On Sobolev spaces and density theorems on Finsler manifolds
| Autor: | Bidabad, Behroz, Shahi, Alireza |
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| Rok vydání: | 2020 |
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| Zdroj: | On Sobolev spaces and density theorems on Finsler manifolds, AUT J. Math. Com. 1 2020, 37-45 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.22060/ajmc.2018.3039 |
| Popis: | 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 $H_1^p (M)$. 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 $W \subset M$ 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 $k\leq r$. Finally, several examples are illustrated and sharpness of the inequality $k\leq r$ is shown. Comment: Published in AUT Journal of Math. Comp. arXiv admin note: text overlap with arXiv:1310.8027 |
| Databáze: | arXiv |
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