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