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

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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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Authors and Affiliations

  1. Faculté des Sciences de Tunis, Laboratoire d’Analyse Mathématques et Applications LR11ES11, Université de Tunis El Manar, 2092 El Manar I, Tunis, Tunisia

    Chaabane Rejeb

  2. Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350, Université de Tours, Campus de Grandmont, 37200, Tours, France

    Chaabane Rejeb

  3. Université de Sherbrooke, Sherbrooke, Canada

    Chaabane Rejeb

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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 55, 251–275 (2021). https://doi.org/10.1007/s11118-020-09856-2

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