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Two-scale convergence. (English) Zbl 1061.35015
In this paper the authors give in a clear way a self-contained presentation of two-scale convergence, introduced by one of the authors [G. Nguetseng, SIAM J. Math. Anal. 20, No. 3, 608–623 (1989; Zbl 0688.35007)] and so-called later by Allaire. A family in $$\varepsilon > 0$$ of functions $$(u_{\varepsilon}) \subset L^p(\Omega)$$ is said to two-scale converge to a function $$u \in L^p(\Omega \times [0,1]^n)$$ ($$\Omega$$ open subset of $${\mathbb R}^n$$) if $\lim_{\varepsilon \to 0} \int_{\Omega} u_{\varepsilon}(x) \phi(x,x/\varepsilon) \, dx = \int_{\Omega} \int_{[0,1]^n} u(x,y) \phi(x,y) \, dy \, dx$ for every $$\phi \in L^{p'}(\Omega ; C_{per}([0,1]^n))$$ ($$p' = p/(p-1)$$, $$p>1$$ and $$C_{per}$$ is the set of continuous and periodic functions on $$[0,1]^n$$). Clearly this can be easily extended to more than two scales (multi-scale convergence). Some classical applications are shown, as homogenization for the family of linear elliptic operators $$- \text{ div} (A(x/\varepsilon) \cdot D)$$ defined in $$H^1_0(\Omega)$$ ($$A$$ definite positive and bounded matrix) and for the family of nonlinear elliptic operators $$u \mapsto - \text{ div} (A(x/\varepsilon, \cdots, x/\varepsilon^n, Du)$$ using multi-scale convergence. Moreover it is shown that correctors can be obtained using two-scale convergence. Finally a guide to the literature is presented.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35B40 Asymptotic behavior of solutions to PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
reiterated homogenization