Homogenization of Thin Structures by Two-Scale Method with Respect to Measures

To the aim of studying the homogenization of low-dimensional periodic structures, we identify each of them with a periodic positive measure μ on . We introduce a new notion of two-scale convergence for a sequence of functions v_ ∈ L ᵖ_ (; ᵈ), where is an open bounded subset of , and the measures μ _ are the -scalings of μ, namely, μ_ (B) := ⁿ μ ( ^ -1B). Enforcing the concept of tangential calculus with respect to measures and related periodic Sobolev spaces, we prove a structure theorem for all the possible two-scale limits reached by the sequences ( u_, ∇ u ₎ when _\ ⊂ C ¹₀ () satisfy the boundedness condition _ ∫ _ || ᵖ + |∇ | ᵖ \, d < + ∞ and when the measure μ satisfies suitable connectedness properties. This leads us to deduce the homogenized density of a sequence of energies of the form ∫ _ j (, ∇ u) \, d, where j(y,z) is a convex integrand, periodic in y, and satisfying a p-growth condition. The case of two parameter integrals is also investigated, in particular for what concerns the commutativity of the limit process.