Two-scale convergence of first-order operators

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 W-1,W-p(R-N) is then studied: if u(epsilon) -> mu weakly in W-1,W-p(R-N), a sequence {u(1 epsilon)} is constructed such that u(1 epsilon)(x) -> u(1)(x, y) and del u(epsilon)(x) -> del u(x) + del(y)u(1)(x, y) weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces W-1,W-p(R-N)(N), L-rot(2) (R-3)(3), L-div(2) (R-N)(N), L-div(2) (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 del x [A(u(epsilon)(x), x, x/epsilon)center dot del x u(epsilon)] = f.