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MATHEMATICA BOHEMICA, Vol. 127, No. 2, pp. 283-292 (2002)
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# Localization effects for eigenfunctions near to the edge of a thin domain

## Serguei A. Nazarov

* Serguei A. Nazarov*, Center for Techno-Mathematics $&$ Scientific Computing Laboratory, Harrow School of Computer Science, Westminster University, London, UK, Watford Road, Northwick Park, Harrow Campus, Harrow HA1 3TP; e-mail: ` serna@snark.ipme.ru`

**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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