On uniqueness of recovering the parameters of the Maxwell system via dynamical boundary data

M. I. Belishev ; V. M. Isakov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 31, Tome 285 (2002), pp. 15-32

Résumé

The paper deals with the problem of determination of the parameters (functions) $\varepsilon$, $\mu$ of the Maxwell dynamical system \begin{align*} \varepsilon E_t=\operatorname{rot}H, \quad \mu H_t=-\operatorname{rot}E \quad\text{в}\quad \Omega\times(0,T); \\ |_{t=0}=0, \quad H|_{t=0}=0 \quad\text{в}\quad \Omega; \\ {\tan}=f \quad\text{на}\quad \partial\Omega\times[0,T] \end{align*} (tan is the tangent component; $E=E^f(x,t)$, $H=H^f(x,t)$ is the solution) through the response operator $R^T\colon f\to\nu\times H^f|_{\partial\Omega\times[0,T]}$ ($\nu$ is normal). The parameters determine the velocity $c=(\varepsilon\mu)^{-\frac12}$, the $c$-metric $ds^2=c^{-2}|dx|^2$, and the time $T_*=\max\limits_\Omega\operatorname{dist}_c(\cdot,\partial\Omega)$. We show that, for any fixed $T>T_*$, the operator $R^{2T}$ determines $\varepsilon,\mu$ in $\Omega$ uniquely.