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A general convergence result for a functional related to the theory of homogenization. (English) Zbl 0688.35007
Let $$C_ p$$ denote the usual Banach space of continuous scalar periodic functions on $${\mathbb{R}}^ N$$, $${\mathcal K}({\mathbb{R}}^ N;C_ p)$$ the space of continuous functions of $${\mathbb{R}}^ N$$ into $$C_ p$$ with compact supports. For $$u_{\epsilon}\in L^ 2(\Omega)$$ $$(\epsilon >0$$; $$\Omega$$ a bounded open set in $${\mathbb{R}}^ N$$, $$\Omega$$ independent of $$\epsilon)$$ the functional $w\mapsto F(w)=\int_{\Omega}u_{\epsilon}(x)w(x,x/\epsilon)dx,\quad w\in {\mathcal K}({\mathbb{R}}^ N;C_ p),$ is considered.
Assuming that the sequence $$\{u_{\epsilon}\}$$ $$(\epsilon >0)$$ remains in a bounded subset of $$L^ 2(\Omega)$$ yields a function $$u_ 0\in L^ 2(\Omega;L^ 2_ p)$$ $$(L^ 2_ p$$ is the Hilbert space of the $$v\in L^ 2_{loc}({\mathbb{R}}^ N)$$, v periodic) and a subsequence from $$\{u_{\epsilon}\}$$ such that, as $$\epsilon$$ $$\downarrow 0$$, $F_{\epsilon}(w)\to \int_{\Omega \times Y}u_ 0(x,y)w(x,y)dx dy\quad \forall w\in {\mathcal K}({\mathbb{R}}^ N;C_ p),$ where $$Y=]-,[^ N$$. Finally, the use of multiple-scale expansions in homogenization is justified, and a new approach is proposed for the mathematical analysis of homogenization problems.
Reviewer: G.Nguetseng

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 41A35 Approximation by operators (in particular, by integral operators)
##### Keywords:
homogenization; convergence; multiple-scale expansions
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