scholarly journals On new surface-localized transmission eigenmodes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Youjun Deng ◽  
Yan Jiang ◽  
Hongyu Liu ◽  
Kai Zhang

<p style='text-indent:20px;'>Consider the transmission eigenvalue problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (\Delta+k^2\mathbf{n}^2) w = 0, \ \ (\Delta+k^2)v = 0\ \ \mbox{in}\ \ \Omega;\quad w = v, \ \ \partial_\nu w = \partial_\nu v\ \ \mbox{on} \ \partial\Omega. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>It is shown in [<xref ref-type="bibr" rid="b16">16</xref>] that there exists a sequence of eigenfunctions <inline-formula><tex-math id="M1">\begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id="M2">\begin{document}$ k_m\rightarrow \infty $\end{document}</tex-math></inline-formula> such that either <inline-formula><tex-math id="M3">\begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> are surface-localized, depending on <inline-formula><tex-math id="M5">\begin{document}$ \mathbf{n}&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M6">\begin{document}$ 0&lt;\mathbf{n}&lt;1 $\end{document}</tex-math></inline-formula>. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions <inline-formula><tex-math id="M7">\begin{document}$ (w_m, v_m)_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> associated with <inline-formula><tex-math id="M8">\begin{document}$ k_m\rightarrow \infty $\end{document}</tex-math></inline-formula> such that both <inline-formula><tex-math id="M9">\begin{document}$ \{w_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \{v_m\}_{m\in\mathbb{N}} $\end{document}</tex-math></inline-formula> are surface-localized, no matter <inline-formula><tex-math id="M11">\begin{document}$ \mathbf{n}&gt;1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ 0&lt;\mathbf{n}&lt;1 $\end{document}</tex-math></inline-formula>. Though our study is confined within the radial geometry, the construction is subtle and technical.</p>

2020 ◽  
Vol 36 (10) ◽  
pp. 105002
Author(s):  
S A Buterin ◽  
A E Choque-Rivero ◽  
M A Kuznetsova

2017 ◽  
Vol 82 (5) ◽  
pp. 1013-1042 ◽  
Author(s):  
Xiaofei Li ◽  
Jingzhi Li ◽  
Hongyu Liu ◽  
Yuliang Wang

Abstract This article is concerned with the invisibility cloaking in electromagnetic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. Our study is based on an interior transmission eigenvalue problem. We propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Maxwell–Herglotz approximation of the associated interior transmission eigenfunctions. We provide the mathematical design of the cloaking device and sharply quantify the cloaking performance.


Sign in / Sign up

Export Citation Format

Share Document