Geometric and Functional Analysis

, Volume 26, Issue 3, pp 909–925 | Cite as

Ratios of harmonic functions with the same zero set

  • Alexander Logunov
  • Eugenia Malinnikova


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.

Keywords and phrases

Harmonic functions Divisors of harmonic functions Nodal set Gradient estimates Łojasiewicz exponent 

Mathematics Subject Classification



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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Chebyshev LaboratorySt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  3. 3.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

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