# Seminario: “The spectrum of the Dirichlet-to-Neumann operator for submanifolds of \$\mathbb{R}^n\$ with fixed boundary”

## Martedì 12 Febbraio 2019, ore 12:15 - Aula 1BC50 - Bruno Colbois

ARGOMENTI: Seminars

Martedì 12 Febbraio 2019 alle ore 12:15 in Aula 1BC50, Bruno Colbois (Université de Neuchâtel, CH) terrà un seminario dal titolo “The spectrum of the Dirichlet-to-Neumann operator for submanifolds of \$\mathbb{R}^n\$ with fixed boundary”.

Abstract
We consider the spectrum of the Dirichlet-to-Neumann operator on a family of compact submanifold of \$\mathbb{R}^n\$ with fixed boundary. I will first recall briefly the problem of finding geometric bounds for the spectrum of a Laplace-type operator. For the Dirichlet-to-Neumann operator, in the particular case of revolution submanifolds, we can obtain very precise bounds for all the eigenvalues. In the general case, I will explain how to find upper bounds depending only on the volume of the submanifold and also how to construct examples with arbitrarily small eigenvalues. I will profit of the talk to present a couple of open questions concerning the spectrum of the Dirichlet-to-Neumann operator that are easy to formulate. They are a good illustration of the kind of questions that are of interest in this area.