{"ID":2883224,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.08874","arxiv_id":"2508.08874","title":"A Bourgain-Brezis-Mironescu result for fractional thin films","abstract":"We consider the limit of squared $H^s$-Gagliardo seminorms on thin domains of the form $Ω_\\varepsilon=ω\\times(0,\\varepsilon)$ in $\\mathbb R^d$. When $\\varepsilon$ is fixed, multiplying by $1-s$ such seminorms have been proved to converge as $s\\to 1^-$ to a dimensional constant $c_d$ times the Dirichlet integral on $Ω_\\varepsilon$ by Bourgain, Brezis and Mironescu. In its turn such Dirichlet integrals divided by $\\varepsilon$ converge as $\\varepsilon\\to 0$ to a dimensionally reduced Dirichlet integral on $ω$. We prove that if we let simultaneously $\\varepsilon\\to 0$ and $s\\to 1$ then these squared seminorms still converge to the same dimensionally reduced limit when multiplied by $(1-s) \\varepsilon^{2s-3}$, independently of the relative converge speed of $s$ and $\\varepsilon$. This coefficient combines the geometrical scaling $\\varepsilon^{-1}$ and the fact that relevant interactions for the $H^s$-Gagliardo seminorms are those at scale $\\varepsilon$. We also study the usual membrane scaling, obtained by multiplying by $(1-s)\\varepsilon^{-1}$, which highlighs the {\\em critical scaling} $1-s\\sim|\\log\\varepsilon|^{-1}$, and the limit when $\\varepsilon\\to 0$ at fixed $s$.","short_abstract":"We consider the limit of squared $H^s$-Gagliardo seminorms on thin domains of the form $Ω_\\varepsilon=ω\\times(0,\\varepsilon)$ in $\\mathbb R^d$. When $\\varepsilon$ is fixed, multiplying by $1-s$ such seminorms have been proved to converge as $s\\to 1^-$ to a dimensional constant $c_d$ times the Dirichlet integral on $Ω_\\...","url_abs":"https://arxiv.org/abs/2508.08874","url_pdf":"https://arxiv.org/pdf/2508.08874v2","authors":"[\"Andrea Braides\",\"Margherita Solci\"]","published":"2025-08-12T12:02:48Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.OC\"]","methods":"[]","has_code":false}