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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2412.06437 |
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| _version_ | 1866915068605628416 |
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| author | Henrot, Antoine Lemenant, Antoine Privat, Yannick |
| author_facet | Henrot, Antoine Lemenant, Antoine Privat, Yannick |
| contents | In this article, we address the problem of determining a domain in $\mathbb{R}^N$ that minimizes the first eigenvalue of the Lamé system under a volume constraint. We begin by establishing the existence of such an optimal domain within the class of quasi-open sets, showing that in the physically relevant dimensions $N = 2$ and $3$, the optimal domain is indeed an open set. Additionally, we derive both first and second-order optimality conditions. Leveraging these conditions, we demonstrate that in two dimensions, the disk cannot be the optimal shape when the Poisson ratio is below a specific threshold, whereas above this value, it serves as a local minimizer. We also extend our analysis to show that the disk is nonoptimal for Poisson ratios $ν$ satisfying $ν\leq 0.4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimization of the first eigenvalue for the Lamé system Henrot, Antoine Lemenant, Antoine Privat, Yannick Analysis of PDEs Spectral Theory 49Q10, 47A10, 74B05, 74P10 In this article, we address the problem of determining a domain in $\mathbb{R}^N$ that minimizes the first eigenvalue of the Lamé system under a volume constraint. We begin by establishing the existence of such an optimal domain within the class of quasi-open sets, showing that in the physically relevant dimensions $N = 2$ and $3$, the optimal domain is indeed an open set. Additionally, we derive both first and second-order optimality conditions. Leveraging these conditions, we demonstrate that in two dimensions, the disk cannot be the optimal shape when the Poisson ratio is below a specific threshold, whereas above this value, it serves as a local minimizer. We also extend our analysis to show that the disk is nonoptimal for Poisson ratios $ν$ satisfying $ν\leq 0.4$. |
| title | Minimization of the first eigenvalue for the Lamé system |
| topic | Analysis of PDEs Spectral Theory 49Q10, 47A10, 74B05, 74P10 |
| url | https://arxiv.org/abs/2412.06437 |