Explicit formulas for the Green's function and the Bergman kernel for monogenic functions in annular shaped domains in \(\mathbb{R}^{n+1}\)
Given the harmonic Green's function of a domain, its hypercomplex Bergman kernel for the corresponding class of functions of Clifford or quaternionic analysis can be obtained by applying from the left- and from the right-hand side the proper differential Cauchy-Riemann, or Dirac, operators. Thus the authors first obtain explicit representation formulas for the harmonic Green's function for orthogonal sectors of the annulus of the unit ball in \(\mathbb R^n\), and then construct the respective Bergman kernels. This applies in order to give an explicit analytic representation of the solutions to an \(n\)-dimensional Dirichlet problem in annular shaped domains arising in the context of heat conduction.
scientific article; zbMATH DE number 5800221
scientific article