退化抛物方程

Preface

1. Elliptic equations： Harnack 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

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 7

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 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 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 1 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

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

l. Introduction

2. The intrinsic Hamack inequality

3. Local comparison functions

4. Proof of Theorem 2.1

5. Proof of Theorem 2.2

6. Global versus local estimates

7. Global Hamack 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 Cauchy problem with compactly supported initial data

13. Bibliographical notes

VII. Hamack 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 Hamack inequality for all

5. Sup-estimates for

6. Local subsolution.

7. Time expansion of positivity

8. Space-time configurations

9. Proof of the Hamack inequality

10. Proof of Theorem 1.2

11. Bibliographical notes

VIII. Degenerate and singular parabolic systems

1. Introduction

2. Boundedness of weak solutions

3. Weak differentiability of Du and energy estimates for IOul

4. Boundedness of lOut. Qualitative estimates

5. Quantitative sup-bounds of

6. General structures

7. Bibliographical notes

IX. Parabolic p-systems： Hiolder continuity of Du

1. The main theorem

2. Estimating the oscillation of Du

3. Hlder 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 l.1-（i）

9. Proof of Proposition 1.l-（ii）

10. Proof of Proposition 1.1-（iii）

11. Proof of Proposition 1.1 concluded

12. Proof of Proposition

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 IDwl （the case p > 2 ）

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

7. Comparing w and v （the case max 1

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。目次：函数空间；弱解和局部能量估计；退化抛物方程的Holder连续性；奇异抛物方程解的Holder连续性；弱解有界性；Harnack 估计：p]2；Harnack 估计和；退化和奇异抛物系统；抛物p系统：Du的Holder连续性；抛物p系统：边界奇异；ΣT中的非负解：p]2； ΣT中的非负解：1[p[2。
　　本书适用于数学专业的研究生和科研人员。