Boundary regularity for nonlocal elliptic equations over Reifenberg flat domains

TL;DR

This paper establishes sharp boundary regularity results for solutions to nonlocal elliptic equations on Reifenberg flat domains, showing solutions are nearly $C^{s}$ regular under certain flatness conditions.

Contribution

It provides the first sharp boundary regularity estimates for nonlocal elliptic equations on Reifenberg flat sets, extending classical results to more irregular domains.

Findings

Solutions are $C^{s- ext{epsilon}}$ regular up to the boundary.

Regularity holds under small flatness parameters.

The proof uses barrier construction and comparison principles.

Abstract

We prove sharp boundary regularity of solutions to nonlocal elliptic equations arising from operators comparable to the fractional Laplacian over Reifenberg flat sets and with null exterior condition. More precisely, if the operator has order $2s$ then the solution is $C^{s-\varepsilon}$ regular for all $\varepsilon>0$ provided the flatness parameter is small enough. The proof relies on an induction argument and its main ingredients are the construction of a suitable barrier and the comparison principle.