Higher order parabolic boundary Harnack inequality in $C^1$ and $C^{k,\alpha}$ domains

TL;DR

This paper establishes a higher order boundary Harnack inequality for solutions to second order parabolic equations in moving domains with $C^1$ and $C^{k,eta}$ regularity, showing the quotient of solutions inherits boundary smoothness.

Contribution

It introduces a higher order boundary Harnack inequality in parabolic settings for $C^1$ and $C^{k,eta}$ domains, extending boundary regularity results.

Findings

Boundary quotient of solutions is as smooth as the boundary

New proof of higher order free boundary regularity in parabolic obstacle problem

Extension of boundary Harnack inequality to moving domains

Abstract

We study the boundary behaviour of solutions to second order parabolic linear equations in moving domains. Our main result is a higher order boundary Harnack inequality in $C^1$ and $C^{k,\alpha}$ domains, providing that the quotient of two solutions vanishing on the boundary of the domain is as smooth as the boundary. As a consequence of our result, we provide a new proof of higher order regularity of the free boundary in the parabolic obstacle problem.