Rectifiability of harmonic measure in domains with porous boundaries

Abstract

We show that if $n\geq 1$, $\Omega\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partial\Omega$ is a set of finite and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $\omega$ is absolutely continuous with respect to $H^{n}$, then $\omega|_E$ is concentrated on an $n$-rectifiable set.