Duality properties of metric Sobolev spaces and capacity
We study the properties of the dual Sobolev space $H^{-1,q}(\mathbb{X})= \big(H^{1,p}(\mathbb{X})\big)'$ on a complete extended metric-topological measure space $\mathbb{X}=(X,\tau,\rm{d},\rm{m})$ for $p\in (1,\infty)$. We will show that a crucial role is played by the strong closure $H_{{\rm{p...
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| Format: | Article |
| Language: | English |
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AIMS Press
2021-10-01
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| Series: | Mathematics in Engineering |
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| Online Access: | https://www.aimspress.com/article/10.3934/mine.2021001/fulltext.html |
| _version_ | 1818384634524729344 |
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| author | Luigi Ambrosio Giuseppe Savaré |
| author_facet | Luigi Ambrosio Giuseppe Savaré |
| author_sort | Luigi Ambrosio |
| collection | DOAJ |
| description | We study the properties of the dual Sobolev space $H^{-1,q}(\mathbb{X})= \big(H^{1,p}(\mathbb{X})\big)'$ on a complete extended metric-topological measure space $\mathbb{X}=(X,\tau,\rm{d},\rm{m})$ for $p\in (1,\infty)$. We will show that a crucial role is played by the strong closure $H_{{\rm{pd}}}^{ - 1,q}\left( {\mathbb{X}} \right)$ of $L^q(X,\rm{m})$ in the dual $H^{-1,q}(\mathbb{X})$, which can be identified with the predual of $H^{1,p}(\mathbb{X})$. We will show that positive functionals in $H^{-1,q}(\mathbb{X})$ can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure $\mu$ with finite dual Sobolev energy, Cap<sub><em>p</em></sub>-negligible sets are also $\mu$-negligible and good representatives of Sobolev functions belong to $L^1(X,\mu)$. We eventually show that the Newtonian-Sobolev capacity Cap<sub><em>p</em></sub> admits a natural dual representation in terms of such a class of Radon measures. |
| first_indexed | 2024-12-14T03:25:23Z |
| format | Article |
| id | doaj.art-ea50ca17899a4bb69cc16375cd012ab0 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-12-14T03:25:23Z |
| publishDate | 2021-10-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-ea50ca17899a4bb69cc16375cd012ab02022-12-21T23:18:53ZengAIMS PressMathematics in Engineering2640-35012021-10-013113110.3934/mine.2021001Duality properties of metric Sobolev spaces and capacityLuigi Ambrosio0Giuseppe Savaré11 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy2 Dipartimento di Matematica, Università di Pavia Via Ferrata 1, 27100 Pavia, Italy 3 Institute for Advanced Study, Technische Universität München, Lichtenbergstrasse 2, Garching, GermanyWe study the properties of the dual Sobolev space $H^{-1,q}(\mathbb{X})= \big(H^{1,p}(\mathbb{X})\big)'$ on a complete extended metric-topological measure space $\mathbb{X}=(X,\tau,\rm{d},\rm{m})$ for $p\in (1,\infty)$. We will show that a crucial role is played by the strong closure $H_{{\rm{pd}}}^{ - 1,q}\left( {\mathbb{X}} \right)$ of $L^q(X,\rm{m})$ in the dual $H^{-1,q}(\mathbb{X})$, which can be identified with the predual of $H^{1,p}(\mathbb{X})$. We will show that positive functionals in $H^{-1,q}(\mathbb{X})$ can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure $\mu$ with finite dual Sobolev energy, Cap<sub><em>p</em></sub>-negligible sets are also $\mu$-negligible and good representatives of Sobolev functions belong to $L^1(X,\mu)$. We eventually show that the Newtonian-Sobolev capacity Cap<sub><em>p</em></sub> admits a natural dual representation in terms of such a class of Radon measures.https://www.aimspress.com/article/10.3934/mine.2021001/fulltext.htmlmetric sobolev spacescapacitymodulus of a family of rectifiable curvesdynamic transport plansdual cheeger energycapacitary measures |
| spellingShingle | Luigi Ambrosio Giuseppe Savaré Duality properties of metric Sobolev spaces and capacity Mathematics in Engineering metric sobolev spaces capacity modulus of a family of rectifiable curves dynamic transport plans dual cheeger energy capacitary measures |
| title | Duality properties of metric Sobolev spaces and capacity |
| title_full | Duality properties of metric Sobolev spaces and capacity |
| title_fullStr | Duality properties of metric Sobolev spaces and capacity |
| title_full_unstemmed | Duality properties of metric Sobolev spaces and capacity |
| title_short | Duality properties of metric Sobolev spaces and capacity |
| title_sort | duality properties of metric sobolev spaces and capacity |
| topic | metric sobolev spaces capacity modulus of a family of rectifiable curves dynamic transport plans dual cheeger energy capacitary measures |
| url | https://www.aimspress.com/article/10.3934/mine.2021001/fulltext.html |
| work_keys_str_mv | AT luigiambrosio dualitypropertiesofmetricsobolevspacesandcapacity AT giuseppesavare dualitypropertiesofmetricsobolevspacesandcapacity |