Multiscale analysis and homogenization of nonlocal thin films

In this paper, we introduce a nonlocal, variational model for thin films. We consider convolution-type functionals defined on a thin domain whose thickness is of order $\gamma$, where the effective interactions range between points is of order $\varepsilon$. We study the $\Gamma$-convergence of these energies, as both parameters vanish, to a local integral functional defined on a lower-dimensional domain. In the periodic homogenization setting, the limit energy density is characterized by an asymptotic formula that depends on the interplay between $\varepsilon$ and $\gamma$. Under suitable assumptions, this formula exhibits a separation of scales effect, namely, the limit energy can be obtained by performing two successive $\Gamma$-limits, first letting one parameter tend to zero while keeping the other fixed.