Asymptotics of the eigenvalues of elliptic systems with fast oscillating coefficients

TL;DR

This paper develops asymptotic expansions for eigenvalues and eigenfunctions of elliptic systems with rapidly oscillating coefficients, providing insights into their behavior as they approach the homogenized system's eigenvalues.

Contribution

It introduces a comprehensive method to derive asymptotic expansions for eigenvalues and eigenfunctions in elliptic systems with fast oscillating coefficients.

Findings

Asymptotic expansions for eigenvalues near homogenized system eigenvalues

Complete asymptotic series for associated eigenfunctions

Convergence results for the expansions

Abstract

We consider singularly perturbed second order elliptic system in the whole space with fast oscillating coefficients. We construct the complete asymptotic expansions for the eigenvalues converging to the isolated ones of the homogenized system, as well as the complete asymptotic expansions for the associated eigenfunctions.