Global gradient bounds for the parabolic p-Laplacian system

A by now classical result due to DiBenedetto states that the spatial gradient of solutions to the parabolic p-Laplacian system is locally Holder continuous in the interior. However, the boundary regularity is not yet well understood. In this paper, we prove a boundary L-infinity-estimate for the spatial gradient Du of solutions to the parabolic p-Laplacian system partial derivative tu - div (vertical bar Du vertical bar(p-2) Du) = 0 in Omega x (0,T) with non-homogeneous boundary data for p >= 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.