We define and characterize weak and strong two-scale convergence in ${L}^{p}$, ${C}^{0}$ 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}, publisher = {EDP-Sciences}, volume = {12}, number = {3}, year = {2006}, doi = {10.1051/cocv:2006012}, mrnumber = {2224819}, zbl = {1110.35009}, language = {en}, url = {http://archive.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 - http://archive.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, Volume 12 (2006) no. 3, pp. 371-397. doi : 10.1051/cocv:2006012. http://archive.numdam.org/articles/10.1051/cocv:2006012/

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