Higher regularity of the free boundary in the parabolic Signorini problem

TL;DR

This paper proves higher regularity of the quotient of caloric functions near a slit and applies this to show the free boundary in the parabolic thin obstacle problem is infinitely smooth in space and time.

Contribution

It establishes higher regularity results for quotients of caloric functions and demonstrates the smoothness of the free boundary in the parabolic thin obstacle problem.

Findings

Quotients of caloric functions are $H^{k+ eta}$ regular near a slit.

Free boundary in the parabolic thin obstacle problem is $C^{ty}$ smooth.

Results extend regularity theory for parabolic free boundary problems.

Abstract

We show that the quotient of two caloric functions which vanish on a portion of an $H^{k+ \alpha}$ regular slit is $H^{k+ \alpha}$ at the slit, for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+\alpha}$ if the slit 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 [DSS14a], whose ideas inspired us. As an application, we show that the free boundary near a regular point of the parabolic thin obstacle problem studied in [DGPT] with zero obstacle is $C^{\infty}$ regular in space and time.