Homogenisation for a statinory Maxwell system

TL;DR

This paper investigates the homogenization of the stationary Maxwell system with rapidly oscillating periodic coefficients, providing asymptotic descriptions of solutions and resolvent behavior outside the spectrum.

Contribution

It introduces a novel analysis of the Maxwell system with large-scale periodic variations in material properties, including asymptotic solutions and resolvent estimates.

Findings

Asymptotic behavior of solutions outside the spectrum is characterized.

Resolvent operator behavior is described with controlled remainders.

The analysis handles large-distance variations in material parameters.

Abstract

We study homogenization problem for the stationary Maxwell system. It is supposed that the magnetic permeability and the dielectric permittivity locally close to fast-oscillating (with respect to some small parameter) periodic functions which can change the form on rather big distances. An asymptotic behavior to solutions of the Maxwell system outside of its spectrum is obtained. We also describe asymptotic behavior of resolvent with control of the remainder in terms of some appropriate operator norms.