H\"older gradient estimates for parabolic homogeneous p-Laplacian equations

TL;DR

This paper establishes interior H"older continuity estimates for the spatial gradient of viscosity solutions to a class of parabolic p-Laplacian equations, which are relevant in stochastic game models and non-divergence form PDEs.

Contribution

It provides the first interior H"older gradient estimates for solutions to the parabolic homogeneous p-Laplacian equation in non-divergence form.

Findings

Proved interior H"older estimates for the spatial gradient of solutions.

Applicable to equations arising from stochastic tug-of-war games.

Bridges gap between stochastic game models and PDE regularity theory.

Abstract

We prove interior H\"older estimates for the spatial gradient of viscosity solutions to the parabolic homogeneous $p$-Laplacian equation \[ u_t=|\nabla u|^{2-p} \mbox{ div} (|\nabla u|^{p-2}\nabla u), \] where $1<p<\infty$. This equation arises from tug-of-war-like stochastic games with noise. It can also be considered as the parabolic $p$-Laplacian equation in non divergence form.