Ratios of harmonic functions with the same zero set

TL;DR

This paper investigates ratios of harmonic functions sharing the same zero set, establishing Harnack inequalities and gradient estimates that depend only on the zero set and the number of nodal domains, in any dimension.

Contribution

It proves Harnack and gradient estimates for ratios of harmonic functions with identical zero sets, with explicit dependence on the zero set and nodal domains in two dimensions.

Findings

Established Harnack inequality for ratios of harmonic functions

Derived gradient estimates depending on zero set and nodal domains

Constants depend only on the zero set and number of nodal domains in 2D

Abstract

We study the ratio of harmonic functions $u,v$, which have the same zero set $Z$ in the unit ball $B\subset \mathbb{R}^n$. The ratio $f=u/v$ can be extended to a real analytic nowhere vanishing function in $B$. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set $K\subset B$ we show that $\sup_K|f|\le C_1\inf_K|f|$ and $\sup_K\left|\nabla f\right|\le C_2 \inf_K|f|$, where $C_1$ and $C_2$ depend on $K$ and $Z$ only. In dimension two we specify the dependence of the constants on $Z$ in these inequalities by showing that only the number of nodal domains of $u$, i.e. the number of connected components of $B\setminus Z$, plays a role.