Bound states in the infinite Fichera layer
Catégorie d'évènement : Séminaire EDP, Analyse et Applications (Metz) Date/heure : 17 novembre 2017 10:45-11:30 Lieu : Oratrice ou orateur : Monique Dauge Résumé :The problem under consideration belongs to the wide family of quantum wave guides. Such guides are unbounded domains endowed with a simple structure at infinity. For instance, in two dimensions, they coincide outside of a compact set with two half-strips of constant width. The Dirichlet Laplace operator in these guides has a non empty essential spectrum. The game is to investigate the presence of discrete spectrum under the threshold of the essential spectrum. In two dimensions the situation is well-known: For any wave guide of constant width and non identically zero curvature, bound states do exist. In three dimensions, recent results provide the existence of infinitely many bound states in conical layers with smooth profiles. In this talk we address the archetypic non smooth conical layer, which we name after Fichera. It can be viewed as an octant from which is removed another octant translated from the first one along the diagonal line of coordinates. We characterize the essential spectrum and prove that the discrete spectrum has at most a finite number of elements. Numerical computations tend to prove that there is exactly one bound state. We mention various generalizations of this result. From a joint work with Yvon Lafranche (Rennes) and Thomas Ourmières-Bonafos (Paris-Sud).
Singularities of the Maxwell equations in polyhedral domains with impedance boundary conditions
Catégorie d'évènement : Séminaire EDP, Analyse et Applications (Metz) Date/heure : 17 novembre 2017 11:30-12:15 Lieu : Oratrice ou orateur : Serge Nicaise Résumé :We will first review some variational formulations of the time-harmonic Maxwell equations with impedance boundary conditions in smooth and non-smooth domains. Secondly, the singularities of this system in polyhedral domains will be described. The talk is based on joint works with M. Costabel (Rennes), M. Dauge (Rennes) and J. Tomezyk (Valenciennes).