Perturbations of elliptic operators in chord arc domains

Abstract

We study the boundary regularity of solutions to divergence form operators which are small perturbations of operators for which the boundary regularity of solutions is known. An operator is a small perturbation of another operator if the deviation function of the coefficients satisfies a Carleson measure condition with small norm. We extend Escauriaza's result on Lipschitz domains to chord arc domains with small constant. In particular we prove that if $L_1$ is a small perturbation of $L_0$ and $\log k_0$ has small BMO norm so does $\log k_1$. Here $k_i$ denotes the density of the elliptic measure of $L_i$ with respect to the surface measure of the boundary of the domain.