Localization of Eigenfunctions of the Dirichlet Problem near a Contour at the Boundary of a Thin Domain

We consider the spectral Dirichlet problem for the Laplace operator in a thinthree-dimensional domain of variable thickness which admits a maximum value on a smoothclosed contour either inside the longitudinal cross-section or at its boundary. We find asymptoticexpansions of the eigenvalues which involve the eigenvalues of the harmonic oscillator on the lineor half-line as well as of a certain second-order ordinary differential equation on the contour. Theeigenfuctions are localized in a neighborhood of the contour.