We consider the boundary value problem for the Laplace operator in a two-dimensional domain perforated along a part of the boundary under the assumption that the cavity diameter can be much less than the distance between the cavities. The homogeneous Neumann condition is imposed on the exterior boundary, whereas the homogeneous Dirichlet condition is stated on the boundary of small cavities. We establish types of the homogenized boundary value problems depending on the ratio of the diameter of small cavities to the small distance between the cavities. We prove the convergence of eigenvalues and eigenfunctions to eigenelements of the homogenized (limit) problems. Bibliography: 21 titles. Illustrations: 2 figures.