Abstract: It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain $\Omega _h$ is localized either at the whole lateral surface $\Gamma _h$ of the domain, or at a point of $\Gamma _h$, while the eigenfunction decays exponentially inside $\Omega _h$. Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.
Keywords: spectral problem, thin domain, boundary layer, trapped mode, localized eigenfunction
Classification (MSC2000): 74B05, 74E10, 35B40
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