### Workshop on stability of functional inequalities and applications

The workshop will take place on June 13-15 2018, at the Institut de Mathematiques de Toulouse, Universite Paul Sabatier. It is supported by the Labex CIMI and the ANR project “Entropies, Flots, Inegalites”.

#### Speakers

-Marco Barchiesi (Università di Napoli)

-Ivan Bardet (IHES)

-Lorenzo Brasco (Università di Ferrara)

-Thomas Courtade (UC Berkeley)

-Jean Dolbeault (Universite Paris Dauphine)

-Sabine Jansen (Universität München)

-Joseph Lehec (Universite Paris Dauphine)

-Chaowei Zhang (Universite Blaise Pascal)

-Simon Zugmeyer (Universite Lyon 1)

#### Preliminary timetable

Wednesday June 13th

- 2-3:10 pm: Barchiesi
- 3:40-4:50pm: Zugmeyer

Thursday June 14th

- 9-10:10am: Jansen
- 10:40-11:50am: Lehec
- 2-2:35pm: Zhang
- 2:35-3:10pm: Bardet
- 3:40-4:50: Brasco

Friday June 15th

- 9-10:10am: Courtade
- 10:40-11:50am: Dolbeault

#### Abstracts

*Marco Barchiesi*: Stability of the Gaussian Isoperimetric Problem

I will present an analysis of the sets that minimize the gaussian perimeter plus the norm of the barycenter. These two terms are in competition, and in general the solutions are not the half-spaces. In fact we prove that when the volume is close to one, the solutions are the strips centered in the origin. As a corollary, we obtain that the symmetric strip is the solution of the Gaussian isoperimetric problem among symmetric sets when the volume is close to one. Co-Author: Vesa Julin

*Ivan Bardet*: Introduction to non-commutative hypercontractivity and log-Sobolev inequality

Hypercontractivity and log-Sobolev Inequalities also appear in non-commutative analysis, where they are defined with respect to certain interpolating familly of Lp spaces on non-commutative operator algebras. The basic example is the Fermionic semigroup, studied by Gross and then by Carlen and Lieb, a non-commutative counterpart of the Ornstein Uhlenbeck semigroup. We will discuss the similarities and highlight the difference with the classical (non-commutative) theory and, if time allows it, present some recent progress.

*Lorenzo Brasco*: Stability for Faber-Krahn inequalities

Among N-dimensional open sets with given measure, balls (uniquely) minimize the first eigenvalue of the Laplacian with homogeneous Dirichlet boundary conditions. We review this classical result and discuss some of its applications. Then we show how this can be enhanced by means of a quantitative stability estimate. The resulting inequality, first conjectured by Nadirashvili and Bhattacharya & Weitsman, is sharp.

The results presented are contained in a paper in collaboration with Guido De Philippis and Bozhidar Velichkov.

*Thomas Courtade*: Deficit estimates for the Shannon-Stam inequality

The Shannon-Stam inequality states that the so-called 'entropy power' is superadditive on convolution of densities, and plays a foundational role in information theory. We discuss two different deficit estimates for this inequality. The first, based on ideas from optimal transport, establishes a dimension-free stability estimate when the densities are log-concave. The second estimate, also dimension-free, has a variational form and leads to interesting improvements of the Shannon-Stam inequality, as well as Gross' LSI.

*Jean Dolbeault*: Reverse Hardy-Littlewood-Sobolev inequalities

This lecture is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities with a power law kernel with positive exponent. In the range of the admissible parameters, the main issue is to characterize the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with nonlinear diffusion equations involving mean field drifts. J.A. Carrillo, M. Delgadino, R. Frank and F. Hoffmann have been involved in this research project.

*Sabine Jansen*: Metastability and free energy landscape for the Widom-Rowlinson model

The Widom-Rowlinson model is one of the few continuum models of statistical mechanics for which the existence of a phase transition is rigorously proven. It is a Gibbs modification of a Poisson point process: around each point, a ball of fixed radius is drawn; the energy is the area covered by the union of balls (the area of the “halo”). We investigate the model in a finite box as the intensity of the underlying Poisson point process goes to infinity. Our primary goal is to understand metastable behavior of a Markovian birth and death process (or exponentially small eigenvalues of the associated infinitesimal generator) for which the Gibbs measure is reversible. In order to do that, we need to understand first the effective free energy landscape for a suitably chosen order parameter. The talk will explain some partial results and how they relate to large deviations and central limit theorems in stochastic geometry. It is based on joint work in progress with Frank den Hollander, Roman Kotecky and Elena Pulvirenti.

*Joseph Lehec*: Stability in the logarithmic Sobolev inequality

I will review recent developments regarding stability estimates in the logarithmic Sobolev inequality for the Gaussian measure. Maybe I’ll also give a new result.

*Chaoen Zhang*: Mean-field type functional inequalities and convergence to equilibrium for McKean-Vlasov equation.
Abstract: I will present some results on functional inequalities for McKean-Vlasov equation and its application to convergence to equilibrium. We study the particle systems of mean-field type and their invariant measures. For these invariant measures, Poincaré inequalities and logarithmic Sobolev inequalities are established under certain assumptions with constants independent of the number of dimensions. By propagation of chaos, we then deduce convergence to equilibrium in quadratic Wasserstein distance.

*Simon Zugmeyer*: From the Borell-Brascamp-Lieb inequality to sharp trace-Sobolev inequalities

Sharp Sobolev inequalities were first obtained through the calculus of variations, notably by Talenti and Aubin. More recently, it has been seen that such inequalities may be reached within the framework of the Brunn-Minkowski theory. This makes sense, since for instance, it is well known that the isoperimetric inequality is equivalent to the standard sharp Sobolev inequality in the Euclidean space (for p=1). Expanding on an idea by Nazaret, I will show how to use the Borell-Brascamp-Lieb inequality as a tool to prove sharp (trace-)Sobolev inequalities on domains such as convex cones.

#### Participants

-Franck Barthe (Toulouse)

-Jerome Bertrand (Toulouse)

-Ángela Capel (ICMAT)

-Patrick Cattiaux (Toulouse)

-Djalil Chafai (Dauphine)

-Josephine Evans (Cambridge)

-Max Fathi (Toulouse)

-Nathael Gozlan (Paris-Descartes)

-Arnaud Guillin (Clermont-Ferrand)

-Aldéric Joulin (Toulouse)

-Michel Ledoux (Toulouse)

-Jason Ledwidge (Tubingen)

-Yan Pautrat (Orsay)

-Clément Pellegrini (Toulouse)

-Cambyse Rouze (Cambridge)

-Nikita Simonov (Madrid)

-Marta Sztrelecka (Warsaw)

-Michal Sztrelecki (Warsaw)

If you plan on attending this workshop, please contact the organizers at max[dot]fathi[at]math[dot]univ-toulouse.fr.