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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2510.09346 |
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| _version_ | 1866915544629772288 |
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| author | Verzini, Gianmaria |
| author_facet | Verzini, Gianmaria |
| contents | We consider a shape optimization problem for the persistence threshold of a biological species dispersing in a periodically fragmented environment, the unknown shape corresponding to the portion of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the periodic Laplacian, with respect to a bang-bang indefinite weight. For such problem, we exploit some recent results obtained in the framework of Dirichlet or Neumann boundary conditions, to provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes.
First, we show that the optimal favorable zone shrinks to a connected, convex, nearly spherical set, in $C^{1,1}$ sense. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the $C^{1,α}$ sense, for every $α<1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shape optimization of a small favorable region in a periodically fragmented environment Verzini, Gianmaria Analysis of PDEs We consider a shape optimization problem for the persistence threshold of a biological species dispersing in a periodically fragmented environment, the unknown shape corresponding to the portion of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the periodic Laplacian, with respect to a bang-bang indefinite weight. For such problem, we exploit some recent results obtained in the framework of Dirichlet or Neumann boundary conditions, to provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes. First, we show that the optimal favorable zone shrinks to a connected, convex, nearly spherical set, in $C^{1,1}$ sense. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the $C^{1,α}$ sense, for every $α<1$. |
| title | Shape optimization of a small favorable region in a periodically fragmented environment |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.09346 |