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

We study the asymptotic behavior of the eigenvalues $\beta^\varepsilon$ and the associated eigenfunctions of an $\varepsilon$-dependent Steklov type eigenvalue problem posed in a bounded domain $\Omega$ of $\R^2$, when $\varepsilon \to 0$. The eigenfunctions $u^\varepsilon$ being harmonic functions inside $\Omega$, the Steklov condition is imposed on segments $T^\varepsilon$ of length $O(\varepsilon)$ periodically distributed on a fixed part $\Sigma$ of the boundary $\partial \Omega$; a homogeneous Dirichlet condition is imposed outside. The homogenization of this problem as $\varepsilon \to 0$ involves the study of the spectral local problem posed in the unit reference domain, namely the half-band $G=(-P/2,P/2)\times (0,+\infty)$ 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 $\beta^\varepsilon\varepsilon$, and the associated eigenfunctions by means of those of the local problem.