On the normalized $p$-parabolic equation in arbitrary domains

TL;DR

This paper investigates the boundary regularity of the normalized p-parabolic equation in arbitrary domains, introducing new criteria, constructing solutions via Perron's method, and analyzing the behavior as p approaches infinity.

Contribution

It provides a comprehensive classification of boundary points, establishes a Petrovsky criterion, and identifies a fundamental solution for the normalized p-parabolic equation.

Findings

Classification of boundary regular points using barrier functions

Establishment of a Petrovsky criterion for the equation

Identification of a fundamental solution and analysis of p→∞ behavior

Abstract

The boundary regularity for the normalized $p$-parabolic equation $u_t =\frac{1}{p}|Du|^{2-p}\Delta_pu$ is studied. Perron's method is used to construct solutions in arbitrary domains. We classify the regular boundary points in terms of barrier functions, and prove an Exterior Sphere result. A fundamental solution is identified. A Petrovsky criterion is established, and we examine the convergence of solutions as $p \to \infty$.