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Gradient Bounds for Elliptic Problems Singular at the Boundary

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dc.contributor.authorTommaso, Leonori
dc.contributor.authorPorretta, Alessio
dc.date.accessioned2024-11-21T10:47:10Z
dc.date.available2024-11-21T10:47:10Z
dc.date.issued2011-07-05
dc.description.abstractLet 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.en
dc.description.versionversión final
dc.identifier.citationLeonori, T., Porretta, A. Gradient Bounds for Elliptic Problems Singular at the Boundary. Arch Rational Mech Anal 202, 663–705 (2011). https://doi.org/10.1007/s00205-011-0436-9
dc.identifier.doihttps://doi.org/10.1007/s00205-011-0436-9
dc.identifier.issn0003-9527; e-ISSN: 1432-0673
dc.identifier.urihttps://hdl.handle.net/20.500.14468/24467
dc.journal.titleArchive for Rational Mechanics and Analysis
dc.journal.volume202
dc.language.isoen
dc.page.final705
dc.page.initial663
dc.publisherSpringer Nature
dc.relation.centerFacultades y escuelas::Facultad de Ciencias
dc.relation.departmentMatemáticas Fundamentales
dc.rightsinfo:eu-repo/semantics/openAccess
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/4.0/deed.es
dc.subject12 Matemáticas
dc.subject.keywordsviscosity solutionen
dc.subject.keywordselliptic problemen
dc.subject.keywordsneumann conditionen
dc.subject.keywordstransport termen
dc.subject.keywordselliptic regularityen
dc.titleGradient Bounds for Elliptic Problems Singular at the Boundaryen
dc.typeartículoes
dc.typejournal articleen
dspace.entity.typePublication
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