Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities
Résumé
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit stability measurements, i.e., bound with explicit constants distances to the manifold of optimal functions. Various results are obtained for the Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the Gaussian generalized Poincaré inequalities and for the Gagliardo-Nirenberg inequalities. As a consequence, faster convergence rates in diffusion equations (fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained.
Mots clés
- Gaussian isoperimetric inequality
- stability
- porous medium equation
- extremal functions
- improved functional inequalities
- entropy power
- generalized Fisher information
- Gagliardo-Nirenberg inequalities
- interpolation
- entropy -- entropy production inequalities
- rates of convergence
- optimal constants
- relative entropy
- generalized Poincaré inequalities
- logarithmic Sobolev inequality
- Sobolev inequality
- fast diffusion equation
- Ornstein-Uhlenbeck equation
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