Boundary regularity for nonlocal operators with kernels of variable orders

Abstract

We study the boundary regularity of solutions of the Dirichlet problem for the nonlocal operator with a kernel of variable orders. Since the order of differentiability of the kernel is not represented by a single number, we consider the generalized Holder space. We prove that there exists a unique viscosity solution of Lu = f in D, u = 0 in R-n\D, where D is a bounded C-1,C-1 open set, and that the solution u satisfies u is an element of C-V(D) and u/V(d(D)) is an element of C-alpha(D) with the uniform estimates, where V is the renewal function and d(D)(x) = dist(x, partial derivative D).