Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911521508950016 |
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| author | Braides, Andrea Pinamonti, Andrea Solci, Margherita |
| author_facet | Braides, Andrea Pinamonti, Andrea Solci, Margherita |
| contents | We study the behaviour of fractional Sobolev spaces $H^s(Ω_\varepsilon)$ with $s\in(0,1/2)$ defined on ``thin films'' $Ω_\varepsilon=ω\times (0,\varepsilon)$ in $\mathbb R^d$, and prove that they tend to the space $H^{s+\frac12}(ω)$ as $\varepsilon\to 0$. This is made precise by using a notion of dimension-reduction convergence, with respect to which suitably scaled Gagliardo seminorms define equicoercive functionals. Asymptotic results are proved for $s\to 0^+$ and $s\to 1/2^-$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_10620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime Braides, Andrea Pinamonti, Andrea Solci, Margherita Analysis of PDEs 49J45, 35R11, 73K10 We study the behaviour of fractional Sobolev spaces $H^s(Ω_\varepsilon)$ with $s\in(0,1/2)$ defined on ``thin films'' $Ω_\varepsilon=ω\times (0,\varepsilon)$ in $\mathbb R^d$, and prove that they tend to the space $H^{s+\frac12}(ω)$ as $\varepsilon\to 0$. This is made precise by using a notion of dimension-reduction convergence, with respect to which suitably scaled Gagliardo seminorms define equicoercive functionals. Asymptotic results are proved for $s\to 0^+$ and $s\to 1/2^-$. |
| title | Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime |
| topic | Analysis of PDEs 49J45, 35R11, 73K10 |
| url | https://arxiv.org/abs/2512.10620 |