H\"older regularity for the gradient of the inhomogeneous parabolic normalized $p$-Laplacian

TL;DR

This paper establishes local Hölder continuity for the gradient of solutions to a parabolic normalized p-Laplacian equation with a source term, using an iterative improvement of flatness method.

Contribution

It provides the first regularity result for the gradient of solutions to inhomogeneous normalized p-Laplacian evolution equations.

Findings

Proved local Hölder regularity of the gradient

Applied an improvement of flatness technique

Results are relevant for stochastic tug-of-war game models

Abstract

In this paper we study an evolution equation involving the normalized $p$-Laplacian and a bounded continuous source term. The normalized $p$-Laplacian is in non divergence form and arises for example from stochastic tug-of-war games with noise. We prove local $C^{\alpha, \frac{\alpha}{2}}$ regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.