AUT Journal of Mathematics and Computing (Feb 2020)

On Sobolev spaces and density theorems on Finsler manifolds

  • Behroz Bidabad,
  • Alireza Shahi

DOI
https://doi.org/10.22060/ajmc.2018.3039
Journal volume & issue
Vol. 1, no. 1
pp. 37 – 45

Abstract

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Here, a natural extension of Sobolev spaces is defined for a Finsler structure $F$ and it is shown that the set of all real $C^{\infty}$ 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 $\Delta u=f$ can be approximated by $C^{\infty}$ 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 $k\leq r$. Finally, several examples are illustrated and sharpness of the inequality $k\leq r$ is shown.

Keywords