H\"older gradient estimates for a class of singular or degenerate parabolic equations

TL;DR

This paper establishes interior H"older continuity for the spatial gradients of viscosity solutions to a class of singular or degenerate parabolic equations, extending regularity results for the parabolic p-Laplacian.

Contribution

It provides new interior gradient estimates for solutions to a broad class of singular or degenerate parabolic equations, including cases with different $ abla u$-dependent degeneracies.

Findings

Proves interior H"older estimates for spatial gradients.

Includes regularity results for both divergence and non-divergence form equations.

Extends previous work on parabolic p-Laplacian equations.

Abstract

We prove interior H\"older estimate for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation $$ u_t=|\nabla u|^{\kappa}\mbox{div} (|\nabla u|^{p-2}\nabla u), $$ 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}.