Towards a two-scale calculus
Università degli Studi di Trento,
Dipartimento di Matematica,
via Sommarive 14,
38050 Povo (Trento), Italia; Visintin@science.unitn.it
Revised: 5 April 2005
We define and characterize weak and strong two-scale convergence in Lp, C0 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.
Mathematics Subject Classification: 35B27 / 35J20 / 74Q / 78M40
Key words: Two-scale convergence / two-scale decomposition / Sobolev spaces / homogenization.
© EDP Sciences, SMAI, 2006