Mathematical Notes

, Volume 59, Issue 3, pp 247–252 | Cite as

The final version of the mean value theorem for harmonic functions

  • V. V. Volchkov


We construct examples of nonharmonic functions satisfying the mean value equation for some set of spheres. These results permit us to obtain the two-circle theorem in its definitive form.


Harmonic Function Final Version Definitive Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Flatto, “The converse of Gauss's theorem for harmonic functions,”J. Differential Equations,1, No. 4, 483–490 (1965).zbMATHMathSciNetGoogle Scholar
  2. 2.
    L. Zalcman, “Offbeat integral geometry,”Amer. Math. Monthly,87, No. 3, 161–175 (1980).zbMATHMathSciNetGoogle Scholar
  3. 3.
    L. Zalcman, “Mean values and differential equations,”Israel J. Math.,14, 339–352 (1973).zbMATHMathSciNetGoogle Scholar
  4. 4.
    C. A. Berenstein and L. Zalcman, “Pompeiu's problem on symmetric spaces,”Comment. Math. Helv.,55, 593–621 (1980).MathSciNetGoogle Scholar
  5. 5.
    C. A. Berenstein and R. Gay, “A local version of the two-circle theorem,”Israel J. Math.,55, 267–388 (1986).MathSciNetGoogle Scholar
  6. 6.
    V. V. Volchkov, “New two-circle theorems in the theory of harmonic functions,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],58, No. 1, 41–49 (1994).Google Scholar
  7. 7.
    C. A. Berenstein and D. Struppa, “Complex analysis and convolution equations,” in:Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki. Fundamental'nye Napravleniya [in Russian], Vol. 54, VINITI, Moscow (1989), pp. 5–111.Google Scholar
  8. 8.
    N. Ya. Vilenkin,Special Functions and Group Representation Theory [in Russian], Nauka, Moscow (1991).Google Scholar
  9. 9.
    B. G. Korenev,Introduction to the Theory of Bessel Functions [in Russian], Nauka, Moscow (1971).Google Scholar
  10. 10.
    R. Courant,Methods of Mathematical Physics, Vol. II, J. Wiley, New York (1966).Google Scholar
  11. 11.
    E. Stein and G. Weiss,Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton (1971).Google Scholar
  12. 12.
    S. Helgason,Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York (1983).Google Scholar
  13. 13.
    S. Lang, SL2(ℝ), Addison-Wesley, Reading, Mass. (1975).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • V. V. Volchkov
    • 1
  1. 1.Donetsk State UniversityUSSR

Personalised recommendations