On Sobolev spaces and density theorems on Finsler manifolds
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, t...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Amirkabir University of Technology
2020-02-01
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| Series: | AUT Journal of Mathematics and Computing |
| Subjects: | |
| Online Access: | https://ajmc.aut.ac.ir/article_3039_bcbcb1f45609881ba462e01ecc38e982.pdf |
| _version_ | 1797307136231866368 |
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| author | Behroz Bidabad Alireza Shahi |
| author_facet | Behroz Bidabad Alireza Shahi |
| author_sort | Behroz Bidabad |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T00:51:55Z |
| format | Article |
| id | doaj.art-68b3810b2b8745d4b57d95ebf177794f |
| institution | Directory Open Access Journal |
| issn | 2783-2449 2783-2287 |
| language | English |
| last_indexed | 2024-03-08T00:51:55Z |
| publishDate | 2020-02-01 |
| publisher | Amirkabir University of Technology |
| record_format | Article |
| series | AUT Journal of Mathematics and Computing |
| spelling | doaj.art-68b3810b2b8745d4b57d95ebf177794f2024-02-14T19:33:06ZengAmirkabir University of TechnologyAUT Journal of Mathematics and Computing2783-24492783-22872020-02-0111374510.22060/ajmc.2018.30393039On Sobolev spaces and density theorems on Finsler manifoldsBehroz Bidabad0Alireza Shahi1Department of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, IranDepartment of Mathematics and Computer Science, Amirkabir University of Technology, 424, Hafez Ave., Tehran 15914, IranHere, 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.https://ajmc.aut.ac.ir/article_3039_bcbcb1f45609881ba462e01ecc38e982.pdfdensity theoremsobolev spacesdirichlet problemfinsler space |
| spellingShingle | Behroz Bidabad Alireza Shahi On Sobolev spaces and density theorems on Finsler manifolds AUT Journal of Mathematics and Computing density theorem sobolev spaces dirichlet problem finsler space |
| title | On Sobolev spaces and density theorems on Finsler manifolds |
| title_full | On Sobolev spaces and density theorems on Finsler manifolds |
| title_fullStr | On Sobolev spaces and density theorems on Finsler manifolds |
| title_full_unstemmed | On Sobolev spaces and density theorems on Finsler manifolds |
| title_short | On Sobolev spaces and density theorems on Finsler manifolds |
| title_sort | on sobolev spaces and density theorems on finsler manifolds |
| topic | density theorem sobolev spaces dirichlet problem finsler space |
| url | https://ajmc.aut.ac.ir/article_3039_bcbcb1f45609881ba462e01ecc38e982.pdf |
| work_keys_str_mv | AT behrozbidabad onsobolevspacesanddensitytheoremsonfinslermanifolds AT alirezashahi onsobolevspacesanddensitytheoremsonfinslermanifolds |