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

Abstract

Let ${\Delta }_k$ be the Dunkl Laplacian relative to a fixed root system $\mathcal{R}$ in ${\mathbb{R}}^d$, $d\ge 2$, 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.