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- W3105834265 abstract "We study the ratio of harmonic functions $u,v$, which have the same zero set $Z$ in the unit ball $Bsubset mathbb{R}^n$. 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 $Ksubset B$ we show that $sup_K|f|le C_1inf_K|f|$ and $sup_Kleft|nabla fright|le C_2 inf_K|f|$, 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 $Bsetminus Z$, plays a role." @default.
- W3105834265 created "2020-11-23" @default.
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- W3105834265 date "2016-06-01" @default.
- W3105834265 modified "2023-10-10" @default.
- W3105834265 title "Ratios of harmonic functions with the same zero set" @default.
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- W3105834265 doi "https://doi.org/10.1007/s00039-016-0369-4" @default.
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