Dimension reduction of fractional Sobolev seminorms on thin domains

We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films ${\mathrm{\Omega }}_{\text{e}}=\omega \times (0,\text{e})$. The first relevant order is ${\text{e}}^{1-2s}$, at which the corresponding limit captures the vertical fractional oscillations through one-dimensional sections. The second relevant order produces dimension-reduction regimes that undergo a qualitative transition at the critical exponent $s=\frac{1}{2}$. For $s<\frac{1}{2}$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is ${\text{e}}^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\frac{1}{2}$ in the differentiability index. At the critical exponent $s=1\mathrm{/}2$, the dimension-reduction scale is ${\text{e}}^2\mathrm{\mid }\log \text{e}\mathrm{\mid }$, and the limit is {\em local}, with dominant interactions at scales between $\text{e}$ and $1$, producing a Dirichlet-type limit on $\omega$. For $s>\frac{1}{2}$, the dominant contribution is instead driven by interactions at distances of order $\epsilon$, the dimension-reduction scale is ${\text{e}}^{3-2s}$, and the second-order $\mathrm{\Gamma }$-limit is still local. We also study the case $s=s_{\text{e}}\to 1^{-}$, showing a Bourgain–Brezis–Mironescu-type result.