École d’été 1

Calcul des variations, transport optimal, théorie géométrique de la mesure: théorie et applications

 du 27 juin au 1 juillet 2016

À Lyon
ICJ, Université Claude Bernard Lyon 1, Villeurbanne, France

Cette école est organisée dans le cadre de la période thématique du même nom structurée en 2 écoles d’été et une conférence internationale.

Trois conférences sont au programme :

Shape optimization of spectral functionals 

In these lectures, isoperimetric type inequalities involving the spectrum of the Laplace operator (with some boundary conditions) will be seen from a shape optimisation point of view. Depending on the boundary conditions, the analysis of those problems (existence of solution, regularity, qualitative properties) is either related to a free boundary problem of Alt-Caffarelli type, or to a free discontinuity problem. I will make an introduction to this topic and present recent results, with a focus on Robin boundary conditions. In particular I will detail a monotonicity formula which is the key point for the (Ahlfors) regularity of the optimal sets.

In these lectures, isoperimetric type inequalities involving the spectrum of the Laplace operator (with some boundary conditions) will be seen from a shape optimisation point of view. Depending on the boundary conditions, the analysis of those problems (existence of solution, regularity, qualitative properties) is either related to a free boundary problem of Alt-Caffarelli type, or to a free discontinuity problem. I will make an introduction to this topic and present recent results, with a focus on Robin boundary conditions. In particular I will detail a monotonicity formula which is the key point for the (Ahlfors) regularity of the optimal sets.

The selection principle: the use of regularity theory in proving quantitative inequalities

I will first introduce the topic of quantitative inequalities and give some examples. Then I will present a general technique to derive them based on the regularity theory for solutions of variational problems. The course will be mainly focused on the (quantitative) isoperimetric inequality and on the (quantitative) Faber-Krahn inequality

I will first introduce the topic of quantitative inequalities and give some examples. Then I will present a general technique to derive them based on the regularity theory for solutions of variational problems. The course will be mainly focused on the (quantitative) isoperimetric inequality and on the (quantitative) Faber-Krahn inequality.

Optimal transport, optimal curves, optimal flows

The course will consist in an introduction to optimal transport theory with a special attention to the comparison between Eulerian and Lagrangian point of views, and between statical and dynamical approaches. Optimal flow versions of some issues of the problem will also be presented, and this will lead at the end of the course to the study of some traffic equilibrium problems.
The course will consist of four lectures, roughly divided as follows:

  1. Basic theory of Optimal Transport

The problems by Monge and Kantorovich.
Convex duality and Kantorovich potentials.
Existence of optimal maps (Brenier Theorem) for strictly convex costs.

  1. Wasserstein distances

Definitions of the distances W_p induced by optimal transport costs.
The duality between W_1 and Lipschitz functions and the topology induced by the distances W_p.
The continuity equation and the curves in the space W_p.

  1. Curves of measures and geodesics in the Wasserstein space

From measures on curves to curves of measures and back.
Constant-speed geodesics in the Wasserstein space.
The Benamou-Brenier dynamical formulation of optimal transport.

  1. Minimal flows

An Eulerian formulation of the Monge problem with cost |x-y| (p=1): the Beckmann problem.
From measures on curves to vector flows and back.
Extensions to traffic congestion models.

 

Revenir à la page de la période thématique

 


Site de l’école d’été