The singular strata of a free-boundary problem for harmonic measure

We obtain quantitative estimates on the fine structure of the singular set of the mutual boundary $\partial {\mathrm{\Omega }}^{\pm }$ for pairs of complementary domains ${\mathrm{\Omega }}^{+},{\mathrm{\Omega }}^{-}\subset {ℝ}^n$ which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary $\partial {\mathrm{\Omega }}^{\pm }$.