A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman

Autores
Martinez, Sandra Rita; Wolanski, Noemi Irene
Año de publicación
2009
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div ( g(|∇uε|) |∇uε| ∇uε) = βε(uε), uε ≥ 0. A solution to (Pε) is a function uε ∈ W1,G(Ω)∩L∞(Ω) such that Ω g(|∇uε|) ∇uε |∇uε| ∇ϕ dx = − Ω ϕ βε(uε) dx for every ϕ ∈ C∞0 (Ω). Here βε(s) = 1 ε β s ε , with β ∈ Lip(R), β > 0 in (0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C1,α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [Comm. Partial Differential Equations, 16 (1991), pp. 311-361].
Fil: Martinez, Sandra Rita. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Wolanski, Noemi Irene. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Materia
Free boundaries
Orlicz spaces
Singular perturbation
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acceso abierto
Condiciones de uso
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Consejo Nacional de Investigaciones Científicas y Técnicas
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spelling A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of LiebermanMartinez, Sandra RitaWolanski, Noemi IreneFree boundariesOrlicz spacesSingular perturbationhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div ( g(|∇uε|) |∇uε| ∇uε) = βε(uε), uε ≥ 0. A solution to (Pε) is a function uε ∈ W1,G(Ω)∩L∞(Ω) such that Ω g(|∇uε|) ∇uε |∇uε| ∇ϕ dx = − Ω ϕ βε(uε) dx for every ϕ ∈ C∞0 (Ω). Here βε(s) = 1 ε β s ε , with β ∈ Lip(R), β > 0 in (0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C1,α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [Comm. Partial Differential Equations, 16 (1991), pp. 311-361].Fil: Martinez, Sandra Rita. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Wolanski, Noemi Irene. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaSociety for Industrial and Applied Mathematics2009-01info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/245145Martinez, Sandra Rita; Wolanski, Noemi Irene; A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman; Society for Industrial and Applied Mathematics; Siam Journal On Mathematical Analysis; 41; 1; 1-2009; 318-3590036-1410CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://epubs.siam.org/doi/10.1137/070703740info:eu-repo/semantics/altIdentifier/doi/10.1137/070703740info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-22T11:41:31Zoai:ri.conicet.gov.ar:11336/245145instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-10-22 11:41:31.735CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
title A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
spellingShingle A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
Martinez, Sandra Rita
Free boundaries
Orlicz spaces
Singular perturbation
title_short A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
title_full A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
title_fullStr A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
title_full_unstemmed A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
title_sort A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman
dc.creator.none.fl_str_mv Martinez, Sandra Rita
Wolanski, Noemi Irene
author Martinez, Sandra Rita
author_facet Martinez, Sandra Rita
Wolanski, Noemi Irene
author_role author
author2 Wolanski, Noemi Irene
author2_role author
dc.subject.none.fl_str_mv Free boundaries
Orlicz spaces
Singular perturbation
topic Free boundaries
Orlicz spaces
Singular perturbation
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div ( g(|∇uε|) |∇uε| ∇uε) = βε(uε), uε ≥ 0. A solution to (Pε) is a function uε ∈ W1,G(Ω)∩L∞(Ω) such that Ω g(|∇uε|) ∇uε |∇uε| ∇ϕ dx = − Ω ϕ βε(uε) dx for every ϕ ∈ C∞0 (Ω). Here βε(s) = 1 ε β s ε , with β ∈ Lip(R), β > 0 in (0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C1,α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [Comm. Partial Differential Equations, 16 (1991), pp. 311-361].
Fil: Martinez, Sandra Rita. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Wolanski, Noemi Irene. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
description In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div ( g(|∇uε|) |∇uε| ∇uε) = βε(uε), uε ≥ 0. A solution to (Pε) is a function uε ∈ W1,G(Ω)∩L∞(Ω) such that Ω g(|∇uε|) ∇uε |∇uε| ∇ϕ dx = − Ω ϕ βε(uε) dx for every ϕ ∈ C∞0 (Ω). Here βε(s) = 1 ε β s ε , with β ∈ Lip(R), β > 0 in (0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C1,α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [Comm. Partial Differential Equations, 16 (1991), pp. 311-361].
publishDate 2009
dc.date.none.fl_str_mv 2009-01
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/11336/245145
Martinez, Sandra Rita; Wolanski, Noemi Irene; A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman; Society for Industrial and Applied Mathematics; Siam Journal On Mathematical Analysis; 41; 1; 1-2009; 318-359
0036-1410
CONICET Digital
CONICET
url http://hdl.handle.net/11336/245145
identifier_str_mv Martinez, Sandra Rita; Wolanski, Noemi Irene; A Singular Perturbation Problem for a Quasi-Linear Operator Satisfying the Natural Growth Condition of Lieberman; Society for Industrial and Applied Mathematics; Siam Journal On Mathematical Analysis; 41; 1; 1-2009; 318-359
0036-1410
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://epubs.siam.org/doi/10.1137/070703740
info:eu-repo/semantics/altIdentifier/doi/10.1137/070703740
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
publisher.none.fl_str_mv Society for Industrial and Applied Mathematics
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.982451

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