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Parabolic methods for ultraspherical interpolation inequalities

hal.structure.identifierCEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
dc.contributor.authorDolbeault, Jean
HAL ID: 87
ORCID: 0000-0003-4234-2298
hal.structure.identifierSchool of Mathematical Science [Claremont]
dc.contributor.authorZhang, An
dc.date.accessioned2022-12-08T13:38:28Z
dc.date.available2022-12-08T13:38:28Z
dc.date.issued2023
dc.identifier.issn1078-0947
dc.identifier.urihttps://basepub.dauphine.psl.eu/handle/123456789/23321
dc.language.isoenen
dc.subjectGagliardo-Nirenberg-Sobolev inequalities
dc.subjectCaffarelli-Kohn-Nirenberg inequalities
dc.subjectinterpolation
dc.subjectsphere
dc.subjectflows
dc.subjectoptimal constants
dc.subjectweights
dc.subjectultraspherical operator
dc.subjectcarré du champ method
dc.subjectentropy methods
dc.subjectnonlinear parabolic equations
dc.subjectporous media
dc.subjectfast diffusion
dc.subjectregularity
dc.subject.ddc515en
dc.titleParabolic methods for ultraspherical interpolation inequalities
dc.typeArticle accepté pour publication ou publié
dc.description.abstractenThe carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties.
dc.relation.isversionofjnlnameDiscrete and Continuous Dynamical Systems. Series A
dc.relation.isversionofjnlvol43
dc.relation.isversionofjnlissue3&4
dc.relation.isversionofjnldate2023
dc.relation.isversionofjnlpages1347-1365
dc.relation.isversionofdoi10.3934/dcds.2022080
dc.subject.ddclabelAnalyseen
dc.relation.forthcomingnonen
dc.description.ssrncandidatenon
dc.description.halcandidatenon
dc.description.readershiprecherche
dc.description.audienceInternational
dc.relation.Isversionofjnlpeerreviewedoui
dc.date.updated2023-01-20T14:20:22Z
hal.author.functionaut
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