Homogenization of thin structures by two-scale method with respect to measures (Q2706230)

The authors study the asymptotic behavior of a sequence functional of the form NEWLINE\[NEWLINE J_\varepsilon (u) = \int_\Omega j({{x}\over{\varepsilon}}, \nabla u) d\mu_\varepsilon, \quad u\in { C}^1_0(\Omega),NEWLINE\]NEWLINE where \(\mu\) is a given periodic Radom measure on \({\mathbb R}^n\), \(\Omega\) is a bounded open subset of \({\mathbb R}^n\), the integrand \(j = j(y,z)\) is assumed to be periodic \(\mu\)-measurable in \(y\) and convex with a \(p\)-growth condition in \(z\), \(\varepsilon\) is a positive parameter tending to zero, and \(\mu_\varepsilon\) is the rescaled measure \(\mu_\varepsilon (B):= \varepsilon^n\mu({B\over\varepsilon})\). In the case considered in this paper of thin periodic structures, another parameter \(\delta\) is considered, which corresponds to the thickness of the region occupied by the material; a double passage to the limit is taken, as the small parameters \(\varepsilon\) and \(\delta\) both tend to zero. A notion of two-scale convergence is introduced with respect to measures; a structure result is proved for the two-scale possible limits of a sequence \(\{ (u_\varepsilon, \nabla u_\varepsilon) \}\), when \(u_\varepsilon \in { C}^1_0(\Omega) \) satisfy the uniform estimate \( \int ( |u_\varepsilon|^p + |\nabla u_\varepsilon|^p) d\mu\varepsilon \leq M\). The homogenized energy density is deduced in connection with \(J_\varepsilon\) in terms of local unit-cell problem; further, when fattened structures are considered, the limit process as the two parameters \(\varepsilon\) (of periodicity) and \(\delta\) (of thickness) tend to zero, is shown to be commutative.