A-Quasiconvexity: Relaxation and Homogenization

Andrea Braides; Irene Fonseca; Giovanni Leoni

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 5, page 539-577
  • ISSN: 1292-8119

Abstract

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Integral representation of relaxed energies and of Γ-limits of functionals ( u , v ) Ω f ( x , u ( x ) , v ( x ) ) d x 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.

How to cite

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Braides, Andrea, Fonseca, Irene, and Leoni, Giovanni. "A-Quasiconvexity: Relaxation and Homogenization." ESAIM: Control, Optimisation and Calculus of Variations 5 (2010): 539-577. <http://eudml.org/doc/197360>.

@article{Braides2010,
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. },
author = {Braides, Andrea, Fonseca, Irene, Leoni, Giovanni},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {$\{\cal A\}$-quasiconvexity; equi-integrability; Young measure; relaxation; Γ-convergence; homogenization.; Young measures; Gamma-convergence; differential constraint; integral representation of the relaxed energy},
language = {eng},
month = {3},
pages = {539-577},
publisher = {EDP Sciences},
title = {A-Quasiconvexity: Relaxation and Homogenization},
url = {http://eudml.org/doc/197360},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Braides, Andrea
AU - Fonseca, Irene
AU - Leoni, Giovanni
TI - A-Quasiconvexity: Relaxation and Homogenization
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 5
SP - 539
EP - 577
AB - 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.
LA - eng
KW - ${\cal A}$-quasiconvexity; equi-integrability; Young measure; relaxation; Γ-convergence; homogenization.; Young measures; Gamma-convergence; differential constraint; integral representation of the relaxed energy
UR - http://eudml.org/doc/197360
ER -

References

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