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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.