On the second order regularity of solutions to the parabolic $p$-Laplace equation

TL;DR

This paper establishes the second order Sobolev regularity of solutions to the parabolic p-Laplace equation, demonstrating that a specific derivative exists and belongs to a local L^2 space, with the range of s being optimal.

Contribution

It proves the existence of second order derivatives of a transformed gradient for solutions to the parabolic p-Laplace equation, with sharp conditions on the parameter s.

Findings

The derivative D(|Du|^{(p-2+s)/2}Du) exists as a function.

This derivative belongs to L^2_{loc} for s > -1.

The range of s for which regularity holds is sharp.

Abstract

In this paper, we study the second order Sobolev regularity of solutions to the parabolic $p$-Laplace equation. For any $p$-parabolic function $u$, we show that $D(|Du|^{\frac{p-2+s}{2}}Du)$ exists as a function and belongs to $L^{2}_{\text{loc}}$ with $s>-1$ and $1<p<\infty$. The range of $s$ is sharp.