Abstract

This is a common work with Lucas Chesnel and Sonia Fliss. We consider the Maxwell’sequations with perfect electric conductor boundary conditions in three-dimensional unbounded domainswhich are the union of a bounded resonator and one or several semi-infinite waveguides. We areinterested in the existence of electromagnetic trapped modes, that is L2 solutions of the problemwithout source term. These trapped modes are associated to eigenvalues of the Maxwell’s operator,that can be either below the continuous spectrum or embedded in it. First for homogeneous waveguides,we present different families of geometries for which we can prove the existence of eigenvalues. Thenwe show that certain non homogeneous waveguides with local perturbations of the dielectric constantscan support trapped modes. Let us mention that certain mechanisms we propose are very specific toMaxwell’s equations and have no equivalent in the classical proofs of existence of trapped modes forthe scalar Dirichlet or Neumann Laplacians.