Parabolic methods for ultraspherical interpolation inequalities
Dolbeault, Jean; Zhang, An (2023), Parabolic methods for ultraspherical interpolation inequalities, Discrete and Continuous Dynamical Systems. Series A, 43, 3&4, p. 1347-1365. 10.3934/dcds.2022080
TypeArticle accepté pour publication ou publié
Journal nameDiscrete and Continuous Dynamical Systems. Series A
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
School of Mathematical Science [Claremont]
Abstract (EN)The 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.
Subjects / KeywordsGagliardo-Nirenberg-Sobolev inequalities; Caffarelli-Kohn-Nirenberg inequalities; interpolation; sphere; flows; optimal constants; weights; ultraspherical operator; carré du champ method; entropy methods; nonlinear parabolic equations; porous media; fast diffusion; regularity
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