Journal of Mathematical Sciences

, Volume 122, Issue 5, pp 3459–3469

On the Uniqueness of the Recovery of Parameters of the Maxwell System from Dynamical Boundary Data


  • M. I. Belishev
    • St.Petersburg Department of the Steklov Mathematical Institute
  • V. M. Isakov
    • Department of Mathematics and StatisticsWichita State University

DOI: 10.1023/B:JOTH.0000034024.38243.02

Cite this article as:
Belishev, M.I. & Isakov, V.M. Journal of Mathematical Sciences (2004) 122: 3459. doi:10.1023/B:JOTH.0000034024.38243.02


The paper deals with the problem of recovering the parameters (functions) \(\varepsilon\) and \(\mu\) of the Maxwell dynamical system
$$\varepsilon E_t = {\text{rot}}\,H,\quad \mu H_t = - {\text{rot}}\,E\quad in\quad \Omega \times \left( {0,T} \right);$$
$$E\left| {_{t = 0} = 0,\quad H} \right|_{t = 0} = 0\quad in\quad \Omega ;$$
$$E_{\tan } = f\quad on\quad \partial \Omega \times \left[ {0,T} \right]$$
(tan is the tangent component; \(E = E^f \left( {x,t} \right),\;H = H^f \left( {x,t} \right)\) is a solution) by the response operator \(R^T :f \to \nu \times H^f \left| {_{\partial \Omega \times \left[ {0,T} \right]} } \right.\) (\(\nu\) is the normal). The parameters determine the velocity \(c = \left( {\varepsilon \mu } \right)^{ - \frac{1}{2}}\), the c-metric \(ds^2 = c^{ - 2} \left| {dx} \right|^2\), and the time \(T_* = \mathop {\max }\limits_\Omega \;{\text{dist}}_c \left( { \cdot ,\partial \Omega } \right)\). It is shown that for any fixed \(T > T\), the operator \(R^{2T}\) determines \(\varepsilon\) and \(\mu\) in \(\Omega\) uniquely. Bibliography: 15 titles.

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© Plenum Publishing Corporation 2004