Explicit Formulas for the Green’s Function and the Bergman Kernel for Monogenic Functions in Annular Shaped Domains in $${\mathbb{R}^{n+1}}$$

Abstract

By applying a reflection principle we set up fully explicit representation formulas for the harmonic Green’s function for orthogonal sectors of the annulus of the unit ball of \({\mathbb{R}^n}\). From the harmonic Green’s function we then can determine the Bergman kernel function of Clifford holomorphic functions by applying an appropriate vector differentiation. As a concrete application we give an explicit analytic representation formula of the solutions to an n-dimensional Dirichlet problem in annular shaped domains that arises in the context of heat conduction.