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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.19339 |
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| _version_ | 1866909705339666432 |
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| author | Noris, Benedetta Siclari, Giovanni Verzini, Gianmaria |
| author_facet | Noris, Benedetta Siclari, Giovanni Verzini, Gianmaria |
| contents | We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $Ω$ having prescribed volume and contained in a fixed box $D$; equivalently, we look for the best way to remove a compact set (obstacle) $K\subset\overline{D}$ of Lebesgue measure $|K|=\varepsilon$, $0<\varepsilon<|D|$, in order to minimize the first Dirichlet eigenvalue of the set $Ω= D \setminus K$.
In the small volume regime $\varepsilon\to0$, we prove that the optimal obstacles accumulate, in a suitable sense, to points of $\partial D$ where $|\nabla ϕ_0|$ is minimal, where $ϕ_0$ denotes the first eigenfunction of the Dirichlet Laplacian on $D$. Moreover, we provide a fairly detailed description of the convergence of the optimal eigenvalues, eigenfunctions and free boundaries. Our results are based on sharp estimates of the optimal eigenvalues, in terms of a suitable notion of relative capacity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19339 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle Noris, Benedetta Siclari, Giovanni Verzini, Gianmaria Analysis of PDEs We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $Ω$ having prescribed volume and contained in a fixed box $D$; equivalently, we look for the best way to remove a compact set (obstacle) $K\subset\overline{D}$ of Lebesgue measure $|K|=\varepsilon$, $0<\varepsilon<|D|$, in order to minimize the first Dirichlet eigenvalue of the set $Ω= D \setminus K$. In the small volume regime $\varepsilon\to0$, we prove that the optimal obstacles accumulate, in a suitable sense, to points of $\partial D$ where $|\nabla ϕ_0|$ is minimal, where $ϕ_0$ denotes the first eigenfunction of the Dirichlet Laplacian on $D$. Moreover, we provide a fairly detailed description of the convergence of the optimal eigenvalues, eigenfunctions and free boundaries. Our results are based on sharp estimates of the optimal eigenvalues, in terms of a suitable notion of relative capacity. |
| title | Miminization of the first eigenvalue of the Dirichlet Laplacian with a small volume obstacle |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.19339 |