Non convex homogenization problems for singular structures

We prove a homogenization theorem for non-convex functionals depending on vector-valued functions, defined on Sobolev spaces with respect to oscillating measures. The proof combines the use of the localization methods of Γ-convergence with a ‘discretization’ argument, which allows to link the oscillating energies to functionals defined on a single Lebesgue space, and to state the hypothesis of p-connectedness of the underlying periodic measure in a handy way. Introduction. In this paper we consider homogenization problems on singular structures, with in mind the model case of an energy defined on smooth functions over a periodic piecewise-smooth k-dimensional manifold E. Starting from such a geometry, after the usual homogenization scaling we are led to dealing with functionals of the form εn−k ∫ Ω∩εE f (x ε ,Du ) dH(x), (1) where f is a Borel function, one-periodic in the first variable. Here H denotes the k-dimensional Hausdorff measure, and the factor εn−k follows from the scaling properties ofH. In order that the functional above be well defined one can consider it as defined only on smooth functions. Note that if we denote by με the measure εn−kHk restricted to εE, then this integral can be equivalently written as ∫ Ω f (x ε ,Du ) dμε. (2) Following the choice made by several authors (see e.g. Bouchitté and Fragalà [8], Zhikov, [14, 15], Pastukhova [16, 17], etc.) the study of these types of problems can be set in a more general framework by fixing a general 1-periodic measure μ and defining με(B) = εμ (1 ε B ) . 2000 Mathematics Subject Classification. Primary: 74Q15, 49J45; Secondary: 35B27.