Mathematics

We consider the problem of the homogenization of non−local quadratic energies defined on δ−periodic disconnected sets defined by a double integral, depending on a kernel concentrated at scale ε. For kernels with unbounded support we show that we may have three regimes: 

(i) ε ≪ δ, for which the Γ−limit even in the strong topology of L² is 0. 

(ii) 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 κ.

(iii) δ ≪ ε, for which the 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 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 ε → 0.

 A slightly more complex description is necessary for case (ii) if the kernel is compactly supported. This is a joint work with A. Braides and C. Trifone