Some properties of two-scale convergence

Augusto Visintin

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 2, page 93-107
  • ISSN: 1120-6330

Abstract

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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 ⁢ R N , L r ⁢ o ⁢ t 2 ⁢ R 3 3 , L d ⁢ i ⁢ v 2 ⁢ R 3 3 and W 2 , p ⁢ 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.

How to cite

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Visintin, Augusto. "Some properties of two-scale convergence." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.2 (2004): 93-107. <http://eudml.org/doc/252290>.

@article{Visintin2004,
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.},
author = {Visintin, Augusto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Two-scale convergence; Two-scale decomposition; Sobolev spaces; two-scale convergence; two-scale decomposition},
language = {eng},
month = {6},
number = {2},
pages = {93-107},
publisher = {Accademia Nazionale dei Lincei},
title = {Some properties of two-scale convergence},
url = {http://eudml.org/doc/252290},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Visintin, Augusto
TI - Some properties of two-scale convergence
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/6//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 2
SP - 93
EP - 107
AB - 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.
LA - eng
KW - Two-scale convergence; Two-scale decomposition; Sobolev spaces; two-scale convergence; two-scale decomposition
UR - http://eudml.org/doc/252290
ER -

References

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