The Petrovski\u{\i} criterion and barriers for degenerate and singular p-parabolic equations

TL;DR

This paper establishes precise Petrovskii criteria for p-parabolic equations in both degenerate and singular cases, and demonstrates that boundary regularity cannot be solely characterized by barrier existence.

Contribution

It provides sharp criteria for boundary regularity in p-parabolic equations and shows the limitations of barrier-based characterizations.

Findings

Sharp Petrovskii criteria for p>2 and 1<p<2 cases

Existence of irregular boundary points with barriers

Regularity cannot be fully characterized by barriers

Abstract

In this paper we obtain sharp Petrovskii criteria for the p-parabolic equation, both in the degenerate case p>2 and the singular case 1<p<2. We also give an example of an irregular boundary point at which there is a barrier, thus showing that regularity cannot be characterized by the existence of just one barrier.