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

Abstract

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain Ω ⊂ R 2 , for a vectorial elliptic operator −∇ · A e (·)∇ with e-periodic coefficients. We analyse the asymptotics of the eigen- values λ e,k when e → 0, the mode k being fixed. A first-order asymptotic expansion is proved for λ e,k 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 Proc. Roy. Soc. Edinburgh Sect. A 127(6) (1997), 1263-1299 restricted to scalar operators and convex polygonal domains with sides of rational slopes. We take advantage of the recent progress due to Gerard-Varet and Masmoudi (J. Eur. Math. Soc. 13 (2011), 1477-1503; Acta Math. 209 (2012), 133-178) in the homogenization of boundary layer type systems.