Optimal regularity for supercritical parabolic obstacle problems

We study the obstacle problem for parabolic operators of the type (Formula presented.), where L is an elliptic integro-differential operator of order 2s, such as (Formula presented.), in the supercritical regime (Formula presented.). The best result in this context was due to Caffarelli and Figalli,...

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Autores: Ros-Oton, X., Torres-Latorre, C.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2023
País:España
Institución: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:2072/537457
Acceso en línea:http://hdl.handle.net/2072/537457
Access Level:acceso abierto
Palabra clave:parabolic operators, supercritical regime, elliptic setting
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spelling Optimal regularity for supercritical parabolic obstacle problemsRos-Oton, X.Torres-Latorre, C.parabolic operators, supercritical regime, elliptic settingWe study the obstacle problem for parabolic operators of the type (Formula presented.), where L is an elliptic integro-differential operator of order 2s, such as (Formula presented.), in the supercritical regime (Formula presented.). The best result in this context was due to Caffarelli and Figalli, who established the (Formula presented.) regularity of solutions for the case (Formula presented.), the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C1, 1 in space and time, and that this is optimal. We also deduce the (Formula presented.) regularity of the free boundary. Moreover, at all free boundary points (Formula presented.), we establish the following expansion: (Formula presented.) with (Formula presented.), (Formula presented.) and (Formula presented.). © 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.Both authors were supported by the European Research Council (ERC) under the Grant Agreement No 801867, and the AEI project PID2021‐125021NAI00 (Spain). X.R. was supported by AGAUR Grant 2021 SGR 00087 (Catalunya), by AEI Grant RED2022‐134784‐T funded by MCIN/AEI/10.13039/501100011033 (Spain), and by the AEI through the María de Maeztu Program for Centres and Units of Excellence in R&D (CEX2020‐001084‐M).John Wiley and Sons Inc2023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersion42 p.application/pdfhttp://hdl.handle.net/2072/537457RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésCommunications on Pure and Applied MathematicsL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5374572026-05-29T05:05:01Z
dc.title.none.fl_str_mv Optimal regularity for supercritical parabolic obstacle problems
title Optimal regularity for supercritical parabolic obstacle problems
spellingShingle Optimal regularity for supercritical parabolic obstacle problems
Ros-Oton, X.
parabolic operators, supercritical regime, elliptic setting
title_short Optimal regularity for supercritical parabolic obstacle problems
title_full Optimal regularity for supercritical parabolic obstacle problems
title_fullStr Optimal regularity for supercritical parabolic obstacle problems
title_full_unstemmed Optimal regularity for supercritical parabolic obstacle problems
title_sort Optimal regularity for supercritical parabolic obstacle problems
dc.creator.none.fl_str_mv Ros-Oton, X.
Torres-Latorre, C.
author Ros-Oton, X.
author_facet Ros-Oton, X.
Torres-Latorre, C.
author_role author
author2 Torres-Latorre, C.
author2_role author
dc.subject.none.fl_str_mv parabolic operators, supercritical regime, elliptic setting
topic parabolic operators, supercritical regime, elliptic setting
description We study the obstacle problem for parabolic operators of the type (Formula presented.), where L is an elliptic integro-differential operator of order 2s, such as (Formula presented.), in the supercritical regime (Formula presented.). The best result in this context was due to Caffarelli and Figalli, who established the (Formula presented.) regularity of solutions for the case (Formula presented.), the same regularity as in the elliptic setting. Here we prove for the first time that solutions are actually more regular than in the elliptic case. More precisely, we show that they are C1, 1 in space and time, and that this is optimal. We also deduce the (Formula presented.) regularity of the free boundary. Moreover, at all free boundary points (Formula presented.), we establish the following expansion: (Formula presented.) with (Formula presented.), (Formula presented.) and (Formula presented.). © 2023 The Authors. Communications on Pure and Applied Mathematics published by Courant Institute of Mathematics and Wiley Periodicals LLC.
publishDate 2023
dc.date.none.fl_str_mv 2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/2072/537457
url http://hdl.handle.net/2072/537457
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Communications on Pure and Applied Mathematics
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 42 p.
application/pdf
dc.publisher.none.fl_str_mv John Wiley and Sons Inc
publisher.none.fl_str_mv John Wiley and Sons Inc
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
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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