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112 | 2013
Annuaire du Collège de France 2011-2012
Résumé des cours et travaux 112e année
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Stabilité dans les inégalités fonctionnelles, transport optimal et EDP

Résumé des conférences du Collège de France (2011-2012)
Alessio Figalli
p. 841-842

Texte intégral

1Geometric and functional inequalities play a crucial role in several problems arising in the calculus of variations, partial differential equations, geometry, etc. More recently, there has been a growing interest in the study of the stability for such inequalities. The basic question one wants to address is the following: suppose we have a functional inequality for which we know the sets of functions for which equality is attained. Can we prove, in some quantitative way, that if a function “almost attains the equality” then it is close (in some suitable sense) to a minimizer?

2In recent years there have been several results in this direction, showing sharp stability results for the Euclidean/anisotropic isoperimetric inequality, the Brunn-Minkowski inequality on convex sets, the Gaussian isoperimetric inequality, Sobolev and Gagliardo-Nirenberg inequalities, etc. The aim of these lectures is to show different ways to attack these problems, together with some applications of these results.

3More precisely, we first focused on the isoperimetric inequality showing three different approaches to prove a sharp stability result: via symmetrization techniques, via optimal transport theory, and via a second order stability analysis. Then we applied this result to two different variational problems: the first one concerns the shape of liquid drops/crystals under the action of an exterior potential force; the second one is the study of the isoperimetric problem on manifolds with density when the volume is small. We then focused on another family of inequalities: Sobolev and Gagliardo-Nirenberg. We discussed the stability issue for them, and as an application we obtained a quantitative rate of convergence to a stationary state for the critical mass Keller-Segel system.

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Cours et travaux du Collège de France. Annuaire 112e année, Collège de France, Paris, avril 2013, p. 841-842. ISBN 978-2-7226-0198-7

Référence électronique

Alessio Figalli, « Stabilité dans les inégalités fonctionnelles, transport optimal et EDP », L’annuaire du Collège de France [En ligne], 112 | 2013, mis en ligne le 28 août 2013, consulté le 20 février 2018. URL : http://journals.openedition.org/annuaire-cdf/1116

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Auteur

Alessio Figalli

Professeur à l’université du Texas, Austin (États-Unis)

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Droits d’auteur

Collège de France

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