ABSTRACT

For electrodynamic equations related to non-conducting dispersible medium we consider the inverse problem of recovering two variable coefficients from a given phaseless information of solutions to the equations. One of these coefficients is the permittivity while the second one characterizes the time dispersion of the medium. We suppose that unknown coefficients differ from given constants inside of a compact domain Ω. A plane electromagnetic wave going in the direction ν from infinity fall down on this domain and modulus of the electric strength is measured on a part of the boundary of Ω for all $\nu \in {\mathbb{S}}^2$. The inverse problem consists in determining unknown functions from this information. We reduce the inverse problem to two problems: (1) the inverse kinematic problem for recovering the refractive index and (2) the integral geometry problem for recovering the second coefficient related to the dispersion. An uniqueness theorem for the first problem is stated on the base of known results. The second problem differs from have studied by the more general weight function and it is still open. Then we demonstrate that under some natural assumption the weight function uniformly close to 1. Replacing the weight function by 1, we obtain the integral geometry problem for which the uniqueness theorem and stability estimate are established and some numerical algorithms are proposed.

摘要

对于与非导电可分散介质相关的电动力学方程，我们考虑从方程解的给定无相信息中恢复两个可变系数的逆问题。这些系数之一是介电常数，而第二个系数表征介质的时间色散。我们假设未知系数不同于紧域 Ω 内的给定常数。从无穷远处沿ν方向行进的平面电磁波落在该域上，并且在所有 Ω 边界的一部分上测量电气强度的模量$\nu \in {\mathbb{小号}}^2$. 逆问题在于从该信息中确定未知函数。我们将逆问题简化为两个问题：（1）用于恢复折射率的逆运动学问题和（2）用于恢复与色散相关的第二个系数的积分几何问题。第一个问题的唯一性定理是在已知结果的基础上陈述的。第二个问题不同于已经研究过的更一般的权重函数，它仍然是开放的。然后我们证明了在一些自然假设下，权函数一致地接近1。将权函数替换为1，我们得到了积分几何问题，建立了唯一性定理和稳定性估计，并提出了一些数值算法。