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Parabolic Systems with Polynomial Growth and Regularity

About this Title

Frank Duzaar, Department Mathematik, Universität Erlangen–Nürnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Giuseppe Mingione, Dipartimento di Matematica, Università di Parma, Viale Usberti 53/a, Campus, 43100 Parma, Italy and Klaus Steffen, Mathematisches Institut, Heinrich Heine Universität Düsseldorf, Universitätstr.1 D-40225, Düsseldorf, Germany

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 214, Number 1005
ISBNs: 978-0-8218-4967-5 (print); 978-1-4704-0622-6 (online)
Published electronically: March 10, 2011
MathSciNet review: 2866816
Keywords:Parabolic systems, regularity, higher integrability, singular sets
MSC: Primary 35D10, 35K92

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Table of Contents


  • Introduction
  • Chapter 1. Results
  • Chapter 2. Basic material, assumptions
  • Chapter 3. The $A$-caloric approximation lemma
  • Chapter 4. Partial regularity
  • Chapter 5. Some basic regularity results and a priori estimates
  • Chapter 6. Dimension estimates
  • Chapter 7. Hölder continuity of $u$
  • Chapter 8. Non-linear Calderón-Zygmund theory


We establish a series of optimal regularity results for solutions to general non-linear parabolic systems

$\displaystyle u_t- \operatorname{div} a(x,t,u,Du)+H=0 , $

under the main assumption of polynomial growth at rate $p$ i.e.

$\displaystyle \vert a(x,t,u,Du)\vert\leq L(1+\vert Du\vert^{p-1}) ,\qquad p \geq 2 \;. $

We 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 here achieved.

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