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Gradient Bounds for Elliptic Problems Singular at the Boundary

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dc.contributor.authorPorretta, Alessio
dc.contributor.authorLeonori, Tommaso
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.provenanceMade available in DSpace on 2024-11-21T10:47:10Z (GMT). No. of bitstreams: 1 GRADIENT BOUNDS. Leonori_T.pdf: 490472 bytes, checksum: 3855c41065d42ec3b7f79cdcf67a6cd4 (MD5) Previous issue date: 2011-07-05en
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.centerFacultad 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.typejournal articleen
dspace.entity.typePublication
relation.isAuthorOfPublication07099038-eae7-41de-9f88-7ce9062e30f1
relation.isAuthorOfPublication.latestForDiscovery07099038-eae7-41de-9f88-7ce9062e30f1
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