{"ID":2830603,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.09400","arxiv_id":"2512.09400","title":"Geometric properties of optimizers for the maximum gradient of the torsion function","abstract":"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.","short_abstract":"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 maxi...","url_abs":"https://arxiv.org/abs/2512.09400","url_pdf":"https://arxiv.org/pdf/2512.09400v2","authors":"[\"Krzysztof Burdzy\",\"Ilias Ftouhi\",\"Phanuel Mariano\"]","published":"2025-12-10T07:54:47Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.OC\"]","methods":"[]","has_code":false}