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

Abstract

We assume that ${\Omega }_1,{\Omega }_2\subset {\mathbb{R}}^{n+1}$, $n\ge 1$ 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 ${\Omega }_2$.

In this version we corrected an important typo in the definition of joint big pieces of chord arc subdomains. To appear in Transactions of the AMS