Two Phase Free Boundary Problem for Poisson Kernels
Classical Analysis and ODEs
2020-11-11 v3 Analysis of PDEs
Metric Geometry
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.
Cite
@article{arxiv.1908.03033,
title = {Two Phase Free Boundary Problem for Poisson Kernels},
author = {Simon Bortz and Max Engelstein and Max Goering and Tatiana Toro and Zihui Zhao},
journal= {arXiv preprint arXiv:1908.03033},
year = {2020}
}
Comments
48 pages. Final version including many suggestions by the referee(s). To appear in IUMJ