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

TL;DR

This paper investigates the boundary regularity of solutions to nonlocal elliptic Dirichlet problems with $C^{0,eta}$ exterior data in non-smooth domains, establishing optimal Hölder regularity results that extend previous smooth domain findings.

Contribution

It extends boundary regularity results for nonlocal elliptic equations to cases with less smooth exterior data and non-smooth domains, providing optimal Hölder regularity estimates.

Findings

Established optimal Hölder regularity of solutions up to the boundary.

Extended previous results from smooth to non-smooth domains.

Generalized boundary regularity theory for nonlocal elliptic operators.

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 H\"older regularity of $u$ up to the boundary. Our results extend previous results of Grubb for $C^\infty$ domains $\Omega$.