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**The periodic unfolding method in homogenization.**
*(English)*
Zbl 1167.49013

Summary: The periodic unfolding method was introduced in 2002 in [D. Cioranescu, A. Damlamian and G. Griso, C. R., Math., Acad. Sci. Paris 335, No. 1, 99–104 (2002; Zbl 1001.49016)] (with the basic proofs in [A. Damlamian, Multi-scale problems and asymptotic analysis. Proceedings of the midnight sun Narvik conference, Narvik 2004. Tokyo: Gakkōtosho, GAKUTO Int. Ser., Math. Sci. Appl. 24, 119–136 (2006; Zbl 1204.35038)]). 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 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, loc. cit.], and where the unfolding method has been successfully applied.