  This paper is concerned with the homogenization of the Dirichlet eigenvalue
problem, posed in a bounded domain $\Omega\subset\mathbb R^2$, for a vectorial
elliptic operator $-\nabla\cdot A^\epsilon(\cdot)\nabla$ with
$\epsilon$-periodic coefficients. We analyse the asymptotics of the eigenvalues
$\lambda^{\epsilon,k}$ when $\epsilon\rightarrow 0$, the mode $k$ being fixed.
A first-order asymptotic expansion is proved for $\lambda^{\epsilon,k}$ in the
case when $\Omega$ is either a smooth uniformly convex domain, or a convex
polygonal domain with sides of slopes satisfying a small divisors assumption.
Our results extend those of Moskow and Vogelius restricted to scalar operators
and convex polygonal domains with sides of rational slopes. We take advantage
of the recent progress due to G\'erard-Varet and Masmoudi in the homogenization
of boundary layer type systems.
