The tusk condition and Petrovski criterion for the normalized $p\mspace{1mu}$-parabolic equation

TL;DR

This paper extends boundary regularity results, including the tusk condition and Petrovski criterion, from the heat equation to the normalized p-parabolic equation, establishing conditions for boundary point regularity and H"older continuity.

Contribution

It generalizes the tusk condition and Petrovski criterion to the normalized p-parabolic equation, providing new regularity criteria and continuity results.

Findings

Tusk condition guarantees boundary regularity for the normalized p-parabolic equation.

A sharp Petrovski criterion determines the regularity of boundary points.

Regularity is influenced by multiplying one side of the equation by a constant.

Abstract

We study boundary regularity for the normalized $p\mspace{1mu}$-parabolic equation in arbitrary bounded domains. Effros and Kazdan (Indiana Univ. Math. J. 20 (1970), 683-693) showed that the so-called tusk condition guarantees regularity for the heat equation. We generalize this result to the normalized $p\mspace{1mu}$-parabolic equation, and also obtain H\"older continuity. The tusk condition is a parabolic version of the exterior cone condition. We also obtain a sharp Petrovski criterion for the regularity of the latest moment of a domain. This criterion implies that the regularity of a boundary point is affected if one side of the equation is multiplied by a constant.