On new surface-localized transmission eigenmodes

Consider the transmission eigenvalue problem $$(\Delta+k^2\mathbf{n}^2) w=0,\ \ (\Delta+k^2)v=0\ \ \mbox{in}\ \ \Omega;\quad w=v,\ \ \partial_\nu w=\partial_\nu v=0\ \ \mbox{on} \ \partial\Omega.$$ It is shown in [12] that there exists a sequence of eigenfunctions $(w_m, v_m)_{m\in\mathbb{N}}$ associated with $k_m\rightarrow \infty$ 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 $k_m\rightarrow \infty$ 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.