Ratios of harmonic functions with the same zero set

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.