Calculus of Variations and Geometric Measure Theory
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F. Duzaar - G. Mingione - K. Steffen

Parabolic systems with polynomial growth and regularity

created by mingione on 29 Sep 2012
modified on 27 Mar 2017


Published Paper

Inserted: 29 sep 2012
Last Updated: 27 mar 2017

Journal: Memoirs Amer. Math. Soc.
Volume: 214
Number: 1005
Pages: 128
Year: 2011


We establish a series of optimal regularity results for solutions to general non-linear parabolic systems \[ u_t- div \ a(x,t,u,Du)+H=0 \] under the main assumption of polynomial growth at rate $p$ i.e. \[ |a(x,t,u,Du)| \leq L(1+|Du|^{p-1})\,,\qquad p \geq 2 \;. \] 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 Calder\'on-Zygmund estimates for non-homogeneous problems are here achieved.

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