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

Audrito, Alessandro; Ros-Oton, Xavier

doi:10.1090/proc/15121

The Dirichlet problem for nonlocal elliptic operators with $C^{0,\alpha }$ exterior data
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by Alessandro Audrito and Xavier Ros-Oton PDF

Proc. Amer. Math. Soc. 148 (2020), 4455-4470 Request permission

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 $\Omega$.

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