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Normalized System for the Super Laplace Operator

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Abstract

In this paper, 0-normalized system for the super Laplace operator (that is Laplace operator in superspace) is established. According to this system, we obtain Almansi type decomposition of polyharmonic functions in superspace. Besides, we set up the relationship between the Riquier problem and the Dirichlet problem in superspace. In the end, the connection between harmonic functions and solutions to the Helmholtz equation in superspace is investigated.

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References

  1. F. Brack, R. Delanghe, F. Sommen, Clifford Analysis. Research Notes in Mathematics, 76, Pitman (Advanced Publishing Program), Boston, MA, 1982.

  2. Gilbert R. P., Buchanan J. L.: First order elliptic system, A function theoretic approach. Academic Press, New York (1983)

    Google Scholar 

  3. J. Ryan, Introductory Clifford Analysis. Lectures on Clifford (Geometric) Algebras and Applications, Birkhäuser, Boston, MA, p. xvii, 221, 2004.

  4. Wen G. C.: Clifford analysis and elliptic systerm, hyperbolic systerm of first order equation. World Scientific, Singapore (1991)

    Google Scholar 

  5. Qiao Y. Y., Xie Y. H.: A nonlinear boundary value problem in Clifford analysis. Adv. Appl. Clifford Algebra 11, 260–276 (2001)

    MathSciNet  Google Scholar 

  6. Huang S., Qiao Y. Y., Wen G.C.: Real and Complex Clifford Analysis. Springer, New York (2005)

    Google Scholar 

  7. F. A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables. Moskov. Gos. Univ. Moscow, 1983.

  8. Leĭites D. A.: Introduction to the theory of supermanifolds. Usp. Mat. Nauk 35, 3–57 (1980)

    MathSciNet  Google Scholar 

  9. DeWitt B.: Supermanifolds Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1984)

    Google Scholar 

  10. Rogers A.: A global theory of supermanifolds. J. Math. Phys. 21, 1352–1365 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Batchelor M.: Two approaches to supermanifolds. Trans. Amer. Math. Soc. 258(1), 257–270 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. De Bie H., Sommen F.: Correct rules for Clifford calculus on superspace. Adv. Appl. Clifford Alg. 17, 357–382 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. De Bie H., Sommen F.: Fundamental solutions for the super Laplace and Dirac operators and all their natural powers. J. Math. Anal. Appl. 338, 1320–1328 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. H. De Bie and F. Sommen, Fischer decompositions in superspace. Function Spaces in Complex and Clifford Analysis, National University Publishers Hanoi (2008), 170–188.

  15. K. Coulembier, H. De Bie, and F. Sommen, Integration in superspace using distribution theory. J. Phys. A: Math. Theor. 42 (2009), 395206 (23pp).

    Google Scholar 

  16. B. A. Bondarenko, Operatornye Algoritmy v Differenzial Uravneni’nyah. Fan, Tashkent, 1984 (in Russian).

  17. V. V. Karachik, Polynomial solutions to the systems of partial differential equations with constant coefficients. Yokohama Math. J. 47 (2000), 121–142.

    Google Scholar 

  18. Karachik V. V. Normalized system of functions with respect to the Laplace operator and its applications. J. Math. Anal. Appl. 287 (2003), 577–592.

    Article  MathSciNet  MATH  Google Scholar 

  19. Karachik V. V.: On an Expansion of Almansi Type. Mat. Zametki 83(3), 370–380 (2008)

    Article  MathSciNet  Google Scholar 

  20. L. Liu and G. B. Ren, Normalized System for Wave and Dunkl Operators. Taiwanese J. Math, Vol. 14, No. 2 (2010), 675–683.

  21. N. Aronszajn, T. M. Creese and L. J. Lipkin, Polyharmonic Functions. Oxford Mathematics Monographs (The Clarendon Press, Oxford University Press, New York, 1983).

  22. Almansi E.: Sull’integrazione dell’equazione differenziale Δ2m u = 0. Ann. Mat. Pura. Appl. (ser 3) 2, 1–51 (1898)

    Google Scholar 

  23. Malonek Helmuth R., Ren G. B.: Almansi-type theorems in Clifford analysis. Math. Meth. Appl. Sci. 25, 1541–1552 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yude Bu, Du J. Y.: The RH boundary value problem of the k-monogenic functions. J. Math. Anal. Appl. 347, 633–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. H. F. Yuan, Y. Y. Qiao and H. J. Yang, Decomposition of k-monogenic functions in superspace. Complex Variables and Elliptic Equations, Available online: 21 Nov 2011, 1-16.

  26. M. Nicolescu, Sur le probléme de Riquier. Comptes Rendus, Vol. CXCIV, 1932.

  27. Nicolescu M.: Les fonctions polyharmoniques. Hermann, Paris (1936)

    Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, P. R. China

    Yuying Qiao & Hongfen Yuan

  2. College of Science, Hebei University of Engineering, Handan, 056038, P. R. China

    Hongfen Yuan

  3. College of Science, Hebei University of Science and Technology, Shijiazhuang, 050018, P. R. China

    Heju Yang

Authors
  1. Yuying Qiao
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  2. Hongfen Yuan
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  3. Heju Yang
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Correspondence to Hongfen Yuan.

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Qiao, Y., Yuan, H. & Yang, H. Normalized System for the Super Laplace Operator. Adv. Appl. Clifford Algebras 22, 1109–1128 (2012). https://doi.org/10.1007/s00006-012-0331-y

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  • DOI: https://doi.org/10.1007/s00006-012-0331-y

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