关于新的表面局域传输本征模
On new surface-localized transmission eigenmodes
摘要：
考虑传输本征值问题\[ (Δ+k^2\mathbf{n}^2) w=0,\ \ (Δ+k^2)v=0\ \ \mbox{in}\ \ Ω;\quad w=v,\ \ \partial_νw=\partial_νv=0\ \ \mbox{on} \ \partialΩ. \]在[12]中示出存在与$k_m\rightarrow \infty$相关联的本征函数序列$(w_m, v_m)_{m\in\mathbb{N}}$，使得$\{w_m\}_{m\in\mathbb{N}}$或$\{v_m\}_{m\in\mathbb{N}}$是表面局部化的，这取决于$\mathbf{n}>1$或$0<\mathbf{n}<1$。本文通过构造一系列与$k_m\rightarrow \infty$相关的传输本征函数$(w_m, v_m)_{m\in\mathbb{N}}$，发现了一种新的表面局域传输本征模，使得无论是$\mathbf{n}>1$还是$0<\mathbf{n}<1$，$\{w_m\}_{m\in\mathbb{N}}$和$\{v_m\}_{m\in\mathbb{N}}$都是表面局域的.虽然我们的研究仅限于放射状的几何形状，但其结构是微妙的和技术性的。
英文摘要：
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 $k_m\rightarrow \infty$ 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.
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