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 ${\Omega }_{\epsilon }=\omega \times (0,\epsilon )$ in ${\mathbb{R}}^d$. When $\epsilon$ 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 ${\Omega }_{\epsilon }$ by Bourgain, Brezis and Mironescu. In its turn such Dirichlet integrals divided by $\epsilon$ converge as $\epsilon \to 0$ to a dimensionally reduced Dirichlet integral on $\omega$. We prove that if we let simultaneously $\epsilon \to 0$ and $s\to 1$ then these squared seminorms still converge to the same dimensionally reduced limit when multiplied by $(1-s){\epsilon }^{2s-3}$, independently of the relative converge speed of $s$ and $\epsilon$. This coefficient combines the geometrical scaling ${\epsilon }^{-1}$ and the fact that relevant interactions for the $H^s$-Gagliardo seminorms are those at scale $\epsilon$. We also study the usual membrane scaling, obtained by multiplying by $(1-s){\epsilon }^{-1}$, which highlighs the {\em critical scaling} $1-s\sim |\log \epsilon {|}^{-1}$, and the limit when $\epsilon \to 0$ at fixed $s$.

A more complete version of the paper can be found at arXiv:2603.13968 .That version also completes arXiv:2512.10620