On new surface-localized transmission eigenmodes

Consider the transmission eigenvalue problem $$(\mathrm{\Delta }+k^2{\mathbf{n}}^2)w=0,(\mathrm{\Delta }+k^2)v=0in\mathrm{\Omega };w=v,{\mathrm{\partial }}_{\nu }w={\mathrm{\partial }}_{\nu }v=0on\mathrm{\partial }\mathrm{\Omega }\mathrm{.}$$ It is shown in [12] that there exists a sequence of eigenfunctions $(w_m,v_m{)}_{m\in \mathbb{N}}$ associated with $k_m\to \mathrm{\infty }$ such that either $\{w_m{\}}_{m\in \mathbb{N}}$ or $\{v_m{\}}_{m\in \mathbb{N}}$ are surface-localized, depending on $\mathbf{n}>1$ or $0<\mathbf{n}<1$. In this paper, we discover a new type of surface-localized transmission eigenmodes by constructing a sequence of transmission eigenfunctions $(w_m,v_m{)}_{m\in \mathbb{N}}$ associated with $k_m\to \mathrm{\infty }$ such that both $\{w_m{\}}_{m\in \mathbb{N}}$ and $\{v_m{\}}_{m\in \mathbb{N}}$ are surface-localized, no matter $\mathbf{n}>1$ or $0<\mathbf{n}<1$. Though our study is confined within the radial geometry, the construction is subtle and technical.