Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions

TL;DR

This paper applies geometric measure theory to analyze the size and structure of sets where interior and exterior harmonic measures are mutually absolutely continuous in higher-dimensional Euclidean spaces.

Contribution

It introduces the use of GMT tools to study harmonic measure relationships in higher dimensions, advancing understanding of boundary structures.

Findings

Characterization of boundary structures via GMT

Insights into the size of mutually absolutely continuous sets

Extension of harmonic measure theory to higher dimensions

Abstract

In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the inside and outside harmonic measures of domains in n-dimensional Euclidean space, with n bigger than or equal to 3.