Boundary Regularity for the $\infty$-Heat Equation

TL;DR

This paper investigates the boundary regularity of solutions to the normalized infinity-heat equation in arbitrary domains, introducing new criteria and methods for understanding boundary behavior.

Contribution

It provides a comprehensive characterization of boundary regularity, including barrier functions, an exterior ball condition, and a Petrovsky-like criterion for the normalized infinity-heat equation.

Findings

Characterization of regular boundary points using barrier functions

Establishment of an exterior ball condition for boundary regularity

Development of a Petrovsky-like criterion for the equation

Abstract

We study the boundary regularity for the normalised $\infty$-heat equation $u_t = \Delta_{\infty}^Nu$ in arbitrary domains. Perron's Method is used for constructing solutions. We characterize regular boundary points with barrier functions, and prove an Exterior Ball result. A Petrovsky-like criterion is established.