Skip to main content
SpringerLink
Log in

Panharmonic Functions: Mean Value Properties and Related Topics

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Recently obtained results about m-dimensional panharmonic functions, i.e., solutions to the modified Helmholtz equation, are presented in a systematic form. They concern various mean value properties. Their corollaries are considered along with converse and inverse of these properties, and relations between panharmonic and harmonic functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. J. Duffin, “Yukawan potential theory,” J. Math. Anal. Appl. 35, 105–130 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Yukawa, “On the interaction of elementary particles,” Proc. Phys.-Math. Soc. Japan 17, 48–57 (1935).

  3. I. N. Sneddon, “A relation between the solutions of the half-space Dirichlet problems for Helmholtz’s equation in ℝn and Laplace’s equation in ℝn+1,” J. Eng. Math. 8, 177–180 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Lukeš, I. Netuka, and J. Veselý, “Choquet’s theory and the Dirichlet problem,” Expo. Math. 20, No. 3, 229–254 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Neumann, Allgemeine Untersuchungen über das Newton’sche Princip der Fernwirkungen, Teubner, Leipzig (1896).

    MATH  Google Scholar 

  6. M. T. Boudjelkha and J. B. Diaz, “Half space and quarter space Dirichlet problems for the partial differential equation Δu − λ2u = 0. I,” Appl. Anal. 1, 297–324 (1971/2).

  7. L. R. Bragg and J. W. Dettman, “Function theories for the Yukawa and Helmholtz equations,” Rockey Mt. J. Math. 25, No.. 3, 887–917 (1995).

    MathSciNet  MATH  Google Scholar 

  8. R. J. Duffin, “Hilbert transforms in Yukawan potential theory,” Proc. Natl. Acad. Sci. USA 69, 3677–3679 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. L Schiff and W. J. Walker, “A Bieberbach condition for a class of pseudo-analytic functions,” J. Math. Anal. Appl. 146, No. 2, 570–579 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  10. L. A. Caffarelli and W. Littman, “Representation formulas for solutions to Δu − u = 0 in ℝn,” MAA Stud. Math. 23, 249–263 (1982).

    MathSciNet  Google Scholar 

  11. N. Kuznetsov, “Mean value properties of solutions to the Helmholtz and modified Helmholtz equations,” J. Math. Sci. 257, No. 5, 673–683 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Kuznetsov, “Asymptotic mean value properties of meta- and panharmonic functions,” J. Math. Sci. 259, No.2, 205–209 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Kuznetsov, “Characterization of balls via solutions of the modified Helmholtz equation,” C. R., Math.., Acad. Sci. Paris 359, No. 8, 945–948 (2021).

  14. N. Kuznetsov, “Metaharmonic functions: mean flux theorem, its converse and related properties,” St. Petersbg. Math. J. 33, No. 2, 243–254 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Kuznetsov, “Inverse mean value property of solutions to the modified Helmholtz equation,” St. Petersbg. Math. J. 33, No. 6, 943–948 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  16. N. Kuznetsov, “‘Potato kugel’ for nuclear forces and a small one for acoustic waves,” J. Math. Sci. 267, No. 3, 375–381 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  17. N. Kuznetsov, “On relations between harmonic functions and solutions of the modified Helmholtz equation,” J. Math. Sci. 267, No. 4, 494–500 (2022).

    Article  Google Scholar 

  18. N. Kuznetsov, “Mean value properties of harmonic functions and related topics (a survey),” J. Math. Sci. 242, No. 2, 177–199 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  19. C. F. Gauss, Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrats der Entfernung wirkenden Anziehungs- und, Wiedmannschen Buchhandlung, Leipzig (1840).

    MATH  Google Scholar 

  20. F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York (1955).

    MATH  Google Scholar 

  21. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge (1944).

    MATH  Google Scholar 

  22. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin etc. (2001).

    Book  MATH  Google Scholar 

  23. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Vol. 2: Special Functions, Gordon & Breach, New York etc. (1986).

  24. W. Blaschke, “‘Ein Mittelwertsatz und eine kennzeichnende Eigenschaft des logarithmischen Potentials,” Leipz. Ber. 68, 3–7 (1916).

    MATH  Google Scholar 

  25. I. Priwaloff, “Sur les fonctions harmoniques,” Moscou, Rec. Math. 32, 464–471 (1925).

    Google Scholar 

  26. S. Zaremba, “Contributions à la théorie d’une équation fonctionelle de la physique,” Palermo Rend. 19, 140–150 (1905).

    Article  MATH  Google Scholar 

  27. I. Netuka and J. Veselý, “Mean value property and harmonic functions,” In: Classical and Modern Potential Theory and Applications, pp. 359–398, Kluwer, Dordrecht (1994).

  28. M. Brelot, Éléments de la Théorie Classique du Potential, CDU, Paris (1961).

    MATH  Google Scholar 

  29. V. S. Vladimirov, Equations of Mathematical Physics, Marcel Dekker, New York (1971).

    MATH  Google Scholar 

  30. O. D. Kellogg, Foundations of Potential Theory, Springer, Berlin (1929).

    Book  MATH  Google Scholar 

  31. S. G. Mlkhlin, Mathematical Physics. An Advanced Course, North-Holland, Amsterdam etc. (1970).

  32. E. F. Beckenbach and M. Reade, “Mean values and harmonic polynomials,” Trans. Am. Math. Soc. 53, 230–238 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  33. N. Wiener, “The Dirichlet problem,” J. Math. Massachusetts 3, 127–147 (1924).

    MATH  Google Scholar 

  34. O. A. Oleinik, “On the Dirichlet problem for equations of elliptic type” [in Russian], Mat. Sb. 24, 3–14 (1949).

    Google Scholar 

  35. G. Tautz, “Zur Theorie der ersten Randwertaufgaben,” Math. Nachr. 2, 279–303 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  36. C. Miranda, Partial Differential Equations of Elliptic Type, Springer, Berlin et al. (1970).

    Book  MATH  Google Scholar 

  37. D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, London (2001).

    Book  MATH  Google Scholar 

  38. B. Epstein, “On the mean-value property of harmonic functions,” Proc. Am. Math. Soc. 13, 830 (1962).

    MathSciNet  MATH  Google Scholar 

  39. B. Epstein and M. M. Schiffer, “On the mean-value property of harmonic functions,” J. Anal. Math. 14, 109–111 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  40. W. Hansen and I. Netuka, “Inverse mean value property of harmonic functions,” Math. Ann. 297, No. 1, 147–156 (1993); “Corrigendum,” Ibid., 303, No. 2, 373–375 (1995).

  41. A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics. A Unified Introduction with Applications, Birkh¨auser, Basel (1988).

  42. D. Aharonov, M. M. Schiffer, and L. Zalcman, “Potato kugel,” Israel J. Math. 40, 331–339 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  43. M. M. Schiffer, Selected Papers. Vol. 2, Springer, New York etc. (2014).

    MATH  Google Scholar 

  44. G. Cupini and E. Lanconelli, “On the harmonic characterization of domains via mean value formulas,” Matematiche 75, No. 1, 331–352 (2020).

    MathSciNet  MATH  Google Scholar 

  45. F. Riesz, “Sur les functions subharmoniques et leur rapport à la théorie du potentiel. II,” Acta Math. 54, 321–360 (1930).

    Article  MathSciNet  MATH  Google Scholar 

  46. W. K. Hayman and P. B. Kennedy, Subharmonic Functions. Vol. I, Academic Press, London etc. (1976).

    MATH  Google Scholar 

  47. F. Treves, Basic Linear Partial Differential Equations, Academic Press, New York etc. (1975).

    MATH  Google Scholar 

  48. A. Bennett, “Symmetry in an overdetermined fourth order elliptic boundary value problem,” SIAM J. Math. Anal. 17, 1354–1358 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  49. P. Freitas and J. P. Matos, “On the characterization of harmonic and subharmonic functions via mean-value properties,” Potential Anal. 32, No. 2, 189–200 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  50. G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer, Berlin (1993).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, RAS, V.O., Bol’shoy pr. 61, St Petersburg, 199178, Russia

    N. Kuznetsov

Authors
  1. N. Kuznetsov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. Kuznetsov.

Additional information

To Vladimir Maz’ya, in remembrance of our collaboration

Translated from Problemy Matematicheskogo Analiza 120, 2023, pp. 51-69.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuznetsov, N. Panharmonic Functions: Mean Value Properties and Related Topics. J Math Sci 269, 53–76 (2023). https://doi.org/10.1007/s10958-023-06254-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-023-06254-y

Search

Navigation