Capacitary inradius and Poincaré-Sobolev inequalities
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911894285058048 |
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| author | Bozzola, Francesco Brasco, Lorenzo |
| author_facet | Bozzola, Francesco Brasco, Lorenzo |
| contents | We prove a two-sided estimate on the sharp $L^p$ Poincaré constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case $p=2$, in the subconformal regime. We cover the whole range of $p$, by allowing in particular the extremal cases $p=1$ (Cheeger's constant) and $p=N$ (conformal case), as well. We also discuss the more general case of the sharp Poincaré-Sobolev embedding constants and get an analogous result. Finally, we present a brief discussion on the superconformal case, as well as some examples and counter-examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Capacitary inradius and Poincaré-Sobolev inequalities Bozzola, Francesco Brasco, Lorenzo Analysis of PDEs Functional Analysis We prove a two-sided estimate on the sharp $L^p$ Poincaré constant of a general open set, in terms of a capacitary variant of its inradius. This extends a result by Maz'ya and Shubin, originally devised for the case $p=2$, in the subconformal regime. We cover the whole range of $p$, by allowing in particular the extremal cases $p=1$ (Cheeger's constant) and $p=N$ (conformal case), as well. We also discuss the more general case of the sharp Poincaré-Sobolev embedding constants and get an analogous result. Finally, we present a brief discussion on the superconformal case, as well as some examples and counter-examples. |
| title | Capacitary inradius and Poincaré-Sobolev inequalities |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2405.19786 |