A-Quasiconvexity: Relaxation and Homogenization

Andrea Braides
S.I.S.S.A.
Trieste, Italy

and

Irene Fonseca
Department of Mathematical Sciences
Carnegie-Mellon University
Pittsburgh, PA, U.S.A.

and

Giovanni Leoni
Dipartimento di Scienze e Tecnologie Avanzate
Università del Piemonte Orientale
Alessandria, Italy


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 W1,p are recovered.


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