Boundary Hölder regularity for the fractional Laplacian over Reifenberg flat domains via ABP maximum principle

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 $\alpha$-Hölder 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.

To appear in the proceedings of the conference "Geometric Measure Theory and Applications 2024", held in Cortona (Italy) in June 2024. The proceedings will be published in the Springer INdAM Series