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研究生:林奕亘
研究生(外文):Yi-Hsuan Lin
論文名稱: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
論文名稱(外文):The development of the Enclosure Method in an Anisotropic Background and the Strong Unique Continuation for the Elasticity with Residual Stress
指導教授:王振男
指導教授(外文):Jenn-Nan Wang
口試委員:夏俊雄陳俊全林景隆林太家
口試日期:2015-12-23
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:英文
論文頁數:124
中文關鍵詞:包圍重構法非等向性殘留應力強唯一連續性
外文關鍵詞:Enclosure methodanisotorpicresidual stressstrong unique continuation property
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  • 被引用被引用:0
  • 點閱點閱:269
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  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
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.


目 錄
口試委員審定書………………………………………………………i
誌謝……………………………………………………………………ii
中文摘要………………………………………………………………iii
英文摘要………………………………………………………………iv
1 Preliminaries 1
2 The enclosure method for the second order elliptic equations 4
2.1 Calderón''s problem 4
2.2 Ideas of the enclosure method 6
2.3 Complex geometric optics solutions and related topics 8
2.4 The enclosure-type method: Second order anisotropic elliptic equations 12
3 The enclosure method for the Maxwell system . 38
3.1 Basic properties for the Maxwell system 38
3.2 Enclosing unknown obstacles in the isotropic media 41
3.3 Constructing CGO solutions 42
3.4 Proof of Theorem 3.3 45
3.5 Enclosing unknown obstacles in the anisotropic media 51
3.6 A new reduction method: From anisotropic Maxwell system to the second order strongly elliptic system 55
3.7 Constructing of oscillating-decaying solutions for the anisotropic Maxwell system 67
3.8 Proof of Theorem 3.13 76
4 Strong unique continuation for a residual stress system with Gevrey Coefficients 100
4.1 SUCP for the elliptic equation 100
4.2 Basic properties for the Gevrey class 102
4.3 SUCP for the residual stress system with Gevrey coefficients 103
4.4 Reduction to a fourth order elliptic system 106
4.5 The asymptotic behavior of u near 0 112
4.6 Proof of the main theorem 113
5 Future work 117
5.1 Fundamental solutions for the anisotropic Maxwell system 117
5.2 More Lp estimates for the anisotropic Maxwell system 118
5.3 Strong unique continuation for the general second order elliptic system 118
Bibliography 119

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