On Sobolev spaces and density theorems on Finsler manifolds
| Autor: | Behroz Bidabad, Alireza Shahi |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | AUT Journal of Mathematics and Computing, Vol 1, Iss 1, Pp 37-45 (2020) |
| Druh dokumentu: | article |
| ISSN: | 2783-2449 2783-2287 |
| 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 $H^p_1(M)$. 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. |
| Databáze: | Directory of Open Access Journals |
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