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

Chaabane Rejeb (1, 2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FST - Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis

Let ∆ k be the Dunkl Laplacian relative to a fixed root system R in R d , d ≥ 2, and to a nonnegative multiplicity function k on R. Our first purpose in this paper is to solve the ∆ k-Dirichlet problem for annular regions. Secondly, we introduce and study the ∆ k-Green function of the annulus and we prove that it can be expressed by means of ∆ k-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for ∆ k-subharmonic functions and we study positive continuous solutions for a ∆ k-semilinear problem.

Mots clés: Dunkl-Laplace operator; Poisson kernel; Green function; Dirichlet problem; spherical harmonics; Newton kernel

Domaines: Equations aux dérivées partielles [math.AP]

Chaabane Rejeb. Green function and Poisson kernel associated to root systems for annular regions. 2019. ⟨hal-02005375⟩