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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:Vortragsankündigung\nIm Rahmen des Oberseminars Analysis/Nume
 rik sprichtHerr Dylan Machado(ENSTA Paris)\nüber\nSpectral analysis of a 
 closed perturbed waveguide filled with a hyperbolic medium\nAbstract: We i
 nvestigate the spectrum of the first-order operator that governs the propa
 gationof electromagnetic waves in the case of a closed locally perturbed w
 aveguide filled with ahyperbolic medium. The perturbation is localized on 
 the boundary and we imposehomogeneous Dirichlet boundary conditions.From a
  physics perspective\, this type of medium falls within the class of hyper
 bolicelectromagnetic metamaterials. They are artificial media which exhibi
 t a lot of interesting andpromising physics phenomena such as negative ref
 raction\, rainbow-trapping and enhancedspontaneous emission.In practice\, 
 the simplest mathematical model stems from electromagnetic wave propagatio
 n ina strongly magnetized cold plasma. It is described by 2D Maxwell equat
 ions which\, in theirtime-harmonic form\, may be reduced to the Klein-Gord
 on equation for a certain range offrequencies\, namely the hyperbolic freq
 uencies. Hence\, the nature of the problem is purelyhyperbolic.Firstly\, w
 e prove a limiting absorption principle (LAP) for the hyperbolic frequenci
 es satisfyinga geometric condition on the perturbation. The latter is rela
 ted to the interplay between thevelocities of the boundary and of the prop
 agating waves\, which\, in our case\, depends on thefrequency. This range 
 of frequency is called the subsonic regime. The key idea of our work isthe
  use of the well-posedness and energy estimates of the Klein-Gordon equati
 on in boundedtime-dependent domains.The second part of the talk will be de
 voted to the 'supersonic' regime: the frequencies for whichthe boundary mo
 ves faster than the waves. In particular\, the Klein-Gordon equation in ar
 bitrarytime-dependent domains is no longer well-posed for homogeneous Diri
 chlet boundaryconditions. Nevertheless\, we are still able to prove a LAP 
 by using the weak observabilityresults of waves and exchanging the role of
  the time and space variables. \nDer Vortrag findet statt amDonnerstag\, 
 den 07.05.2026um 14.30 – 15.30 Uhr im Raum W01 1-117\nInteressierte sind
  herzlich eingeladen.
X-ALT-DESC;FMTTYPE=text/html:<h2 class="text-center">Vortragsankündigung<
 /h2>\n<h3 class="text-center"><br><strong>Im Rahmen des Oberseminars Analy
 sis/Numerik spricht<br></strong><i><strong>Herr Dylan Machado</strong></i>
 <strong><br>(ENSTA Paris)</strong></h3>\n<p class="text-center"><br />übe
 r</p>\n<h2 class="text-center"><br><strong>Spectral analysis of a closed p
 erturbed waveguide filled with a hyperbolic medium</strong></h2>\n<p class
 ="text-center"><br />Abstract: We investigate the spectrum of the first-or
 der operator that governs the propagation<br />of electromagnetic waves in
  the case of a closed locally perturbed waveguide filled with a<br />hyper
 bolic medium. The perturbation is localized on the boundary and we impose<
 br />homogeneous Dirichlet boundary conditions.<br />From a physics perspe
 ctive\, this type of medium falls within the class of hyperbolic<br />elec
 tromagnetic metamaterials. They are artificial media which exhibit a lot o
 f interesting and<br />promising physics phenomena such as negative refrac
 tion\, rainbow-trapping and enhanced<br />spontaneous emission.<br />In pr
 actice\, the simplest mathematical model stems from electromagnetic wave p
 ropagation in<br />a strongly magnetized cold plasma. It is described by 2
 D Maxwell equations which\, in their<br />time-harmonic form\, may be redu
 ced to the Klein-Gordon equation for a certain range of<br />frequencies\,
  namely the hyperbolic frequencies. Hence\, the nature of the problem is p
 urely<br />hyperbolic.<br />Firstly\, we prove a limiting absorption princ
 iple (LAP) for the hyperbolic frequencies satisfying<br />a geometric cond
 ition on the perturbation. The latter is related to the interplay between 
 the<br />velocities of the boundary and of the propagating waves\, which\,
  in our case\, depends on the<br />frequency. This range of frequency is c
 alled 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 bound
 ed<br />time-dependent domains.<br />The second part of the talk will be d
 evoted to the 'supersonic' regime: the frequencies for which<br />the boun
 dary moves faster than the waves. In particular\, the Klein-Gordon equatio
 n in arbitrary<br />time-dependent domains is no longer well-posed for hom
 ogeneous Dirichlet boundary<br />conditions. Nevertheless\, we are still a
 ble to prove a LAP by using the weak observability<br />results of waves a
 nd exchanging the role of the time and space variables.<br />&nbsp\;</p>\n
 <h4 class="text-center"><strong>Der Vortrag findet statt am<br>Donnerstag\
 , den 07.05.2026<br>um 14.30 – 15.30 Uhr im Raum W01 1-117</strong></h4>
 \n<p class="text-center"><br />Interessierte sind herzlich eingeladen.</p>
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