# Trimester Seminar

## Venue: HIM, Poppelsdorfer Allee 45, Lecture Hall

## Wednesday, February 20th, 3 p.m.

**Sobolev homeomorphic extensions**

Speaker: Aleksis Koski (University of Jyväskylä)

#### Abstract

In the mathematical theory of nonlinear elasticity one typically represents elastic bodies as domains in Euclidean space, and the main object of study are deformations (mappings) between two such bodies. The class of acceptable deformations one considers usually consists of Sobolev homeomorphisms between the respective domains, for example, with some given boundary values. It is hence a fundamental question in this theory to ask whether a given boundary map admits a homeomorphic extension in the Sobolev class or not. We share some recent developments on this subject, including sharp existence results and counterexamples.

## Monday, February 4th, 3 p.m.

**Dependence with respect to the data in incompressible optimal transport**

Speaker: Aymeric Baradat (Ecole Polytechnique)

#### Abstract

Incompressible optimal transport (or Brenier model) is a minimization problem introduced by Brenier in 89 in order to describe the behavior of an incompressible and inviscid fluid in a Lagrangian way. The data of the problem is the joint law of the initial and final positions of the particles, and the dynamics is guided by the Lagrange multiplier corresponding to the incompressibility constraint: the pressure field. In this talk, I will present a positive and a negative result concerning the continuous dependence of the pressure field with respect to the data. The negative part is related to the question of ill-posedness of the so-called kinetic Euler equation, a kinetic PDE known in plasma physics as the limit of the Vlasov-Poisson equation in a quasineutral regime.