Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime

We study the behaviour of fractional Sobolev spaces $H^s({\mathrm{\Omega }}_{\epsilon })$ with $s\in (0,1\mathrm{/}2)$ defined on ``thin films’’ ${\mathrm{\Omega }}_{\epsilon }=\omega \times (0,\epsilon )$ in ${\mathbb{R}}^d$, and prove that they tend to the space $H^{s+\frac{1}{2}}(\omega )$ as $\epsilon \to 0$. 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 $s\to 1\mathrm{/}{2}^{-}$.