Dimension reduction of fractional Sobolev seminorms on thin domains

We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $\Omega_\e=\omega\times(0,\e)$. The first relevant order is $\e^{1-2s}$, 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 $s=\tfrac12$. For $s<\tfrac12$, the dominant contribution is driven by interactions at finite planar distance, and the dimension-reduction scale is $\e^2$. In this regime, the limit is a lower-dimensional \emph{fractional} energy with an effective gain of $\tfrac12$ in the differentiability index. At the critical exponent $s=1/2$, the dimension-reduction scale is $\e^{2}|\log\e|$, and the limit is {\em local}, with dominant interactions at scales between $\e$ and $1$, producing a Dirichlet-type limit on $\omega$. For $s>\tfrac12$, the dominant contribution is instead driven by interactions at distances of order $\varepsilon$, the dimension-reduction scale is $\e^{3-2s}$, and the second-order $\Gamma$-limit is still local. We also study the case $s=s_\e\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result.