Uniform estimates for transmission problems with high contrast in heat conduction and electromagnetism

TL;DR

This paper establishes uniform a priori estimates for transmission problems involving high contrast in heat conduction and electromagnetism, particularly when material coefficients differ greatly, using asymptotic series and field decomposition techniques.

Contribution

It introduces a method to derive uniform estimates for high contrast transmission problems, including Maxwell equations, with applications to asymptotic expansion convergence.

Findings

Uniform estimates for scalar transmission problems with large coefficient ratios.

Decomposition technique for electric field in Maxwell transmission problems.

Convergence proof of asymptotic expansion as conductivity approaches infinity.

Abstract

In this paper we prove uniform a priori estimates for transmission problems with constant coefficients on two subdomains, with a special emphasis for the case when the ratio between these coefficients is large. In the most part of the work, the interface between the two subdomains is supposed to be Lipschitz. We first study a scalar transmission problem which is handled through a converging asymptotic series. Then we derive uniform a priori estimates for Maxwell transmission problem set on a domain made up of a dielectric and a highly conducting material. The technique is based on an appropriate decomposition of the electric field, whose gradient part is estimated thanks to the first part. As an application, we develop an argument for the convergence of an asymptotic expansion as the conductivity tends to infinity.