A parabolic analogue of the higher-order comparison theorem of De Silva and Savin

TL;DR

This paper establishes regularity results for the quotient of caloric functions near the boundary of certain domains, extending higher-order comparison theorems to the parabolic setting and providing counterexamples at boundary corners.

Contribution

It introduces a parabolic analogue of the higher-order comparison theorem, showing boundary regularity of caloric function quotients in $H^{k+\alpha}$ domains and space-time $C^{1,\alpha}$ domains.

Findings

Quotients of caloric functions are $H^{k+\alpha}$ up to the boundary for $k\geq 2$.

For $k=1$, the quotient is $H^{1+\alpha}$ in space-time $C^{1,\alpha}$ domains.

Counterexamples show regularity fails at non-lateral boundary points.

Abstract

We show that the quotient of two caloric functions which vanish on a portion of the lateral boundary of a $H^{k+ \alpha}$ domain is $H^{k+ \alpha}$ up to the boundary for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+\alpha}$ if the domain is assumed to be space-time $C^{1, \alpha}$ regular. This can be thought of as a parabolic analogue of a recent important result in [DS1], and we closely follow the ideas in that paper. We also give counterexamples to the fact that analogous results are not true at points on the parabolic boundary which are not on the lateral boundary, i.e., points which are at the corner and base of the parabolic boundary.