Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey

This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of a...

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Autor: Cabré Vilagut, Xavier|||0000-0001-5682-3135
Formato: artículo
Fecha de publicación:2017
País:España
Recursos:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/100681
Acesso em linha:https://hdl.handle.net/2117/100681
https://dx.doi.org/10.1007/s11401-016-1067-0
Access Level:acceso abierto
Palavra-chave:Isoperimetric inequalities
Principal eigenvalue
Wulff shapes
ABP estimate
Desigualtats isoperimètriques
Àrees temàtiques de la UPC::Matemàtiques i estadística
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spelling Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A surveyCabré Vilagut, Xavier|||0000-0001-5682-3135Isoperimetric inequalitiesIsoperimetric inequalitiesPrincipal eigenvalueWulff shapesABP estimateDesigualtats isoperimètriquesÀrees temàtiques de la UPC::Matemàtiques i estadísticaThis paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.Peer Reviewed20172017-01-0520172017-02-08journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/100681https://dx.doi.org/10.1007/s11401-016-1067-0reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2http://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/1006812026-05-27T15:37:01Z
dc.title.none.fl_str_mv Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
title Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
spellingShingle Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
Cabré Vilagut, Xavier|||0000-0001-5682-3135
Isoperimetric inequalities
Isoperimetric inequalities
Principal eigenvalue
Wulff shapes
ABP estimate
Desigualtats isoperimètriques
Àrees temàtiques de la UPC::Matemàtiques i estadística
title_short Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
title_full Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
title_fullStr Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
title_full_unstemmed Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
title_sort Isoperimetric, Sobolev, and eigenvalue inequalities via the Alexandroff-Bakelman-Pucci method: A survey
dc.creator.none.fl_str_mv Cabré Vilagut, Xavier|||0000-0001-5682-3135
author Cabré Vilagut, Xavier|||0000-0001-5682-3135
author_facet Cabré Vilagut, Xavier|||0000-0001-5682-3135
author_role author
dc.subject.none.fl_str_mv Isoperimetric inequalities
Isoperimetric inequalities
Principal eigenvalue
Wulff shapes
ABP estimate
Desigualtats isoperimètriques
Àrees temàtiques de la UPC::Matemàtiques i estadística
topic Isoperimetric inequalities
Isoperimetric inequalities
Principal eigenvalue
Wulff shapes
ABP estimate
Desigualtats isoperimètriques
Àrees temàtiques de la UPC::Matemàtiques i estadística
description This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First, the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.
publishDate 2017
dc.date.none.fl_str_mv 2017
2017-01-05
2017
2017-02-08
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/100681
https://dx.doi.org/10.1007/s11401-016-1067-0
url https://hdl.handle.net/2117/100681
https://dx.doi.org/10.1007/s11401-016-1067-0
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

http://creativecommons.org/licenses/by-nc-nd/3.0/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

http://creativecommons.org/licenses/by-nc-nd/3.0/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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