A General Convergence Result for a Functional Related to the Theory of Homogenization

The convergence, as $\varepsilon \downarrow 0$, of the functional $F_\varepsilon (\Psi ) = \int _{\mathbb{R}^N } u_\varepsilon (x)\Psi (x,{x / \varepsilon })$ associated with a given $L^2 $ function $u_\varepsilon $ with support in a fixed compact set is studied. The test functions $\Psi (x,y)$ are continuous on $\mathbb{R}^N \times \mathbb{R}^N $ and periodic in y. A convergence theorem is proved under the weaker assumption that $u_\varepsilon $ remains in a bounded subset of $L^2 $. Finally, the use of multiple-scale expansions in homogenization is justified, and a new approach is proposed for the mathematical analysis of homogenization problems.