Boundary localization of transmission eigenfunctions in spherically stratified media

Abstract

Consider the transmission eigenvalue problem for $\mathit{u}\in {\mathit{H}}^1\mathrm{(}\mathrm{\Omega }\mathrm{)}$ and $\mathit{v}\in {\mathit{H}}^1\mathrm{(}\mathrm{\Omega }\mathrm{)}$ : $$\begin{array}{c}\left\{\begin{array}{cc}\nabla ·\mathrm{(}\mathit{\sigma }\nabla \mathit{u}\mathrm{)}+{\mathit{k}}^2{\mathbf{n}}^2\mathit{u}=0\hfill & \text{in }\mathrm{\Omega }\mathrm{,}\hfill \\ \mathrm{\Delta }\mathit{v}+{\mathit{k}}^2\mathit{v}=0\hfill & \text{in }\mathrm{\Omega }\mathrm{,}\hfill \\ \mathit{u}=\mathit{v}\mathrm{,}\mathit{\sigma }\frac{\partial \mathit{u}}{\partial \mathit{\nu }}=\frac{\partial \mathit{v}}{\partial \mathit{\nu }}\hfill & \text{on }\partial \mathrm{\Omega }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where Ω is a ball in ${\mathbb{R}}^{\mathit{N}}$ , $\mathit{N}=2\mathrm{,}3$ . If σ and n are both radially symmetric, namely they are functions of the radial parameter r only, we show that there exists a sequence of transmission eigenfunctions ${\{{\mathit{u}}_{\mathit{m}}\mathrm{,}{\mathit{v}}_{\mathit{m}}\}}_{\mathit{m}\in \mathbb{N}}$ associated with ${\mathit{k}}_{\mathit{m}}\to +\infty$ as $\mathit{m}\to +\infty$ such that the ${\mathit{L}}^2$ -energies of ${\mathit{v}}_{\mathit{m}}$ ’s are concentrated around $\partial \mathrm{\Omega }$ . If σ and n are both constant, we show the existence of transmission eigenfunctions ${\{{\mathit{u}}_{\mathit{j}}\mathrm{,}{\mathit{v}}_{\mathit{j}}\}}_{\mathit{j}\in \mathbb{N}}$ such that both ${\mathit{u}}_{\mathit{j}}$ and ${\mathit{v}}_{\mathit{j}}$ are localized around $\partial \mathrm{\Omega }$ . Our results extend the recent studies in (SIAM J. Imaging Sci. 14 (2021), 946–975; Chow et al.). Through numerics, we also discuss the effects of the medium parameters, namely σ and n, on the geometric patterns of the transmission eigenfunctions.