Calculus of Variations and Geometric Measure Theory

G. Bouchitté - I. Fragalà

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

created on 04 Apr 2000
modified on 10 Dec 2003


Published Paper

Inserted: 4 apr 2000
Last Updated: 10 dec 2003

Journal: SIAM J. Math. Anal.
Volume: 32
Number: 6
Pages: 1198-1226
Year: 2001


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