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

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are C1,α, then they are C∞. This completes the study of regular points, initiated in [5]. In order to achieve this, we need to establish optimal boundary regularity estimates for solutions to linear nonlocal equations in Ck,α domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for k=∞) and by the second author and Serra (for k=1)