On ratios of harmonic functions

Alexander Logunov, Eugenia Malinnikova

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


Let u and v be harmonic functions in Ω⊂Rn 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 supK≤|f|≤CinfK≤|f|andsupK≤|∇;f|≤CinfK≤|f| for any compact subset K of Ω, where the constant C depends on K, Z, Ω 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≥4).

Original languageEnglish (US)
Pages (from-to)241-262
Number of pages22
JournalAdvances in Mathematics
StatePublished - Apr 9 2015

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • 31B05
  • 35B09
  • 35B50
  • Boundary Harnack principle
  • Gradient estimates
  • Harmonic functions
  • Harmonic polynomials
  • Harnack inequality
  • Maximum principle
  • Nodal set


Dive into the research topics of 'On ratios of harmonic functions'. Together they form a unique fingerprint.

Cite this