Two phase free boundary problem for Poisson kernels
We provide a potential theoretic characterization of vanishing chord-arc domains under minimal assumptions. In particular we show that, in the appropriate class of domains, 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. A striking feature of this work is that we make (almost) no a priori topological assumptions on our domain in contrast with Bortz-Hofmann PAMS 16 and Kenig-Toro Crelle 06.