On new surface-localized transmission eigenmodes

Abstract

<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}>1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M6">\begin{document}$ 0<\mathbf{n}<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}>1 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M12">\begin{document}$ 0<\mathbf{n}<1 $\end{document}</tex-math></inline-formula>. Though our study is confined within the radial geometry, the construction is subtle and technical.</p>