Uniform estimates for transmission problems in electromagnetism with high contrast in magnetic permeabilities

TL;DR

This paper establishes uniform estimates for electromagnetic transmission problems with high magnetic permeability contrast, providing insights into solution behavior and multiscale expansions in such media.

Contribution

It introduces uniform a priori estimates for Maxwell solutions in high contrast magnetic materials and derives multiscale expansions of the magnetic field.

Findings

Uniform estimates hold for Lipschitz interfaces.

Magnetic field admits a multiscale expansion in powers of 1/√μ_r.

Profiles decay rapidly inside the magnetic conductor.

Abstract

We consider the time-harmonic Maxwell equations set on a domain made up of two subdomains that represent a magnetic conductor and a non-magnetic material, and we assume that the relative magnetic permeability $\mu_{r}$ between the two materials is high. We prove uniform a priori estimates for Maxwell solutions when the interface between the two subdomains is supposed to be Lipschitz. Assuming smoothness for the interface between the subdomains, we prove also that the magnetic field possesses a multiscale expansion in powers of $1/ \sqrt{\mu_{r}}$ with profiles rapidly decaying inside the magnetic conductor.