Hölder gradient estimates for a class of singular or degenerate parabolic equations

Abstract

We prove interior Hölder estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$ u_t=|\nabla u|^κ\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ where $p\in (1,\infty )$ and $\kappa \in (1-p,\infty ).$ This includes the from $L^{\infty }$ to $C^{1,\alpha }$ regularity for parabolic $p$-Laplacian equations in both divergence form with $\kappa =0$, and non-divergence form with $\kappa =2-p$. This work is a continuation of a paper by the last two authors \cite{JS}.

29 pages