Natural second-order regularity for parabolic systems with operators having $(p,\delta )$-structure and depending only on the symmetric gradient

Abstract

In this paper we consider parabolic problems with stress tensor depending only on the symmetric gradient. By developing a new approximation method (which allows to use energy-type methods typical for linear problems) we provide an approach to obtain global regularity results valid for general potential operators with $(p,\delta )$-structure, for all $p>1$ and for all $\delta >0$. In this way we prove ``natural'' second order spatial regularity -- up to the boundary -- in the case of homogeneous Dirichlet boundary conditions. The regularity results, are presented with full details for the parabolic setting in the case $p>2$. However, the same method also yields regularity in the elliptic case and for $1<p\le 2$, thus proving in a different way results already known.