Optimal regularity for nonlocal elliptic equations and free boundary problems

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp \mathrm{\mid }y{\mathrm{\mid }}^{-n-2s}$. 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 $K(y)\asymp \mathrm{\mid }y{\mathrm{\mid }}^{-n-2s}$. 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 $C^s$ regularity and non-degeneracy estimates.