Multiscale homogenization of non-local energies of convolution-type
Giuseppe Cosma Brusca
TL;DR
The paper analyzes convolution-type non-local functionals with dual vanishing scales ε and δ and establishes a Γ-convergence framework that reveals a separation of localization and homogenization effects. The limit functional depends on the ratio λ = lim ε/δ, yielding three regimes (subcritical, critical, supercritical) with explicit non-local cell formulas for the limit density f_λ. The authors develop truncated-to-full Γ-convergence arguments, derive sharp lower and upper bounds, and address subtle issues in the supercritical case under precise structural hypotheses. The results provide a rigorous multiscale description of how fine periodic structures interact with localization in non-local energies, with broad implications for homogenization of non-local variational problems.
Abstract
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 $λ\in(0,+\infty)$.