On ratios of harmonic functions II

Alexander Logunov, Eugenia Malinnikova

Published 2015-06-26Version 1

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: 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 only on $K,Z$. In dimension two we specify these estimates by showing that only the number of nodal domains of $u$ plays a role.