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

Abstract

We assume that Ω 1 , Ω 2 ⊂ R n + 1 \Omega _1, \Omega _2 \subset \mathbb {R}^{n+1} , n ≥ 1 n \geq 1 , are two disjoint domains whose complements satisfy the capacity density condition and where the intersection of their boundaries F F has positive harmonic measure. Then we show that in a fixed ball B B centered on F F , if the harmonic measure of Ω 1 \Omega _1 satisfies a scale invariant A ∞ A_\infty -type condition with respect to the harmonic measure of Ω 2 \Omega _2 in B B , then there exists a uniformly n n -rectifiable set Σ \Sigma so that the harmonic measure of Σ ∩ F \Sigma \cap F contained in B B is bounded below by a fixed constant independent of B 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 Ω 1 \Omega _1 and Ω 2 \Omega _2 are complementary NTA domains, we obtain a characterization of the A ∞ A_\infty condition between the respective harmonic measures of Ω 1 \Omega _1 and Ω 2 \Omega _2 .