An inhomogeneous polyharmonic Dirichlet problem with boundary data in the upper half-plane

Abstract

In this paper, an inhomogeneous Dirichlet problem with data for the polyharmonic equation is investigated in the upper half-plane. By using higher order Poisson kernels and Pompeiu operators, which are respectively due to Du et al. [Du Z, Qian T, Wang J. polyharmonic Dirichlet problems in regular domains II: the upper half plane. J Differ Equ. 2012;252:1789–1812] as well as Begehr and Hile [Begehr H, Hile G. A hierarchy of integral operators. Rocky Mountain J Math. 1997;27:669–706], an integral representation for the unique solution is given under certain assumptions.

Keywords:

• Dirichlet problem
• polyharmonic equation
• higher order Poisson kernels
• higher order Pompeiu operators
• inhomogeneous
• non-tangential maximal functions

AMS Subject Classifications:

• 31B10
• 31B30

Acknowledgements

This work was carried out when the first named author visited Department of Mathematics, Temple University by the invitation from Prof. Irina Mitrea and the support from State Scholarship Fund Award of China. He was also partially supported by the NNSF [grant number 11126065], [grant number 11401254]. The second named author was partially supported by SRF of NJUPT [grant number NY208070]. The first named author thanks for the hospitality of Department of Mathematics of Temple University, and a lot of help given by Prof. Irina Mitrea. All the authors appreciate the referee for careful reading and good suggestions which are helpful to improve this paper.

Notes

No potential conflict of interest was reported by the authors.

Dedicate to Professor Dr Heinrich Begehr for his lasting contributions to the theory of integral representations.