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 H\"older space. We prove that there exists a unique viscosity solution of $Lu = f$ in $D$, $u=0$ in $\mathbb{R}^n \setminus D$, where $D$ is a bounded $C^{1,1}$ open set, and that the solution $u$ satisfies $u \in C^V(D)$ and $u/V(d_D) \in C^\alpha (D)$ with the uniform estimates, where $V$ is the renewal function and $d_D(x) = \mbox{dist}(x, \partial D)$.