Ratios of harmonic functions with the same zero set

Alexander Logunov, Eugenia Malinnikova

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We study the ratio of harmonic functions u,v which have the same zero set Z in the unit ball B⊂ Rn. 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⊂ B we show that sup K| f| ≤ C1inf K| f| and sup K|∇ f| ≤ C2inf K|f| , where C1 and C2 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\ Z, plays a role.

Original languageEnglish (US)
Pages (from-to)909-925
Number of pages17
JournalGeometric and Functional Analysis
Issue number3
StatePublished - Jun 1 2016

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology


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


Dive into the research topics of 'Ratios of harmonic functions with the same zero set'. Together they form a unique fingerprint.

Cite this