On Sobolev spaces and density theorems on Finsler manifolds

Document Type : Original Article

Authors

1 Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)

2 Faculty of Mathematics and computer science, Amirkabir University of Technology

Abstract

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 $H_1^p (M)$.
As a consequence, the weak solutions $u$ of the Dirichlet equation $Δu=f$ can be approximated by $C^∞$ functions with compact support on $M$.
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 $k≤r$. Finally, several examples are illustrated and sharpness of the inequality $k≤r$ is shown.

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