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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.

First-order expansion for the Dirichlet eigenvalues of an elliptic system with oscillating coefficients