Homogenization of thin structures by two-scale method with respect to measures

To the aim of studying the homogenization of low-dimensional periodic struc tures, we identify each of them with a periodic positive measure mu on R-n. We introduce a new notion of two-scale convergence for a sequence of funct ions v(epsilon) is an element of L-mu epsilon(p) (Omega; R-d), where Omega is an open bounded subset of R-n, and the measures mu (epsilon) are the eps ilon -scalings of mu, namely, mu (epsilon) (B) : = epsilon (n) mu (epsilon (-1) B). Enforcing the concept of tangential calculus with respect to measu res and related periodic Sobolev spaces, we prove a structure theorem for a ll the possible two-scale limits reached by the sequences (mu (epsilon), de l mu (epsilon)) when {mu (epsilon)} subset of C-0(1)(Omega) satisfy the bou ndedness condition sup(epsilon)integral (Omega)\u(epsilon)\(p) + \delu(epsi lon)\(p) d mu (epsilon) < + <infinity> and when the measure mu satis es sui table connectedness properties. This leads us to deduce the homogenized den sity of a sequence of energies of the form integral (Omega)j(x/epsilon, del u) d mu (epsilon), where j(y, z) is a convex integrand, periodic in y, and satisfying p-growth condition. The case of two parameter integrals is also investigated, in particular for what concerns the commutativity of the limi t process.