Skip to main content

On Dirichlet to Neumann and Robin to Neumann operators suitable for reflecting harmonic functions subject to a non-homogeneous condition on an arc

Analysis and Mathematical Physics volume 9, pages 729–745 (2019)Cite this article

Abstract

According to the Schwarz symmetry principle, every harmonic function vanishing on a real-analytic curve has an odd continuation, while a harmonic function satisfying homogeneous Neumann condition has an even continuation. Using a technique of Dirichlet to Neumann and Robin to Neumann operators, we derive reflection formulae for non-homogeneous Neumann and Robin conditions from a reflection formula subject to a non-homogeneous Dirichlet condition.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Khavinson, D., Shapiro, H.S.: Remarks on the reflection principles for harmonic functions. J. Anal. Math. 54, 60–76 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  2. Ebenfelt, P., Khavinson, D.: On point to point reflection of harmonic functions across real analytic hypersurfaces in \(\mathbb{R}^n\). J. Anal. Math. 68, 145–182 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  3. Study, E.: Einige elementare Bemerkungen über den Prozess der analytischen Fortsetzung. Math. Ann. 63, 239–245 (1907)

    MATH  MathSciNet  Article  Google Scholar 

  4. Davis, Ph: The Schwarz Function and Its Applications. Carus Mathematical Monographs. MAA, Washington (1979)

    Google Scholar 

  5. Lewy, H.: On the reflection laws of second order differential equations in two independent variables. Bull. Am. Math. Soc. 65, 37–58 (1959)

    MathSciNet  MATH  Article  Google Scholar 

  6. Garabedian, P.R.: Partial differential equations with more than two independent variables in the complex domain. J. Math. Mech. 9, 241–271 (1960)

    MathSciNet  MATH  Google Scholar 

  7. Aberra, D., Savina, T.V.: The Schwarz reflection principle for polyharmonic functions in \({\mathbb{R}}^2\). Complex Var. Theory Appl. 41(1), 27–44 (2000)

    MATH  MathSciNet  Google Scholar 

  8. Savina, T.V., Sternin, BYu., Shatalov, V.E.: On the reflection law for the Helmholtz equation. Dokl. Math. 45(1), 42–45 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Savina, T.V.: On non-local reflection for elliptic equation of the second order in \({\mathbb{R}}^2\) (the Dirichlet condition). Trans. Am. Math. Soc. 364(5), 2443–2460 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  10. López, R.R.: On reflection principles supported on a final set. J. Math. Anal. Appl. 351, 556–566 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  11. Khavinson, D., Lundberg, E.: Linear Holomorphic Partial Differential Equations and Classical Potential Theory. Linear Holomorphic Partial Differential Equations and Classical Potential Theory, vol. 232. American Mathematical Soceity, Providence (2018)

    MATH  Book  Google Scholar 

  12. Beznea, L., Pascu, M.N., Pascu, N.R.: An equivalence between the Dirichlet and the Neumann problem for the Laplace operator. Potential Anal. 44, 655–672 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  13. Evans, L.C.: Partial Differential Equations. Graduate Studies, vol. 19, second edn. American Mathematical Society, Providence (2010)

    Google Scholar 

  14. Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, New York (1969)

    MATH  Google Scholar 

  15. Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Algebraic Asp Dirichlet Probl Ration Data. Quadrature domains and their applications, Operator Theory Advances and Applications 156, 151–172 (2005)

    MATH  Article  Google Scholar 

  16. Khavinson, D., Shapiro, H.S.: Dirichlet’s problem when the data is an entire function. Bull. Lond. Math. Soc. 24, 456–468 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  17. Armitage, D.H.: The Dirichlet problem when the boundary function is entire. J. Math. Anal. Appl. 291(2), 565–577 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  18. Chamberland, M., Siegel, D.: Polynomial solutions to Dirichlet problems. Proc. Am. Math. Soc. 129, 211–217 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  19. Khavinson, D., Lundberg, E., Render, H.: Dirichlet’s problem with entire data posed on an ellipsoidal cylinder. Potential Anal. 46(1), 55–62 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  20. Khavinson, D., Lundberg, E., Render, H.: The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions. Can. Math. Bull. 60(1), 146–153 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  21. Khavinson, D., Stylianopoulos, N.: Recurrence relations for orthogonal polynomials and algebraicity of solutions of the Dirichlet problem. In: Laptev, A. (ed.) Around the Research of Vladimir Maz’ya II, Partial Differential Equations. International Mathematical Series, vol. 12, pp. 219–228. Springer, Berlin (2010)

    MATH  Google Scholar 

  22. Putinar, M., Stylianopoulos, N.: Finite-term relations for planar orthogonal polynomials. Complex Anal. Oper. Theory 1(3), 447–456 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  23. Render, H.: Real Bargmann spaces, Fischer decompositions and sets of uniqueness for polyharmonic functions. Duke Math. J. 142, 313–352 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  24. Ebenfelt, P.: Singularities encountered by the analytic continuation of solutions to Dirichlet’s problem. Complex Var. 20, 75–91 (1992)

    MathSciNet  MATH  Google Scholar 

  25. Savina, T.V.: A reflection formula for the Helmholtz equation with the Neumann condition. Comput. Math. Math. Phys. 39(4), 652–660 (1999)

    MathSciNet  Google Scholar 

  26. Savina, T.V.: From reflections to a uniform elliptic growth. J. Math. Anal. Appl. (2019). https://doi.org/10.1016/j.jmaa.2019.05.021

Download references

Acknowledgements

The authors are very grateful to the anonymous referee for the suggestions, which helped to improve the paper. The second author would like to thank the organizers of the International Conference on Complex Analysis, Potential Theory and Applications in honour of Professor Stephen J Gardiner on the occasion of his 60th Birthday and Professor Lucian Beznea for providing the authors with his interesting papers.

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio University, Athens, OH, USA

    Murdhy Aldawsari & Tatiana Savina

Authors
  1. Murdhy Aldawsari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tatiana Savina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatiana Savina.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Aldawsari, M., Savina, T. On Dirichlet to Neumann and Robin to Neumann operators suitable for reflecting harmonic functions subject to a non-homogeneous condition on an arc. Anal.Math.Phys. 9, 729–745 (2019). https://doi.org/10.1007/s13324-019-00314-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13324-019-00314-w

Keywords

  • Schwarz symmetry principle
  • Dirichlet to Neumann operator
  • Robin to Neumann operator
  • Analytic continuation