Consider the transmission eigenvalue problem [ (Delta+k^2mathbf{n}^2) w=0, (Delta+k^2)v=0 mbox{in} Omega;quad w=v, partial_ u w=partial_ u v=0 mbox{on} partialOmega. ] It is shown in [12] that there exists a sequence of eigenfunctions $(w_m, v_m)_{minmathbb{N}}$ associated with $k_mrightarrow infty$ such that either ${w_m}_{minmathbb{N}}$ or ${v_m}_{minmathbb{N}}$ 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 $(w_m, v_m)_{minmathbb{N}}$ associated with $k_mrightarrow infty$ such that both ${w_m}_{minmathbb{N}}$ and ${v_m}_{minmathbb{N}}$ 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.
Download