Optimal regularity for supercritical parabolic obstacle problems

TL;DR

This paper proves that solutions to supercritical parabolic obstacle problems with fractional operators are more regular than previously known, achieving $C^{1,1}$ regularity in space and time, and describes the free boundary behavior.

Contribution

It establishes the optimal $C^{1,1}$ regularity for solutions in supercritical regimes, surpassing previous elliptic regularity results, and characterizes free boundary expansions.

Findings

Solutions are $C^{1,1}$ in space and time.

Free boundary is $C^{1,eta}$ regular.

Precise expansion at free boundary points.

Abstract

We study the obstacle problem for parabolic operators of the type $\partial_t + L$, where $L$ is an elliptic integro-differential operator of order $2s$, such as $(-\Delta)^s$, in the supercritical regime $s \in (0,{1/2})$. The best result in this context was due to Caffarelli and Figalli, who established the $C^{1,s}_x$ regularity of solutions for the case $L = (-\Delta)^s$, the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually \textit{more} regular than in the elliptic case. More precisely, we show that they are $C^{1,1}$ in space and time, and that this is optimal. We also deduce the $C^{1,\alpha}$ regularity of the free boundary. Moreover, at all free boundary points $(x_0,t_0)$, we establish the following expansion: $$(u - \varphi)(x_0+x,t_0+t) = c_0(t - a\cdot x)_+^2 + O(t^{2+\alpha}+|x|^{2+\alpha}),$$ with $c_0 > 0$, $\alpha > 0$ and $a \in \mathbb R^n$.