On new surface-localized transmission eigenmodes

Abstract

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

References in corpus (1)

• Addendum to: "On vanishing near corners of transmission eigenfunctions"

Cited by in corpus (1)

• Geometrical and topological properties of transmission resonance and artificial mirage