TY - JOUR ID - 3039 TI - On Sobolev spaces and density theorems on Finsler manifolds JO - AUT Journal of Mathematics and Computing JA - AJMC LA - en SN - 2783-2449 AU - Bidabad, Behroz AU - Shahi, Alireza AD - Department of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, Iran Y1 - 2020 PY - 2020 VL - 1 IS - 1 SP - 37 EP - 45 KW - Density theorem KW - Sobolev spaces KW - Dirichlet problem KW - Finsler space DO - 10.22060/ajmc.2018.3039 N2 - 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. UR - https://ajmc.aut.ac.ir/article_3039.html L1 - https://ajmc.aut.ac.ir/article_3039_bcbcb1f45609881ba462e01ecc38e982.pdf ER -