Eigenvalues for Maxwell's equations with dissipative boundary conditions

TL;DR

This paper analyzes the spectral properties of Maxwell's equations with dissipative boundary conditions, establishing bounds on eigenvalues in the complex plane for the associated semigroup generator under certain boundary conditions.

Contribution

It provides a detailed eigenvalue localization result for Maxwell's equations with dissipative boundary conditions, extending understanding of spectral behavior in exterior domains.

Findings

Eigenvalues lie within specific regions in the complex plane.

Eigenvalues are confined to regions depending on boundary dissipation parameter.

Results hold for all eigenvalues with sufficiently large imaginary parts.

Abstract

Let $V(t) = e^{tG_b},\: t \geq 0,$ be the semigroup generated by Maxwell's equations in an exterior domain $\Omega \subset {\mathbb R}^3$ with dissipative boundary condition $E_{tan}- \gamma(x) (\nu \wedge B_{tan}) = 0, \gamma(x) > 0, \forall x \in \Gamma = \partial \Omega.$ We prove that if $\gamma(x)$ is nowhere equal to 1, then for every $0 < \epsilon \ll 1$ and every $N \in {\mathbb N}$ the eigenvalues of $G_b$ lie in the region $\Lambda_{\epsilon} \cup {\mathcal R}_N,$ where $\Lambda_{\epsilon} = \{ z \in {\mathbb C}:\: |\Re z | \leq C_{\epsilon} (|\Im z|^{\frac{1}{2} + \epsilon} + 1), \: \Re z < 0\},$ ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$