On ratios of harmonic functions

Let $u$ and $v$ be harmonic in $Ω\subset \mathbb{R}^n$ functions with the same zero set $Z$. We show that the ratio $f$ of such functions is always well-defined and is real analytic. Moreover it satisfies the maximum and minimum principles. For $n=3$ we also prove the Harnack inequality and the gradient estimate for the ratios of harmonic functions, namely ${ \sup\limits_{K} |f| \leq C \inf\limits_{K}| f| \quad \& \quad \sup\limits_{K} |\nabla f| \leq C \inf\limits_{K}| f| }$ for any compact subset $K$ of $Ω$, where the constant $C$ depends on $K$, $Z$, $Ω$ only. In dimension two the first inequality follows from the boundary Harnack principle and the second from the gradient estimate recently obtained by Mangoubi. It is an open question whether these inequalities remain true in higher dimensions ($n \geq 4$).

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