In this paper, we investigate solutions of the hyperbolic Poisson equation Δ hu(x) = ψ(x) , where ψ∈ L∞(Bn, Rn) and (Formula Presented.)Δhu(x)=(1-|x|2)2Δu(x)+2(n-2)(1-|x|2)∑i=1nxi∂u∂xi(x)is the hyperbolic Laplace operator in the n-dimensional space Rn for n≥ 2. We show that if n≥ 3 and u∈ C2(Bn, Rn) ∩ C(Bn¯ , Rn) is a solution to the hyperbolic Poisson equation, then it has the representation u= Ph[ ϕ] - Gh[ ψ] provided that u∣Sn-1=ϕ and ∫Bn(1-|x|2)n-1|ψ(x)|dτ(x)<∞. Here Ph and Gh denote Poisson and Green integrals with respect to Δ h, respectively. Furthermore, we prove that functions of the form u= Ph[ ϕ] - Gh[ ψ] are Lipschitz continuous.
|Journal||Calculus of Variations and Partial Differential Equations|
|State||Published - 1 Feb 2018|
- Primary: 31B05
- Secondary: 31C05