Preprint

Inserted: 14 mar 2026
Last Updated: 14 mar 2026

Year: 2026

Abstract:

We study the asymptotic behaviour of Gagliardo seminorms in $H^s$ defined on thin films $\Omega_\varepsilon=\omega\times(0,\varepsilon)$. The first relevant order is $\varepsilon^{1-2s}$, 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 $\varepsilon^2$. 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 $\varepsilon^{2}
\log\varepsilon
$, and the limit is {\em local}, with dominant interactions at scales between $\varepsilon$ 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 $\varepsilon^{3-2s}$, and the second-order $\Gamma$-limit is still local. We also study the case $s=s_\varepsilon\to 1^-$, showing a Bourgain--Brezis--Mironescu-type result.

Download:

• Dimensional_reduction-2.pdf