Optimal regularity for supercritical parabolic obstacle problems

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