A TWO-PHASE FREE BOUNDARY PROBLEM FOR HARMONIC MEASURE AND UNIFORM RECTIFIABILITY

摘要

We assume that Omega(1), Omega(2) 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 Omega(1) satisfies a scale invariant A(infinity)-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 boolean AND 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 characterization of the A(infinity) condition between the respective harmonic measures of Omega(1) and Omega(2).