On Sobolev spaces and density theorems on Finsler manifolds

Abstract

Let $(M,F)$ be a $C^{\infty }$ Finsler manifold, $p\ge 1$ 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 $k\le r$. This work is an extension of some density theorems of T. Aubin on Riemannian manifolds.

13 pages

References in corpus (1)

• On Compact Finsler Spaces of Positive Constant Curvature

Cited by in corpus (1)

• The Schwarzian derivative on Finsler manifolds of constant curvature