article

Title: The limit behavior of a family of variational multiscale problems

Authors: Margarida Baía and Irene Fonseca

Issue: Volume 56 (2007), Issue 1, 1-50

Abstract:

$$\Gamma$-convergence techniques combined with techniques of $2$-scale convergence are used to give a characterization of the behavior as $\epsilon$ goes to zero of a family of integral functionals defined on $L^{p}(\Omega; \mathbb{R}^{d})$ by \[ \mathcal{I}_{\epsilon}(u) := \begin{cases} \int_{\Omega} f \left( x, \frac{x}{\epsilon}, \nabla u(x) \right) dx & \mbox{ if } u \in W^{1,p}(\Omega; \mathbb{R}^{d}),\\ \infty & \mbox{otherwise}, \end{cases} \] under periodicity (and nonconvexity) hypothesis, standard $p$-coercivity and $p$-growth conditions with $p > 1$. Uniform continuity with respect to the $x$ variable, as it is customary in the existing literature, is not required.$