数学真正意义上研究退化和奇异抛物偏微分方程是近些年才开始的，起源于60年代中叶DeGiorgi，Moser，Ladyzenskajia和Ural'tzeva这些人的工作。本书是近些年来该领域的进展的综述。其基本思想来自上个世纪90年代作者在波恩大学的Lipschitz讲义。

Preface

 1. Elliptic equations: Hamack estimates and Holder continuity

 2. Parabolic equations: Hamack estimates and Holder continuity

 3. Parabolic equations and systems

 4. Main results

Ⅰ. Notation and function spaces

 1. Some notation

 2. Basic facts about W1,and W2

 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 levels k and the constant

 6. Bibliographical notes

Ⅲ. Holder continuity of solutions of degenerate

parabolic equations

 1. The regularity theorem

 2. Preliminaries

 3. The main proposition

 4. The first alternative

 5. The first altemative 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 to t = 0

 12. Regularity up to ST. Dirichlet data

 13. Regularity at ST. Variational data

 14. Remarks on stability

 15. Bibliographical notes

Ⅳ. Holder 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

 15. Bibliographical notes

Ⅴ. Boundedness of weak solutions

 1. Introduction

 2. Quasilinear parabolic equations

 3. Sup-bounds

 4. Homogeneous structures. The degenerate case p > 2

 5. Homogeneous structures. The singular case 1 < p < 2

 6. Energy estimates

 7. Local iterative inequalities

 8. Local iterative inequalities

 9. Global iterative inequalities

 10. Homogeneous structures and 1 <p<max

 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 case p>2

 1. Introduction

 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. i concluded

 12. The Canchy problem with compactly supported initial data

 13. Bibliographical notes

Ⅶ. Harnack estimates and extinction profile for

singular equations

 1. The Hamack inequality

 2. Extinction in finite time (bounded domains)

 3. Extinction in finite time (in RN)

 4. An integral Harnack inequality for all 1 < p < 2

 5. Sup-estimates for < p < 2

 6. Local subsolutions

 7. Time expansion of positivity

 8. Space-time configurations

 9. Proof of the Harnack inequality

 10. Proof of Theorem 1.2

 11. Bibliographical notes

Ⅷ. Degenerate and singular parabolic systems

 1. Introduction

 2. Bonndedness of weak solutions

 3. Weak differentiability of |Du|: Du and energy estimates for |Du|

 4. Bonndedness of |Du|. Qualitative estimates

 5. Quantitative sup-bounds of |Du|

 6. General structures

 7. Bibliographical notes

Ⅸ. Parabolic p-systems: Holder continuity of Du

 1. The main theorem

 2. Estimating the oscillation of Du

 3. Holder continuity of Du (the case p > 2 )

 4. Holder continuity of Du (the case 1 < p < 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

 15. Bibliographical notes

Ⅹ Parabolic p-systems: boundary regularity

 1. Introduction

 2. Flattening the boundary

 3. An iteration lemma

 4. Comparing w and v (the case p > 2)

 5. Estimating the local average of Ho (thecase p > 2)

 6. Estimating the local averages of w (the case p > 2)

 7. Comparing w and v

 8. Estimating the local average of |Dw|

 9. Bibfiographieal notes

Ⅺ. Non-negative solutions in ∑T. The case p>2

 1. Introduction

 2. Behaviour of non-negative solutions as |x|

 3. Proof of (2.4)

 4. Initial traces

 5. Estimating

 6. Uniqueness for data in L(RN)

 7. Solving the Cauchy problem

 8. Bibliographical notes

Ⅻ. Non-negative solutions in ∑r. The case 1<p<2

 1. Introduction

 2. Weak solutions

 3. Estimating |Da|

 4. The weak Harnack inequality and initial traces

 5. 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 the t variable

 11. More on the time-compactness

 12. The limiting process

 13. Bounded solutions. A counterexample

 14. Bibliographical notes

Bibliography