A new approach for solving a Dirichlet problem for polyharmonic functions

Abstract

In this article, the Dirichlet problem for polyharmonic functions investigated in (Begehr, H., Wang, Y. and Du, J., A Dirichlet problem for polyharmonic functions (In press)) is also solved by a new approach different from the reflection method used in (Begehr, H., Wang, Y. and Du, J., A Dirichlet problem for polyharmonic functions (In press)). The explicit expression of the unique solution for the Dirichlet problem of triharmonic functions is obtained by using the so-called weak decomposition of polyharmonic functions and turning the problem into an equivalent Dirichlet boundary value problem for polyanalytic functions. In contrast to the boundary condition according to Begehr, Wang and Du, the requirement of smoothness for the given functions is reduced.

This article is dedicated to Professor Lu Jianke on the occasion of his 85th birthday.

Keywords:

• Polyharmonic function
• Triharmonic function
• Trianalytic function
• Dirichlet problem

AMS Subject Classifications: :

• 30G30
• 45E05

Acknowledgements

This project was supported by NNSF of China (10471107) and the Tianyuan Math. Fund of China (10626039).

Notes

This article is dedicated to Professor Lu Jianke on the occasion of his 85th birthday.