Towards a two-scale calculus

Visintin, Augusto

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 3, pp. 371-397.

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.

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