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

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 $\Omega_2$.