Analytic and geometric aspects of the elliptic measure on non-smooth domains

Abstract

Harmonic/elliptic measure arises naturally in probability and in the study of boundary value problems for elliptic operators. It has attracted the attention of many mathematicians to study the relationship between the harmonic/elliptic measure ω of a given domain and its surface measure σ, in particular, whether or not they are absolutely continuous with each other. We focus on two aspects of this subject: 1) getting an equivalent characterization of the quantitative absolute continuity between these two measures, i.e. ω ∈ A∞(σ), in terms of the PDE solvability of the corresponding Dirichlet problem; 2) studying what the regularity of the elliptic measure (with respect to the surface measure) can tell us about the geometric structure of the domain, such as the rectifiability of the boundary. We combine tools from PDE, harmonic analysis and geometric measure theory to answer these two questions.