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 $W^{1,p}({\mathbb{R}}^N)$ is then studied: if $u_{\epsilon }\to u$ weakly in $W^{1,p}({\mathbb{R}}^N)$, a sequence $\{u_{1\epsilon }\}$ is constructed such that $u_{1\epsilon }(x)\to u_1(x,y)$ and $\nabla u_{\epsilon }(x)\to \nabla u(x)+{\nabla }_y{u}_1(x,y)$ weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces $W^{1,p}({\mathbb{R}}^N{)}^N$, $L_{\mathrm{rot}}^2({\mathbb{R}}^3{)}^3$, $L_{\mathrm{div}}^2({\mathbb{R}}^N{)}^N$, $L_{\mathrm{div}}^2({\mathbb{R}}^N{)}^{N^2}$. 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 $\nabla \times [A(u_{\epsilon }(x),x,\frac{x}{\epsilon })\cdot \nabla \times u_{\epsilon }]=f$.