Two-scale convergence of first-order operators

Abstract

Nguetseng's notion of two-scale convergence and some of its main properties are first shortly reviewed. The (weak) two-scale limit of the gradient of bounded sequences of W1'P(ℝN) is then studied: if uε → u weakly in W1,p(ℝN), a sequence {u1ε} is constructed such that u1ε(x) → u 1(x,y) and ∇uε(x) → ∇u(x) → ∇u(x) + ∇yu1(x,y) weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces W1,p(ℝN)N, Lrot2(ℝ3)3, Ldiv2(Rdbl;N)N, Ldiv2(ℝ N)N2. The application to the two-scale limit of some classical equations of electromagnetism and continuum mechanics is outlined. These results are then applied to the homogenization of quasilinear elliptic equations like ∇ × [A(uε(x), x, x/ε) · ∇ × uε] = f. © European Mathematical Society.