Homogenization of Non-Local Energies on Disconnected Sets

We consider the problem of the homogenization of non-local quadratic energies defined on $\delta$-periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale $\varepsilon$. For kernels with unbounded support we show that we may have three regimes: (i) $\varepsilon<\!<\delta$, for which the $\Gamma$-limit even in the strong topology of $L^2$ is $0$; (ii) $\frac\varepsilon\delta\to\kappa$, 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 $\kappa$; (iii) $\delta<\!<\varepsilon$, for which the $\Gamma$-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 $\delta\to 0$ (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 $\varepsilon\to0$. A slightly more complex description is necessary for case (ii) if the kernel is compactly supported.