A two-phase free boundary problem for harmonic measure and uniform rectifiability

TL;DR

This paper establishes a link between harmonic measure conditions and the geometric structure of boundary sets, showing that under certain conditions, the boundary is uniformly rectifiable without requiring doubling measures.

Contribution

It proves that scale-invariant $A_ abla$-type conditions imply uniform rectifiability of boundary sets in a two-phase harmonic measure problem, without assuming measure doubling.

Findings

Existence of a uniformly rectifiable set within boundary intersections.

Harmonic measures satisfy $A_ abla$-type conditions without doubling.

Geometric characterization of harmonic measure relationships in NTA domains.

Abstract

We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we show that in a fixed ball $B$ centered on $F$, if the harmonic measure of $\Omega_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the harmonic measure of $\Omega_2$ in $B$, then there exists a uniformly $n$-rectifiable set $\Sigma$ so that the harmonic measure of $\Sigma \cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of this result is that the harmonic measures do not need to satisfy any doubling condition. In the particular case that $\Omega_1$ and $\Omega_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective harmonic harmonic measures of $\Omega_1$ and $\Omega_2$.