Local regularity for quasi-linear parabolic equations in non-divergence form

TL;DR

This paper establishes local regularity results, including Hölder and Lipschitz estimates, for viscosity solutions of non-divergence form quasi-linear parabolic equations of p-Laplacian type, addressing both degenerate and singular cases.

Contribution

It introduces new regularity estimates for viscosity solutions of non-divergence form p-Laplacian type equations, including gradient Hölder regularity in degenerate cases.

Findings

Proved local Hölder and Lipschitz estimates for solutions.

Established gradient Hölder regularity in degenerate cases.

Combined method of alternatives with flatness improvement techniques.

Abstract

We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local H\"older and Lipschitz estimates for the solutions. In the degenerate case, we prove the H\"older regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.