Eigenvalues for Maxwell's equations with dissipative boundary conditions

Abstract

Let $V(t)=e^{t{G}_b},t\ge 0,$ 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 ${\Lambda }_{\epsilon }=\{z\in \mathbb{C}:|\Re z|\le C_{\epsilon }(|\Im z{|}^{\frac{1}{2}+\epsilon }+1),\Re z<0\},$ ${\mathcal{R}}_N=\{z\in \mathbb{C}:|\Im z|\le C_N(|\Re z|+1{)}^{-N},\Re z<0\}.$

arXiv commentto appear in Asymptotic Analysis