Estimates on the generalized critical strata of Green's function
Sean McCurdy · arXiv (Cornell University) · 2019
In this paper, we obtain quantitative estimates on the fine structure of Green's functions for pairs of complementary domains, $\Omega^+, \Omega^- \subset \mathbb{R}^n$ which arise in a class of two-sided free boundary problems for harmonic measure. These estimates give new insight into the structure of the mutual boundary, $\partial \Omega^{\pm},$ and on how critical set of the Green's functions approach the boundary. These estimates are not obtainable by naively combining boundary and interior estimates.