Boundary rectifiability and elliptic operators with $W^{1,1}$ coefficients

TL;DR

This paper demonstrates that for elliptic operators with $W^{1,1}$ coefficients in a uniform domain, the $A_ abla$ property of the elliptic measure implies boundary rectifiability, linking measure properties to geometric structure.

Contribution

It establishes a connection between the $A_ abla$ property of elliptic measures and boundary rectifiability for operators with $W^{1,1}$ coefficients, extending previous results.

Findings

$A_ abla$ property implies boundary rectifiability

Absolute continuity of surface measure ensures rectifiability

Additional regularity results for continuous coefficients

Abstract

We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $\Omega$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such operator implies that $\Omega$ is a set of locally finite perimeter whose boundary, $\partial\Omega$, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional information about $\Omega$.