Boundary regularity for the Poisson equation in reifenberg-flat domains

Abstract

This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -Δu = f & \text{in} Ω u= 0 & \text{on} \partial Ω$$ where $f$ belongs to some $L^p(\Omega )$ and $\Omega$ is a Reifenberg-flat domain of ${\mathbb{R}}^n.$ More precisely, we prove that given an exponent $\alpha \in (0,1)$, there exists an $\epsilon >0$ such that the solution $u$ to the previous system is locally Hölder continuous provided that $\Omega$ is $(\epsilon ,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.