A Bourgain-Brezis-Mironescu result for fractional thin films
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arXiv
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| Format: | Preprint |
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2025
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| author | Braides, Andrea Solci, Margherita |
| author_facet | Braides, Andrea Solci, Margherita |
| contents | 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 $s$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_08874 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Bourgain-Brezis-Mironescu result for fractional thin films Braides, Andrea Solci, Margherita Analysis of PDEs Optimization and Control 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 $s$. |
| title | A Bourgain-Brezis-Mironescu result for fractional thin films |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2508.08874 |