On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem

We study the asymptotic behavior of the eigenvalues beta(epsilon) and the associated eigenfunctions of an epsilon-dependent Steklov type eigenvalue problem posed in a bounded domain Omega of R-2, when epsilon --> 0. The eigenfunctions u(epsilon) being harmonic functions inside Omega, the Steklov condition is imposed on segments T-epsilon of length O(epsilon) periodically distributed on a fixed part Sigma of the boundary partial derivative Omega; a homogeneous Dirichlet condition is imposed outside. The homogenization of this problem as epsilon --> 0 involves the study of the spectral local problem posed in the unit reference domain, namely the half-band G = (-P/2, P/2) x (0, +infinity) with P a fixed number, with periodic conditions on the lateral boundaries and mixed boundary conditions of Dirichlet and Steklov type respectively on the segment lying on {y(2) = 0}. We characterize the asymptotic behavior of the low frequencies of the homogenization problem, namely of beta(epsilon) epsilon, and the associated eigenfunctions by means of those of the local problem.