On Lipschitz Continuity and Smoothness Up to the Boundary of Solutions of Hyperbolic Poisson’s Equation

Abstract

We solve the Dirichlet problem u|Bn=φ, for hyperbolic Poisson’s equation Δ hu= μ where φ∈ L1(∂Bn) and μ is a measure that satisfies a growth condition. Next we present a short proof for Lipschitz continuity of solutions of certain hyperbolic Poisson’s equations, previously established at Chen et al. (Calc Var 57:13, 2018. https://doi.org/10.1007/s00526-017-1290-x). In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz’s potential. Also, local Hölder continuity is proved for solution of certain hyperbolic Poisson’s equations. We show that, if u is hyperbolic harmonic in the upper half-space, then ∂u∂y(x0,y)→0,y→0+ , when boundary function f of the functions u is differentiable at the boundary point x . As a corollary, we show C1(Hn¯) smoothness of a hyperbolic harmonic function, which is reproduced from the Cc1(Rn-1) boundary values.