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

TL;DR

This paper proves new regularity results for obstacle problems involving general integro-differential operators, showing that free boundaries are smooth and solutions have optimal regularity near regular points, extending previous results beyond the fractional Laplacian.

Contribution

It establishes the regularity of free boundaries and solutions for obstacle problems with general integro-differential operators, not just the fractional Laplacian, using purely nonlocal methods.

Findings

Free boundary is $C^{1, extgamma}$ near regular points.

Solutions are $C^{1,s}$ near regular points.

Results extend to fully nonlinear integro-differential operators.

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.