A shape optimization problem in cylinders and related overdetermined problems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909722332889088 |
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| author | Caldiroli, Paolo Iacopetti, Alessandro Pacella, Filomena |
| author_facet | Caldiroli, Paolo Iacopetti, Alessandro Pacella, Filomena |
| contents | In this paper, we study a shape optimization problem for the torsional energy associated with a domain contained in an infinite cylinder, under a volume constraint. We prove that a minimizer exists for all fixed volumes and show some of its geometric and topological properties. As this issue is closely related to the question of characterizing domains in cylinders that admit solutions to an overdetermined problem, our minimization result allows us to deduce interesting consequences in that direction. In particular, we find that, for some cylinders and some volumes, the ``trivial" domain given by a bounded cylinder is not the only domain where the overdetermined problem has a solution. Moreover, it is not even a minimizer, which indicates that solutions with flat level sets are not always the best candidates for optimizing the torsional energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A shape optimization problem in cylinders and related overdetermined problems Caldiroli, Paolo Iacopetti, Alessandro Pacella, Filomena Analysis of PDEs 35N25, 49Q10 In this paper, we study a shape optimization problem for the torsional energy associated with a domain contained in an infinite cylinder, under a volume constraint. We prove that a minimizer exists for all fixed volumes and show some of its geometric and topological properties. As this issue is closely related to the question of characterizing domains in cylinders that admit solutions to an overdetermined problem, our minimization result allows us to deduce interesting consequences in that direction. In particular, we find that, for some cylinders and some volumes, the ``trivial" domain given by a bounded cylinder is not the only domain where the overdetermined problem has a solution. Moreover, it is not even a minimizer, which indicates that solutions with flat level sets are not always the best candidates for optimizing the torsional energy. |
| title | A shape optimization problem in cylinders and related overdetermined problems |
| topic | Analysis of PDEs 35N25, 49Q10 |
| url | https://arxiv.org/abs/2409.09448 |