# Towards a two-scale calculus

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PDF/03/cocv0520.pdf.url ESAIM: COCV ESAIM: Control, Optimisation and Calculus of Variations July 2006, Vol. 12, 371–397 www.edpsciences.org/cocv DOI: 10.1051/cocv:2006012 TOWARDS A TWO-SCALE CALCULUS Augusto Visintin1 Abstract. We define and characterize weak and strong two-scale convergence in Lp, C0 and other spaces via a transformation of variable, extending Nguetseng’s definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzela`, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey. Mathematics Subject Classification. 35B27, 35J20, 74Q, 78M40. Received September 9, 2004. Revised April 5, 2005. Introduction Let Ω be a domain of RN (N ≥ 1), and set Y := [0, 1[N . In the seminal work [25], Nguetseng introduced the following concept: a bounded sequence {uε} of L2(Ω) is said (weakly) two-scale convergent to u ∈ L2(Ω×Y ) if and only if lim ε→0 ∫ Ω uε(x) ψ ( x, x ε ) dx = ∫∫ Ω×Y u(x, y) ψ(x, y) dxdy, (1) for any smooth function ψ : RN×RN → R that is Y -periodic w.r.t. the second argument. It should be noticed that uε : Ω→ R for any ε, whereas u : Ω×Y → R. This notion was then analyzed in detail and applied to a number of problems by Allaire [1] and others. It can account for occurrence of a fine-scale periodic structure, and indeed has been and is still extensively applied to homogenization, see e.g. [2, 5, 8, 13, 17, 20, 21, 26, 35, 36], just to mention some papers of a growing literature. In the framework of periodic homogenization, two-scale convergence can represent an alternative to the classic energy method of Tartar, see e.g. [3, 7, 16, 19, 24, 28–31]. Extensions to the nonperiodic setting have been proposed by Casado-Diaz and Gayte [11, 12] and by Nguetseng

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