Boundary Hölder Regularity for Elliptic Equations on Reifenberg Flat Domains

Abstract

In this paper, we investigate the boundary Hölder 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.