We define and characterize weak and strong two-scale convergence in , 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.
Keywords: two-scale convergence, two-scale decomposition, Sobolev spaces, homogenization
@article{COCV_2006__12_3_371_0,
author = {Visintin, Augusto},
title = {Towards a two-scale calculus},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {371--397},
year = {2006},
publisher = {EDP Sciences},
volume = {12},
number = {3},
doi = {10.1051/cocv:2006012},
mrnumber = {2224819},
zbl = {1110.35009},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv:2006012/}
}
TY - JOUR AU - Visintin, Augusto TI - Towards a two-scale calculus JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2006 SP - 371 EP - 397 VL - 12 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv:2006012/ DO - 10.1051/cocv:2006012 LA - en ID - COCV_2006__12_3_371_0 ER -
Visintin, Augusto. Towards a two-scale calculus. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 371-397. doi: 10.1051/cocv:2006012
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