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Parabolic Systems with Polynomial Growth and Regularity

Frank Duzaar Universität Erlangen-Nürnberg, Erlangen, Germany
Giuseppe Mingione Università di Parma, Parma, Italy
Klaus Steffen Heinrich-Heine-Universität, Düsseldorf, Germany
Available Formats:
Electronic ISBN: 978-1-4704-0622-6
Product Code: MEMO/214/1005.E
118 pp
List Price: $75.00 MAA Member Price:$67.50
AMS Member Price: $45.00 Click above image for expanded view Parabolic Systems with Polynomial Growth and Regularity Frank Duzaar Universität Erlangen-Nürnberg, Erlangen, Germany Giuseppe Mingione Università di Parma, Parma, Italy Klaus Steffen Heinrich-Heine-Universität, Düsseldorf, Germany Available Formats:  Electronic ISBN: 978-1-4704-0622-6 Product Code: MEMO/214/1005.E 118 pp  List Price:$75.00 MAA Member Price: $67.50 AMS Member Price:$45.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2142011
MSC: Primary 35;

The authors establish a series of optimal regularity results for solutions to general non-linear 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. $|a(x,t,u,Du)|\leq L(1+|Du|^{p-1}), p \geq 2.$ 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 Calderón-Zygmund estimates for non-homogeneous 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. Non-linear Calderón-Zygmund theory
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Volume: 2142011
MSC: Primary 35;

The authors establish a series of optimal regularity results for solutions to general non-linear 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. $|a(x,t,u,Du)|\leq L(1+|Du|^{p-1}), p \geq 2.$ 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 Calderón-Zygmund estimates for non-homogeneous 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. Non-linear Calderón-Zygmund theory
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