International audienceThe periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo, 2006, pp. 119-136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The first idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in L(p) spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104], and where the unfolding method has been successfully applied

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Abstract.In a previous article about the homogenization of the classical problem of diffusion in a bounded domain with sufficiently smooth boundary we proved that the error is of order ε 1/2 . Now, for an open set Ω with sufficiently smooth boundary (C

We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis x 3) and we investigate the limit behavior of this problem as the periodicity ε and the radius r of the rods tend to zero (see fig.1 below). We use a decomposition of the displacement field in the rods of the form u = U + u where the principal part U is a field which is piecewise constant with respect to the variables (x 1 , x 2) (and then naturally extended on a fixed domain), while the perturbation u remains defined on the oscillating domain containing the rods. We derive estimates of U and u in term of the total elastic energy. This allows to obtain a priori estimates on u without solving the delicate question of the dependence, with respect to ε and r, of the constant in Korn's inequality in such an oscillating domain. To deal with the field u, we use a version of an unfolding operator which permits both to rescale all the rods and to work on the same fixed domain as for U to carry out the homogenization process. The above decomposition also helps in passing to the limit and to identify the limit junction conditions between the rods and the 3d plate. Résumé Nous considérons un ensemble de poutresélastiques périodiquement distribuées sur une plaqueélastique 3d (toutes d'axe x 3) et nous analysons le comportement limite de ce problème lorsque la périodicité ε et le rayon r des poutres tendent vers zéro. Nous introduisons une décomposition du champ de déplacement de la forme u = U + u dans laquelle la partie principale U est un champ constant par morceau par rapport aux variables (x 1 , x 2) (et qui s'étend donc naturellement sur un domaine fixe), alors que la perturbation u reste un champ défini sur le domaine oscillant qui représente les poutres. Nous donnons des estimations de U et u en fonction de l'énergieélastique totale. Ceci permet d'obtenir des estimations a priori de u sans chercheràévaluer la dépendance, par rapportà ε et r, de la constante de l'inégalité de Korn pour un tel domaine oscillant. Pour traiter le champ u, nous utilisons une version d'opérateur d'éclatement qui permet simultanément de redimensionner toutes les poutres et de travailler sur le