Some properties of two-scale convergence

Visintin, Augusto

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 15 (2004) no. 2, pp. 93-107

Résumé

We reformulate and extend G. Nguetseng’s notion of two-scale convergence by means of a variable transformation, and outline some of its properties. We approximate two-scale derivatives, and extend this convergence to spaces of differentiable functions. The two-scale limit of derivatives of bounded sequences in the Sobolev spaces $W^{1,p}(\mathbb{R}^{N})$, $L^{2}_{rot}(\mathbb{R}^{3})^{3}$, $L^{2}_{div}(\mathbb{R}^{3})^{3}$ and $W^{2,p}(\mathbb{R}^{N})$ is then characterized. The two-scale limit behaviour of the potentials of a two-scale convergent sequence of irrotational fields is finally studied.