JournalszaaVol. 26 , No. 2DOI 10.4171/zaa/1315

Two-Scale Convergence of First-Order Operators

  • Augusto Visintin

    Università di Trento, Italy
Two-Scale Convergence of First-Order Operators cover

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 W1,p(\erreN)W^{1,p}(\erre^N) is then studied: if u\epsuu_\eps\to u weakly in W1,p(\erreN)W^{1,p}(\erre^N), a sequence {u1\eps}\{u_{1\eps}\} is constructed such that u1\eps(x)u1(x,y)u_{1\eps}(x)\to u_1(x,y) and u\eps(x)u(x)+yu1(x,y)\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 W1,p(\erreN)NW^{1,p}(\erre^N)^N, L\mbox\eightrmrot2(\erre3)3L^2_{\mbox{\eightrm rot}}(\erre^3)^3, L\mbox\eightrmdiv2(\erreN)NL^2_{\mbox{\eightrm div}}(\erre^N)^N, L\mbox\eightrmdiv2(\erreN)N2L^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  ⁣× ⁣[A(u\eps(x),x,x\eps) ⁣ ⁣ ⁣× ⁣u\eps]=f\nabla \!\times\! \big[A(u_\eps(x), x,\frac{x}{\eps}) \!\cdot\! \nabla \!\times\! u_\eps\big] = f.