The Singular Strata of a Free-Boundary problem for harmonic measure

Abstract

In this paper, we obtain \textit{quantitative} estimates on the fine structure of the singular set of the mutual boundary $\partial {\Omega }^{\pm }$ for pairs of complementary domains, ${\Omega }^{+},{\Omega }^{-}\subset {\mathbb{R}}^n$ 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, $\partial {\Omega }^{\pm }.$

50 pages

References in corpus (2)

• Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps
• Structure of sets which are well approximated by zero sets of harmonic polynomials