Cylindrical estimates for the Cheeger constant and applications
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909379555491840 |
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| author | Pratelli, Aldo Saracco, Giorgio |
| author_facet | Pratelli, Aldo Saracco, Giorgio |
| contents | We prove a lower bound for the Cheeger constant of a cylinder $Ω\times (0,L)$, where $Ω$ is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the first Dirichlet eigenvalue of the $p$-Laplacian and the $p$-th power of the Cheeger constant, within the class of bounded convex sets in any $\mathbb{R}^N$. This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani-Buttazzo-Prinari (Ann. Mat. Pura Appl. (2023)). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_09864 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cylindrical estimates for the Cheeger constant and applications Pratelli, Aldo Saracco, Giorgio Analysis of PDEs Spectral Theory Primary: 49Q10. Secondary: 49R05, 35P15 We prove a lower bound for the Cheeger constant of a cylinder $Ω\times (0,L)$, where $Ω$ is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the first Dirichlet eigenvalue of the $p$-Laplacian and the $p$-th power of the Cheeger constant, within the class of bounded convex sets in any $\mathbb{R}^N$. This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani-Buttazzo-Prinari (Ann. Mat. Pura Appl. (2023)). |
| title | Cylindrical estimates for the Cheeger constant and applications |
| topic | Analysis of PDEs Spectral Theory Primary: 49Q10. Secondary: 49R05, 35P15 |
| url | https://arxiv.org/abs/2402.09864 |