# Workshop on PDE and related topics

### Time and location:

16.-30. September 2016, Department of mathematics, TU Chemnitz

### Contents:

We will have several long-term guests whose visiting periods will overlap during a small workshop.

### Accomodation:

Due to an event, it might not be easy to find accomodation during the time in Chemnitz.

Here, you will find some useful information on how to come to the departement, on hotels etc. We also encourage you to check on portals such as airbnb.

### Confirmed Participants:

 Name Time Room Ivan Veselic (TU Chemnitz) uncertainty principles, harmonich analysis, Schrödinger equation September 16-30 Martin Tautenhahn (TU Chemnitz) unique continuation properties of eigenfunctions, random Schrödinger operators Matthias Täufer (TU Chemnitz) unique continuation properties of eigenfunctions, random Schrödinger operators Michela Egidi (TU Chemnitz) Fourier analysis Mohamad Haidar (Jacobs University Bremen) Nonlinear PDE September 19-24 707 (from 19 to 21 September: Room 704) Reinhard Stahn (TU Dresden) Control of the wave equation using resolvent estimates September 19-25 707 (from 19 to 21 September: Room 704) Ivica Nakic (University of Zagreb, Croatia) damping of vibrational systems, spectral and perturbation theory of linear operators and pencils, control theory September 16-30 608 Denis Borisov (Bashkir State Pedagogical University, Russia), PDE, spectral theory, analytic families of self-adjoint operators September 15-28 703 Martin Lazar (University of Dubrovnik, Croatia) Microlocal defect tools (H_measures etc), Control of parameter dependent problems September 25-30 637 Thomas Kalmes (TU Chemnitz) partial differential equations, theory of distributions September 26-30 Irena Brdar (University of Dubrovnik, Croatia) PDEs September 19-30 715 Christian Rose (TU Chemnitz) Geometric Analysis September 26-30 Jussi Behrndt (TU Graz) September 19

### Schedule

Five slots per week will be reserved for lectures given by the participants. Talks will take place in room 41/705.

 Mon. 19 Sept. Tue. 20 Sept. Wed. 21 Sept. Thu. 22 Sept. Fri. 23 Sept. 9:30-13:00 - - - Mohamad Haidar (10:30), Reinhard Stahn (11:30), Matthias Täufer (informal talk, 11:00) 14:00-17:00 Jussi Behrndt (14:00) Martin Tautenhahn (14:00), Reinhard Stahn (15:00) - Ivica Nakic (14:00), Matthias Täufer (14:30) Denis Borisov (14:00)

 Mon. 26 Sept. Tue. 27 Sept. Wed. 28 Sept. Thu. 29 Sept. Fri. 30 Sept. 11:30-12:30 - - - - - 14:00-17:00 Martin Lazar (14:00), Thomas Kalmes (15:00) Christian Rose (16:00) - - - Lecture

The remaining time slots are open for discussions and/or research.

The remaining time slots are open for discussions and/or research.

### Talks and abstracts

##### Jussi Behrndt: Dirichlet-to-Neumann maps and the spectral data of Schrödinger operators, Monday, 19 September, 14:00, Location: Reichenhainer Str. 41, Room 705

Abstract: In this talk we discuss how the spectral data of selfadjoint Schrödinger operators on bounded or unbounded domains can be described with an associated Dirichlet-to-Neumann map. In particular, a characterization of the isolated and embedded eigenvalues, the corresponding eigenspaces, as well as the continuous and absolutely continuous spectrum in terms of the limiting behaviour of the Dirichlet-to-Neumann map is obtained. The results are natural multidimensional analogs of classical facts from singular Sturm-Liouville theory and can also be viewed as mild uniqueness results in the context of the classical Calderon problem.

##### Dennis Borisov: Initial length scale estimates for waveguides with small random perturbations

Abstract: We consider a waveguide with small random perturbation. The waveguides is modeled by an infinite multi-dimensional layer, in which Schroedinger operator subject to Dirichlet or Neumann condition is considered. In the waveguide we choose a periodic lattice which splits the waveguide into a family of periodicity cells. To each of such cells, we associated a random variable and assume that all random variables are independent and identically distributed. The perturbations are described by an abstract symmetric operator acting in each of the periodicity cells and depending on a parameter. As a parameter, we choose a random variable associated with a cell and this variable is multiplied by a global small parameter. We consider two main cases assuming that the described random perturbation shifts the bottom of the unperturbed spectrum up or down. In both cases we establish
an initial length scale estimate for our model. General results are accompanied by examples of particular perturbations, both new and studied before.

Abstract

##### Mohamad Haidar: On the non-relativistic limit for the semi-linear Klein-Gordon equation

Abstract: We extend an iterative fast-slow  construction, which was already used for a class of singularly perturbed ODEs,  to be applied on the semi-linear Klein-Gordon equation.  We construct an approximate system for the slow motion, whose nonlinear part is  an asymptotic series in $\epsilon$  with coefficient functions recursively defined, up to a small remainder with respect to $\epsilon$. We prove that the   solutions of the slow systems shadow solutions of the Klein-Gordon equation at the corresponding order over  a finite interval of time.

### Kontakt

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund

Sie finden uns auf dem sechsten Stock des Mathetowers.

#### Sekretariat

Janine Textor (Raum M 620)

Tel.: (0231) 755-3063
Fax: (0231) 755-5219
Mail: janine.textor@tu-dortmund.de