Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries

Abstract

We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $x_0\in\partial\{u=\varphi\}$: (i) either $u(x)-\varphi(x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+\alpha})$ for some $c>0$, (ii) or $u(x)-\varphi(x)=o(|x-x_0|^{1+s+\alpha})$, where $d$ is the distance to the contact set $\{u=\varphi\}$. Moreover, we show that the set of free boundary points $x_0$ satisfying (i) is open, and that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near those points. These results were only known for the fractional Laplacian \cite{CSS}, and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators.