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Homogenization and two-scale convergence

Abstract : Following an idea of G. Nguetseng, we define a notion of "two-scale" convergence, which is aimed to a better description of sequences of oscillating functions. Bounded sequences in $L^2(\Omega)$ are proved to be relatively compact with respect to this new type of convergence. We also establish a corrector-type theorem (i.e. which permits, in some cases, to replace a sequence by its "two-scale" limit, up to a strongly convergent remainder in $L^2(\Omega)$). These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, we propose a new method for proving the convergence of homogenization processes, which is an alternative to the so-called energy method of L. Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and non-linear second-order elliptic equations.
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Grégoire Allaire. Homogenization and two-scale convergence. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 1992, 23 (6), pp.1482-1518. ⟨hal-01111805⟩

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