Date/heure
22 février 2024
14:15 - 15:15
Oratrice ou orateur
Elmar Schrohe (Hanovre)
Catégorie d'évènement Séminaire Théorie de Lie, Géométrie et Analyse
Résumé
A plasmon of a bounded domain $\Omega\subseteq\mathbb R^n$ is a nontrivial bounded function on $\mathbb R^n\setminus \partial \Omega$ which is continuous at $\partial \Omega$ and whose interior and exterior normal derivative at $\partial \Omega$ have a constant ratio.
This ratio is called a plasmonic eigenvalue of $\Omega$.
Our longterm term goal is to understand this problem on a manifold with fibered cusp singularities. A prototypical example would be the complement of two touching strictly convex domains in $\mathbb R^n$.
The problem requires a precise analysis of the Dirichlet-to-Neumann operator in this setting. In a first step, we consider the Calderón projector for general elliptic differential operators of arbitrary order associated with this type of singularity, so-called $\phi$-differential operators. We show that the Calderón projector is a $\phi$-pseudodifferential operator in the sense of Mazzeo and Melrose. Next we study the Dirichlet-to-Neumann operator for Laplacians associated with fibered cusp metrics and obtain that it also is a $\phi$-pseudodifferential operator of order one.
This is a report on ongoing work with Karsten Fritzsch and Daniel Grieser.