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A uniqueness theorem for a transmission problem in inverse electromagnetic scattering. (English) Zbl 0805.35154

The author studies an inverse problem for the inhomogeneous Maxwell equations. It is assumed that there exists a bounded set $D\subset {ℝ}^{3}$ such that outside of $D$ the electric permittivity $\epsilon$ and the magnetic permeability $\mu$ are constant, and that these quantities and the electric conductivity $\sigma$ change discontinuously across $\partial D$ and are inhomogeneous in $D$. It is moreover assumed that $\sigma$ vanishes outside $D$, but is not identically zero inside $D$.

The author proves in a first step that if for two given inhomogeneous media the far field patterns coincide for all incoming plane waves, then the jump discontinuities $\partial {D}_{1}$ and $\partial {D}_{2}$ coincide. Starting from this result, he proves in a second step that under weak conditions also the quantities $\mu ,\epsilon$ and $\sigma$ are uniquely defined by the far field pattern of the solution.

The proof of this second result is based on ideas from the fundamental paper of J. Sylvester and G. Uhlmann [Ann. Math., II. Ser. 125, 153-169 (1987; Zbl 0625.35078)].

##### MSC:
 35R30 Inverse problems for PDE 35P25 Scattering theory (PDE) 35Q60 PDEs in connection with optics and electromagnetic theory 78A45 Diffraction, scattering (optics)