On Lipschitz continuity of solutions of hyperbolic Poisson's equation

Abstract

In this paper, we investigate solutions of the hyperbolic Poisson equation ${\Delta }_{h}u(x)=\psi (x)$, where $\psi \in L^{\infty }({\mathbb{B}}^n,{\mathbb{R}}^n)$ and \[ Δ_{h}u(x)= (1-|x|^2)^2Δu(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \] is the hyperbolic Laplace operator in the $n$-dimensional space ${\mathbb{R}}^n$ for $n\ge 2$. We show that if $n\ge 3$ and $u\in C^2({\mathbb{B}}^n,{\mathbb{R}}^n)\cap C(\overline{{\mathbb{B}}^n},{\mathbb{R}}^n)$ is a solution to the hyperbolic Poisson equation, then it has the representation $u=P_h[\varphi ]-G_h[\psi ]$ provided that $u{\mid }_{{\mathbb{S}}^{n-1}}=\varphi$ and ${\int }_{{\mathbb{B}}^n}(1-|x{|}^2{)}^{n-1}|\psi (x)|d\tau (x)<\infty$. Here $P_h$ and $G_h$ denote Poisson and Green integrals with respect to ${\Delta }_h$, respectively. Furthermore, we prove that functions of the form $u=P_h[\varphi ]-G_h[\psi ]$ are Lipschitz continuous.

32 pages

Cited by in corpus (4)

• Khavinson problem for hyperbolic harmonic mappings in Hardy space
• Schwarz lemma for hyperbolic harmonic mappings in the unit ball
• Optimal estimates for hyperbolic harmonic mappings in Hardy space
• Modulus of continuity and Heinz-Schwarz type inequalities of solutions to biharmonic equations