Lp Polyharmonic Dirichlet Problems in the Upper Half Plane

DOI: 10.4236/apm.2015.514077   PDF   HTML   XML   4,644 Downloads   5,021 Views  


In this article, a class of Dirichlet problem with Lp 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.

Share and Cite:

Pan, K. (2015) Lp Polyharmonic Dirichlet Problems in the Upper Half Plane. Advances in Pure Mathematics, 5, 828-834. doi: 10.4236/apm.2015.514077.

Conflicts of Interest

The authors declare no conflicts of interest.


[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.
[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.
[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.
[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.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.