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 C 1 , 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 C k , 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). [ABSTRACT FROM AUTHOR]