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.

中文翻译：

---

确定电动方程中各向异性电导率的问题

摘要

对于一个电动方程组，要考虑确定各向异性电导率的反问题。假设电导率由对角矩阵σ（x）= \（{\ text {diag}}（{{\ sigma} _ {1}}（x），{{\ sigma} _ {2} }（X）\），σ ₃（X）），与\（\西格马（X）= 0 \）域Ω之外= \（\ {X \在{{\ mathbb {R}} ^ {3} } | \ left | x \ right | <R \} \），\（R> 0 \）和介质的介电常数ε和磁导率μ在\（{{\ mathbb {R}} ^ {3}} \）。考虑了来自无穷大并撞击在Ω中的不均匀性的平面波。为了确定未知函数\（{{\ sigma __ {1}}（x）\），\（{{\ sigma} _ {2}}（x）\）和\（{{\ sigma } _ {3}}（x）\），在域Ω的边界S上给出了与电强度矢量有关的信息。结果表明，该信息将反问题简化为X射线断层扫描的三个相同问题。