Radiation condition for a non-smooth interface between a dielectric and a metamaterial

Anne-Sophie Bonnet-Ben Dhia (1) , Lucas Chesnel (1) , Xavier Claeys (2)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 DMIA - Département de Mathématiques, Informatique, Automatique

We study a 2D scalar harmonic wave transmission problem between a classical dielectric and a medium with a real valued negative permittivity/permeability which models an ideal metamaterial. When the interface between the two media has a corner, according to the value of the contrast (ratio) of the physical constants, this non-coercive problem can be ill-posed (not Fredholm) in $H^1$. This is due to the degeneration of the two dual singularities which then behave like $r^{\pm i\eta}=e^{\pm i\eta\ln\,r}$ with $\eta\in\mathbb{R}^{\ast}$. This apparition of propagative singularities is very similar to the apparition of propagative modes in a waveguide for the classical Helmholtz equation with Dirichlet boundary condition, the contrast playing the role of the wavenumber. In this work, we derive for our problem a functional framework by adding to $H^1$ one of these propagative singularities. Well-posedness is then obtained by imposing a radiation condition, justified by means of a limiting absorption principle, at the corner between the two media.

Keywords: radiation condition; Interface problem; metamaterial; Mellin transform

Domains: Mathematics [math]; Analysis of PDEs [math.AP]

Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Xavier Claeys. Radiation condition for a non-smooth interface between a dielectric and a metamaterial. Mathematical Models and Methods in Applied Sciences, 2013. ⟨hal-00651008⟩