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On new surface-localized transmission eigenmodes

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  • 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\ \ \mbox{on} \ \partial\Omega. $

    It is shown in [16] 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.

    Mathematics Subject Classification: Primary: 35P25, 78A46; Secondary: 35Q60, 78A05.

    Citation:

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  • Figure 1.  Schematic illustration of $ \int_{0}^{1}f(r)dr $ which is bigger than the area of the triangle under the tangent of $ f(1) $

    Figure 2.  Graphical illustration of the classification of transmission eigenvalues

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