In this talk we discuss several recent results on the nonlocal one-phase free boundary problem.
We show optimal regularity and non-degeneracy of solutions, and establish an improvement of flatness result, as well as higher regularity of the free boundary near regular points.
This allows us to prove smoothness of the free boundary in an open, dense set. Moreover, we prove that the free boundary is smooth in two dimensions. We study this problem for general integro-differential operators of order 2s.
Our technique is purely nonlocal and deeply connected to nonlocal equations with local boundary conditions.
This talk is based on recent joint works together with Xavier Ros-Oton.