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

TL;DR

This paper proves that free boundaries in obstacle problems for integro-differential operators are infinitely smooth once they are initially $C^{1,eta}$, by establishing optimal boundary regularity estimates for solutions in $C^{k,eta}$ domains.

Contribution

It extends regularity results for free boundaries in nonlocal obstacle problems from $C^{1,eta}$ to $C^$, completing the understanding of regular points.

Findings

Free boundaries are $C^$ if initially $C^{1,eta}$.

Established optimal boundary regularity estimates for solutions.

Extended previous results to higher regularity of free boundaries.

Abstract

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators. Our main result establishes that, once free boundaries are $C^{1,\alpha}$, then they are $C^\infty$. 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 $C^{k,\alpha}$ domains. These new estimates are the core of our paper, and extend previously known results by Grubb (for $k=\infty$) and by the second author and Serra (for $k=1$).