Green function and Poisson kernel associated to root systems for annular regions

Chaabane Rejeb

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arXiv · 2304.10172

Green function and Poisson kernel associated to root systems for annular regions

Chaabane Rejeb

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Abstract

Let $\Delta_k$ be the Dunkl Laplacian relative to a fixed root system $\mathcal{R}$ in $\mathbb{R}^d$, $d\geq2$, and to a nonnegative multiplicity function $k$ on $\mathcal{R}$. Our first purpose in this paper is to solve the $\Delta_k$-Dirichlet problem for annular regions. Secondly, we introduce and study the $\Delta_k$-Green function of the annulus and we prove that it can be expressed by means of $\Delta_k$-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for $\Delta_k$-subharmonic functions and we study positive continuous solutions for a $\Delta_k$-semilinear problem.

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Chaabane Rejeb. 2023-04-20. Green function and Poisson kernel associated to root systems for annular regions. https://arxiv.org/abs/2304.10172

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