Multiscale analysis and homogenization of nonlocal thin films

Abstract

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 ${\hat{I}}^3$, where the effective interactions range between points is of order $\epsilon$. We study the $\hat{I}“$-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 $\epsilon$ and ${\hat{I}}^3$. Under suitable assumptions, this formula exhibits a separation of scales effect, namely, the limit energy can be obtained by performing two successive $\hat{I}“$-limits, first letting one parameter tend to zero while keeping the other fixed.