包圍重構法在非等向性介質下的發展與殘留應力的彈性系統之強唯一連續性

Abstract

這篇論文的目的是在三維中非等向性介質下重構可穿透與不可穿透障礙物。我們將會示範如何利用包圍法重構對於以下兩種數學模型：非等向性的橢圓方程以及非等向性的馬克士威方程。到目前為止，對於非等向性的數學模型，沒有可以利用的複幾何光學解用來重構未知障礙物。因此我們將會使用另一種特別解：震盪遞減解使用在我們的逆問題之中。 特別的，在這篇文章中，我們會介紹一種新的轉換法，把非等向性的馬克士威方程轉變成一個二階線性強橢圓系統。這個方法是用來建構非等向性的馬克士威方程的震盪遞減解。而在此篇文章的最後，我們將會討論強唯一連續性質對於Gevrey係數的殘留應力系統。

The goal of this dissertation is to develop reconstruction schemes to determine penetrable and impenetrable obstacles in a region in 3-dimensional in an anisotropic background. We demonstrate the enclosure-type method for two different mathematical models: The anisotropic elliptic equation and the anisotropic Maxwell system. So far, in the anisotropic case, there are no complex geometrical optics solutions which we can use to reconstruct the unknown obstacles in a given medium. Therefore, we use another special type solution: the oscillating decaying solutions, which are useful in our inverse problems. In particular, for the anisotropic Maxwell system model, we also introduce a new reduction method to transform the Maxwell system into a second order strongly elliptic system. This reduction method is the main tool to construct the oscillating decaying solutions for the anisotropic Maxwell system. In addition, we prove the strong unique continuation for a residual stress system with Gevrey coefficients.