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UID:calendarize-spectral-analysis-of-a-closed-perturbed-waveguide-filled-w
 ith-a-hyperbolic-medium
DTSTAMP:20260420T091257Z
DTSTART:20260507T123000Z
DTEND:20260507T133000Z
SUMMARY:Spectral analysis of a closed perturbed waveguide filled with a hy
 perbolic medium
DESCRIPTION:Lecture announcement\nIn the context of the advanced seminar A
 nalysis/NumericsMr Dylan Machado(ENSTA Paris)\nabout\nSpectral analysis of
  a closed perturbed waveguide filled with a hyperbolic medium\nAbstract: W
 e investigate the spectrum of the first-order operator that governs the pr
 opagationof electromagnetic waves in the case of a closed locally perturbe
 d waveguide filled with ahyperbolic medium. The perturbation is localized 
 on the boundary and we imposehomogeneous Dirichlet boundary conditions.Fro
 m a physics perspective\, this type of medium falls within the class of hy
 perbolicelectromagnetic metamaterials. They are artificial media which exh
 ibit a lot of interesting andpromising physics phenomena such as negative 
 refraction\, rainbow-trapping and enhancedspontaneous emission.In practice
 \, the simplest mathematical model stems from electromagnetic wave propaga
 tion ina strongly magnetized cold plasma. It is described by 2D Maxwell eq
 uations which\, in theirtime-harmonic form\, may be reduced to the Klein-G
 ordon equation for a certain range offrequencies\, namely the hyperbolic f
 requencies. Hence\, the nature of the problem is purelyhyperbolic.Firstly\
 , we prove a limiting absorption principle (LAP) for the hyperbolic freque
 ncies satisfyinga geometric condition on the perturbation. The latter is r
 elated to the interplay between thevelocities of the boundary and of the p
 ropagating waves\, which\, in our case\, depends on thefrequency. This ran
 ge 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 equ
 ation in boundedtime-dependent domains.The second part of the talk will be
  devoted to the 'supersonic' regime: the frequencies for whichthe boundary
  moves faster than the waves. In particular\, the Klein-Gordon equation in
  arbitrarytime-dependent domains is no longer well-posed for homogeneous D
 irichlet boundaryconditions. Nevertheless\, we are still able to prove a L
 AP by using the weak observabilityresults of waves and exchanging the role
  of the time and space variables. \nThe lecture will take place onThursday
 \, 07.05.2026at 14.30 - 15.30 in room W01 1-117\nInterested parties are co
 rdially invited.
X-ALT-DESC;FMTTYPE=text/html:<h2 class="text-center">Lecture announcement<
 /h2>\n<h3 class="text-center"><br><strong>In the context of the advanced s
 eminar Analysis/Numerics<br></strong><i><strong>Mr Dylan Machado</strong><
 /i><strong><br>(ENSTA Paris)</strong></h3>\n<p class="text-center"><br />a
 bout</p>\n<h2 class="text-center"><br><strong>Spectral analysis of a close
 d perturbed waveguide filled with a hyperbolic medium</strong></h2>\n<p cl
 ass="text-center"><br />Abstract: We investigate the spectrum of the first
 -order operator that governs the propagation<br />of electromagnetic waves
  in the case of a closed locally perturbed waveguide filled with a<br />hy
 perbolic medium. The perturbation is localized on the boundary and we impo
 se<br />homogeneous Dirichlet boundary conditions.<br />From a physics per
 spective\, this type of medium falls within the class of hyperbolic<br />e
 lectromagnetic metamaterials. They are artificial media which exhibit a lo
 t of interesting and<br />promising physics phenomena such as negative ref
 raction\, rainbow-trapping and enhanced<br />spontaneous emission.<br />In
  practice\, the simplest mathematical model stems from electromagnetic wav
 e propagation in<br />a strongly magnetized cold plasma. It is described b
 y 2D Maxwell equations which\, in their<br />time-harmonic form\, may be r
 educed to the Klein-Gordon equation for a certain range of<br />frequencie
 s\, namely the hyperbolic frequencies. Hence\, the nature of the problem i
 s purely<br />hyperbolic.<br />Firstly\, we prove a limiting absorption pr
 inciple (LAP) for the hyperbolic frequencies satisfying<br />a geometric c
 ondition on the perturbation. The latter is related to the interplay betwe
 en the<br />velocities of the boundary and of the propagating waves\, whic
 h\, in our case\, depends on the<br />frequency. This range of frequency i
 s called the subsonic regime. The key idea of our work is<br />the use of 
 the well-posedness and energy estimates of the Klein-Gordon equation in bo
 unded<br />time-dependent domains.<br />The second part of the talk will b
 e devoted to the 'supersonic' regime: the frequencies for which<br />the b
 oundary moves faster than the waves. In particular\, the Klein-Gordon equa
 tion in arbitrary<br />time-dependent domains is no longer well-posed for 
 homogeneous Dirichlet boundary<br />conditions. Nevertheless\, we are stil
 l able to prove a LAP by using the weak observability<br />results of wave
 s and exchanging the role of the time and space variables.<br /> </p>\n<h4
  class="text-center"><strong>The lecture will take place on<br>Thursday\, 
 07.05.2026<br>at 14.30 - 15.30 in room W01 1-117</strong></h4>\n<p class="
 text-center"><br />Interested parties are cordially invited.</p>
LOCATION:
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