Regularity results for nonlocal equations and applications

We introduce the concept of $C^{m,\alpha}$-nonlocal operators, extending the notion of second order elliptic operator in divergence form with $C^{m,\alpha}$-coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the H\"older continuity of the gradient of the solutions in the case of $C^{0,\alpha}$-coefficients and the classical Shauder estimates for $C^{m+1,\alpha}$-coefficients. We further apply the regularity results for $C^{m,\alpha}$-nonlocal operators to derive optimal higher order regularity estimates of Lipschitz graphs with prescribed Nonlocal Mean Curvature. Applications to nonlocal equation on manifolds are also provided.