Electronic ISBN:  9781470406226 
Product Code:  MEMO/214/1005.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 214; 2011; 118 ppMSC: Primary 35;
The authors establish a series of optimal regularity results for solutions to general nonlinear parabolic systems \[ u_t \mathrm{div} \ a(x,t,u,Du)+H=0,\] under the main assumption of polynomial growth at rate \(p\) i.e. \[a(x,t,u,Du)\leq L(1+Du^{p1}), p \geq 2.\] They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first CalderónZygmund estimates for nonhomogeneous problems are achieved here.

Table of Contents

Chapters

Acknowledgments

Introduction

1. Results

2. Basic material, assumptions

3. The $A$caloric approximation lemma

4. Partial regularity

5. Some basic regularity results and a priori estimates

6. Dimension estimates

7. Hölder continuity of $u$

8. Nonlinear CalderónZygmund theory


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 Table of Contents

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The authors establish a series of optimal regularity results for solutions to general nonlinear parabolic systems \[ u_t \mathrm{div} \ a(x,t,u,Du)+H=0,\] under the main assumption of polynomial growth at rate \(p\) i.e. \[a(x,t,u,Du)\leq L(1+Du^{p1}), p \geq 2.\] They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first CalderónZygmund estimates for nonhomogeneous problems are achieved here.

Chapters

Acknowledgments

Introduction

1. Results

2. Basic material, assumptions

3. The $A$caloric approximation lemma

4. Partial regularity

5. Some basic regularity results and a priori estimates

6. Dimension estimates

7. Hölder continuity of $u$

8. Nonlinear CalderónZygmund theory