Geometric properties of optimizers for the maximum gradient of the torsion function
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914205072883712 |
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| author | Burdzy, Krzysztof Ftouhi, Ilias Mariano, Phanuel |
| author_facet | Burdzy, Krzysztof Ftouhi, Ilias Mariano, Phanuel |
| contents | Consider $J(Ω):= \|\nabla u_Ω\|_\infty/\sqrt{|Ω|} $ and $J_P(Ω):= \|\nabla u_Ω\|_\infty/P(Ω) $, where $Ω$ is a planar convex domain, $u_Ω$ is the torsion function, $P(Ω)$ is the perimeter of $Ω$ and $|Ω|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_Ω|$ attains its maximum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09400 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric properties of optimizers for the maximum gradient of the torsion function Burdzy, Krzysztof Ftouhi, Ilias Mariano, Phanuel Analysis of PDEs Optimization and Control Consider $J(Ω):= \|\nabla u_Ω\|_\infty/\sqrt{|Ω|} $ and $J_P(Ω):= \|\nabla u_Ω\|_\infty/P(Ω) $, where $Ω$ is a planar convex domain, $u_Ω$ is the torsion function, $P(Ω)$ is the perimeter of $Ω$ and $|Ω|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_Ω|$ attains its maximum. |
| title | Geometric properties of optimizers for the maximum gradient of the torsion function |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2512.09400 |