Webrelaunch 2020

Forschungsseminar (Dauerveranstaltung)

Vorträge im Wintersemester 2021/2022

Die Vorträge finden in Präsenz im Seminarraum 2.067 im Mathematikgebäude 20.30 statt.
Bei jeder Sitzung besteht eine 2G-Nachweispflicht sowie Maskenpflicht. Außerdem werden an jedem Termin die Kontaktdaten aller Teilnehmenden erfasst.

16.11.2021, 14:00 Uhr Robert Schippa (Karlsruhe)

Resolvent estimates for time-harmonic Maxwell's equations
We prove resolvent estimates for time-harmonic Maxwell's equations in L^p-spaces with pointwise, spatially homogeneous, and possibly anisotropic material laws. These allow for the proofs of Limiting Absorption Principles and construction of solutions. In the fully anisotropic case, which is joint work with Rainer Mandel, the construction relies on new Bochner-Riesz estimates with negative index for non-elliptic surfaces.


arXiv:2103.16951: Resolvent estimates for time-harmonic Maxwell's equations in the partially anisotropic case
arXiv:2103.17176: Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces

30.11.2021, 14:00 Uhr Konstantin Zerulla (Karlsruhe)

Time integration of Maxwell equations with low regularity
The Peaceman-Rachford alternating direction implicit (ADI) scheme is very attractive for the time discretization of linear Maxwell equations on cuboids. Indeed, it is numerically stable and has optimal linear complexity. During the error analysis, it however turns out that the accuracy of the scheme heavily depends on the regularity of the solution of the Maxwell equations.
In the talk, we study linear time-dependent Maxwell equations on a heterogeneous cuboid, modelling a rectangular waveguide. The medium is assumed to consist of several small subcuboids with different material properties. Here the solution of the Maxwell equations is not H^1-regular. We hence construct a new directional splitting scheme for the time discretization, and we provide a rigorous time-discrete error estimate for the scheme in L^2.

03.12.2021, 10:00 Uhr in Zoom Jean-Claude Cuenin (Loughborough)

Schrödinger operators with complex potentials
I will report on recent progress concerning eigenvalues of Schrödinger operators with complex potentials. We are interested in the magnitude and distribution of eigenvalues, and we seek bounds that only depend on an L^p norm of the potential.
These questions are well understood for real potentials, but completely new phenomena arise for complex potentials. I will explain how techniques from harmonic analysis, particularly those related to Fourier restriction theory, can be used to prove upper and lower bounds. We will also discuss some open problems. The talk is based on recent joint work with Sabine Bögli (Durham).

07.12.2021, 14:00 Uhr Lucrezia Cossetti (Karlsruhe)

Eigenvalue bounds and spectral stability of Lamé operator with complex potential
In this talk I will show how to get quantitative bounds on the location of eigenvalues, both discrete and embedded, of the non-self-adjoint Lamé operator of elasticity in terms of suitable norms of the potential. In order to do that we will use a nowadays well-oiled machinery based on the use of the Birman-Schwinger principle together with suitable uniform resolvent estimates. To emphasise the challenging feature of problems involving non-self-adjoint operators, we will show how, in the self-adjoint framework, such spectral enclosures are easily obtained as a consequence of the variational characterisation of the spectrum (no-longer available in a complex-valued context) and Sobolev inequalities.
The talk is based on the following two works:
L. Cossetti, Bounds on eigenvalues for perturbed Lamé operators with complex potentials, Math. Eng. 4 (2021), 5, 1-29
B. Cassano, L. Cossetti, L. Fanelli, Eigenvalue bounds and spectral stability of Lamé operator with complex potentials, Journal of Differential Equations 298 (2021), 528-559

18.01.2022, 14:00 Uhr Peer Christian Kunstmann (Karlsruhe)

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25.01.2022, 14:00 Uhr Yonas Mesfun (Karlsruhe)

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01.02.2022, 14:00 Uhr Patrick Tolksdorf (Mainz)

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