Two-Scale Convergence of First-Order Operators

Abstract

Nguetseng's notion of {\it 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 $W^{1,p}(\erre^N)$ is then studied: if $u_\eps\to u$ weakly in $W^{1,p}(\erre^N)$, a sequence $\{u_{1\eps}\}$ is constructed such that $u_{1\eps}(x)\to u_1(x,y)$ and $\nabla u_\eps(x)\to \nabla u(x) + \nabla_y u_1(x,y)$ weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces $W^{1,p}(\erre^N)^N$, $L^2_{\mbox{\eightrm rot}}(\erre^3)^3$, $L^2_{\mbox{\eightrm div}}(\erre^N)^N$, $L^2_{\mbox{\eightrm div}}(\erre^N)^{N^2}$. 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 $\nabla \!\times\! \big[A(u_\eps(x), x,\frac{x}{\eps}) \!\cdot\! \nabla \!\times\! u_\eps\big] = f$.