Gradient regularity for a singular parabolic equation in non-divergence form

TL;DR

This paper proves interior gradient regularity for viscosity solutions of a class of singular non-divergence parabolic equations, employing approximation and perturbation techniques to handle the singularity.

Contribution

It establishes new interior Hölder regularity results for the gradient of solutions to singular parabolic equations in non-divergence form.

Findings

Proved interior Hölder regularity of the gradient.

Developed two alternative methods: iteration with approximation and small perturbation.

Extended regularity theory to a class of singular parabolic equations.

Abstract

In this paper we consider viscosity solutions of a class of non-homogeneous singular parabolic equations $$\partial_t u-|Du|^\gamma\Delta_p^N u=f,$$ where $-1<\gamma<0$, $1<p<\infty$, and $f$ is a given bounded function. We establish interior H\"older regularity of the gradient by studying two alternatives: The first alternative uses an iteration which is based on an approximation lemma. In the second alternative we use a small perturbation argument.