On Lipschitz continuity and smoothness up to the boundary of solutions of hyperbolic Poisson's equation

TL;DR

This paper investigates the boundary regularity and continuity properties of solutions to hyperbolic Poisson's equation, providing new proofs and conditions for Lipschitz and Hölder continuity, and analyzing boundary behavior of hyperbolic harmonic functions.

Contribution

It offers a short proof of Lipschitz continuity for hyperbolic Poisson solutions and explores alternative assumptions related to Riesz potentials, advancing understanding of boundary regularity.

Findings

Solutions are Lipschitz continuous under certain conditions.

Local Hölder continuity is established for specific hyperbolic Poisson equations.

Boundary derivatives of hyperbolic harmonic functions tend to zero at boundary points.

Abstract

We solve the Dirichlet problem $\left.u\right|_{\mathbb{B}^n}=\varphi,$ for hyperbolic Poisson's equation $\Delta_h u=\mu$ where $\varphi\in L_1(\partial \mathbb{B}^n)$ and $\mu$ 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 \cite{ChenRas}. In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz's potential. Also, local H\"{o}lder 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 $\frac{\partial u}{\partial y}(x_0,y)\to 0, y\to 0^+$, when boundary function $f$ of the functions $u$ is differentiable at the boundary point $x_0$. As a corollary, we show $C^1(\overline{\mathbb{H}^n})$ smoothness of a hyperbolic harmonic function, which is reproduced from the $C_c^1(\mathbb{R}^{n-1})$ boundary values.