# Trimester Seminar

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

## Tuesday, April 9th, 5 p.m.

**Optimization of resonances in composite structures**

Speaker: Illia Karabash (University Bonn)

#### Abstract

We start from the optical engineering motivation of optimization problems for resonances and eigenvalues of non-Hermitian operators. It is planned to discuss numerical and analytical difficulties in the study of extremal composite structures of resonators. A recently developed theory for rigorous derivation and numerical solution of associated Euler-Lagrange equations will be presented for the case of layered optical cavities. We also discuss first steps in the multidimensional theory and connections with the topics of full topology optimization and symmetry breaking. The talk is partially based on the joint works (Albeverio, Karabash, 2017), (Karabash, Logachova, Verbytskyi, 2017), (Karabash, Koch, Verbytskyi, 2018), and (Eller, Karabash, 2019).

## Wednesday, March 6th, 3 p.m.

**Modelling pattern formation on the surface of a ferrofluid**

Speaker: Athanasios Stylianou (University Kassel)

#### Abstract

The talk deals with patterns appearing on the free surface of a ferromagnetic fluid placed in a vertical magnetic field, undergoing a so-called Rosensweig instability. We present some old results concerning existence and stability of periodic structures as well as a new existence theory for static solitons and for the associated free boundary problem.

## Wednesday, February 27th, 3 p.m.

**Sobolev regularity of of infinity Laplace equations in the plane and its application**

Speaker: Yi Zhang (University Bonn)

#### Abstract

The infinity Laplacian, introduced by Aronsson in 1960's, is a highly degenerate nonlinear second elliptic partial differential operator. The interior regularity of infinity Laplace equations is one of the main questions in this field. In 2005 $C_{loc}^1$-regularity of planar infinity harmonic functions is proved by Savin, and later in 2008 by Evans and Savin the $C_{loc}^{1,\,\alpha}$-regularity. The everywhere differentiability is proved by Evans and Smart in 2011. In this talk, I present my joint work with Prof. Koch and Prof. Zhou showing that $|Du|^2\ $in $ W^{1,\,2}_{loc}$-regularity for infinity Laplace equations with some other sharp regularity results, together with my resent joint work with Prof. Zhou which gives an alternative proof of $C_{loc}^1$-regularity of planar infinity harmonic functions.

## 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.