L^p Polyharmonic Dirichlet Problems in the Upper Half Plane

Kanda Pan^*

Department of Mathematics, Jinan University, Guangzhou, China.

DOI: 10.4236/apm.2015.514077                

Abstract

In this article, a class of Dirichlet problem with L^p boundary data for poly-harmonic function in the upper half plane is mainly investigated. By introducing a sequence of kernel functions called higher order Poisson kernels and a hierarchy of integral operators called higher order Pompeiu operators, we obtain a main result on integral representation solution as well as the uniqueness of the polyharmonic Dirichlet problem under a certain estimate.

Keywords

Dirichlet Problem, Polyharmonic Function, Higher Order Poisson Kernels, Higher Order Pompeiu Operators, Non-Tangential Maximal Function, Uniqueness

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Aronszajn, N., Cresse, T. and Lipkin, L. (1983) Polyharmonic Functions, Oxford Math. Clarendon, Oxford.
[2]	Goursat, E. (1898) Sur I’équation ΔΔu = 0. Bulletin de la Société Mathématique de France, 26, 236-237.
[3]	Vekua, I.N. (1976) On One Method of Solving the First Biharmonic Boundary Value Problem and the Dirichlet Problem. American Mathematical Society Translations, 104, 104-111.
[4]	Begehr, H., Du, J. and Wang, Y. (2008) A Dirichlet Problem for Polyharmonic Functions. Annali di Matematica Pura ed Applicata, 187, 435-457. http://dx.doi.org/10.1007/s10231-007-0050-5
[5]	Begehr, H. and Gaertner, E. (2007) A Dirichlet Problem for the Inhomogeneous Polyharmonic Equations in the Upper Half Plane. Georgian Mathematical Journal, 14, 33-52.
[6]	Verchota, G.C. (2005) The Biharmonic Neumann Problem in Lipschitz Domain. Acta Mathematica, 194, 217-279. http://dx.doi.org/10.1007/BF02393222
[7]	Du, Z. (2008) Boundary Value Problems for Higher Order Complex Differential Equations. Doctoral Dissertation, Freie Universität Berlin, Berlin.
[8]	Du, Z., Qian, T. and Wang, J.X. (2012) Polyharmonic Dirichlet Problem in Regular Domain: The Upper Half Plane. Journal of Differential Equations, 252, 1789-1812. http://dx.doi.org/10.1016/j.jde.2011.08.024
[9]	Stein, E.M. and Weiss, G. (1971) Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton, New Jersey.
[10]	Garnett, J. (2007) Bounded Analytic Functions. Springer, New York.
[11]	Begehr, H. and Hile, G.N. (1997) A Hierarchy of Integral Operators. Rocky Mountain Journal of Mathematics, 27, 669-706. http://dx.doi.org/10.1216/rmjm/1181071888