Homogenization of non-local energies on disconnected sets

We consider the problem of the homogenization of non-local quadratic energies defined on -periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale. For kernels with unbounded support we show that we may have three regimes: (i) <\!<, for which the -limit even in the strong topology of L² is 0; (ii), 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 ; (iii) <\!<, for which the -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 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 0. A slightly more complex description is necessary for case (ii) if the kernel is compactly supported.