Let Δ_k 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 Δ_k-Dirichlet problem for annular regions. Secondly, we introduce and study the Δ_k-Green function of the annulus and we prove that it can be expressed by means of Δ_k-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for Δ_k-subharmonic functions and we study positive continuous solutions for a Δ_k-semilinear problem.

中文翻译：

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与环形区域根系统相关的Green函数和Poisson核

让Δ _ķ是Dunkl拉普拉斯相对于固定的根系统\（\ mathcal {R} \）在\（\ mathbb {R} ^ {d} \） ，d ≥2，以及一个非负多重功能ķ上\ （\ mathcal {R} \）。本文的第一个目的是解决环形区域的_Δk -Dirichlet问题。其次，我们引入和研究的Δ _ķ环的-绿色功能，我们证明，它可通过Δ来表达_ķ -spherical谐波。作为应用，我们得到泊松詹森公式Δ _ķ -subharmonic功能和我们研究了Δ正连续解_ķ-半线性问题。