Dirichlet problem for a generalized inhomogeneous polyharmonic equation in an annular domain

Abstract

In this article, we investigate the solvability of the Dirichlet problems in ring domains for elliptic linear complex partial differential equations having polyharmonic operators as main parts. First, we give higher order Green functions as fundamental solutions of the homogeneous problems using the iteration of harmonic Green functions for ring domains. Second, we introduce some classes of operators related to Dirichlet problems together with their basic properties. Next, we transform the original problems into equivalent singular integral equations. Finally, solvability of the problems is discussed by defining the adjoint problems and using Fredholm alternative.

Keywords:

• Dirichlet problem
• Green function
• higher order Poisson equation
• polyharmonic equation
• doubly connected domain

AMS Subject Classifications::

• 31A30
• 31A10
• 31A25

Acknowledgements

The authors thank the referees for suggestions to improve this article.