$A_\infty$ implies NTA for a class of variable coefficient elliptic operators

TL;DR

This paper demonstrates that for a class of variable coefficient elliptic operators, the $A_ abla$ property of their elliptic measure implies the domain is NTA, extending the understanding of boundary regularity and measure properties.

Contribution

It establishes a new implication from the $A_ abla$ property of elliptic measure to the NTA condition for variable coefficient elliptic operators.

Findings

$A_ abla$ property implies NTA domain for certain elliptic operators.

The converse (NTA implies $A_ abla$) was previously known.

Results connect elliptic measure properties with geometric domain conditions.

Abstract

We consider a certain class of second order, variable coefficient divergence form elliptic operators, in a uniform domain $\Omega$ with Ahlfors regular boundary, and we show that the $A_\infty$ property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact NTA (and hence chord-arc). The converse was already known, and follows from work of Kenig and Pipher.