Geometric and Functional Analysis

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

Ratios of harmonic functions with the same zero set

Article
  • 185 Downloads

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 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 \({B\setminus Z}\), plays a role.

Keywords and phrases

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

Mathematics Subject Classification

31B05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agr15.
    M. Agranovsky. Ruled nodal surfaces of Laplace eigenfunctions and injectivity sets for the spherical mean Radon transform in \({\mathbb{R}^3}\) Journal of Spectral Theory (to appear). arXiv:1504.01250.
  2. AG01.
    Armitage D. H., Gardiner S. J.: Classical Potential Theory. Springer, Berlin (2001)CrossRefMATHGoogle Scholar
  3. BR11.
    Bourgain J., Rudnick Z.: On the nodal sets of toral eigenfunction. Inventiones Mathematicae 185, 199–237 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. BFHK76.
    D. A. Brannan, W. H. J. Fuchs, W. K. Hayman and Ü. Kuran. A characterization of harmonic polynomials in the plane. Proceedings of London Mathematical Society, (3)32 (1976), 213–229.Google Scholar
  5. Han07.
    Han Q.: Nodal sets of harmonic functions. Pure and Applied Mathematics Quarterly 3, 647–688 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. KP02.
    Kranz S. G., Parks H. R.: A Primer of Real Analytic Functions, 2nd edn. Birkhäuser Verlag, Basel (2002)CrossRefGoogle Scholar
  7. LM15.
    Logunov A., Malinnikova E.: On ratios of harmonic functions. Advances in Mathematics 274, 241–262 (2015)MathSciNetCrossRefGoogle Scholar
  8. Loj59.
    Łojasiewicz S.: Sur le problème de la division (French). Studia Mathematica 18, 87–136 (1959)MathSciNetMATHGoogle Scholar
  9. Man14.
    Mangoubi D.: A gradient estimate for harmonic functions sharing the same zeros. Electronic Research Announcements in Mathematical Sciences 21, 62–71 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. Nad91.
    Nadirashvili N.: Metric properties of eigenfunctions of the Laplace operator on manifolds. Annales de l’institut Fourier (Grenoble) 41, 259–265 (1991)MathSciNetCrossRefMATHGoogle Scholar
  11. Nad99.
    Nadirashvili N.: Harmonic functions with bounded number of nodal domains. Applied Analysis 71, 187–196 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. NPS05.
    Nazarov F., Polterovich L., Sodin M.: Sign and area in nodal geometry of Laplace eigenfunctions. American Journal of Mathematics 127, 879–910 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. Rob39.
    Robertson M. S.: The variation of the sign of V for an analytic function U + iV. Duke Mathematical Journal 5, 512–519 (1939)MathSciNetCrossRefMATHGoogle Scholar
  14. Ste86.
    Stephenson K.: Analytic functions sharing level curves and tracts. Annals of Mathematics 123(2), 107–144 (1986)MathSciNetCrossRefMATHGoogle Scholar

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

Personalised recommendations