A two phase elliptic singular perturbation problem with a forcing term

Autores
Lederman, C.; Wolanski, N.
Año de publicación
2006
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(s, ε)), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and ∫ β (s) d s = M. The functions uε and fε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit (ε → 0) and we show that limit functions are solutions to the two phase free boundary problem:Δ u = f χ{u ≢ 0} in  Ω {set minus} ∂ {u > 0},| ∇ u+ |2 - | ∇ u- |2 = 2 M on  Ω ∩ ∂ {u > 0}, where f = lim fε, in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case fε ≡ 0. The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary. © 2006 Elsevier SAS. All rights reserved.
Fil:Lederman, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fuente
J. Math. Pures Appl. 2006;86(6):552-589
Materia
Combustion
Free boundary problem
Regularity
Two phase
Viscosity solutions
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by/2.5/ar
Repositorio
Biblioteca Digital (UBA-FCEN)
Institución
Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
OAI Identificador
paperaa:paper_00217824_v86_n6_p552_Lederman

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oai_identifier_str paperaa:paper_00217824_v86_n6_p552_Lederman
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repository_id_str 1896
network_name_str Biblioteca Digital (UBA-FCEN)
spelling A two phase elliptic singular perturbation problem with a forcing termLederman, C.Wolanski, N.CombustionFree boundary problemRegularityTwo phaseViscosity solutionsWe study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(s, ε)), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and ∫ β (s) d s = M. The functions uε and fε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit (ε → 0) and we show that limit functions are solutions to the two phase free boundary problem:Δ u = f χ{u ≢ 0} in  Ω {set minus} ∂ {u > 0},| ∇ u+ |2 - | ∇ u- |2 = 2 M on  Ω ∩ ∂ {u > 0}, where f = lim fε, in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case fε ≡ 0. The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary. © 2006 Elsevier SAS. All rights reserved.Fil:Lederman, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.2006info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttp://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_LedermanJ. Math. Pures Appl. 2006;86(6):552-589reponame:Biblioteca Digital (UBA-FCEN)instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesinstacron:UBA-FCENenginfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/2.5/ar2025-09-29T13:42:49Zpaperaa:paper_00217824_v86_n6_p552_LedermanInstitucionalhttps://digital.bl.fcen.uba.ar/Universidad públicaNo correspondehttps://digital.bl.fcen.uba.ar/cgi-bin/oaiserver.cgiana@bl.fcen.uba.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:18962025-09-29 13:42:50.721Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesfalse
dc.title.none.fl_str_mv A two phase elliptic singular perturbation problem with a forcing term
title A two phase elliptic singular perturbation problem with a forcing term
spellingShingle A two phase elliptic singular perturbation problem with a forcing term
Lederman, C.
Combustion
Free boundary problem
Regularity
Two phase
Viscosity solutions
title_short A two phase elliptic singular perturbation problem with a forcing term
title_full A two phase elliptic singular perturbation problem with a forcing term
title_fullStr A two phase elliptic singular perturbation problem with a forcing term
title_full_unstemmed A two phase elliptic singular perturbation problem with a forcing term
title_sort A two phase elliptic singular perturbation problem with a forcing term
dc.creator.none.fl_str_mv Lederman, C.
Wolanski, N.
author Lederman, C.
author_facet Lederman, C.
Wolanski, N.
author_role author
author2 Wolanski, N.
author2_role author
dc.subject.none.fl_str_mv Combustion
Free boundary problem
Regularity
Two phase
Viscosity solutions
topic Combustion
Free boundary problem
Regularity
Two phase
Viscosity solutions
dc.description.none.fl_txt_mv We study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(s, ε)), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and ∫ β (s) d s = M. The functions uε and fε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit (ε → 0) and we show that limit functions are solutions to the two phase free boundary problem:Δ u = f χ{u ≢ 0} in  Ω {set minus} ∂ {u > 0},| ∇ u+ |2 - | ∇ u- |2 = 2 M on  Ω ∩ ∂ {u > 0}, where f = lim fε, in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case fε ≡ 0. The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary. © 2006 Elsevier SAS. All rights reserved.
Fil:Lederman, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Wolanski, N. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
description We study the following two phase elliptic singular perturbation problem:Δ uε = βε (uε) + fε, in Ω ⊂ RN, where ε > 0, βε (s) = frac(1, ε) β (frac(s, ε)), with β a Lipschitz function satisfying β > 0 in (0, 1), β ≡ 0 outside (0, 1) and ∫ β (s) d s = M. The functions uε and fε are uniformly bounded. One of the motivations for the study of this problem is that it appears in the analysis of the propagation of flames in the high activation energy limit, when sources are present. We obtain uniform estimates, we pass to the limit (ε → 0) and we show that limit functions are solutions to the two phase free boundary problem:Δ u = f χ{u ≢ 0} in  Ω {set minus} ∂ {u > 0},| ∇ u+ |2 - | ∇ u- |2 = 2 M on  Ω ∩ ∂ {u > 0}, where f = lim fε, in a viscosity sense and in a pointwise sense at regular free boundary points. In addition, we show that the free boundary is smooth and thus limit functions are classical solutions to the free boundary problem, under suitable assumptions. Some of the results obtained are new even in the case fε ≡ 0. The results in this paper also apply to other combustion models. For instance, models with nonlocal diffusion and/or transport. Several of these applications are discussed here and we get, in some cases, the full regularity of the free boundary. © 2006 Elsevier SAS. All rights reserved.
publishDate 2006
dc.date.none.fl_str_mv 2006
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_Lederman
url http://hdl.handle.net/20.500.12110/paper_00217824_v86_n6_p552_Lederman
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/2.5/ar
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/2.5/ar
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv J. Math. Pures Appl. 2006;86(6):552-589
reponame:Biblioteca Digital (UBA-FCEN)
instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron:UBA-FCEN
reponame_str Biblioteca Digital (UBA-FCEN)
collection Biblioteca Digital (UBA-FCEN)
instname_str Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron_str UBA-FCEN
institution UBA-FCEN
repository.name.fl_str_mv Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
repository.mail.fl_str_mv ana@bl.fcen.uba.ar
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