Lp Polyharmonic Dirichlet Problems in the Upper Half Plane
Author(s)
Kanda Pan*
ABSTRACT
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.
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.
KEYWORDS
Dirichlet Problem, Polyharmonic Function, Higher Order Poisson Kernels, Higher Order Pompeiu Operators, Non-Tangential Maximal Function, Uniqueness
Dirichlet Problem, Polyharmonic Function, Higher Order Poisson Kernels, Higher Order Pompeiu Operators, Non-Tangential Maximal Function, Uniqueness
Cite this paper
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.
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.
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
[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