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 where 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 characterization of the $A_\infty$ condition between the respective harmonic measures of $\Omega _1$ and $\Omega _2$.