The Zaremba problem in two-dimensional Lipschitz graph domains

We study the Zaremba problem, or mixed problem associated to the Laplace operator, in two-dimensional Lipschitz graph domains with mixed Dirichlet and Neumann boundary data in Lebesgue and Lorentz spaces. We obtain an explicit value such that the Zaremba problem is solvable in for Lp and in the Lore...

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Detalles Bibliográficos
Autores: Carro Rossell, María Jesús, Luque Martínez, Teresa Elvira, Naibo, V.
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Universidad Complutense de Madrid (UCM)
Repositorio:Docta Complutense
Idioma:inglés
OAI Identifier:oai:docta.ucm.es:20.500.14352/130468
Acceso en línea:https://hdl.handle.net/20.500.14352/130468
Access Level:acceso abierto
Palabra clave:Finite Hilbert transform.
Lipschitz graph domain
Mixed problem
Muckenhoupt weights
Weighted Lebesgue spaces
Zaremba problem
Matemáticas (Matemáticas)
Ecuaciones diferenciales
12 Matemáticas
1206.02 Ecuaciones Diferenciales
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spelling The Zaremba problem in two-dimensional Lipschitz graph domainsCarro Rossell, María JesúsLuque Martínez, Teresa ElviraNaibo, V.Finite Hilbert transform.Lipschitz graph domainMixed problemMuckenhoupt weightsWeighted Lebesgue spacesZaremba problemMatemáticas (Matemáticas)Ecuaciones diferenciales12 Matemáticas1206.02 Ecuaciones DiferencialesWe study the Zaremba problem, or mixed problem associated to the Laplace operator, in two-dimensional Lipschitz graph domains with mixed Dirichlet and Neumann boundary data in Lebesgue and Lorentz spaces. We obtain an explicit value such that the Zaremba problem is solvable in for Lp and in the Lorentz space L(p,1). Applications include those where the domain is a cone with vertex at the origin and, more generally, a Schwarz–Christoffel domain. The techniques employed are based on results of the Zaremba problem in the upper half-plane, the use of conformal maps and the theory of solutions to the Neumann problem. For the case when the domain is the upper half-plane, we work in the weighted setting, establishing conditions on the weights for the existence of solutions and estimates for the non-tangential maximal function of the gradient of the solution. In particular, in the unweighted case, where known examples show that the gradient of the solution may fail to be squared-integrable, we prove restricted weak-type estimates.American Mathematical SocietyUniversidad Complutense de Madrid20252025-01-0120252025-01-01journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/130468reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)InglésengAgencia Estatal de Investigación http://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PID2020-113048GB-I00 ESPACIOS DE FUNCIONES Y TECNICAS DE ACOTACION DE OPERADORES EN ANALISISopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/1304682026-06-02T12:44:21Z
dc.title.none.fl_str_mv The Zaremba problem in two-dimensional Lipschitz graph domains
title The Zaremba problem in two-dimensional Lipschitz graph domains
spellingShingle The Zaremba problem in two-dimensional Lipschitz graph domains
Carro Rossell, María Jesús
Finite Hilbert transform.
Lipschitz graph domain
Mixed problem
Muckenhoupt weights
Weighted Lebesgue spaces
Zaremba problem
Matemáticas (Matemáticas)
Ecuaciones diferenciales
12 Matemáticas
1206.02 Ecuaciones Diferenciales
title_short The Zaremba problem in two-dimensional Lipschitz graph domains
title_full The Zaremba problem in two-dimensional Lipschitz graph domains
title_fullStr The Zaremba problem in two-dimensional Lipschitz graph domains
title_full_unstemmed The Zaremba problem in two-dimensional Lipschitz graph domains
title_sort The Zaremba problem in two-dimensional Lipschitz graph domains
dc.creator.none.fl_str_mv Carro Rossell, María Jesús
Luque Martínez, Teresa Elvira
Naibo, V.
author Carro Rossell, María Jesús
author_facet Carro Rossell, María Jesús
Luque Martínez, Teresa Elvira
Naibo, V.
author_role author
author2 Luque Martínez, Teresa Elvira
Naibo, V.
author2_role author
author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv Finite Hilbert transform.
Lipschitz graph domain
Mixed problem
Muckenhoupt weights
Weighted Lebesgue spaces
Zaremba problem
Matemáticas (Matemáticas)
Ecuaciones diferenciales
12 Matemáticas
1206.02 Ecuaciones Diferenciales
topic Finite Hilbert transform.
Lipschitz graph domain
Mixed problem
Muckenhoupt weights
Weighted Lebesgue spaces
Zaremba problem
Matemáticas (Matemáticas)
Ecuaciones diferenciales
12 Matemáticas
1206.02 Ecuaciones Diferenciales
description We study the Zaremba problem, or mixed problem associated to the Laplace operator, in two-dimensional Lipschitz graph domains with mixed Dirichlet and Neumann boundary data in Lebesgue and Lorentz spaces. We obtain an explicit value such that the Zaremba problem is solvable in for Lp and in the Lorentz space L(p,1). Applications include those where the domain is a cone with vertex at the origin and, more generally, a Schwarz–Christoffel domain. The techniques employed are based on results of the Zaremba problem in the upper half-plane, the use of conformal maps and the theory of solutions to the Neumann problem. For the case when the domain is the upper half-plane, we work in the weighted setting, establishing conditions on the weights for the existence of solutions and estimates for the non-tangential maximal function of the gradient of the solution. In particular, in the unweighted case, where known examples show that the gradient of the solution may fail to be squared-integrable, we prove restricted weak-type estimates.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025-01-01
2025
2025-01-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/20.500.14352/130468
url https://hdl.handle.net/20.500.14352/130468
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://dx.doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020 PID2020-113048GB-I00 ESPACIOS DE FUNCIONES Y TECNICAS DE ACOTACION DE OPERADORES EN ANALISIS
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
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
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
dc.source.none.fl_str_mv reponame:Docta Complutense
instname:Universidad Complutense de Madrid (UCM)
instname_str Universidad Complutense de Madrid (UCM)
reponame_str Docta Complutense
collection Docta Complutense
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,811543

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