Localization effect for Dirichlet eigenfunctions in thin non-smooth domains

Nazarov, S.; Pérez, E.; Taskinen, J.

doi:10.1090/tran/6625

Localization effect for Dirichlet eigenfunctions in thin non-smooth domains
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by S. A. Nazarov, E. Pérez and J. Taskinen PDF

Trans. Amer. Math. Soc. 368 (2016), 4787-4829 Request permission

Abstract:

We study the localization effect for the eigenfunctions of the Laplace-Dirichlet problem in a thin three-dimensional plate with curved non-smooth bases. We show that the eigenfunctions are localized at the thickest region, or the longest traverse axis, of the plate and that the magnitude of the eigenfunctions decays exponentially as a function of the distance to this axis. We consider some extensions like mixed boundary value problems in thin domains. The obtained asymptotic formulas for eigenfunctions prove the existence of gaps in the essential spectrum of the Dirichlet Laplacian in an unbounded double-periodic curved piecewise smooth thin layer.

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