Multiscale homogenization of non-local energies of convolution-type

We analyze a family of non-local integral functionals of convolution-type depending on two small positive parameters $\varepsilon,\delta$: the first rules the length-scale of the non-local interactions and produces a `localization' effect as it tends to $0$, the second is the scale of oscillation of a finely inhomogeneous periodic structure in the domain. We prove that a separation of the two scales occurs and that the interplay between the localization and homogenization effects in the asymptotic analysis is determined by the parameter $\lambda$ defined as the limit of the ratio $\varepsilon/\delta$. We compute the $\Gamma$-limit of the functionals with respect to the strong $L^p$-topology for each possible value of $\lambda$ and detect three different regimes, the critical scale being obtained when $\lambda\in(0,+\infty)$.