Problem of Determining the Anisotropic Conductivity in Electrodynamic Equations

Abstract

Abstract For a system of electrodynamic equations, the inverse problem of determining an anisotropic conductivity is considered. It is supposed that the conductivity is described by a diagonal matrix σ(x) = $${\text{diag}}({{\sigma }_{1}}(x),{{\sigma }_{2}}(x)$$, σ3(x)) with $$\sigma (x) = 0$$ outside of the domain Ω = $$\{ x \in {{\mathbb{R}}^{3}}|\left| x \right| < R\} $$, $$R > 0$$, and the permittivity ε and the permeability μ of the medium are positive constants everywhere in $${{\mathbb{R}}^{3}}$$. Plane waves coming from infinity and impinging on an inhomogeneity localized in Ω are considered. For the determination of the unknown functions $${{\sigma }_{1}}(x)$$, $${{\sigma }_{2}}(x)$$, and $${{\sigma }_{3}}(x)$$, information related to the vector of electric intensity is given on the boundary S of the domain Ω. It is shown that this information reduces the inverse problem to three identical problems of X-ray tomography.