Two-scale convergence ∗ Dag Lukkassen † , Gabriel Nguetseng ‡ and Peter Wall § (Dedicated to the memory of Jacques-Louis Lions 1928-2001) Abstract This paper is devoted to the properties and applications of two-scale convergence introduced by Nguetseng in 1989. In a self-contained way, we present the details of the basic ideas in this theory. Moreover, we give an overview of the main homogenization problems which have been studied by this technique. We also bridge gaps in previous presentations, make generalizations and give alternative proofs. AMS Subject Classification (2000): 35B27, 35B40 Keywords: Homogenization, two-scale convergence, multiscale convergence, reiterated homogenization. 1 Introduction This paper is devoted to a special type of convergence in L p spaces. Let Ω be an open bounded subset in R N , Y the unit cube in R N and (ε) a sequence of positive numbers converging to 0. In 1989 G. Nguetseng, see [58], proved that for each bounded sequence (u ε ) in L 2 (Ω) there exists a subsequence, still indexed by ε, and a u ∈ L 2 (Ω × Y ) such that Z Ω u ε (x)φ(x, x ε ) dx → Z Ω Z Y u(x, y)φ(x, y) dydx, (1) for every sufficiently smooth φ(x, y) which is Y -periodic in y. Nguetseng also proved that for a bounded sequence (u ε ) in W 1,2 (Ω) there exist functions u ∈ L 2 (Ω × Y ) and u 1 ∈ L 2 (Ω; W 1,2 per (Y )) such that, up to a subsequence, u ε → u weakly in W 1,2 (Ω), Z Ω ³ Du ε (x), Φ(x, x ε ) ´ dx → Z Ω Z Y (Du(x)+ D y u 1 (x, y), Φ(x, y)) dydx, ∗ Published in: Int. J. of Pure and Appl. Math. 2, 1, 35-86, 2002. † Narvik University College, P.O.B. 385 N-8505 Narvik, Norway, dl@hin.no, http://hin.no/~dl/ ‡ Department of Mathematics, University of Yaounde1, P.O.B. 812 Yaounde, Cameroun, gnguets@uycdc.uninet.cm § Department of Mathematics, Luleå University, S-97187 Luleå, Sweden, wall@sm.luth.se, http://www.sm.luth.se/~wall 1