Perturbations of elliptic operators in chord arc domains

TL;DR

This paper investigates how small perturbations of elliptic operators affect boundary regularity in chord arc domains, extending previous results to more general geometries and establishing stability of elliptic measure densities.

Contribution

It extends Escauriaza's boundary regularity results from Lipschitz to chord arc domains for small operator perturbations, demonstrating stability of elliptic measure properties.

Findings

Boundary regularity is preserved under small perturbations in chord arc domains.

Logarithm of elliptic measure densities remains small in BMO norm after perturbation.

Results generalize known Lipschitz domain theorems to more complex geometries.

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.