On Sobolev spaces and density theorems on Finsler manifolds

TL;DR

This paper extends density theorems for Sobolev spaces on Finsler manifolds, showing smooth compactly supported functions are dense, enabling approximation of solutions to boundary value problems in this setting.

Contribution

It generalizes Aubin's density theorems from Riemannian to Finsler manifolds, establishing density of smooth functions in Sobolev spaces on Finsler structures.

Findings

Density of smooth compactly supported functions in $H_1^p(M)$.

Approximation of solutions to Dirichlet problems by smooth functions.

Density of functions in $C^r(W) \cap C^0(\overline{W})$ in $H_k^p(W)$.

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.