Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Hoai Minh Nguyen, Swarnendu Sil

TL;DR

The paper addresses the stability of time-harmonic Maxwell fields in media with anisotropic sign-changing coefficients, motivated by negative-index metamaterials. It develops two complementary analytical frameworks to prove the limiting absorption principle and well-posedness: a Fourier-analytic approach based on ADNII complementing conditions, and a variational/reflection approach that leverages energy estimates and local chart techniques. The authors derive general, local criteria on the coefficients that guarantee uniform bounds and convergence of the perturbed solutions as loss vanishes, along with precise statements for radiating solutions and uniqueness. These results unify and extend the mathematical understanding of Maxwell systems with sign-changing media, offering practically checkable conditions and illuminating connections to resonance, cloaking, and metamaterial stability in electromagnetic applications.

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.

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

TL;DR

The paper addresses the stability of time-harmonic Maxwell fields in media with anisotropic sign-changing coefficients, motivated by negative-index metamaterials. It develops two complementary analytical frameworks to prove the limiting absorption principle and well-posedness: a Fourier-analytic approach based on ADNII complementing conditions, and a variational/reflection approach that leverages energy estimates and local chart techniques. The authors derive general, local criteria on the coefficients that guarantee uniform bounds and convergence of the perturbed solutions as loss vanishes, along with precise statements for radiating solutions and uniqueness. These results unify and extend the mathematical understanding of Maxwell systems with sign-changing media, offering practically checkable conditions and illuminating connections to resonance, cloaking, and metamaterial stability in electromagnetic applications.

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.

Paper Structure

This paper contains 10 sections, 19 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Fourier approach for the Cauchy problems
  4. Preliminaries
  5. Proof of Theorem \ref{['thm1']}
  6. Variational approach for the Cauchy problems
  7. Some useful lemmas
  8. Proof of Theorem \ref{['thm2']}
  9. Some applications of Theorem \ref{['thm2']}
  10. Proof of Proposition \ref{['pro-opt']}

Key Result

Theorem 2.1

Let $0 < \delta < 1$, $J \in [L^2(\mathbb{R}^3)]^3$ with $\hbox{supp} J \subset B_{R_0}$, and let $(E_\delta, H_\delta) \in [H_{\operatorname{loc}} (\operatorname{curl}, \mathbb{R}^3)]^2$ be the unique radiating solution of Main-eq-delta. Assume that $\varepsilon^+, \, \mu^+ \in C^1(\bar{D}_{-\tau}) for some positive constant $C_R$ independent of $\delta$ and $J$. Moreover, $(E_\delta, H_\delta)$

Theorems & Definitions (39)

  • Definition 2.1: Agmon, Douglis, Nirenberg ADNII
  • Remark 2.1
  • Theorem 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Definition 2.2
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 29 more