BMO solvability and the $A_{\infty}$ condition of the elliptic measure in uniform domains

TL;DR

This paper establishes that in uniform domains with Ahlfors regular boundaries, the BMO solvability of divergence form elliptic boundary value problems is equivalent to the elliptic measure belonging to the Muckenhoupt $A_{}$ class, generalizing previous Lipschitz domain results.

Contribution

It proves the equivalence between BMO solvability and $A_{}$ absolute continuity of elliptic measure in uniform domains with Ahlfors regular boundaries, extending prior work beyond Lipschitz domains.

Findings

BMO solvability is equivalent to elliptic measure being in $A_{}$.

Generalization from Lipschitz to uniform domains.

Provides a quantitative criterion for elliptic measure absolute continuity.

Abstract

We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO solvability of such problems is equivalent to a quantitative absolute continuity of the elliptic measure with respect to the surface measure, i.e. $\omega_L\in A_{\infty}(\sigma)$. This generalizes a previous result on Lipschitz domains by Dindos, Kenig and Pipher.