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Some remarks on two-scale convergence and periodic unfolding. (English) Zbl 1265.35017
The author introduces the adjoint definition of the two-scale convergence, based on the periodic unfolding, in comparison with the classical one. Such approach avoids most problems of the admissibility (specific type of regularity) of test functions. The principal part of the paper gives a self-contained introduction to the two-scale convergence. It includes proofs relating to its classical definition and dual definitions, to the periodic unfolding extended by zero, as well as on that (original one) extended by identity, and to its properties in standard Lebesgue spaces \(L^p\) (\(p>1\)) and properties of gradients from corresponding Sobolev spaces \(W^{1,p}\). The last section presents some ideas for generalization of such results to the non-periodic homogenization, referring to the two-scale transform, the integral conservation and the general \(\Sigma \)-convergence by G. Nguetseng and N. Svanstedt [Banach J. Math. Anal. 5, No. 1, 101–135 (2011; Zbl 1229.46035)].
Reviewer: Jiří Vala (Brno)

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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