Existence and regularity of weakly harmonic maps into a Finsler manifold with a special structure

TL;DR

This paper proves the existence and regularity of weakly harmonic maps from low-dimensional Riemannian manifolds into a special class of Finsler manifolds with a specific structure.

Contribution

It establishes existence and interior regularity results for harmonic maps into Finsler manifolds with a particular structure, extending classical results to this setting.

Findings

Existence of harmonic maps under specified conditions.

Interior regularity of these harmonic maps.

Applicable for source manifolds of dimension up to 4.

Abstract

We study Dirichlet problems for harmonic maps from a Riemannian $m$-manifold $(M,g)$ into a Finsler $n$-manifold $(N, F)$. We assume that the dimension of the source manifold $M$ is less than or equal to 4, and that the finsler structure $F(u,X)$ is given as F(u,X)= \sqrt{h_{ij}(u)X^i X^j + {\cal B}(u,X)}, (u\in N, X \in T_uN) where $(h_{ij})$ is a Riemannian metric and ${\cal B}(u,X)$ is a function on $TN$ with positive homogeneity of degree 2 with respect to $X$. Under these assumptions, an existence and interior regularity result will be given.