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 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.