Skip to main content
SpringerLink
Log in

Dirichlet’s Problem with Entire Data Posed on an Ellipsoidal Cylinder

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We consider the Dirichlet problem for Laplace’s equation with real-analytic data posed on the boundary of a cylinder. If the base of the cylinder is an ellipsoid and if the data function is an entire function of order less than 1, we show that there is an entire solution.

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Armitage, D., Gardiner, S.: Classical Potential Theory. Springer, London (2001)

    Book  MATH  Google Scholar 

  3. Brelot, M., Choquet, G.: Polynômes harmoniques et polyharmoniques, Deuxième colloque sur les équations aux dérivées partielles. Bruxelles, 45–66 (1954)

  4. Fischer, E: Uber die Differentiationsprozesse der Algebra. J. für Math. 148, 1–78 (1917)

    MATH  Google Scholar 

  5. Gardiner, S., Render, H.: Harmonic functions which vanish on a cylindrical surface, J. Math. Anal. Appl., available online September 3 2015 doi:10.1016/j.jmaa.2015.08.077 (to appear in print)

  6. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (1977)

    Book  MATH  Google Scholar 

  7. Khavinson, D.: On reflection of harmonic functions in surfaces of revolution. Complex Var. 17, 7–14 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Khavinson, D., Lundberg, E.: The search for singularities of solutions to the Dirichlet problem: Recent developments. CRM Proc. Lect. Notes 51, 121–132 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Khavinson, D., Lundberg, E.: A tale of ellipsoids in potential theory. Not. AMS 61, 148–156 (2014)

    MathSciNet  MATH  Google Scholar 

  10. Khavinson, D., Lundberg, E., Render, H.: The Dirichlet problem for the slab with entire data and a difference equation for harmonic functions. Canad. Math. Bull. (to appear), preprint: arXiv:1602.01823

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

  12. Miyamoto, I.: A type of uniquenes of solutions for the Dirichlet problem on a cylinder. Tôhoku Math. J. 48, 267–292 (1996)

  13. Ramachandran, K.: Asymptotic behavior of positive harmonic functions in certain unbounded domains, Ph.D. thesis, Purdue Univ. (2014)

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

    Article  MATH  Google Scholar 

  15. Render, H.: Cauchy, Goursat, and Dirichlet Problems for holomorphic partial differential equations. Comput. Methods Funct. Theory 10, 519–554 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shapiro, H.S.: An algebraic theorem of E. Fischer and the Holomorphic Goursat Problem. Bull. London Math. Soc 21, 513–537 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yoshida, H.: Harmonic majorization of a subharmonic function on a cone or on a cylinder. Pac. J. Math. 148, 369–395 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Florida Atlantic University, Boca Raton, FL, USA

    Erik Lundberg

  2. University of South Florida, Tampa, FL, USA

    Dmitry Khavinson

  3. University College Dublin, Dublin, Ireland

    Hermann Render

Authors
  1. Dmitry Khavinson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Erik Lundberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hermann Render
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erik Lundberg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khavinson, D., Lundberg, E. & Render, H. Dirichlet’s Problem with Entire Data Posed on an Ellipsoidal Cylinder. Potential Anal 46, 55–62 (2017). https://doi.org/10.1007/s11118-016-9568-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-016-9568-8

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation