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

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.