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⊂R², for a vectorial elliptic operator −∇·A^ε(·)∇ with ε-periodic coefficients. We analyse the asymptotics of the eigenvalues λ^ε,k when ε→0, the mode k being fixed. A first-order asymptotic expansion is proved for λ^ε,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 Gérard-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.