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

We assume that Ω1,Ω2 ⊂ Rn+1, n ≥ 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 Ω1 satisfies a scale invariant A∞-type condition with respect to the harmonic measure of Ω2 in B, then there exists a uniformly n-rectifiable set σ so that the harmonic measure of σ⋒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 Ω1 and Ω2 are complementary NTA domains, we obtain a characterization of the A∞ condition between the respective harmonic measures of Ω1 and Ω2,.