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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.

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