One phase problem for two positive harmonic function: below the codimension $1$ threshold

TL;DR

This paper investigates the geometric and boundary properties of domains in Euclidean space where positive harmonic functions exhibit specific asymptotic behaviors or satisfy boundary Harnack principles, providing insights in special cases.

Contribution

It offers new results characterizing domains based on harmonic function behavior and boundary principles, especially below the codimension one threshold.

Findings

Characterization of domains with Green's function asymptotics

Analysis of boundary Harnack principle implications

Results in special geometric configurations

Abstract

What can be said about the domain $\Om$ in $\bR^n$ for which its Green's function $G(z)$ satisfies $G(z)\asymp \dist (z, \pd\Om)^\delta$? What can we say about $\Om$ if the Boundary Harnack Principle holds in the form $u/v=\text{real analytic}$ on the part $E$ of its boundary? Here $u, v$ are positive harmonic functions on $\Om$ vanishing on $E$. Is this part of the boundary also nice? We discuss these questions below and give answers in very special cases.