Seminar

Analysis & Applications Seminar

The spectrum of the plasmonic problem for polyhedra

Speaker:  Marta De León-Contreras (Universidad de La Laguna)
Date:  Friday, 10 March 2023 - 11:00
Place:  Aula Naranja, ICMAT

Abstract:

In this talk we will see how to describe the spectrum of the plasmonic problem for polyhedra \(\Omega\) in \(\mathbb{R}^3\),

            where the plasmonic problem consists of seeking a potential \(U\),

            \(U(x) = o(1), \qquad  |x| \to \infty,\)

            such that

            \(\Delta U (x) = 0, \qquad x \in \mathbb{R}^3 \setminus \partial \Omega,\)

            and satisfies

            \(\text{Tr}_+ U = \text{Tr}_- U, \qquad \epsilon \left( \frac{\partial}{\partial n} U\right)_+ - \left( \frac{\partial}{\partial n} U\right)_- = g \quad \text{ on } \partial \Omega. \)

            Here, \(\text{Tr}_\pm U\) and \(\left( \frac{\partial}{\partial n} U\right)_\pm\)  denote the interior/exterior traces and limiting outward normal derivatives of \(U\) on \(\partial \Omega\).

            For that aim, we will interpret the plasmonic problem as a spectral problem through an integral operator in the boundary, the direct value of the layer potential, also known as the Neumann–Poincaré operator. Therefore, we will study the spectrum of the double layer potential in infinite polyhedral cones and after that, by using localization techniques, we will get the result for bounded polyhedra.

                        The contents of this talk is based on a joint work with K. M. Perfekt.