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.
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The main result, Theorem 1.2, and Sections 4, 5, 6 were supported by the Russian Science Foundation grant 14-21-00035. Theorem 1.1 and Sections 2, 3 were supported by Project 213638 of the Research Council of Norway.
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Logunov, A., Malinnikova, E. Ratios of harmonic functions with the same zero set. Geom. Funct. Anal. 26, 909–925 (2016). https://doi.org/10.1007/s00039-016-0369-4
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DOI: https://doi.org/10.1007/s00039-016-0369-4