Boundary rectifiability and elliptic operators with $W^{1,1}$ 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 $\Omega$.