ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2006 - Volume 12, Issue 03  

Research Article

Towards a two-scale calculus

Visintin, Augusto

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

Abstract

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.

(Received September 9 2004)

(Revised April 5 2005)

(Online publication June 20 2006)

Key Words:

  • Two-scale convergence;
  • two-scale decomposition;
  • Sobolev spaces;
  • homogenization.

Mathematics Subject Classification:

  • 35B27;
  • 35J20;
  • 74Q;
  • 78M40