KIT - Fakultät für Mathematik -

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Fakultät für Mathematik

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Institute

Institut für Analysis

Arbeitsgruppe Nichtlineare Partielle Differentialgleichungen

Marion Ewald

Marion Ewald

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Speakers Program Abstracts Venue Information

The workshop is organized jointly by the Institute for Analysis at KIT and the Chair for Analysis at RWTH Aachen. It is held on the occasion of Professor Catherine Bandle's 80th birthday and in honor of her manifold contributions to the analysis of PDEs. The workshop brings together internationally renowned experts in the areas of PDEs, calculus of variations, and applications in geometry as well as colleagues and friends.

Speakers

Michiel van den Berg (University of Bristol, UK)

Danielle Hilhorst (Université Paris-Saclay, France)

Bernd Kawohl (University of Cologne)

Vitaly Moroz (Swansea University, UK)

Michael Plum (KIT)

Wolfgang Reichel (KIT)

Alfred Wagner (RWTH Aachen)

Program

Wednesday
March 13, 2024
13:00 - 14:00	Welcome
14:00 - 14:45	van den Berg: On some isoperimetric inequalities for the Newtonian capacity
14:50 - 15:35	Hilhorst: Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile
15:35 - 16:15	Coffee Break
16:15 - 17:00	Plum: A Computer-assisted existence proof for Emden's equation on an unbounded domain via eigenvalue bounds
17:05 - 17:50	Wagner: The second domain variation
19:30	Conference Dinner

Thursday
March 14, 2024
09:00 - 09:45	Reichel: Time-periodic waves in a nonlinear Maxwell model
09:50 - 10:35	Moroz: Normalized solutions and limit profiles of the Gross-Pitaevskii-Poisson equation
10:35 -11:15	Coffee Break
11:15 - 12:00	Kawohl: How to cut a pie
12:10 - 13:00	Farewell

Abstracts

Michiel van den Berg (University of Bristol, UK)
On some isoperimetric inequalities for the Newtonian capacity

It is shown that (i) for non-empty, compact, convex sets in $\mathbb{R}^d,\,d\geq 3$ with a $C^2$ boundary the Newtonian capacity is bounded from above by $(d-2)M(K)$ , where $M(K)>0$ is the integral of the mean curvature over the boundary of $K$ with equality if $K$ is a ball, (ii) for compact, convex sets in $\R^d,\,d\ge 3$ with non-empty interior the Newtonian capacity is bounded from above by $\frac{(d-2)P(K)^2}{d|K|}$ with equality if $K$ is a ball. Here $P(K)$ is the perimeter of $K$ and $|K|$ is its measure. A quantitative refinement of the latter inequality in terms of the Fraenkel asymmetry is also obtained.

Danielle Hilhorst (Université Paris-Saclay, France)
Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile
Joint work with Sabrina Roscani and Piotr Rybka

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct a unique self-similar solution and show that for a large class of initial data, the solution of the time evolution problem converges to this self-similar solution as time tends to infinity. Similar results were already obtained by Bouguezzi, Hilhorst, Miyamoto, and Scheid in the case of Dirichlet data on the fixed boundary. However, they had to show that the space derivative of the solution uniformly converges to its limit. Here, our proof requires much less regularity, which should make our arguments easier to adapt to different settings.

Bernd Kawohl (University of Cologne)
How to cut a pie

Suppose you want to cut a plane bounded convex set $D$ of given area $\pi$ , e.g. a pie with radius one, into two subsets of equal area with a straight or curved cut of minimal length. Now vary $D$ among all plane convex domains of equal area. For which domain is the minimal cut maximal? This question was posed by Polyá in 1958. He also conjectured that the answer is a disc which is bisected by each diameter. If one limits the permissible cuts to straight line segments, however, the answer is quite different. It is the so-called Auerbach triangle. In my talk I try to explain why.

Vitaly Moroz (Swansea University, UK)
Normalized solutions and limit profiles of the Gross-Pitaevskii-Poisson equation

Gross-Pitaevskii-Poisson equation is a nonlocal modification of the Gross-Pitaevskii equation with an attractive Coulomb-like term. It appears in the models of self-gravitating Bose-Einstein condensates proposed in cosmology and astrophysics to describe Cold Dark Matter and Boson Stars. We investigate the existence of prescribed mass (normalised) solutions to the GPP equation, paying special attention to the shape and asymptotic behaviour of the associated mass-energy relation curves and to the limit profiles of solutions at the endpoints of these curves. In particular, we show that after appropriate rescalings, the constructed normalized solutions converge either to a ground state of the Choquard equation, or to a compactly supported radial ground state of the integral Thomas-Fermi equation. In different regimes the constructed normalised solutions include global minima, local but not global minima and unstable mountain-pass type solutions. This is a joint work with Riccardo Molle and Giuseppe Riey.

Michael Plum (KIT)
A Computer-assisted existence proof for Emden's equation on an unbounded domain via eigenvalue bounds
Joint work with Filomena Pacella and Dagmar Rütters

We prove existence and non-degeneracy of a non-trivial solution to Emden's equation $-\Delta u = | u |^3$ on an unbounded $L$ -shaped domain, subject to Dirichlet boundary conditions. Our proof is computer-assisted: Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The most crucial step in this argument is the computation of a norm bound for the inverse of the linearization at the approximate solution. We propose to use eigenvalue bounds for this purpose, which in turn we compute by variational methods supported by a homotopy approach. These eigenvalue bounds also imply non-degeneracy of the solution.

Wolfgang Reichel (KIT)
Time-periodic waves in a nonlinear Maxwell model
Joint work with Simon Kohler (KIT) and Sebastian Ohrem (KIT)

We consider the system of Maxwell equations where the refractive index depends nonlinearly on the amplitude of the electric field $\mathbf{E}$ . Depending on the choice of the model, the governing equation for a second order formulation for the $\mathbf{E}$ field is either

$g(x)\mathbf{E}_{tt}+\nabla\times\nabla\times \mathbf{E}+h(x)(|\mathbf{E}|^2\mathbf{E})_{tt}=0 \quad \mbox{ on } \quad \mathbb{R}^3\times\mathbb{R}$

or

$g(x)\mathbf{E}_{tt}+\nabla\times\nabla\times \mathbf{E}+h(x)\bigl((\kappa\ast |\mathbf{E}|^2)\mathbf{E}\bigr)_{tt}=0 \quad \mbox{ on } \quad \mathbb{R}^3\times\mathbb{R},$

where $g, h, \kappa$ are functions representing material parameters. Using variational principles we show the existence of time-periodic, spatially localized solutions.

Alfred Wagner RWTH Aachen
The second domain variation
Joint work with Catherine Bandle

As a first example we consider the Gamow's model for the stability of liquid droplets. The associated energy consists of the sum of the surface area and the Coulomb energy. While the surface area is minimized for all domains of given volume, the Coulomb energy is maximized in this class. We examine the sphere for its stability.

A second example is the first Steklov eigenvalue of the Bilaplace operator. It is known, that among all planar domains of given area the disc is a critical point for this eigenvalue. On the other hand a long standing conjecture states that in this class, the regular pentagon is the global minimizer. We investigate the stability of the ball.

Venue

Mathematics Building 20.30, Englerstrasse 2, 76131 Karlsruhe.
Scientific Talks: Room 1.067
Coffee Breaks: Room 1.058

Travel Information

WLAN Information

Information

Please contact Wolfgang Reichel wolfgang.reichel@kit.edu.

For administrative questions please contact Marion Ewald marion.ewald@kit.edu.

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