# Two-Scale Convergence of First-Order Operators

### Augusto Visintin

Università di Trento, Italy

## 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 $W_{1,p}(R_{N})$ is then studied: if $u_{ε}→u$ weakly in $W_{1,p}(R_{N})$, a sequence ${u_{1ε}}$ is constructed such that $u_{1ε}(x)→u_{1}(x,y)$ and $∇u_{ε}(x)→∇u(x)+∇_{y}u_{1}(x,y)$ weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces $W_{1,p}(R_{N})_{N}$, $L_{rot}(R_{3})_{3}$, $L_{div}(R_{N})_{N}$, $L_{div}(R_{N})_{N_{2}}$. 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 $∇×[A(u_{ε}(x),x,εx )⋅∇×u_{ε}]=f$.

## Cite this article

Augusto Visintin, Two-Scale Convergence of First-Order Operators. Z. Anal. Anwend. 26 (2007), no. 2, pp. 133–164

DOI 10.4171/ZAA/1315