The Dirichlet problem for nonlocal elliptic operators with $C^{0,\alpha }$ exterior data

Abstract

In this note we study the boundary regularity of solutions to nonlocal Dirichlet problems of the form $ Lu=0$ in $ \Omega $, $ u=g$ in $ \mathbb{R}^N\setminus \Omega $, in non-smooth domains $ \Omega $. When $ g$ is smooth enough, then it is easy to transform this problem into an homogeneous Dirichlet problem with a bounded right-hand side for which the boundary regularity is well understood. Here, we study the case in which $ g\in C^{0,\alpha }$, and establish the optimal Hölder regularity of $ u$ up to the boundary. Our results extend previous results of Grubb for $ C^\infty $ domains $ \Omega $.