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

Jean Dolbeault (1) , Maria J. Esteban (1) , Alessio Figalli (2) , Rupert L. Frank (3) , Michael Loss (4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Department of Mathematics - ETH
3 LMU - Ludwig Maximilian University [Munich] = Ludwig Maximilians Universität München
4 Georgia Institute of Technology [Atlanta]

We prove a sharp quantitative version for the stability 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.

Mots clés: Sobolev inequality; logarithmic Sobolev inequality; stability; rearrangement; Steiner symmetrization

Domaines: Equations aux dérivées partielles [math.AP]

Jean Dolbeault, Maria J. Esteban, Alessio Figalli, Rupert L. Frank, Michael Loss. Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence. 2023. ⟨hal-03780031v4⟩