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

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω ⊂ R, for a vectorial elliptic operator −∇·A(·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λ when ε → 0, the mode k being fixed. A first-order asymptotic expansion is proved for λ in the case when Ω 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 in [17] restricted to scalar operators and convex polygonal domains with sides of rational slopes. We take advantage of the recent progress due to Gérard-Varet and Masmoudi [11, 10] in the homogenization of boundary layer type systems.