Asymptotic behavior of the interior transmission eigenvalues

Orateur: VODEV Georgi
Localisation: Université de Nantes, France
Type: Séminaire problèmes spectraux en physique mathématique
Site: IHP
Salle: 314
Date de début: 19/10/2015 - 14:00
Date de fin: 19/10/2015 - 14:00

We study the transmission eigenvalues associated with a connected d-dimensional domain $\Omega$. Such an eigenvalue $\lambda$ is defined if the following problem admits a nontrivial solution $(u_1(x), u_2(x))$:
(\nabla c_1\nabla+\lambda n_1)u_1=0& in\ \Omega\\
(\nabla c_2\nabla+\lambda n_2)u_2=0& in\ \Omega\\
u_1=u_2,\ c_1\partial_\nu u_1=c_2\partial_\nu u_2& on\ \partial\Omega
Here the functions $c_i(x)$, $n_i(x)$ are smooth and strictly positive on $\Omega$. We will discuss two questions :
1. The localization of the transmission eigenvalues in the complex plane ;
2. The behaviour of the counting function $N(r)=\#\{|\lambda|>r\}$ when $r$ is large. Among others, we prove that $N(r)\sim \tau r^d$, where $\tau$ is given explicitly in terms of the functions $c_i$, $n_i$.