On an interior Calder\'{o}n operator and a related Steklov eigenproblem for Maxwell's equations

TL;DR

This paper investigates a Steklov-type spectral problem for Maxwell's equations related to an interior Calderón operator, providing spectral representations and analyzing the properties of associated boundary maps.

Contribution

It introduces a novel Steklov eigenproblem for Maxwell's equations linked to an interior Calderón operator and develops spectral representations for boundary value solutions.

Findings

Neumann-to-Dirichlet map is compact

Provides a Fourier basis of Steklov eigenfunctions

Spectral representations for trace spaces and solutions

Abstract

We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calder\'{o}n operator and an appropriate Dirichlet-to-Neumann type map. The corresponding Neumann-to-Dirichlet map turns out to be compact and this provides a Fourier basis of Steklov eigenfunctions for the associated energy spaces. With an approach similar to that developed by Auchmuty for the Laplace operator, we provide natural spectral representations for the appropriate trace spaces, for the Calder\'{o}n operator itself and for the solutions of the corresponding boundary value problems subject to electric or magnetic boundary conditions on a cavity.