Green Function and Poisson Kernel Associated to Root Systems for Annular Regions

Abstract

Let Δk be the Dunkl Laplacian relative to a fixed root system \(\mathcal {R}\) in \(\mathbb {R}^{d}\), d ≥ 2, and to a nonnegative multiplicity function k on \(\mathcal {R}\). Our first purpose in this paper is to solve the Δk-Dirichlet problem for annular regions. Secondly, we introduce and study the Δk-Green function of the annulus and we prove that it can be expressed by means of Δk-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for Δk-subharmonic functions and we study positive continuous solutions for a Δk-semilinear problem.

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Acknowledgements

It is a pleasure to thank the referee for the valuable suggestions which improved the presentation of the paper.

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Correspondence to Chaabane Rejeb.

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Rejeb, C. Green Function and Poisson Kernel Associated to Root Systems for Annular Regions. Potential Anal (2020). https://doi.org/10.1007/s11118-020-09856-2

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Keywords

  • Dunkl-Laplace operator
  • Poisson kernel
  • Green function
  • Dirichlet problem
  • Spherical harmonics
  • Newton kernel

Mathematics Subject Classification (2020)

  • Primary: 31B05
  • 31B20
  • 31J05
  • 35J08
  • Secondary: 31C45
  • 46F10
  • 47B39