On Sobolev spaces and density theorems on Finsler manifolds

Abstract

Let $(M,F)$ be a $C^\infty$ Finsler manifold, $p\geq 1$ a real number, $k$ a positive integer and $H_k^p (M)$ a certain Sobolev space determined by a Finsler structure $F$. Here, it is shown that the set of all real $C^{\infty}$ functions with compact support on $M$ is dense in the Sobolev space $H_1^p (M)$. This result permits to approximate certain solution of Dirichlet problem living 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 an extension of some density theorems of T. Aubin on Riemannian manifolds.