Homogenization of convex functionals which are weakly coercive and not equi-bounded from above

This paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q >...

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Detalles Bibliográficos
Autores: Briane, Marc, Casado Díaz, Juan
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2013
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/139236
Acceso en línea:https://hdl.handle.net/11441/139236
https://doi.org/10.1016/J.ANIHPC.2012.10.005
Access Level:acceso abierto
Palabra clave:Homogenization
Convex functionals
Nonlinear elliptic equations
Weak coercivity
Maximum principle
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spelling Homogenization of convex functionals which are weakly coercive and not equi-bounded from aboveBriane, MarcCasado Díaz, JuanHomogenizationConvex functionalsNonlinear elliptic equationsWeak coercivityMaximum principleThis paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q > N−1q>N−1 if N > 2N>2, and q⩾1q⩾1 if N = 2N=2, such that any sequence of bounded energy is compact in W_{0}^{1,q}(\Omega )W01,q(Ω). Under this assumption the Γ-convergence of the functionals for the strong topology of L^{\infty }(\Omega )L∞(Ω) is proved to agree with the Γ-convergence for the strong topology of L^{1}(\Omega )L1(Ω). This leads to an integral representation of the Γ-limit in C_{0}^{1}(\Omega )C01(Ω) thanks to a local convex density. An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects.ElsevierEcuaciones Diferenciales y Análisis NuméricoFQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales2013info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/139236https://doi.org/10.1016/J.ANIHPC.2012.10.005reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésAnnales de l'Institut Henri Poincaré. Analyse non linéaire, 30 (4), 547-571.https://dx.doi.org/10.1016/J.ANIHPC.2012.10.005info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1392362026-06-17T12:51:07Z
dc.title.none.fl_str_mv Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
title Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
spellingShingle Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
Briane, Marc
Homogenization
Convex functionals
Nonlinear elliptic equations
Weak coercivity
Maximum principle
title_short Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
title_full Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
title_fullStr Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
title_full_unstemmed Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
title_sort Homogenization of convex functionals which are weakly coercive and not equi-bounded from above
dc.creator.none.fl_str_mv Briane, Marc
Casado Díaz, Juan
author Briane, Marc
author_facet Briane, Marc
Casado Díaz, Juan
author_role author
author2 Casado Díaz, Juan
author2_role author
dc.contributor.none.fl_str_mv Ecuaciones Diferenciales y Análisis Numérico
FQM309: Control y Homogeneización de Ecuaciones en Derivadas Parciales
dc.subject.none.fl_str_mv Homogenization
Convex functionals
Nonlinear elliptic equations
Weak coercivity
Maximum principle
topic Homogenization
Convex functionals
Nonlinear elliptic equations
Weak coercivity
Maximum principle
description This paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q > N−1q>N−1 if N > 2N>2, and q⩾1q⩾1 if N = 2N=2, such that any sequence of bounded energy is compact in W_{0}^{1,q}(\Omega )W01,q(Ω). Under this assumption the Γ-convergence of the functionals for the strong topology of L^{\infty }(\Omega )L∞(Ω) is proved to agree with the Γ-convergence for the strong topology of L^{1}(\Omega )L1(Ω). This leads to an integral representation of the Γ-limit in C_{0}^{1}(\Omega )C01(Ω) thanks to a local convex density. An example based on a thin cylinder with very low and very large energy densities, which concentrates to a line shows that the loss of the weak coercivity condition can induce nonlocal effects.
publishDate 2013
dc.date.none.fl_str_mv 2013
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/139236
https://doi.org/10.1016/J.ANIHPC.2012.10.005
url https://hdl.handle.net/11441/139236
https://doi.org/10.1016/J.ANIHPC.2012.10.005
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Annales de l'Institut Henri Poincaré. Analyse non linéaire, 30 (4), 547-571.
https://dx.doi.org/10.1016/J.ANIHPC.2012.10.005
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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