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 thin three-dimensional domain of a variable thickness which admits a maximum value at a smooth closed contour either inside the longitudinal cross-section, or at its boundary. We find out asymptotic expansions of the eigenvalues which involve eigenvalues of the harmonic oscillator at the axis or the half-axis as well as of a certain second order ordinary differential equation at the contour. The eigenfuctions are localized in the vicinity of the contour.