A fractional Michael–Simon Sobolev inequality on convex hypersurfaces

The classical Michael–Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional Lp term on the right-hand side which is weighted by the mean curvature of t...

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Authors: Cabré Vilagut, Xavier|||0000-0001-5682-3135, Cozzi, Matteo|||0000-0001-6105-692X, Csató, Gyula
Format: book part
Publication Date:2023
Country:España
Institution:Universitat Politècnica de Catalunya (UPC)
Repository:UPCommons. Portal del coneixement obert de la UPC
Language:English
OAI Identifier:oai:upcommons.upc.edu:2117/387253
Online Access:https://hdl.handle.net/2117/387253
https://dx.doi.org/10.4171/AIHPC/39
Access Level:Open access
Keyword:Inequalities (Mathematics)
Fractional Sobolev inequalities on manifolds
Nonlocal mean curvature
Fractional mean curvature flow
Maximal time of existence
Convexity
Desigualtats (Matemàtica)
Classificació AMS::26 Real functions::26D Inequalities
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
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spelling A fractional Michael–Simon Sobolev inequality on convex hypersurfacesCabré Vilagut, Xavier|||0000-0001-5682-3135Cozzi, Matteo|||0000-0001-6105-692XCsató, GyulaInequalities (Mathematics)Fractional Sobolev inequalities on manifoldsNonlocal mean curvatureFractional mean curvature flowMaximal time of existenceConvexityDesigualtats (Matemàtica)Classificació AMS::26 Real functions::26D InequalitiesÀrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàticaThe classical Michael–Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional Lp term on the right-hand side which is weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo seminorm of the function, as well as its Lp norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.The first and third authors are supported by grants MTM2017-84214-C2-1-P and RED2018-102650-T funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, and are part of the Catalan research group 2017 SGR 1392. The third author is also supported by the MINECO grant MTM2017-83499-P and by the “María de Maeztu” MINECO grant MDM-2014-0445Peer ReviewedElsevier20232023-01-0120232023-05-10book parthttp://purl.org/coar/resource_type/c_3248VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/bookPartapplication/pdfhttps://hdl.handle.net/2117/387253https://dx.doi.org/10.4171/AIHPC/39reponame: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 2013-2016 MTM2017-84214-C2-1-P ECUACIONES EN DERIVADAS PARCIALES: PROBLEMAS DE REACCION-DIFUSION, INTEGRO-DIFERENCIALES Y GEOMETRICOSopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/3872532026-05-27T15:37:01Z
dc.title.none.fl_str_mv A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
title A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
spellingShingle A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
Cabré Vilagut, Xavier|||0000-0001-5682-3135
Inequalities (Mathematics)
Fractional Sobolev inequalities on manifolds
Nonlocal mean curvature
Fractional mean curvature flow
Maximal time of existence
Convexity
Desigualtats (Matemàtica)
Classificació AMS::26 Real functions::26D Inequalities
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
title_short A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
title_full A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
title_fullStr A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
title_full_unstemmed A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
title_sort A fractional Michael–Simon Sobolev inequality on convex hypersurfaces
dc.creator.none.fl_str_mv Cabré Vilagut, Xavier|||0000-0001-5682-3135
Cozzi, Matteo|||0000-0001-6105-692X
Csató, Gyula
author Cabré Vilagut, Xavier|||0000-0001-5682-3135
author_facet Cabré Vilagut, Xavier|||0000-0001-5682-3135
Cozzi, Matteo|||0000-0001-6105-692X
Csató, Gyula
author_role author
author2 Cozzi, Matteo|||0000-0001-6105-692X
Csató, Gyula
author2_role author
author
dc.subject.none.fl_str_mv Inequalities (Mathematics)
Fractional Sobolev inequalities on manifolds
Nonlocal mean curvature
Fractional mean curvature flow
Maximal time of existence
Convexity
Desigualtats (Matemàtica)
Classificació AMS::26 Real functions::26D Inequalities
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
topic Inequalities (Mathematics)
Fractional Sobolev inequalities on manifolds
Nonlocal mean curvature
Fractional mean curvature flow
Maximal time of existence
Convexity
Desigualtats (Matemàtica)
Classificació AMS::26 Real functions::26D Inequalities
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
description The classical Michael–Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional Lp term on the right-hand side which is weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo seminorm of the function, as well as its Lp norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.
publishDate 2023
dc.date.none.fl_str_mv 2023
2023-01-01
2023
2023-05-10
dc.type.none.fl_str_mv book part
http://purl.org/coar/resource_type/c_3248
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/bookPart
format bookPart
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/387253
https://dx.doi.org/10.4171/AIHPC/39
url https://hdl.handle.net/2117/387253
https://dx.doi.org/10.4171/AIHPC/39
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 2013-2016 MTM2017-84214-C2-1-P ECUACIONES EN DERIVADAS PARCIALES: PROBLEMAS DE REACCION-DIFUSION, INTEGRO-DIFERENCIALES Y GEOMETRICOS
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/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 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
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 15,301603

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