Abstract: We consider a two-dimensional eigenvalue problem for Laplace operator with frequent alternation of the boundary-condition type. We study the cases of the averaged problem with the Dirichlet, Neumann, or Robin boundary condition. In the case of the circle and a periodic alternation of the boundary-condition type the complete two-parametrical asymptotics for the eigen-elements of the problem under consideration is constructed. For an arbitrary domain and a quasiperiodic alternation we obtain the leading terms of the asymptotics for the eigen-elements. In the case of a non-periodical alternation and any domain we derive estimates for the eigenvalues and give the conditions under those the leading terms of the asymptotics for the eigen-elements can be constructed.