An inverse boundary value problem in electrodynamics (Q1320607)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ 3\) with a smooth boundary \(\Gamma\) and a connected complement. Let the electric permittivity, conductivity and magnetic permeability be denoted by \(\varepsilon\), \(\sigma\) and \(\mu\) respectively. Let \(E\), \(H\) denote respectively the electric and magnetic fields inside \(\Omega\). Suppose the map \(\Lambda\) that assigns the tangential component of \(E_ \Gamma\) to that of \(H_ \Gamma\) is defined. Assume that outside \(\Omega\), \(\varepsilon\), \(\mu\) are known constants and \(\sigma\) vanishes. Then, it is proved that, under appropriate smoothness and boundedness conditions on \(\varepsilon\), \(\sigma\) and \(\mu\), the knowledge of the map \(\Lambda\) uniquely determines these quantities.