Towards a two-scale calculus

Università degli Studi di Trento, Dipartimento di Matematica, via Sommarive 14, 38050 Povo (Trento), Italia; Visintin@science.unitn.it

Received: 9 September 2004
Revised: 5 April 2005

Abstract

We define and characterize weak and strong two-scale convergence in L^p, 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.