Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequenc...

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Detalles Bibliográficos
Autores: Cabré Vilagut, Xavier|||0000-0001-5682-3135, Urtiaga Erneta, Iñigo|||0000-0002-7306-2961, Felipe Navarro, Juan Carlos|||0000-0001-7630-6661
Tipo de recurso: artículo
Fecha de publicación:2025
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:2117/445889
Acceso en línea:https://hdl.handle.net/2117/445889
https://dx.doi.org/10.1016/j.jfa.2025.111086
Access Level:acceso abierto
Palabra clave:Field of extremals
Nonlocal operators
Calibration
Viscosity solutions
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
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spelling Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theoryCabré Vilagut, Xavier|||0000-0001-5682-3135Urtiaga Erneta, Iñigo|||0000-0002-7306-2961Felipe Navarro, Juan Carlos|||0000-0001-7630-6661Field of extremalsNonlocal operatorsCalibrationViscosity solutionsÀrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàticaIn this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here “minimality” is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions.Peer Reviewed20252025-11-0120252025-11-10journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/445889https://dx.doi.org/10.1016/j.jfa.2025.111086reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2021-123903NB-I00 ECUACIONES EN DERIVADAS PARCIALES: PROBLEMAS DE REACCION-DIFUSION, INTEGRO-DIFERENCIALES, Y DE LA FISICA MATEMATICAMinisterio de Economía y Competitividad http://doi.org/10.13039/501100003329 MDM-2014-0445 Barcelona Graduate School of Mathematics (BGSMath)open accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4458892026-05-27T15:37:01Z
dc.title.none.fl_str_mv Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
title Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
spellingShingle Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
Cabré Vilagut, Xavier|||0000-0001-5682-3135
Field of extremals
Nonlocal operators
Calibration
Viscosity solutions
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
title_short Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
title_full Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
title_fullStr Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
title_full_unstemmed Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
title_sort Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory
dc.creator.none.fl_str_mv Cabré Vilagut, Xavier|||0000-0001-5682-3135
Urtiaga Erneta, Iñigo|||0000-0002-7306-2961
Felipe Navarro, Juan Carlos|||0000-0001-7630-6661
author Cabré Vilagut, Xavier|||0000-0001-5682-3135
author_facet Cabré Vilagut, Xavier|||0000-0001-5682-3135
Urtiaga Erneta, Iñigo|||0000-0002-7306-2961
Felipe Navarro, Juan Carlos|||0000-0001-7630-6661
author_role author
author2 Urtiaga Erneta, Iñigo|||0000-0002-7306-2961
Felipe Navarro, Juan Carlos|||0000-0001-7630-6661
author2_role author
author
dc.subject.none.fl_str_mv Field of extremals
Nonlocal operators
Calibration
Viscosity solutions
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
topic Field of extremals
Nonlocal operators
Calibration
Viscosity solutions
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
description In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here “minimality” is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025-11-01
2025
2025-11-10
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/2117/445889
https://dx.doi.org/10.1016/j.jfa.2025.111086
url https://hdl.handle.net/2117/445889
https://dx.doi.org/10.1016/j.jfa.2025.111086
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2021-123903NB-I00 ECUACIONES EN DERIVADAS PARCIALES: PROBLEMAS DE REACCION-DIFUSION, INTEGRO-DIFERENCIALES, Y DE LA FISICA MATEMATICA
Ministerio de Economía y Competitividad http://doi.org/10.13039/501100003329 MDM-2014-0445 Barcelona Graduate School of Mathematics (BGSMath)
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
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-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
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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score 15,812429

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