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

Source abstract

To the aim of studying the homogenization of low-dimensional periodic structures, we identify each of them with a periodic positive measure $\mu$ on $\ren$. We introduce a new notion of two-scale convergence for a sequence of functions $v_\e \in L ^p_{\me} (\O; \re ^d)$, where $\O$ is an open bounded subset of $\ren$, and the measures $\mu _\e$ are the $\e$-scalings of $\mu$, namely, $\mu_\e (B) := \e ^n \mu (\e ^ {-1}B)$. 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 $( u_\e, \nabla u _\e)$ when $\{u _\e\} \subset {\cal C} ^1_0 (\O)$ satisfy the boundedness condition $\sup _\e \int _{\O} |\ue| ^p + |\nabla \ue| ^p \, d \me < + \infty$ and when the measure $\mu$ satisfies suitable connectedness properties. This leads us to deduce the homogenized density of a sequence of energies of the form $\int _{\O} j (\xe, \nabla u) \, d \me$, 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.