A-Quasiconvexity: Relaxation and Homogenization

Free access article

Issue		ESAIM: COCV Volume 5, 2000


Page(s)		539 - 577
DOI		10.1051/cocv:2000121

DOI: 10.1051/cocv:2000121

ESAIM: COCV, November 2000, Vol. 5, 539-577

A-Quasiconvexity: Relaxation and Homogenization

Andrea Braides
SISSA, Trieste, Italy; (braides@sissa.it)

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

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

Received March 16, 2000. Revised September 26, 2000.

Abstract: Integral representation of relaxed energies and of $\Gamma$ -limits of functionals

$\begin{displaymath}(u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx \end{displaymath}$

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 W^1,p are recovered.

Keywords and phrases: A-quasiconvexity, equi-integrability, Young measure, relaxation, $\Gamma$ -convergence, homogenization.

AMS Subject Classification: 35D99, 35E99, 49J45

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