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

TL;DR

This paper develops a first-order asymptotic expansion for Dirichlet eigenvalues of elliptic systems with oscillating coefficients in bounded domains, extending previous scalar results to vectorial operators and specific domain geometries.

Contribution

It introduces a novel first-order asymptotic expansion for eigenvalues of vectorial elliptic operators with oscillating coefficients, applicable to convex and polygonal domains under certain geometric conditions.

Findings

Established first-order asymptotics for eigenvalues as epsilon approaches zero.

Extended scalar operator homogenization results to vectorial elliptic systems.

Applied recent boundary layer homogenization techniques to eigenvalue problems.

Abstract

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.