The Plasmonic Eigenvalue Problem, the Calderón Projector and the Dirichlet-to-Neumann Operator on Manifolds with Fibered Cusp Singularities

Date/heure
22 février 2024
14:15 - 15:15

Lieu
Salle de séminaires Metz

Oratrice ou orateur
Elmar Schrohe (Hanovre)

Catégorie d'évènement
Séminaire Théorie de Lie, Géométrie et Analyse

Résumé

A plasmon of a bounded domain $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ is a nontrivial bounded function on ${\mathbb{R}}^n\setminus \partial \mathrm{\Omega }$ which is continuous at $\partial \mathrm{\Omega }$ and whose interior and exterior normal derivative at $\partial \mathrm{\Omega }$ have a constant ratio.
This ratio is called a plasmonic eigenvalue of $\mathrm{\Omega }$.

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 $\varphi$-differential operators. We show that the Calderón projector is a $\varphi$-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 $\varphi$-pseudodifferential operator of order one.

This is a report on ongoing work with Karsten Fritzsch and Daniel Grieser.