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Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ apositive integer and $H_k^p (M)$ a certain Sobolev space determined by aFinsler structure $F$.  Here, it is shown that the set of all real $C^{\infty}$ functions withcompact support on $M$ is dense in the Sobolev space $H_1^p (M)$.  This result permits to approximate certain solution of Dirichlet problemliving on $H_1^p (M)$ by $C^ \infty$ functions with compact support on $(M,F)$.  Moreover, let $W \subset M$ be a regular domain with the $C^r$ boundary$\partial W$, then the set of all real functions in $C^r (W) \cap C^0(\overline W)$ is dense in $H_k^p (W)$, where $k\leq r$. This work is anextension of some density theorems of T. Aubin on Riemannian manifolds.

On Sobolev spaces and density theorems on Finsler manifolds