Rectifiability of harmonic measure in domains with porous boundaries

Abstract

We show that if $n\ge 1$, $\Omega \subset {\mathbb{R}}^{n+1}$ 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.

In this version the main theorem is also proved in the planar case

Cited by in corpus (2)

• Uniform domains with rectifiable boundaries and harmonic measure
• Absolute continuity between the surface measure and harmonic measure implies rectifiability