On Sobolev spaces and density theorems on Finsler manifolds - Behroz Bidabad; Alireza Shahi (AUT Journal of Mathematics and Computing, 2020)
Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{∞}$ functions with compact support on a forward geodesically complete Finsler manifold $(M, F),$ is dense in the extended Sobolev space $H^p_1(M)$. As a consequence, the weak solutions u of the Dirichlet equation $Δ u=f$ can be approximated by $C^{∞}$ functions with compact support on $M$. Moreover, let $W\subseteq M$ 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 $H^p_k(W)$, where $k≤ r$. Finally, several examples are illustrated and sharpness of the inequality $k≤ r$ is shown.