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 $\varepsilon,δ$: 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 $λ$ defined as the limit of the ratio $\varepsilon/δ$. We compute the $Γ$-limit of the functionals with respect to the strong $L^p$-topology for each possible value of $λ$ and detect three different regimes, the critical scale being obtained when $λ\in(0,+\infty)$.

💡 Research Summary

The paper investigates the simultaneous homogenization and localization of a family of non‑local integral functionals of convolution type, depending on two small positive parameters ε and δ. The parameter ε controls the interaction length scale, while δ encodes the period of a rapidly oscillating heterogeneous medium. The authors consider the functional

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