Optimal regularity for nonlocal elliptic equations and free boundary problems

In this article we establish for the first time the Cˢ boundary regularity of solutions to nonlocal elliptic equations with kernels K (y) |y|^-n-2s. This was known to hold only when K is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal C^1+s regularity of solutions to obstacle problems for general nonlocal operators with kernels K (y) |y|^-n-2s. Again, this was only known when K is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal Cˢ regularity and non-degeneracy estimates.