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 $\pi$ when the conductive transmission eigenfunctions satisfy certain Herglotz functions approximation properties. That means, as long as the corner singularity is not degenerate, the vanishing property holds if the underlying conductive transmission eigenfunctions can be approximated by a sequence of Herglotz functions under mild approximation rates. Third, the regularity requirements on the interior transmission eigenfunctions in [9] are significantly relaxed in the present study for the conductive transmission eigenfunctions. 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.

中文翻译：

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具有导电边界条件的传输本征函数的几何结构及应用

本文关注的是导电传输本征函数的内在几何结构。在[9]中首次研究了内部传输特征函数的几何特性。在两种情况下表明，内部传输特征函数必须在内角小于 $\pi$ 的域的角落附近局部消失。我们在几个方面显着扩展和概括了这些结果。首先，我们将包含内部传输本征函数的传导传输本征函数视为一种特殊情况。本文中为传导传输本征函数建立的几何结构包括 [9] 中的结果作为特例。第二，当导电传输本征函数满足某些 Herglotz 函数逼近性质时，只要其内角不为 $\pi$，任何角落的导电传输本征函数的消失性质都是成立的。这意味着，只要角奇异点没有退化，如果底层导电传输本征函数可以在温和的近似率下通过一系列 Herglotz 函数来近似，则消失特性成立。第三，在本研究中，对于导电传输本征函数，[9] 中对内部传输本征函数的规律性要求显着放宽。最后，作为获得的几何结果的一个有趣且实际的应用，