactness in the space variables.
• §10. Compactness in thetvariable.
• §11. More on the time—compactness.
• §12. The limiting process.
• §13. Bounded solutions. A counterexample.
• §14. Bibliographical notes.