are obtained when sequences of fields v may develop oscillations and areconstrained to satisfya system of first order linear partial differential equations. Thisframework includes thetreatement of divergence-free fields, Maxwell's equations inmicromagnetics, and curl-freefields. In the latter case classical relaxation theorems in W1,p , arerecovered." name="description" /> Cambridge Journals Online - ESAIM: Control, Optimisation and Calculus of Variations - Abstract - A-Quasiconvexity: Relaxation and Homogenization

ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - 2000 - Volume 5  

Research Article

A-Quasiconvexity: Relaxation and Homogenization

Braides, Andreaa1, Fonseca, Irenea2 and Leoni, Giovannia3

a1 SISSA, Trieste, Italy; braides@sissa.it.

a2 Department of MathematicalSciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A.; fonseca@andrew.cmu.edu.

a3 Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy; leoni@al.unipmn.it.

Abstract

Integral representation of relaxed energies and of Γ-limits of functionals $$ (u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx $$ are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p , are recovered.

(Received March 16 2000)

(Revised September 26 2000)

(Online publication August 15 2002)

Key Words:

  • ${\cal A}$ -quasiconvexity;
  • equi-integrability;
  • Young measure;
  • relaxation;
  • Γ-convergence;
  • homogenization.

Mathematics Subject Classification:

  • 35D99;
  • 35E99;
  • 49J45