Higher regularity of the free boundary in the parabolic Signorini problem

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 $k\ge 2$. 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.

Revised version, to appear in Calculus of Variations and Partial Differential Equations

References in corpus (4)

• Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem
• Higher regularity of the free boundary in the elliptic Signorini problem
• A note on higher regularity boundary Harnack inequality
• Parabolic Obstacle Problems. Quasi-convexity and Regularity