Boundary H\"older Regularity for Elliptic Equations on Reifenberg Flat Domains

Abstract

In this paper, we investigate the boundary H\"{o}lder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<\alpha<1$, there exists $\delta>0$ such that the solution is $C^{\alpha}$ at $x_0\in \partial \Omega$ provided that $\Omega$ is $\delta$-Reifenberg flat at $x_0$ (see Definition 1.1). In particular, for any $0 < \alpha < 1$, if $\partial \Omega$ is $C^1$ and $u=g$ on $\partial \Omega$ with $g\in C^{\alpha}(x_0)$, then $u\in C^{\alpha}(x_0)$. A similar result for the Poisson equation has been proved by Lemenant and Sire, where the Alt-Caffarelli-Friedman's monotonicity formula is used.