[论文解读] On the geometric structures of conductive transmission eigenfunctions and its application

Huaian Diao, Xinlin Cao+1 more|arXiv (Cornell University)|Nov 5, 2018

Numerical methods in inverse problems被引用 1

一句话总结

本文確立了導電傳輸特徵函數的幾何結構，證明其在內角不等於 π 的角附近局部消失，顯著推廣了先前關於內部傳輸特徵函數的研究結果。本文提出新型分析技術，並將成果應用於從單一遠場測量中唯一恢復多邊形導電障礙物及其表面導電參數。

ABSTRACT

This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $\pi$. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures established for the conductive transmission eigenfunctions in this paper include the results in [9] as a special case. Second, the vanishing property of the conductive transmission eigenfunctions is established for any corner as long as its interior angle is not $\pi$. That means, as long as the corner singularity is not degenerate, the vanishing property holds. Third, the regularity requirements on the interior transmission eigenfunctions in [9] are significantly relaxed in the present study for the conductive transmission eigenfunctions. In order to establish the geometric properties for the conductive transmission eigenfunctions, we develop technically new methods and the corresponding analysis is much more complicated than that in [9]. Finally, as an interesting and practical application of the obtained geometric results, we establish a unique recovery result for the inverse problem associated with the transverse electromagnetic scattering by a single far-field measurement in simultaneously determining a polygonal conductive obstacle and its surface conductive parameter.

研究动机与目标

• 研究導電傳輸特徵函數在區域角落附近的幾何行為。
• 將先前關於內部傳輸特徵函數的研究結果推廣至更廣泛的導電傳輸特徵函數類別。
• 降低早期研究中對特徵函數分析所要求的正則性假設。
• 建立基於單一遠場測量的逆散射問題唯一恢復結果。

提出的方法

• 發展針對導電傳輸特徵函數複雜性量身定制的新穎分析技術。
• 將特徵函數消失行為的分析擴展至內角 ≠ π 的角落。
• 應用漸近分析與奇異性分析方法，研究特徵函數在角落奇異點附近的行為。
• 利用推導出的幾何性質構建逆散射問題的唯一性證明。
• 將邊界條件與傳輸條件整合至導電介質的特徵函數框架中。
• 利用消失性質約束逆問題公式中的解空間。

实验结果

研究问题

• RQ1 在何種幾何條件下，導電傳輸特徵函數在區域角落附近消失？
• RQ2 角落的內角如何影響導電傳輸特徵函數的消失行為？
• RQ3 是否可降低特徵函數的正則性要求，同時保持幾何結構結果？
• RQ4 單一遠場測量能否唯一確定多邊形障礙物的形狀及其表面導電參數？
• RQ5 需要哪些新型分析工具，才能將結果從內部傳輸特徵函數推廣至導電傳輸特徵函數？

主要发现

• 無論具體導電參數為何，導電傳輸特徵函數在內角不等於 π 的任何角落附近均局部消失。
• 研究結果通過將先前結果作為特例包含在內，實現了對內部傳輸特徵函數研究的顯著推廣。
• 與先前工作相比，特徵函數的正則性假設被顯著放寬。
• 幾何消失性質適用於所有非退化角落奇異點，不僅限於特定角度。
• 已建立基於單一遠場測量唯一恢復多邊形導電障礙物及其表面導電參數的逆問題唯一恢復結果。
• 分析結果顯示，導電傳輸特徵函數框架所支持的幾何約束強度，超過以往所知。