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Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence

Dolbeault, Jean; Esteban, Maria J.; Figalli, Alessio; Frank, Rupert L.; Loss, Michael (2022), Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence. https://basepub.dauphine.psl.eu/handle/123456789/24067

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Type
Document de travail / Working paper
Date
2022
Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSL
Published in
Paris
Pages
41
Metadata
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Author(s)
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Esteban, Maria J.
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Figalli, Alessio
Department of Mathematics - ETH
Frank, Rupert L.
Mathematisches Institut [München] [LMU]
Loss, Michael
School of Mathematics - Georgia Institute of Technology
Abstract (EN)
We prove a quantitative version of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as, competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement and refined estimates on the Sobolev-functional in the neighborhood of the optimal Aubin-Talenti functions.
Subjects / Keywords
Sobolev inequality; logarithmic Sobolev inequality; stability; rearrangement; Steiner symmetrization