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

Abstract

We study the asymptotic behavior of theeigenvalues $\beta^\varepsilon$ and the associated eigenfunctions of an$\varepsilon$-dependent Steklov type eigenvalue problem posed in abounded domain $\Omega$ of $\R^2$, when $\varepsilon \to 0$. Theeigenfunctions $u^\varepsilon$ 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 theboundary $\partial \Omega$; a homogeneous Dirichlet condition isimposed outside. The homogenization of this problem as $\varepsilon \to0$ involves the study of the spectral local problem posed in theunit reference domain, namely the half-band $G=(-P/2,P/2)\times(0,+\infty)$ with $P$ a fixed number, with periodic conditions onthe lateral boundaries and mixed boundary conditions of Dirichletand Steklov type respectively on the segment lying on$\{y_2=0\}$. We characterize the asymptotic behavior of the lowfrequencies of the homogenization problem, namely of$\beta^\varepsilon\varepsilon$, and theassociated eigenfunctions by means of those of the localproblem.