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Journal Articles Mathematical Models and Methods in Applied Sciences Year : 2013

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

Anne-Sophie Bonnet-Ben Dhia
Lucas Chesnel
• Function : Author
• PersonId : 891776
Xavier Claeys

#### Abstract

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.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

hal-00651008 , version 1 (12-12-2011)

### Identifiers

• HAL Id : hal-00651008 , version 1

### Cite

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⟩

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