Maxwell's equations with a polarization independent wave velocity: direct and inverse problems
Maxwell's equations with a polarization independent wave velocity: direct and inverse problems
复制标题
具有与偏振无关的波速的麦克斯韦方程组：正问题和反问题
DOI：
10.1016/j.matpur.2006.01.008
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
E. Somersalo
中科院分区：
文献类型：
--
作者：
Y. Kurylev;M. Lassas;E. Somersalo
本文研究了一种特殊类型的各向异性介质的时域麦克斯韦方程组，其特征是波的传播速度与偏振无关。特别地，所有各向同性介质都满足该性质。分析的基础上不变的制定系统的电动力学作为一个狄拉克型一阶系统的黎曼3流形。本文的第一部分研究了该系统的性质。第二部分是从动态边界数据中识别黎曼流形M和相应的方程组的反问题。这些数据是边界图和导纳图ZT。物理上，该图对应于在有限时间间隔[0，T]处的边界上的电场和磁场的切向分量的测量。结果表明，当T>0时，ZT决定黎曼流形和电磁参数.当边界数据仅在边界的开放部分给出时，证明了类似的结果。在域R3中，我们描述了保持导纳图ZT的变换组，提供了基本物理问题的非唯一性的完整表征。在各向同性情况下，M ∈ R3，我们证明了在边界的开部上给定的边界数据唯一地确定了区域M、介电常数ε和磁导率μ。
We study Maxwell's equations in time domain for an anisotropic medium of a special type, characterized by the polarization independent velocity of the wave propagation. In particular, this property is satisfied by all isotropic media. The analysis is based on an invariant formulation of the system of electrodynamics as a Dirac type first order system on a Riemannian 3-manifold. We study the properties of this system in the first part of the paper. The second part is devoted to the inverse problem of the identification of the Riemannian manifold M and the corresponding system of equations from the dynamic boundary data. These data are the boundary ∂M and the admittance map ZT. Physically, this map corresponds to the measurements of the tangential components of the electric and magnetic fields on the boundary at a finite time interval [0,T]. It is shown that, for sufficiently large T>0, ZTdetermines the Riemannian manifold and the underlying electromagnetic parameters. Similar results are proven in the case when the boundary data are given only on an open part of the boundary. In domains of R3, we describe the group of transformations which preserve the admittance map ZT, providing a complete characterization of the non-uniqueness of the underlying physical problem. In the isotropic case with M⊂R3, we prove that the boundary data given on an open part of the boundary determine the domain M, the permittivity ε and the permeability μ uniquely.