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
• V. M. Isakov
Article

## Abstract

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.

## Preview

### REFERENCES

1. 1.
M. I. Belishev, “Boundary control in reconstruction of manifolds and metrics,” Inv. Prob., 13,No. 5, R1–R45 (1997).Google Scholar
2. 2.
M. I. Belishev, “On relations between spectral and dynamical inverse data,” PDMI Preprint-17/2000.Google Scholar
3. 3.
M. I. Belishev and A. K. Glasman, “Dynamical inverse problem for the Maxwell system: recovering the velocity in the regular zone (the BC-method),” Algebra Analiz, 12,No. 2, 131–182 (2000).Google Scholar
4. 4.
M. I. Belishev, V. M. Isakov, L. N. Pestov, and V. A. Sharafutdinov, “On the reconstruction of the metric from external electromagnetic measurements,” Dokl. RAN, 372,No. 3, 298–300 (2000).Google Scholar
5. 5.
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Kiev (1965).Google Scholar
6. 6.
M. S. Birman amd M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in the Hilbert Space [in Russian], Leningrad (1980).Google Scholar
7. 7.
M. Eller, V. Isakov, G. Nakamura, and D. Tataru, “Uniqueness and stability in the Cauchy problem for Maxwell's and elasticity systems,” in: Nonlinear Partial Differential Equations and Their Applications, D. Cioranescu and J.-L. Lions (eds.), Studies in Mathematics and Applications, 31, Elsevier Science, North-Holland (2002), pp. 329–351.Google Scholar
8. 8.
A. K. Glasman, “On the regularity of solutions of the Maxwell dynamical system,” PDMI Preprint-17/2001.Google Scholar
9. 9.
V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York (1998).Google Scholar
10. 10.
M. G. Krein, “On the problem of extending the Hermitian-positive continuous functions,” Dokl. Akad. Nauk SSSR, 26,No. 1, 17–21 (1940).Google Scholar
11. 11.
Ya. Kurylev and M. Lassas, “Dynamic inverse problem on a Riemannian manifold for a hyperbolic equation with dissipation and dispersion,” Proc. Edind. Math. Soc. (to appear).Google Scholar
12. 12.
I. Lasiecka, J.-L. Lions, and R. Triggiani, “Nonhomogeneous boundary-value problems for second order hyperbolic operators,” J. Math. Pures Appl., 65, 149–192 (1986).Google Scholar
13. 13.
R. Leis, Initial Boundary-Value Problems in Mathematical Physics, Wiley (1986).Google Scholar
14. 14.
P. Ola, L. Paivarinta, and E. Somersalo, “An inverse boundary-value problem in electrodynamics,” Duke Math. J., 70,No. 3, 611–653 (1993).Google Scholar
15. 15.
D. Tataru, “Unique continuation for solutions of PDE's: between Hörmander's and Holmgren theorem,” Comm. PDE, 20, 855–894 (1995).Google Scholar