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

\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.