Decompositions of functions and Dirichlet problems in the unit disc

作者：Du Zhihua^*; Guo Guoan; Wang Ning

来源：Journal of Mathematical Analysis and Applications, 2010, 362(1): 1-16.

DOI：10.1016/j.jmaa.2009.07.048

• Decomposition of polyharmonic functions
• Polyharmonic functions
• Polyanalytic functions
• Dirichlet problem

In this paper, we establish a decomposition theorem for polyharmonic functions and consider its applications to some Dirichlet problems in the unit disc. By the decomposition, we get the unique solution of the Dirichlet problem for polyharmonic functions (PHD problem) and give a unified expression for a class of kernel functions associated with the solution in the case of the unit disc introduced by Begehr, Du and Wang. In addition, we also discuss some quasi-Dirichlet problems for homogeneous mixed-partial differential equations of higher order. It is worthy to note that the decomposition theorem in the present paper is a natural extension of the Goursat decomposition theorem for biharmonic functions.

• 出版日期2010-2-1
• 单位南京邮电大学; 武汉大学; 暨南大学