Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ spaces

TL;DR

This paper establishes the existence and uniqueness of harmonic maps solving the Dirichlet problem from strongly rectifiable metric spaces into regular balls in CAT(1) spaces, using Korevaar-Schoen energy.

Contribution

It proves the Korevaar-Schoen energy admits a unique minimizer for harmonic maps in this setting, extending previous results to strongly rectifiable spaces.

Findings

Unique minimizer of Korevaar-Schoen energy established

Existence and uniqueness of harmonic maps proven

Extension of harmonic map theory to rectifiable spaces

Abstract

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in $\CAT(1)$ space. Under the setting, we prove that the Korevaar-Schoen energy admits a unique minimizer.