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
- relative entropy
- rates of convergence
- porous medium equation
- Ornstein-Uhlenbeck equation
- fast diffusion equation
- improved functional inequalities
- stability
- entropy power
- generalized Fisher information
- optimal constants
- extremal functions
- entropy -- entropy production inequalities
- interpolation
- Gagliardo-Nirenberg inequalities
- generalized Poincaré inequalities
- Gaussian isoperimetric inequality
- logarithmic Sobolev inequality
- Sobolev inequality
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