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

We consider the spectral Dirichlet problem for the Laplace operator in a thin three-dimensional domain of variable thickness which admits a maximum value on a smooth closed contour either inside the longitudinal cross-section or at its boundary. We find asymptotic expansions of the eigenvalues which involve the eigenvalues of the harmonic oscillator on the line or half-line as well as of a certain second-order ordinary differential equation on the contour. The eigenfuctions are localized in a neighborhood of the contour.