Approximate tangents, harmonic measure and domains with rectifiable boundaries

Mihalis Mourgoglou

Published 2016-02-01Version 1

We show that if $E \subset \mathbb R^d$, $d \geq 2$ is a closed and weakly lower Ahlfors-David $m$--regular set, then the set of points where there exists an approximate tangent $m$-plane, $m \leq d$, can be written as the union of countably many Lipschitz graphs. This implies that any $m$--rectifiable and weak lower Ahlfors-David $m$--regular set $E$, for which $\mathcal H^m|_E$ is locally finite, can be written as the union of countably many Lipschitz graphs up to set of $\mathcal H^m$-measure zero. Moreover, let $\Omega \subset \mathbb R^{n+1}$, $n \geq 1$ be a connected domain with weak lower Ahlfors-David $n$-regular and $n$--rectifiable boundary so that $\mathcal H^n|_E$ be locally finite. If the reduced boundary of $\Omega$ coincides with its topological boundary up to a set of $\mathcal H^n$--measure zero, then $\partial \Omega$ can be covered $\mathcal H^n$--almost everywhere by a countable union of Lipschitz domains which are contained in $\Omega$. This implies that in such a domain the Hausdorff measure $\mathcal H^n$ is absolutely continuous with respect to harmonic measure $\omega_\Omega$.