Limiting absorption principle and well-posedness for the time-harmonic Maxwell equations with anisotropic sign-changing coefficients

TL;DR

This paper establishes conditions under which Maxwell equations with anisotropic, sign-changing coefficients are well-posed and satisfy the limiting absorption principle, aiding the understanding of electromagnetic stability in negative-index metamaterials.

Contribution

It introduces new, locally checkable conditions for well-posedness and the limiting absorption principle in Maxwell equations with anisotropic sign-changing coefficients.

Findings

Derived a priori estimates for Maxwell Cauchy problems.

Established general conditions for stability of electromagnetic fields.

Provided criteria applicable to negative-index metamaterials.

Abstract

We study the limiting absorption principle and the well-posedness of Maxwell equations with anisotropic sign-changing coefficients in the time-harmonic domain. The starting point of the analysis is to obtain Cauchy problems associated with two Maxwell systems using a change of variables. We then derive a priori estimates for these Cauchy problems using two different approaches. The Fourier approach involves the complementing conditions for the Cauchy problems associated with two elliptic equations, which were studied in a general setting by Agmon, Douglis, and Nirenberg. The variational approach explores the variational structure of the Cauchy problems of the Maxwell equations. As a result, we obtain general conditions on the coefficients for which the limiting absorption principle and the well-posedness hold. Moreover, these {\it new} conditions are of a local character and easy to check. Our work is motivated by and provides general sufficient criteria for the stability of electromagnetic fields in the context of negative-index metamaterials.