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Parabolic Systems with Polynomial Growth and Regularity
Frank Duzaar, Universität Erlangen-Nürnberg, Germany, Giuseppe Mingione, Università di Parma, Italy, and Klaus Steffen, Heinrich-Heine-Universität, Düsseldorf, Germany
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Memoirs of the American Mathematical Society
2011; 118 pp; softcover
Volume: 214
ISBN-10: 0-8218-4967-0
ISBN-13: 978-0-8218-4967-5
List Price: US$71
Individual Members: US$42.60
Institutional Members: US$56.80
Order Code: MEMO/214/1005
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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.

Table of Contents

  • Results
  • Basic material, assumptions
  • The \(A\)-caloric approximation lemma
  • Partial regularity
  • Some basic regularity results and a priori estimates
  • Dimension estimates
  • Hölder continuity of \(u\)
  • Non-linear Calderón-Zygmund theory
  • Bibliography
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