Optimal regularity for supercritical parabolic obstacle problems

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 $C_x^{1,s}$ 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: $$(u-\phi )(x_0+x,t_0+t)=c_0(t-a\cdot x{)}_{+}^2+O(t^{2+\alpha }+|x{|}^{2+\alpha }),$$ with $c_0>0$, $\alpha >0$ and $a\in {\mathbb{R}}^n$.