On ratios of harmonic functions

Abstract

Let $u$ and $v$ be harmonic in $\Omega \subset {\mathbb{R}}^n$ 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 $\underset{K}{\sup }|f|\le C\underset{K}{\inf }|f|\&\underset{K}{\sup }|\nabla f|\le C\underset{K}{\inf }|f|$ for any compact subset $K$ of $\Omega$, where the constant $C$ depends on $K$, $Z$, $\Omega$ 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 ($n\ge 4$).

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• Higher order boundary Harnack principle via degenerate equations
• Structure of sets which are well approximated by zero sets of harmonic polynomials
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• Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients
• Higher order boundary Harnack principles in Dini type domains
• Harmonic functions vanishing on a cone