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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 multiplicityfunction $k$ on $\mathcal{R}$. Our first purpose in this paper is to solve the$\Delta_k$-Dirichlet problem for annular regions. Secondly, we introduce andstudy the $\Delta_k$-Green function of the annulus and we prove that it can beexpressed by means of $\Delta_k$-spherical harmonics. As applications, weobtain a Poisson-Jensen formula for $\Delta_k$-subharmonic functions and westudy positive continuous solutions for a $\Delta_k$-semilinear problem.

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