Lecture announcement

In the context of the advanced seminar Analysis/Numerics
Mr Dylan Machado
(ENSTA Paris)

about

Spectral analysis of a closed perturbed waveguide filled with a hyperbolic medium

Abstract: We investigate the spectrum of the first-order operator that governs the propagation
of electromagnetic waves in the case of a closed locally perturbed waveguide filled with a
hyperbolic medium. The perturbation is localized on the boundary and we impose
homogeneous Dirichlet boundary conditions.
From a physics perspective, this type of medium falls within the class of hyperbolic
electromagnetic metamaterials. They are artificial media which exhibit a lot of interesting and
promising physics phenomena such as negative refraction, rainbow-trapping and enhanced
spontaneous emission.
In practice, the simplest mathematical model stems from electromagnetic wave propagation in
a strongly magnetized cold plasma. It is described by 2D Maxwell equations which, in their
time-harmonic form, may be reduced to the Klein-Gordon equation for a certain range of
frequencies, namely the hyperbolic frequencies. Hence, the nature of the problem is purely
hyperbolic.
Firstly, we prove a limiting absorption principle (LAP) for the hyperbolic frequencies satisfying
a geometric condition on the perturbation. The latter is related to the interplay between the
velocities of the boundary and of the propagating waves, which, in our case, depends on the
frequency. This range of frequency is called the subsonic regime. The key idea of our work is
the use of the well-posedness and energy estimates of the Klein-Gordon equation in bounded
time-dependent domains.
The second part of the talk will be devoted to the 'supersonic' regime: the frequencies for which
the boundary moves faster than the waves. In particular, the Klein-Gordon equation in arbitrary
time-dependent domains is no longer well-posed for homogeneous Dirichlet boundary
conditions. Nevertheless, we are still able to prove a LAP by using the weak observability
results of waves and exchanging the role of the time and space variables.

The lecture will take place on
Thursday, 07.05.2026
at 14.30 - 15.30 in room W01 1-117

Interested parties are cordially invited.