The Wiener criterion for nonlocal Dirichlet problems

TL;DR

This paper extends the Wiener criterion to nonlocal Dirichlet problems involving integro-differential operators, providing a characterization of boundary regularity using nonlocal potential theory.

Contribution

It introduces a nonlocal Wiener criterion for boundary regularity of solutions to integro-differential equations, advancing the understanding of nonlocal boundary behavior.

Findings

Established a nonlocal Wiener criterion for boundary regularity.

Connected boundary regularity to nonlocal potential theory.

Provided theoretical foundations for analyzing nonlocal Dirichlet problems.

Abstract

We study the boundary behavior of solutions to the Dirichlet problems for integro-differential operators with order of differentiability $s \in (0, 1)$ and summability $p>1$. We establish a nonlocal counterpart of the Wiener criterion, which characterizes a regular boundary point in terms of the nonlocal nonlinear potential theory.