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Towards a two-scale calculus

Published online by Cambridge University Press:  20 June 2006

Augusto Visintin*
Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento), Italia;
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We define and characterize weak and strong two-scale convergence in Lp , 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.

Research Article
© EDP Sciences, SMAI, 2006

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