Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces

In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets Ω⊂R<sup>n+1</sup>, n≥2, satisfying the capacity density condition, without any further topological assumption...

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Autores: Cao, Mingming, Hidalgo-Palencia, Pablo, Martell, José María, Prisuelos-Arribas, Cruz, Zhao, Zihui
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2025
País:España
Institución:Consejo Superior de Investigaciones Científicas (CSIC)
Repositorio:DIGITAL.CSIC. Repositorio Institucional del CSIC
OAI Identifier:oai:digital.csic.es:10261/422608
Acceso en línea:http://hdl.handle.net/10261/422608
https://api.elsevier.com/content/abstract/scopus_id/85212559355
Access Level:acceso abierto
Palabra clave:Capacity density condition
Elliptic operators
Hölder spaces
Well-posedness of Dirichlet boundary value problems
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spelling Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spacesCao, MingmingHidalgo-Palencia, PabloMartell, José MaríaPrisuelos-Arribas, CruzZhao, ZihuiCapacity density conditionElliptic operatorsHölder spacesWell-posedness of Dirichlet boundary value problemsIn this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets Ω⊂R<sup>n+1</sup>, n≥2, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if Ω is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when Ω is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.The first and second authors are respectively supported by grants RYC2021-032600-I and CEX2019- 000904-S-20-3, both funded by MCIN/AEI/ 10.13039/501100011033. The first, second, and third authors acknowledge financial support from MCIN/AEI/ 10.13039/501100011033 grants CEX2023-001347- S and PID2022-141354NB-I00. The fourth author acknowledges financial support from MCIN/AEI/ 10.13039/501100011033 grant PID2022-141354NB-I00. The last author was partially supported by NSF grants DMS-1361823, DMS-1500098, DMS-1664867, DMS-1902756 and by The Institute for Advanced StudyPeer reviewedElsevierMinisterio de Ciencia e Innovación (España)Martell, José María [0000-0001-6788-4769]Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]202620262025info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Publisher's versioninfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/10261/422608https://api.elsevier.com/content/abstract/scopus_id/85212559355reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Ingléshttps://doi.org/10.1016/j.jfa.2024.110801Síinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/4226082026-05-22T06:33:51Z
dc.title.none.fl_str_mv Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
title Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
spellingShingle Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
Cao, Mingming
Capacity density condition
Elliptic operators
Hölder spaces
Well-posedness of Dirichlet boundary value problems
title_short Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
title_full Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
title_fullStr Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
title_full_unstemmed Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
title_sort Elliptic operators in rough sets and the Dirichlet problem with boundary data in Hölder spaces
dc.creator.none.fl_str_mv Cao, Mingming
Hidalgo-Palencia, Pablo
Martell, José María
Prisuelos-Arribas, Cruz
Zhao, Zihui
author Cao, Mingming
author_facet Cao, Mingming
Hidalgo-Palencia, Pablo
Martell, José María
Prisuelos-Arribas, Cruz
Zhao, Zihui
author_role author
author2 Hidalgo-Palencia, Pablo
Martell, José María
Prisuelos-Arribas, Cruz
Zhao, Zihui
author2_role author
author
author
author
dc.contributor.none.fl_str_mv Ministerio de Ciencia e Innovación (España)
Martell, José María [0000-0001-6788-4769]
Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]
dc.subject.none.fl_str_mv Capacity density condition
Elliptic operators
Hölder spaces
Well-posedness of Dirichlet boundary value problems
topic Capacity density condition
Elliptic operators
Hölder spaces
Well-posedness of Dirichlet boundary value problems
description In this paper we study the Dirichlet problem for real-valued second order divergence form elliptic operators with boundary data in Hölder spaces. Our context is that of open sets Ω⊂R<sup>n+1</sup>, n≥2, satisfying the capacity density condition, without any further topological assumptions. Our main result states that if Ω is either bounded, or unbounded with unbounded boundary, then the corresponding Dirichlet boundary value problem is well-posed; when Ω is unbounded with bounded boundary, we establish that solutions exist, but they fail to be unique in general. These results are optimal in the sense that solvability of the Dirichlet problem in Hölder spaces is shown to imply the capacity density condition. As a consequence of the main result, we present a characterization of the Hölder spaces in terms of the boundary traces of solutions, and obtain well-posedness of several related Dirichlet boundary value problems. All the results above are new even for 1-sided chord-arc domains, and can be extended to generalized Hölder spaces associated with a natural class of growth functions.
publishDate 2025
dc.date.none.fl_str_mv 2025
2026
2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
http://purl.org/coar/resource_type/c_6501
Publisher's version
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10261/422608
https://api.elsevier.com/content/abstract/scopus_id/85212559355
url http://hdl.handle.net/10261/422608
https://api.elsevier.com/content/abstract/scopus_id/85212559355
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://doi.org/10.1016/j.jfa.2024.110801
Sí
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
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:DIGITAL.CSIC. Repositorio Institucional del CSIC
instname:Consejo Superior de Investigaciones Científicas (CSIC)
instname_str Consejo Superior de Investigaciones Científicas (CSIC)
reponame_str DIGITAL.CSIC. Repositorio Institucional del CSIC
collection DIGITAL.CSIC. Repositorio Institucional del CSIC
repository.name.fl_str_mv
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