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

TL;DR

This paper provides quantitative estimates on the fine structure of the singular set of the mutual boundary in free boundary problems involving harmonic measure, offering new insights into the boundary's geometric complexity.

Contribution

It introduces novel quantitative estimates for the singular set of the mutual boundary in two-sided free boundary problems for harmonic measure.

Findings

Quantitative estimates on the singular set structure

Enhanced understanding of boundary regularity

New geometric insights into free boundary problems

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, $\partial \Omega^{\pm}.$