Asymptotics for the solutions of elliptic systems with fast oscillating coefficients

TL;DR

This paper derives the asymptotic behavior and homogenized operator for elliptic systems with rapidly oscillating coefficients, establishing spectrum convergence and providing explicit examples.

Contribution

It introduces a method to obtain the homogenized operator and asymptotic expansions for solutions of elliptic systems with fast oscillating coefficients.

Findings

Homogenized operator derived for the elliptic system

Asymptotic expansion of the resolvent constructed

Spectrum convergence established

Abstract

We consider a singularly perturbed second order elliptic system in the whole space. The coefficients of the systems fast oscillate and depend both of slow and fast variables. We obtain the homogenized operator and in the uniform norm sense we construct the leading terms of the asymptotics expansion for the resolvent of the operator described by the system. The convergence of the spectrum is established. The convergence of the spectrum is established. The examples are given.