Two phase free boundary problem for Poisson kernels
Simon Bortz, Max Engelstein, Max Goering, Tatiana Toro, Zihui Zhao · Indiana University Mathematics Journal · 2022
We provide a potential theoretic characterization of vanishing chordarc domains under mild 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 classical calculus of variations, potential theory and quantitative geometric measure theory to accomplish this.One feature of this work, compared to [KT06] and [BH16], is that a priori we only require that the domains in question are connected.