Gradient Bounds for Elliptic Problems Singular at the Boundary
Let Ω be a bounded smooth domain in RN, N ≧ 2, and let us denote by d(x) the distance function d(x, ∂Ω). We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is where f belongs to W 1,∞ loc (Ω) and is (possibly) singular at ∂Ω, C ε W1...
| Autores: | , |
|---|---|
| Formato: | artículo |
| Fecha de publicación: | 2011 |
| País: | España |
| Recursos: | Universidad Nacional de Educación a Distancia |
| Repositorio: | e-spacio. Repositorio Institucional de la UNED |
| Idioma: | inglés |
| OAI Identifier: | oai:e-spacio.uned.es:20.500.14468/24467 |
| Acesso em linha: | https://hdl.handle.net/20.500.14468/24467 |
| Access Level: | acceso abierto |
| Palavra-chave: | 12 Matemáticas viscosity solution elliptic problem neumann condition transport term elliptic regularity |
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Gradient Bounds for Elliptic Problems Singular at the BoundaryPorretta, AlessioLeonori, Tommaso12 Matemáticasviscosity solutionelliptic problemneumann conditiontransport termelliptic regularityLet Ω be a bounded smooth domain in RN, N ≧ 2, and let us denote by d(x) the distance function d(x, ∂Ω). We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is where f belongs to W 1,∞ loc (Ω) and is (possibly) singular at ∂Ω, C ε W1,∞ (Ω)(with no sign condition) and the field B ε W1,∞ (Ω)N has an outward direction and satisfies B · v ≧ α at ∂Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.Springer Naturee-Spacio UNED20242024-11-2120112011-07-0520112011-07-05journal articlehttp://purl.org/coar/resource_type/c_6501info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/20.500.14468/24467reponame:e-spacio. Repositorio Institucional de la UNEDinstname:Universidad Nacional de Educación a DistanciaInglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-nd/4.0/deed.esoai:e-spacio.uned.es:20.500.14468/244672026-06-06T12:38:31Z |
| dc.title.none.fl_str_mv |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| title |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| spellingShingle |
Gradient Bounds for Elliptic Problems Singular at the Boundary Porretta, Alessio 12 Matemáticas viscosity solution elliptic problem neumann condition transport term elliptic regularity |
| title_short |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| title_full |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| title_fullStr |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| title_full_unstemmed |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| title_sort |
Gradient Bounds for Elliptic Problems Singular at the Boundary |
| dc.creator.none.fl_str_mv |
Porretta, Alessio Leonori, Tommaso |
| author |
Porretta, Alessio |
| author_facet |
Porretta, Alessio Leonori, Tommaso |
| author_role |
author |
| author2 |
Leonori, Tommaso |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
e-Spacio UNED |
| dc.subject.none.fl_str_mv |
12 Matemáticas viscosity solution elliptic problem neumann condition transport term elliptic regularity |
| topic |
12 Matemáticas viscosity solution elliptic problem neumann condition transport term elliptic regularity |
| description |
Let Ω be a bounded smooth domain in RN, N ≧ 2, and let us denote by d(x) the distance function d(x, ∂Ω). We study a class of singular Hamilton-Jacobi equations, arising from stochastic control problems, whose simplest model is where f belongs to W 1,∞ loc (Ω) and is (possibly) singular at ∂Ω, C ε W1,∞ (Ω)(with no sign condition) and the field B ε W1,∞ (Ω)N has an outward direction and satisfies B · v ≧ α at ∂Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian. |
| publishDate |
2011 |
| dc.date.none.fl_str_mv |
2011 2011-07-05 2011 2011-07-05 2024 2024-11-21 |
| 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/20.500.14468/24467 |
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https://hdl.handle.net/20.500.14468/24467 |
| dc.language.none.fl_str_mv |
Inglés eng |
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Inglés |
| language |
eng |
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open access http://purl.org/coar/access_right/c_abf2 info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by-nc-nd/4.0/deed.es |
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open access http://purl.org/coar/access_right/c_abf2 http://creativecommons.org/licenses/by-nc-nd/4.0/deed.es |
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openAccess |
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application/pdf |
| dc.publisher.none.fl_str_mv |
Springer Nature |
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Springer Nature |
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reponame:e-spacio. Repositorio Institucional de la UNED instname:Universidad Nacional de Educación a Distancia |
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Universidad Nacional de Educación a Distancia |
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e-spacio. Repositorio Institucional de la UNED |
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e-spacio. Repositorio Institucional de la UNED |
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1869424009091743744 |
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15.811543 |