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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.09795 |
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| _version_ | 1866913359440379904 |
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| author | Sun, Liming Wang, Lei |
| author_facet | Sun, Liming Wang, Lei |
| contents | We consider a type of Hardy-Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a conformally invariant one using the conformal radius of the domain. The best constant of such inequality on a smooth bounded domain is achieved if and only if the domain is non-convex. In higher dimensions, the best constant is achieved if the domain has negative mean curvature somewhere. If the mean curvature vanishes but is non-umbilic somewhere, we also establish the attainability for some special cases. In the other direction, we also show the best constant is not achieved if the domain is sufficiently close to a ball in $C^2$ sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09795 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Attainability of the best constant of Hardy-Sobolev inequality with full boundary singularities Sun, Liming Wang, Lei Analysis of PDEs 35A23, 35B09, 35B44, 35J75, 35J91 We consider a type of Hardy-Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a conformally invariant one using the conformal radius of the domain. The best constant of such inequality on a smooth bounded domain is achieved if and only if the domain is non-convex. In higher dimensions, the best constant is achieved if the domain has negative mean curvature somewhere. If the mean curvature vanishes but is non-umbilic somewhere, we also establish the attainability for some special cases. In the other direction, we also show the best constant is not achieved if the domain is sufficiently close to a ball in $C^2$ sense. |
| title | Attainability of the best constant of Hardy-Sobolev inequality with full boundary singularities |
| topic | Analysis of PDEs 35A23, 35B09, 35B44, 35J75, 35J91 |
| url | https://arxiv.org/abs/2405.09795 |