Abstract
In this article, the Dirichlet problem of polyharmonic functions is considered. As well the explicit expression of the unique solution to the simple Dirichlet problem for polyharmonic functions is obtained by using the decomposition of polyharmonic functions and turning the problem into an equivalent Riemann boundary value problem for polyanalytic functions, as the approach to find the kernel functions of the solution for the general Dirichlet problem is given.
Article PDF
Similar content being viewed by others
References
Balk M.B. (1991). Polyanalytic Functions. Akademie Verlag, Berlin
Begehr H. (1994). Complex Analytic Methods for Partial Differential Equation. An Introductory Text. World Scientific, Singapore
Begehr, H.: Boundary value problems in complex analysis I., II. Bol. Asoc. Math. Venezolana 12, 65–85, 217–250 (2005)
Begehr H. and Hile G.N. (1997). A hierarchy of integral operators. Rochy Mountains J. Math. 27: 669–706
Begehr H. and Kumar A. (2005). Boundary value problem for inhomogeneous polyanalytic equation. Analysis 25: 55–71
Begehr H., Vu T.N.H. and Zhang Z.-X. (2006). Polyharmonic Dirichlet problems. Proc. Steklov Inst. Math. 255: 13–34
Du J. and Wang Y. (2003). On boundary value problem of polyanalytic function on the real axis. Complex Variables Theory Appl. 48(6): 527–542
Du J. and Wang Y. (2005). Riemann boundary value problem of polyanalytic function and metaanalytic function on the closed curve. Complex Variables Theory Appl. 50(7–11): 521–533
Fatulaev B.F. (2001). The main Haseman type boundary value problem for metaanalytic function in the case of circular domains. Math. Model. Anal. 6(1): 68–76
Lu J. (1993). Boundary Value Problems for Analytic Functions. World Scientific, Singapore
Muskhelishvili, N.I.: Singular Integral Equations, 2nd edn. Noordhoff, Groningen (1968)
Vekua I.N. (1962). Generalized Analytic Functions. Pergamon Press, Oxford
Wang Y. and Du J. (2004). On Riemann boundary value problem of polyanalytic functions on the real axis. Acta. Math. Sci. 24B(4): 663–671
Wang Y. and Du J. (2006). Hilbert boundary value problem of polyanalytic functions on the unit circumference. Complex Var. Elliptic Eqs. 51: 923–943
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by NNSF of China.
Rights and permissions
About this article
Cite this article
Begehr, H., Du, J. & Wang, Y. A Dirichlet problem for polyharmonic functions. Annali di Matematica 187, 435–457 (2008). https://doi.org/10.1007/s10231-007-0050-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-007-0050-5
Keywords
- Polyharmonic function
- Polyanalytic function
- Analytic function
- Kernel function
- Dirichlet problem
- Riemann boundary value problem