A few results on permittivity variations in electromagnetic cavities

TL;DR

This paper investigates how eigenvalues of Maxwell's equations in electromagnetic cavities change with permittivity variations, proving Lipschitz continuity and real analyticity, and showing generic simplicity of eigenvalues.

Contribution

It establishes the Lipschitz continuity and real analyticity of Maxwell eigenvalues with respect to permittivity changes, providing explicit derivatives and generic simplicity results.

Findings

Eigenvalues are Lipschitz continuous in permittivity.

Simple eigenvalues depend analytically on permittivity.

For generic permittivity, all eigenvalues are simple.

Abstract

We study the eigenvalues of time-harmonic Maxwell's equations in a cavity upon changes in the electric permittivity $\varepsilon$ of the medium. We prove that all the eigenvalues, both simple and multiple, are locally Lipschitz continuous with respect to $\varepsilon$. Next, we show that simple eigenvalues and the symmetric functions of multiple eigenvalues depend real analytically upon $\varepsilon$ and we provide an explicit formula for their derivative in $\varepsilon$. As an application of these results, we show that for a generic permittivity all the Maxwell eigenvalues are simple.