A-Quasiconvexity: Relaxation and Homogenization

Author: Braides Andrea   Fonseca Irene   Leoni Giovanni  

Publisher: Edp Sciences

E-ISSN: 1262-3377|5|issue|539-577

ISSN: 1292-8119

Source: ESAIM: Control, Optimisation and Calculus of Variations, Vol.5, Iss.issue, 2010-03, pp. : 539-577

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Abstract

Integral representation of relaxed energies and ofΓ-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 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.