A variational approach to double-porosity problems

In this paper we outline an approach by Γ-convergence to some problems related to ‘double-porosity’ homogenization. Various such models have been discussed in the mathematical literature, the first rigorous result for a linear double-porosity model having been obtained by Arbogast, Douglas and Hornung in [7]. The two-scale convergence approach to double-porosity problems was developed by Allaire in [4]. Successively, a random model and nonlinear models have been studied by Bourgeat, Mikelic and Piatnitski in [9] and by Pankratov and Piatnitski in [21], respectively. Other double-porosity type problems have been considered in [8, 17, 22, 23, 10, 26] and [24]. In our framework, the homogenization process involves the analysis of energies defined on some (mutually disconnected) highly oscillating connected sets (hard components), in whose complement (soft component) an energy density satisfying weaker coerciveness conditions is considered. To be more precise, we fix N ≥ 1 and 1-periodic Lipschitz open connected sets E1, . . . , EN ⊂ R with dist (Ei, Ej) > 0 if i = j. If n = 2 the connectedness condition can be satisfied only if N = 1; note that even this case will give non-trivial results. We also set E0 = R \ (E1 ∪ · · · ∪ EN ); note that we do not make any connectedness assumption on E0, which may be composed only of isolated bounded components if N = 1. For each j = 0, . . . , N we consider energy densities fj : R ×Mm×n → R and ‘low order terms’ gj : R × R → R. We suppose that gj , fj are Borel functions and 1-periodic in the first variable. For the sake of simplicity of presentation we suppose that there exists p > 1 such that all fj satisfy a p-growth condition, each fj is quasiconvex and f0 is positively homogeneous of degree p. In this way, given an open set Ω ⊂ R, we consider the energy