Some properties of two-scale convergence

Visintin, Augusto

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 15 (2004), p. 93-107 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

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.

Résumé

Mediante una trasformazione di variabile, la nozione di convergenza a due scale di G. Nguetseng è qui riformulata ed estesa, ed alcune delle sue proprietà sono presentate. Tale convergenza è quindi estesa a spazi di funzioni differenziabili mediante l’approssimazione delle derivate a due scale. Inoltre si caratterizza il limite a due scale di derivate di successioni limitate negli spazi di Sobolev $W^{1,p}(\mathbb{R}^{N})$, $L^{2}_{rot}(\mathbb{R}^{3})^{3}$, $L^{2}_{div}(\mathbb{R}^{3})^{3}$ e $W^{2,p}(\mathbb{R}^{N})$. Infine si studia il limite a due scale dei potenziali di una successione convergente a due scale di campi irrotazionali.

Détail

Publié le : 2004-06-01

MR 2148538 | Zbl 1225.35031

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