Two-Scale Convergence of First-Order Operators
Augusto Visintin · Zeitschrift für Analysis und ihre Anwendungen · 2007
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_\varepsilon \to u weakly in W^{1,p}(\mathbb R^N) , a sequence \{u_{1\varepsilon}\} is constructed such that u_{1\varepsilon}(x)\to u_1(x,y) and abla u_\varepsilon(x)\to abla u(x) + abla_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^2_{\mathrm{rot}}(\mathbb R^3)^3 , L^2_{\mathrm{div}}(\mathbb R^N)^N , L^2_{\mathrm{div}}(\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 abla \!\times\! \big[A(u_\varepsilon(x), x,\frac{x}{\varepsilon}) \!\cdot\! abla \!\times\! u_\varepsilon\big] = f .