On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type

We consider a two-dimensional spectral problem of Steklov type for the Laplace operator in a domain divided into two parts by a perforated partition with a periodic microstructure. The Steklov boundary condition is imposed on the lateral sides of the perforation, the Neumann condition on the remaining part of the boundary, and the Dirichlet and Neumann conditions on the outer boundary of the domain. We construct and justify two-term asymptotic expressions for the eigenvalues of this problem. We also construct a two-term asymptotic formula for the corresponding eigenfunctions.