# Eigenvalues for Maxwell's equations with dissipative boundary conditions

@article{Colombini2015EigenvaluesFM,
title={Eigenvalues for Maxwell's equations with dissipative boundary conditions},
author={Ferruccio Colombini and Vesselin Petkov and Jeffrey Rauch},
journal={Asymptot. Anal.},
year={2015},
volume={99},
pages={105-124}
}
• Published 8 June 2015
• Mathematics
• Asymptot. Anal.
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_… 6 Citations ## Figures from this paper • Mathematics • 2017 Let $$V(t) = e^{tG_b},\, t \ge 0,$$V(t)=etGb,t≥0, be the semigroup generated by Maxwell’s equations in an exterior domain $$\Omega \subset {\mathbb R}^3$$Ω⊂R3 with dissipative boundary condition • Materials Science Archiv der Mathematik • 2017 Let V(t)=etGb,t≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} • V. Petkov • Mathematics Research in the Mathematical Sciences • 2021 We study the wave equation in the exterior of a bounded domain K with dissipative boundary condition ∂νu-γ(x)∂tu=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} Let Ω = R 3 ∖ K ¯, where K is an open bounded domain with smooth boundary Γ. Let V ( t ) = e t G b , t ⩾ 0, be the semigroup related to Maxwell’s equations in Ω with dissipative boundary condition ν We present a survey of some recent results concerning the location and the Weyl formula for the complex eigenvalues of two non self-adjoint operators. We study the eigenvalues of the generator G of • G. Vodev • Mathematics Research in the Mathematical Sciences • 2021 We introduce an analog of the Dirichlet-to-Neumann map for the Maxwell equation in a bounded domain. We show that it can be approximated by a pseudodifferential operator on the boundary with a We examine the location of the eigenvalues of the generator$G$of a semi-group$V(t) = e^{tG},\: t \geq 0,$related to the wave equation in an unbounded domain$\Omega \subset {\mathbb R}^d$with • Mathematics • 2010 We prove that in contrast to the free wave equation in$\R^3\$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions
• Mathematics
• 2014
Introduction.- Expansion into Wave Functions.- Scattering From a Perfect Conductor.- The Variational Approach to the Cavity Problem.- Boundary Integral Equation Methods for Lipschitz Domains.-
We prove the existence of large regions free of eigenvalues of the interior transmission problem.
A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.