Carleson perturbations of locally Lipschitz elliptic operators

TL;DR

This paper proves that the absolute continuity of elliptic measure with respect to surface measure in certain domains remains stable under specific $L^2$ Carleson perturbations of elliptic operators, given certain coefficient conditions.

Contribution

It introduces a broader class of $L^2$ Carleson perturbations and establishes stability results under these conditions in chord-arc domains.

Findings

Stability of $A_ abla$-absolute continuity under $L^2$ Carleson perturbations.

Extension of perturbation theory to more general $L^2$ Carleson classes.

Applicability to domains with uniform non-tangential access.

Abstract

In one-sided Chord-Arc Domains $\Omega$, we demonstrate that the $A_\infty$-absolute continuity of the elliptic measure with respect to the surface measure remains stable under $L^2$ Carleson perturbations. This stability holds provided that either the elliptic operator $L_0=-\operatorname{div} A_0\nabla$, which is being perturbed, or the perturbed operator $L_1=-\operatorname{div} A_1\nabla$ satisfies the condition $\sup_{X\in \Omega }\operatorname{dist}(X,\partial \Omega)|\nabla A_i(X)| <\infty$ on its coefficients. $L^2$ Carleson perturbations are slightly more general than those previously discussed in the literature. The proof hinges on the availability of a comprehensive elliptic theory and a domain $\Omega$ that allows uniform non-tangential access to any point on its boundary. Consequently, while the current theory of $L^2$ Carleson perturbations can be extended to more general contexts, we have chosen not to do so in order to simplify the presentation.