Two-Scale Convergence of First-Order Operators

Nguetseng's notion of two-scale convergence and some of its main proper- ties are rst shortly reviewed. The (weak) two-scale limit of the gradient of bounded sequences of W 1;p(RN) is then studied: if u" ! u weakly in W 1;p(RN), a sequence fu1"g is constructed such that u1"(x) ! u1(x; y) and ru"(x) ! ru(x) + ryu1(x; y) weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces W 1;p(RN)N, L2rot(R3)3, L2div(R N)N, L2div(RN)N 2 . The ap- plication to the two-scale limit of some classical equations of electromagnetism and continuum mechanics is outlined. These results are then applied to the homogeniza- tion of quasilinear elliptic equations like r A(u"(x); x; x " ) r u" = f.