Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911966821351424 |
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| author | Ferreri, Lorenzo Mazzoleni, Dario Pellacci, Benedetta Verzini, Gianmaria |
| author_facet | Ferreri, Lorenzo Mazzoleni, Dario Pellacci, Benedetta Verzini, Gianmaria |
| contents | We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $Ω\subset \mathbb{R}^{N}$, $N\ge2$, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of $D$ tends to zero, the unique maximum point of $u$, $P\in \partial Ω$, tends to a point of maximal mean curvature of $\partial Ω$. Furthermore, we show that $D$ is the intersection with $Ω$ of a $C^{1,1}$ nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of $D$.
These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17931 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem Ferreri, Lorenzo Mazzoleni, Dario Pellacci, Benedetta Verzini, Gianmaria Analysis of PDEs We complete the study concerning the minimization of the positive principal eigenvalue associated with a weighted Neumann problem settled in a bounded regular domain $Ω\subset \mathbb{R}^{N}$, $N\ge2$, for the weight varying in a suitable class of sign-changing bounded functions. Denoting with $u$ the optimal eigenfunction and with $D$ its super-level set, corresponding to the positivity set of the optimal weight, we prove that, as the measure of $D$ tends to zero, the unique maximum point of $u$, $P\in \partial Ω$, tends to a point of maximal mean curvature of $\partial Ω$. Furthermore, we show that $D$ is the intersection with $Ω$ of a $C^{1,1}$ nearly spherical set, and we provide a quantitative estimate of the spherical asymmetry, which decays like a power of the measure of $D$. These results provide, in the small volume regime, a fully detailed answer to some long-standing questions in this framework. |
| title | Asymptotic location and shape of the optimal favorable region in a Neumann spectral problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.17931 |