# Approximate solutions to the Dirichlet problem for harmonic maps between hyperbolic spaces

## TL;DR

This paper constructs approximate harmonic maps between hyperbolic spaces that extend given boundary functions, providing solutions to the Dirichlet problem with controlled boundary behavior.

## Contribution

It introduces a method to obtain approximate harmonic maps between hyperbolic spaces with prescribed boundary conditions, extending the boundary data by a small parameter.

## Key findings

- Existence of harmonic maps approximating boundary functions
- Boundary maps can be extended continuously into hyperbolic spaces
- Approximate solutions depend on a small parameter psilon"

## Abstract

Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):\partial H^m\to \partial H^n$ between geometric boundaries of $H^m$ and $H^n$. Then for each $\epsilon >0$ there exists a harmonic map $u:H^m\to H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{\partial H^m}=(f^1,...,f^{n-1},\epsilon)$.