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 Hölder space. We prove that there exists a unique viscosity solution of Lu=f in D, u=0 in Rn∖D, where D is a bounded C1,1 open set, and that the solution u satisfies u∈CV(D) and u/V(dD)∈Cα(D) with the uniform estimates, where V is the renewal function and dD(x)=dist(x,∂D).