Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈LN that do not rely on...

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Authors: Boccardo, L., Gómez Castro, David, Díaz Díaz, Jesús Ildefonso
Format: article
Publication Date:2022
Country:España
Institution:Universidad Complutense de Madrid (UCM)
Repository:Docta Complutense
Language:English
OAI Identifier:oai:docta.ucm.es:20.500.14352/72748
Online Access:https://hdl.handle.net/20.500.14352/72748
Access Level:Open access
Keyword:517.95
Ecuaciones diferenciales
Funciones (Matemáticas)
1202.07 Ecuaciones en Diferencias
1202 Análisis y Análisis Funcional
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oai_identifier_str oai:docta.ucm.es:20.500.14352/72748
network_acronym_str ES
network_name_str España
repository_id_str
spelling Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergenceBoccardo, L.Gómez Castro, DavidDíaz Díaz, Jesús Ildefonso517.95Ecuaciones diferencialesFunciones (Matemáticas)1202.07 Ecuaciones en Diferencias1202 Análisis y Análisis FuncionalIn this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈LN that do not rely on the sign of divE. Then, under the assumption that divE≥0 distributionally, we extend the existence theory to |E|∈L2. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of E singular at one point as Ax/|x|2, or towards the boundary as divE∼dist(x,∂Ω)−2−α. In these cases the singularity of E leads to u vanishing to a certain order. In particular, this shows that the strong maximum principle fails in the presence of such singular drift terms E.Universidad Complutense de Madrid20222022-11-2120222022-11-21journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14352/72748reponame:Docta Complutenseinstname:Universidad Complutense de Madrid (UCM)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:docta.ucm.es:20.500.14352/727482026-06-02T12:44:21Z
dc.title.none.fl_str_mv Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
title Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
spellingShingle Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
Boccardo, L.
517.95
Ecuaciones diferenciales
Funciones (Matemáticas)
1202.07 Ecuaciones en Diferencias
1202 Análisis y Análisis Funcional
title_short Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
title_full Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
title_fullStr Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
title_full_unstemmed Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
title_sort Failure of the strong maximum principle for linear elliptic with singular convection of non-negative divergence
dc.creator.none.fl_str_mv Boccardo, L.
Gómez Castro, David
Díaz Díaz, Jesús Ildefonso
author Boccardo, L.
author_facet Boccardo, L.
Gómez Castro, David
Díaz Díaz, Jesús Ildefonso
author_role author
author2 Gómez Castro, David
Díaz Díaz, Jesús Ildefonso
author2_role author
author
dc.contributor.none.fl_str_mv Universidad Complutense de Madrid
dc.subject.none.fl_str_mv 517.95
Ecuaciones diferenciales
Funciones (Matemáticas)
1202.07 Ecuaciones en Diferencias
1202 Análisis y Análisis Funcional
topic 517.95
Ecuaciones diferenciales
Funciones (Matemáticas)
1202.07 Ecuaciones en Diferencias
1202 Análisis y Análisis Funcional
description In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem −div(M(x)∇u)=−div(E(x)u)+f in a bounded domain of RN with N≥3. We are particularly interested in singular E with divE≥0. We start by recalling known existence results when |E|∈LN that do not rely on the sign of divE. Then, under the assumption that divE≥0 distributionally, we extend the existence theory to |E|∈L2. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of E singular at one point as Ax/|x|2, or towards the boundary as divE∼dist(x,∂Ω)−2−α. In these cases the singularity of E leads to u vanishing to a certain order. In particular, this shows that the strong maximum principle fails in the presence of such singular drift terms E.
publishDate 2022
dc.date.none.fl_str_mv 2022
2022-11-21
2022
2022-11-21
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/20.500.14352/72748
url https://hdl.handle.net/20.500.14352/72748
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
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.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 15,301603

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