Limiting Absorption Principle and Well-Posedness for the Time-Harmonic Maxwell Equations with Anisotropic Sign-Changing Coefficients

• Published in: Communications in mathematical physics Vol. 379; no. 1; pp. 145 - 176
• Main Authors: Nguyen, Hoai-Minh, Sil, Swarnendu
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2020; Springer Nature B.V; Springer Verlag
• Subjects: Absorption; Cauchy problems; Classical and Quantum Gravitation; Coefficients; Complex Systems; Constraining; Electromagnetic fields; Elliptic functions; Engineering Sciences; Mathematical analysis; Mathematical and Computational Physics; Mathematical Physics; Mathematics; Maxwell's equations; Metamaterials; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Stability criteria; Theoretical; Well posed problems; negative-index metamaterials; 35B35; well-posedness; 35J05; resonance; Maxwell equations; 78A25. Contents; Cauchy problems; limiting absorption principle; 35B40; Maxwell equations sign-changing coefficients well-posedness limiting absorption principle Cauchy problems resonance negative-index metamaterials AMS subject classifications: 35B34 35B35 35B40 35J05 78A25. Contents; AMS subject classifications: 35B34; sign-changing coefficients
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-020-03805-1

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 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.