Towards a two-scale calculus

Augusto Visintin

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 3, page 371-397
  • ISSN: 1292-8119

Abstract

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We define and characterize weak and strong two-scale convergence in Lp, C0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.

How to cite

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Visintin, Augusto. "Towards a two-scale calculus." ESAIM: Control, Optimisation and Calculus of Variations 12.3 (2006): 371-397. <http://eudml.org/doc/249669>.

@article{Visintin2006,
abstract = { We define and characterize weak and strong two-scale convergence in Lp, C0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey. },
author = {Visintin, Augusto},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Two-scale convergence; two-scale decomposition; Sobolev spaces; homogenization.; two-scale convergence; two-scale decomposition; two-scale convolution},
language = {eng},
month = {6},
number = {3},
pages = {371-397},
publisher = {EDP Sciences},
title = {Towards a two-scale calculus},
url = {http://eudml.org/doc/249669},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Visintin, Augusto
TI - Towards a two-scale calculus
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/6//
PB - EDP Sciences
VL - 12
IS - 3
SP - 371
EP - 397
AB - We define and characterize weak and strong two-scale convergence in Lp, C0 and other spaces via a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions of the classic theorems of Rellich, Sobolev, and Morrey.
LA - eng
KW - Two-scale convergence; two-scale decomposition; Sobolev spaces; homogenization.; two-scale convergence; two-scale decomposition; two-scale convolution
UR - http://eudml.org/doc/249669
ER -

References

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