Two Phase Free Boundary Problem for Poisson Kernels

TL;DR

This paper characterizes vanishing chord-arc domains using potential theory and Poisson kernels, revealing geometric properties from boundary regularity and oscillation measures, with minimal assumptions on domain connectivity.

Contribution

It introduces a new potential theoretic characterization of vanishing chord-arc domains based on oscillation of Poisson kernels, requiring only domain connectivity.

Findings

Poisson kernel oscillation encodes geometric information

Domains with Ahlfors regular boundary are characterized

Connectivity is sufficient for the analysis

Abstract

We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, if a domain has Ahlfors regular boundary, the oscillation of the logarithm of the interior and exterior Poisson kernels yields a great deal of geometric information about the domain. We use techniques from the classical calculus of variations, potential theory, quantitative geometric measure theory to accomplish this. One feature of this work, compared to Bortz-Hofmann PAMS 16 and Kenig-Toro Crelle 06, is that a priori we only require that the domains in question are connected.