Geodesic
Towards a two-scale calculus
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 371-397

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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.

DOI : 10.1051/cocv:2006012
Classification : 35B27, 35J20, 74Q, 78M40
Keywords: two-scale convergence, two-scale decomposition, Sobolev spaces, homogenization 
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     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://geodesic.mathdoc.fr/articles/10.1051/cocv:2006012/}
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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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