Higher regularity of the free boundary in the parabolic Signorini problem

We show that the quotient of two caloric functions which vanish on a portion of an $H^{k+ α}$ regular slit is $H^{k+ α}$ at the slit, for $k \geq 2$. In the case $k=1$, we show that the quotient is in $H^{1+α}$ if the slit is assumed to be space-time $C^{1, α}$ 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.

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