Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacians

Abstract

This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of Rn, regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems.We investigate the Dirichlet problem{Δϑu=0, in Bn,u=ϕ, on Sn−1, whereΔϑ:=(1−|x|2){1−|x|24Δ+ϑ∑j=1nxj∂∂xj+ϑ(n2−1−ϑ)I}.We show that: (i) when ϑ>0, the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when ϑ=0, there exists a Lipschitz continuous function ϕ:Sn−1→R such that the solution is not Lipschitz continuous; (iii) when ϑ<0, the solution is definitely not Lipschitz continuous unless ϕ≡0.