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.
Categories: math.AP