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

We study the obstacle problem for integro-differential operators of order 2s, with s ∈ (0, 1). Our main result establishes that the free boundary is C and u ∈ C near all regular points. Namely, we prove the following dichotomy at all free boundary points x0 ∈ ∂{u = φ}: (i) either u(x)− φ(x) = c d(x) + o(|x− x0|) for some c > 0, (ii) or u(x)− φ(x) = o(|x− x0|), where d is the distance to the contact set {u = φ}. Moreover, we show that the set of free boundary points x0 satisfying (i) is open, and that the free boundary is C and u ∈ C near those points. These results were only known for the fractional Laplacian [CSS08], 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.