• DOI:10.3233/ASY-161377
• Corpus ID: 207589622

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}
}

• F. Colombini, V. Petkov, J. Rauch
• 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 $\Lambda_{\epsilon} = \{ z \in {\mathbb C}:\: |\Re z | \leq C_{\epsilon} (|\Im z|^{\frac{1}{2} + \epsilon} + 1), \: \Re z < 0\},$ ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z < 0\}.$