On an interior Calderón operator and a related Steklov eigenproblem for Maxwell's equations

Abstract

We discuss a Steklov-type problem for Maxwell's equations which is related to an interior Calderón 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 Calderón operator itself and for the solutions of the corresponding boundary value problems subject to electric or magnetic boundary conditions on a cavity.

Submitted for publication to Siam Journal on Mathematical Analysis on 21 March 2019, revised on 12 May 2020, accepted for publication on 16 July 2020