  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.
