Boundary H\"older regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

TL;DR

This paper establishes boundary Hölder regularity for solutions to fractional Laplacian equations on Reifenberg flat domains, using an iterative approach and a nonlocal ABP maximum principle.

Contribution

It introduces a method to prove boundary regularity for fractional Laplacian problems on Reifenberg flat domains, extending classical techniques to nonlocal operators.

Findings

Solutions are Hölder continuous up to the boundary under flatness conditions.

The regularity exponent depends on the fractional order and domain flatness.

A nonlocal ABP maximum principle is developed for the analysis.

Abstract

For $0<s<1$, we consider the nonlocal equation $(-\Delta)^s u = f$ over a Reifenberg flat domain $\Omega$ with $f \in C({\overline{\Omega}})$ and null Dirichlet exterior condition. Given $\alpha \in (0,s)$, we prove that weak solutions are $\alpha$-H\"older continuous up to the boundary when the flatness parameter is small enough. The main ingredients of the proof are an iterative argument and a nonlocal version of the ABP maximum principle.