Divergence form operators in Reifenberg flat domains

TL;DR

This paper investigates the boundary regularity of solutions to divergence form elliptic operators with rough coefficients in non-smooth domains, establishing a link between elliptic measure regularity and domain geometry.

Contribution

It extends known results for the Laplacian to more general divergence form operators with less regular coefficients in irregular domains.

Findings

Elliptic measure regularity is closely related to domain geometry.

Boundary regularity results similar to Laplacian case are established for rough coefficients.

The study applies to operators with $C^{0,eta}$ coefficients or small perturbations of the Laplacian.

Abstract

We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,\alpha}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain.