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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)
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