Homogenization of Non-Local Energies on Disconnected Sets
Andrea Braides, Sergio Scalabrino, Chiara Trifone · Journal of convex analysis · 2026
We consider the problem of the homogenization of non-local quadratic energies defined on \delta δ -periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale \varepsilon ε . For kernels with unbounded support we show that we may have three regimes: (i) \varepsilon<\!<\delta ε < < δ , for which the \Gamma Γ -limit even in the strong topology of L^2 L 2 is 0 0 ; (ii) \frac\varepsilon\delta\to\kappa ε δ → κ , in which the energies are coercive with respect to a convergence of interpolated functions, and the limit is governed by a non-local homogenization formula parameterized by \kappa κ ; (iii) \delta<\!<\varepsilon δ < < ε , for which the \Gamma Γ -limit is computed with respect to a coarse-grained convergence and exhibits a separation-of-scales effect; namely, it is the same as the one obtained by formally first letting \delta\to 0 δ → 0 (which turns out to be a pointwise weak limit, thanks to an iterated use of Jensen's inequality), and then, noting that the outcome is a nonlocal energy studied by Bourgain, Brezis and Mironescu, letting \varepsilon\to0 ε → 0 . A slightly more complex description is necessary for case (ii) if the kernel is compactly supported.