A mean field equation on a torus: one-dimensional symmetry of solutions

We study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and $\oep$ is a rectangle of $\R^2$ with side lengths $1/\epsilon$ and $1$, $0<...

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Detalles Bibliográficos
Autores: Cabré Vilagut, Xavier|||0000-0001-5682-3135, Lucia D'Agostino, Marcello, Sanchón Rodellar, Manuel
Tipo de recurso: artículo
Fecha de publicación:2003
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/909
Acceso en línea:https://hdl.handle.net/2117/909
Access Level:acceso abierto
Palabra clave:Partial differential equations
mean field equation
torus
Equacions en derivades parcials
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions
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spelling A mean field equation on a torus: one-dimensional symmetry of solutionsCabré Vilagut, Xavier|||0000-0001-5682-3135Lucia D'Agostino, MarcelloSanchón Rodellar, ManuelPartial differential equationsmean field equationtorusEquacions en derivades parcialsClassificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic typeClassificació AMS::35 Partial differential equations::35B Qualitative properties of solutionsWe study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and $\oep$ is a rectangle of $\R^2$ with side lengths $1/\epsilon$ and $1$, $0< \epsilon \leq 1$. We establish that every solution depends only on the $x$--variable when $\lambda \leq \lambda^*(\epsilon)$, where $\lambda^*(\epsilon)$ is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters $\epsilon$ and $\lambda$ in which every solution is identically zero. This range is optimal for $\epsilon\leq1/2$.20032003-01-0120072007-05-07journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/909reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/9092026-05-27T15:37:01Z
dc.title.none.fl_str_mv A mean field equation on a torus: one-dimensional symmetry of solutions
title A mean field equation on a torus: one-dimensional symmetry of solutions
spellingShingle A mean field equation on a torus: one-dimensional symmetry of solutions
Cabré Vilagut, Xavier|||0000-0001-5682-3135
Partial differential equations
mean field equation
torus
Equacions en derivades parcials
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions
title_short A mean field equation on a torus: one-dimensional symmetry of solutions
title_full A mean field equation on a torus: one-dimensional symmetry of solutions
title_fullStr A mean field equation on a torus: one-dimensional symmetry of solutions
title_full_unstemmed A mean field equation on a torus: one-dimensional symmetry of solutions
title_sort A mean field equation on a torus: one-dimensional symmetry of solutions
dc.creator.none.fl_str_mv Cabré Vilagut, Xavier|||0000-0001-5682-3135
Lucia D'Agostino, Marcello
Sanchón Rodellar, Manuel
author Cabré Vilagut, Xavier|||0000-0001-5682-3135
author_facet Cabré Vilagut, Xavier|||0000-0001-5682-3135
Lucia D'Agostino, Marcello
Sanchón Rodellar, Manuel
author_role author
author2 Lucia D'Agostino, Marcello
Sanchón Rodellar, Manuel
author2_role author
author
dc.subject.none.fl_str_mv Partial differential equations
mean field equation
torus
Equacions en derivades parcials
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions
topic Partial differential equations
mean field equation
torus
Equacions en derivades parcials
Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type
Classificació AMS::35 Partial differential equations::35B Qualitative properties of solutions
description We study the equation $$-\Delta u=\lambda\left(\frac{e^u}{\int_\oep e^u}- \frac{1}{|\oep|}\right)\quad \text{in }\oep$$ for $u\in E$, where $E = \{ u \in H^1(\oep): u \hbox{ is doubly periodic}, \int_{\oep} u = 0 \}$ and $\oep$ is a rectangle of $\R^2$ with side lengths $1/\epsilon$ and $1$, $0< \epsilon \leq 1$. We establish that every solution depends only on the $x$--variable when $\lambda \leq \lambda^*(\epsilon)$, where $\lambda^*(\epsilon)$ is an explicit positive constant depending on the maximum conformal radius of the rectangle. As a consequence, we obtain an explicit range of parameters $\epsilon$ and $\lambda$ in which every solution is identically zero. This range is optimal for $\epsilon\leq1/2$.
publishDate 2003
dc.date.none.fl_str_mv 2003
2003-01-01
2007
2007-05-07
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/909
url https://hdl.handle.net/2117/909
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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