Title: Explicit formulas for the Green's function and the Bergman kernel for monogenic functions in annular shaped domains in $\mathbb{R}^{n+1}$
Authors: Constales, Denis
Grob, Dennis
Krausshar, Rolf Sören # ×
Issue Date: Sep-2010
Publisher: Revista Matemática Iberoamericana
Series Title: Revista Matemática Iberoamericana vol:58 issue:1-2 pages:173-189
Abstract: In this paper we consider the space of square integrable monogenic functions in annular shaped domains. Monogenic functions are Clifford-algebra valued functions that are in the kernel of the Cauchy-Riemann operator in $\mathbb{R}^{n+1}$.

We give a fully explicit formula for the reproducing kernel,
called Bergman kernel, for orthogonal sectors of the annulus of the unit ball in the framework of the Cauchy-Riemann operator. The annulus of the half-ball, of the quarter-ball and the annulus of the full unit ball are included as special cases.

Our goal is to develop first an explicit formula for the harmonic Green's function of these types of domains in terms of an infinite series over the classical Green's kernel of $\mathbb{R}^n$ summed over a discrete dilatation group. This arises from successively adding up correction terms corresponding to reflections (or inversions) across the different boundaries of the domain. This corresponds to applying the reflection principle from harmonic analysis to this particular class of annular shaped domains. From the harmonic Green's function we then can determine the Bergman kernel function by applying from the right a Cauchy-Riemann operator and from the left its conjugate in the other variable.

As a concrete application we give an explicit analytic
representation formula of the solutions to the three-dimensional Dirichlet problem in annular shaped domains that arise in the context of heat conduction.
Description: IMPACTFACTOR 2007: 0.89
ISSN: 0213-2230
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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