Localization effects for eigenfunctions near to the edge of a thin domain. (English) Zbl 1022.74003

Nazarov, Serguei A.

Math. Bohem. 127, No. 2, 283-292 (2002).

Summary: It is proved that the first eigenfunction of the mixed boundary value problem for 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.

MSC:

74B05	Classical linear elasticity
74E10	Anisotropy in solid mechanics
35B40	Asymptotic behavior of solutions to PDEs
74G10	Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics

Keywords:

spectrum; thin domain; boundary layer; trapped mode; localized eigenfunction; mixed boundary value problem; Laplacian