Existence result for Maxwell’s equations in half-waveguides

Abstract

Maxwell’s equations are considered in a half-waveguide Ω+ := R+ × S ⊂ R3 where S ⊂ R2 is a bounded Lipschitz domain in R2. The electric permittivity ε and the magnetic permeability μ are assumed to be strictly positive and periodic outside a compact set. A standard radiation condition accompanies the equations. We give a result on existence and uniqueness in the form of a Fredholm alternative: When there is no bound state, i.e., no non-trivial solution of the homogeneous problem, then there is a unique solution for every right-hand side.