$V(t) = e^{tG_b},\: t \geq 0,$ denotes the semigroup generated by Maxwell's equations in an exterior domain $\Omega \subset \R^3$ with dissipative boundary condition $E_{tan}- \gamma(x) (\nu \wedge B_{tan}) = 0, \gamma(x) > 0, \forall x \in \Gamma = \pa \Omega.$ We prove that if $\gamma(x)$ is nowhere equal to 1, then for every $0 < \ep \ll 1$ and every $N \in \N$ the eigenvalues of $G_b$ lie in the region $\Lambda_{\ep} \cup {\mathcal R}_N,$ where $\Lambda_{\epsilon} = \{ z \in \C:\: |\re z | \leq C_{\epsilon} (|\im z|^{\frac{1}{2} + \epsilon} + 1), \: \re z < 0\},$ ${\mathcal R}_N = \{z \in \C:\: |\im z| \leq C_N (|\re z| + 1)^{-N},\: \re z < 0\}.$
Eigenvalues for Maxwell's equations with dissipative boundary conditions.
COLOMBINI, FERRUCCIO;
2016-01-01
Abstract
$V(t) = e^{tG_b},\: t \geq 0,$ denotes the semigroup generated by Maxwell's equations in an exterior domain $\Omega \subset \R^3$ with dissipative boundary condition $E_{tan}- \gamma(x) (\nu \wedge B_{tan}) = 0, \gamma(x) > 0, \forall x \in \Gamma = \pa \Omega.$ We prove that if $\gamma(x)$ is nowhere equal to 1, then for every $0 < \ep \ll 1$ and every $N \in \N$ the eigenvalues of $G_b$ lie in the region $\Lambda_{\ep} \cup {\mathcal R}_N,$ where $\Lambda_{\epsilon} = \{ z \in \C:\: |\re z | \leq C_{\epsilon} (|\im z|^{\frac{1}{2} + \epsilon} + 1), \: \re z < 0\},$ ${\mathcal R}_N = \{z \in \C:\: |\im z| \leq C_N (|\re z| + 1)^{-N},\: \re z < 0\}.$I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
