Eigenvalues for Maxwell’s equations with dissipative boundary conditions

Abstract

Let $\mathit{V}\mathrm{(}\mathit{t}\mathrm{)}={\mathit{e}}^{\mathit{t}{\mathit{G}}_{\mathit{b}}}$ , $\mathit{t}⩾0$ , be the semigroup generated by Maxwell’s equations in an exterior domain $\mathrm{\Omega }\subset {\mathbb{R}}^3$ with dissipative boundary condition ${\mathit{E}}_{\tan }-\mathit{\gamma }\mathrm{(}\mathit{x}\mathrm{)}\mathrm{(}\mathit{\nu }\wedge {\mathit{B}}_{\tan }\mathrm{)}=0$ , $\mathit{\gamma }\mathrm{(}\mathit{x}\mathrm{)}\mathrm{>}0$ , $\forall \mathit{x}\in \mathrm{\Gamma }=\mathrm{\partial }\mathrm{\Omega }$ . We prove that if $\mathit{\gamma }\mathrm{(}\mathit{x}\mathrm{)}$ is nowhere equal to 1, then for every $0\mathrm{<}\mathit{ϵ}\ll 1$ and every $\mathit{N}\in \mathbb{N}$ the eigenvalues of ${\mathit{G}}_{\mathit{b}}$ lie in the region ${\mathrm{\Lambda }}_{\mathit{ϵ}}\cup {\mathcal{R}}_{\mathit{N}}$ , where ${\mathrm{\Lambda }}_{\mathit{ϵ}}=\{\mathit{z}\in \mathbb{C}:|Re\mathit{z}|⩽{\mathit{C}}_{\mathit{ϵ}}\mathrm{(}|Im\mathit{z}{|}^{\frac{1}{2}+\mathit{ϵ}}+1\mathrm{)}\mathrm{,}Re\mathit{z}\mathrm{<}0\}$ , ${\mathcal{R}}_{\mathit{N}}=\{\mathit{z}\in \mathbb{C}:|Im\mathit{z}|⩽{\mathit{C}}_{\mathit{N}}{\mathrm{(}|Re\mathit{z}|+1\mathrm{)}}^{-\mathit{N}}\mathrm{,}Re\mathit{z}\mathrm{<}0\}$ .