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

Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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,.
Original languageEnglish
Pages (from-to)4359-4388
Number of pages30
JournalTransactions of the American Mathematical Society
Volume373
Issue number6
DOIs
Publication statusPublished - 1 Jun 2020

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