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 $L^p$ term on the righthand side which is weighted by the mean curvature of...

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Detalhes bibliográficos
Autores: Cabré, Xavier, Cozzi, Matteo, Csató, Gyula
Formato: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2022
País:España
Recursos:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2445/193498
Acesso em linha:https://hdl.handle.net/2445/193498
Access Level:acceso abierto
Palavra-chave:Desigualtats (Matemàtica)
Espais de Sobolev
Conjunts convexos
Geometria diferencial
Inequalities (Mathematics)
Sobolev spaces
Convex sets
Differential geometry
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spelling A fractional Michael-Simon Sobolev inequality on convex hypersurfacesCabré, XavierCozzi, MatteoCsató, GyulaDesigualtats (Matemàtica)Espais de SobolevConjunts convexosGeometria diferencialInequalities (Mathematics)Sobolev spacesConvex setsDifferential geometryThe 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 $L^p$ term on the righthand 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 $L^p$ 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.Elsevier Masson SAS2023202320222023info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2445/193498Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésVersió postprint del document publicat a: https://doi.org/10.4171/AIHPC/39Annales de l'Institut Henri Poincare-Analyse non Lineaire, 2022https://doi.org/10.4171/AIHPC/39(c) Elsevier Masson SAS, 2022info:eu-repo/semantics/openAccessoai:recercat.cat:2445/1934982026-05-29T05:05:01Z
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, Xavier
Desigualtats (Matemàtica)
Espais de Sobolev
Conjunts convexos
Geometria diferencial
Inequalities (Mathematics)
Sobolev spaces
Convex sets
Differential geometry
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, Xavier
Cozzi, Matteo
Csató, Gyula
author Cabré, Xavier
author_facet Cabré, Xavier
Cozzi, Matteo
Csató, Gyula
author_role author
author2 Cozzi, Matteo
Csató, Gyula
author2_role author
author
dc.subject.none.fl_str_mv Desigualtats (Matemàtica)
Espais de Sobolev
Conjunts convexos
Geometria diferencial
Inequalities (Mathematics)
Sobolev spaces
Convex sets
Differential geometry
topic Desigualtats (Matemàtica)
Espais de Sobolev
Conjunts convexos
Geometria diferencial
Inequalities (Mathematics)
Sobolev spaces
Convex sets
Differential geometry
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 $L^p$ term on the righthand 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 $L^p$ 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 2022
dc.date.none.fl_str_mv 2022
2023
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/193498
url https://hdl.handle.net/2445/193498
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.4171/AIHPC/39
Annales de l'Institut Henri Poincare-Analyse non Lineaire, 2022
https://doi.org/10.4171/AIHPC/39
dc.rights.none.fl_str_mv (c) Elsevier Masson SAS, 2022
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Elsevier Masson SAS, 2022
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elsevier Masson SAS
publisher.none.fl_str_mv Elsevier Masson SAS
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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