Arbeitsgruppe Analysis und Partielle Differentialgleichungen

Graduate seminar on Advanced topics in PDE

Organizers

Schedule

This seminar takes place regularly on Fridays, at 14:15. The seminar will take place in person in SemR 0.011. Please join the pdg-l mailing list for further information.

Oct 17 - Hjørdis Schlüter (University of Helsinki)

Title: Boundary determination in anisotropic elasticity

Abstract:

This is joint work with Joonas Ilmavirta, Mikko Salo and Daniel Windisch. I will talk about the elastic wave equation that models waves that propagate through elastic media. We are particularly interested in materials that are highly anisotropic and aim at recovering the density of mass and the so-called stiffness tensor, that capture the elasticity of the material, from exterior measurements. The exterior measurement that we consider is the Dirichlet-to-Neumann map. I will talk about a recent result, where we for generic stiffness tensor fields are able to determine the normalized stiffness tensor (where we are not able to distinguish between density and stiffness tensor) at the boundary of the medium. For this result we need a construction of special solutions and employ tools from algebraic geometry.

Oct 17, 15:45 - Organizational meeting

Oct 24 - Hendrik Baers (University of Bonn) - Joint seminar with Prof. Rüland's research group

Title: Quantitative reduction of a nonlocal Calderón type problem to the local Calderón problem

Abstract:

The Calderón problem is one of the classic examples of an inverse problem. It is about determining the conductivity of a medium by making voltage and current measurements on its boundary. Some of the main questions of interest are about uniqueness and stability of this reconstruction.

In this talk, we will consider the fractional (or nonlocal) and the local formulation of a Calderón type problem and discuss how these two are related. In particular, we will show that stability estimates for the local problem can be transferred to stability estimates for the nonlocal problem. The results presented in this talk are based on joint work with Angkana Rüland.

Oct 31 - Yi Zhang (Chinese Academy of Sciences)

Title: Weak solutions to Serrin's overdetermined problem

Abstract:

Serrin's classical overdetermined theorem states that if a bounded C^2-domain supports a function with a constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, then the domain must be a ball. In this presentation, we explore this problem further, demonstrating that the theorem's conclusion extends to a broader class of domains, including Lipschitz domains. Our method is based on theory from geometric measure theory and harmonic analysis.

Nov 7 - NO SEMINAR

Nov 14 - NO SEMINAR

Nov 21 - Michel Alexis (University of Bonn)

Title: The Steklov problem for orthogonal polynomials on the unit circle generated by an A2 weight

Abstract:

Let $w$ be a weight on the unit circle, and let $\phi_n$, $n \geq 0$, denote the unique sequence of polynomials, each of degree equal its index, orthonormal in $L^{2}$($w$). Steklov famously conjectured that if $w$ is bounded below, then the polynomials all ought to be uniformly bounded above. While false, this conjecture begs the follow-up question: under what regularity conditions on $w$ are the polynomials uniformly bounded in $L^p$($w$) for some $p>2$? Building upon a preliminary answer given by Nazarov which used the contraction principle and basic properties of the Hilbert transform, we provide a positive answer when $w$ is an A2 weight. This is joint work with Alexander Aptekarev and Sergey Denisov.

Nov 28 - Bonn-Cologne Analysis & PDE Day 2025

Location: Hörsaal Mathematisches Institut (Room 203), Universität zu Köln

Schedule and Abstract:

Please use the link to

Our group member Ruoyuan Liu will give a talk in this event.

Dec 5 - Double Session

Talk 1 - 14:15 - Fabian Höfer (University of Münster)

Title: A statistical analogue of soliton resolution for the focusing Schrödinger equation

Abstract:

We study the infinite-volume limit of Gibbs measures for the one-dimensional focusing, mass-subcritical nonlinear Schrödinger equation. Under the critical scaling, we show that the measure concentrates around a single soliton over a Gaussian background.

This program was initiated by Lebowitz-Rose-Speer (1988), who introduced the grand-canonical ensemble for the focusing NLS and conjectured the existence of a phase transition in the infinite-volume limit. This conjecture was partially confirmed in recent work by Tolomeo-Weber (2023). Building on a soliton-Gaussian decomposition, we complete the analysis of the critical regime and show that, in this case, the measure exhibits a soliton resolution.

The talk is based on joint work with Justin Forlano (Monash University) and Leonardo Tolomeo (University of Edinburgh).

Talk 2 (Online) - 15:30 - Matthew Kowalski (University of California, Los Angeles)

Title: On ill-posedness for the dispersion-managed nonlinear Schrödinger equation

Abstract:

The dispersion-managed nonlinear Schrödinger equation models the propagation of pulses through long-haul optical fibers, where the dispersion profile—and hence the focusing or defocusing nature—varies periodically. When the dispersion oscillates rapidly, this leads to the Gabitov–Turitsyn equation, a nonlocal nonlinear Schrödinger equation obtained by evolving the nonlinearity under the linear flow and averaging in time.

Despite substantial attention in physics and numerics, rigorous results for this model are rare and the sharp well-posedness theory has remained largely unclear. In this talk, we present recent results that identify a threshold for well-posedness, below which contraction mapping arguments fail, and a second, distinct threshold below which norm inflation can be shown under suitable restrictions.

Dec 12 - Double Session

Talk 1 - 14:15 - Lorenzo Laneve (Università della Svizzera italiana)

Title: An adversary bound for quantum signal processing

Abstract:

Quantum signal processing (QSP) has emerged as a unifying framework in the context of quantum algorithm design. This technique allows to carry out efficient polynomial transformations of matrices block-encoded in unitaries, involving only one extra qubit. Recent efforts try to extend QSP to the multivariate setting (M-QSP), where multiple matrices are transformed simultaneously. However, this generalization faces problems not encountered in the univariate counterpart: in particular, the class of polynomials achievable by M-QSP seems hard to characterize. In this work we borrow tools from quantum query complexity, namely the state conversion problem and the adversary bound: we first recast QSP as a state conversion problem over the Hilbert space of square-integrable functions on the unit circle. We then show that the adversary bound for a state conversion problem in this space precisely identifies all and only the QSP protocols in the univariate case. Motivated by this first result, we extend the formalism to several variables: the existence of a feasible solution to the adversary bound implies the existence of a M-QSP protocol, and the computation of a protocol of minimal space is reduced to a rank minimization problem involving the feasible solution space of the adversary bound.

Talk 2 - 15:20 - Yongming Li (Texas A&M University)

Title: Asymptotic stability of solitary waves for the 1D focusing cubic Schrödinger equation

Abstract:

In this talk we present a perturbative proof of the asymptotic stability of the solitary wave solutions for the 1D focusing cubic Schrödinger equation under small perturbations in weighted Sobolev spaces. The strategy of our proof is based on the space-time resonances approach based on the distorted Fourier transform and modulation techniques to capture the asymptotic behavior of the solution. A major difficulty throughout the nonlinear analysis is the slow local decay of the radiation term caused by the threshold resonances in the spectrum of the linearized operator around the solitary wave. The presence of favorable null structures in the quadratic terms mitigates this problem through the use of normal form transformations.