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

We assume that ${\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2\subset {\mathbb{R}}^{n+1}$, $n\ge 1$ 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 ${\mathrm{\Omega }}_1$ satisfies a scale invariant $A_{\mathrm{\infty }}$-type condition with respect to the harmonic measure of ${\mathrm{\Omega }}_2$ in $B$, then there exists a uniformly $n$-rectifiable set $\mathrm{\Sigma }$ so that the harmonic measure of $\Sigma \cap F$containedin$B$isboundedbelowbyafixedconstantindependentof$B$. A remarkable feature of this result is that the harmonic measures do not need to satisfy any doubling condition. In the particular case that ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_{\mathrm{\infty }}$ condition between the respective harmonic harmonic measures of ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$.