Advertisement

Advances in Applied Clifford Algebras

, Volume 22, Issue 3, pp 563–575 | Cite as

A Goursat Decomposition for Polyharmonic Functions in Euclidean Space

  • Fred Brackx
  • Richard Delanghe
  • Hennie De Schepper
  • Vladimir Souček
Article
  • 141 Downloads

Abstract

The Goursat representation formula in the complex plane, expressing a real-valued biharmonic function in terms of two holomorphic functions and their anti-holomorphic complex conjugates, is generalized to Euclidean space, expressing a real-valued polyharmonic function of order p in terms of p the so called monogenic functions of Clifford analysis.

Mathematics Subject Classification (2010)

30G35 

Keywords

Goursat decomposition polyharmonic function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Almansi E.: Sull’ integrazione dell’ equazione differenziale Δ2m u =  0. Ann. Mat. Pura Appl. 3(2), 1–51 (1899)Google Scholar
  2. 2.
    Aronszajn N., Creese T.M., Lipkin L.J.: Polyharmonic Functions. Clarendon Press, Oxford (1983)zbMATHGoogle Scholar
  3. 3.
    Bock S., Gürlebeck K.: On a Spatial Generalization of the Kolosov- Muskhelishvili Formulae. Math. Methods Appl. Sci. 32(2), 223–240 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Bogashov F.A.: Representation of a Biharmonic Function in the Complex Space \({\mathbb{C}^2}\) . Russian Acad. Sci. Dokl. Math. 48(2), 259–262 (1994)MathSciNetGoogle Scholar
  5. 5.
    Brackx F., Delanghe R., Sommen F.: Clifford Analysis. Pitman Publishers, Boston-London-Melbourne (1982)zbMATHGoogle Scholar
  6. 6.
    Brackx F., Delanghe R., Sommen F.: On Conjugate Harmonic Functions in Euclidean Space. Math. Methods Appl. Sci. 25, 1553–1562 (2002)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Delanghe R., Sommen F., Souček V.: Clifford Algebra and Spinor-Valued Functions. Kluwer Academic Publishers, Dordrecht (1992)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fischer E.: Über die Differentiationsprozesse der Algebra. J. für Math. 148, 1–78 (1917)zbMATHGoogle Scholar
  9. 9.
    Ph. Frank and R. V. Mises (eds.), Die Differential- und Integralgleichungen der Mechanik und Physik. Vol. 1, Dover, N.Y., 1961.Google Scholar
  10. 10.
    Goursat E.: Sur l’équation Δ Δ u =  0. Bull. Soc. Math. France 26, 236–237 (1898)MathSciNetzbMATHGoogle Scholar
  11. 11.
    M. Krakowski and A. Charnes, Stokes’ Paradox and Biharmonic Flows. Report 37, Carnegie Institute of Technology, Department of Mathematics, Pittsburgh, PA, 1953.Google Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Fred Brackx
    • 1
  • Richard Delanghe
    • 1
  • Hennie De Schepper
    • 1
  • Vladimir Souček
    • 2
  1. 1.Clifford Research Group, Faculty of Engineering and ArchitectureGhent UniversityGentBelgium
  2. 2.Mathematical Institute, Faculty of Mathematics and PhysicsCharles University PraguePragueCzech Republic

Personalised recommendations