Global gradient bounds for the parabolic p-Laplacian system

Abstract

A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic $p$-Laplacian system is locally H\"older continuous in the interior. However, the boundary regularity is not yet well understood. In this paper we prove a boundary $L^\infty$-estimate for the spatial gradient $Du$ of solutions to the parabolic $p$-Laplacian system \begin{equation*} \partial_t u - \Div \big(|Du|^{p-2}Du\big) = 0 \quad\mbox{in $\Omega\times(0,T)$} \end{equation*} for $p\ge 2$, together with a quantitative estimate. In particular, this implies the global Lipschitz regularity of solutions. The result continues to hold for the so called asymptotically regular parabolic systems.