Limiting Absorption Principle and Well-Posedness for the Time-Harmonic Maxwell Equations with Anisotropic Sign-Changing
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Communications in
Mathematical Physics
Limiting Absorption Principle and Well-Posedness for the Time-Harmonic Maxwell Equations with Anisotropic Sign-Changing Coefficients Hoai-Minh Nguyen1 , Swarnendu Sil2 1 Department of Mathematics, EPFL SB CAMA, Station 8, 1015 Lausanne, Switzerland.
E-mail: [email protected]
2 Forschungsinstitut für Mathematik, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland.
E-mail: [email protected] Received: 21 August 2019 / Accepted: 23 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 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.
Contents 1. 2. 3.
Introduction . . . . . . . . . . . . . . . . . . Statement of the Main Results . . . . . . . . . Fourier Approach for the Cauchy Problems . . 3.1 Preliminaries . . . . . . . . . . . . . . . 3.2 Proof of Theorem 2.1 . . . . . . . . . . . 4. Variational Approach for the Cauchy Problems 4.1 Some useful lemmas . . . . . . . . . . . 4.2 Proof of Theorem 2.2 . . . . . . . . . . . 4.3 Some applications of Theorem 2.2 . . . . A. Proof of Proposition 3.1 . . . . . . . . . . . .
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H.-M. Nguyen, S. Sil
1. Introduction Negative-index metamaterials are artificial structures whose refractive index has a negative value over some frequency range. Their existence was postulated by Veselago in 1964 [35] and confirmed experimentally by Shelby, Smith, and Schultz in 2001 [33]. Negative-index metamaterial research has been a very active topic of investigation not only because of potentially interesting applications, but also because of challenges involved in understanding their peculiar properties due to the sign-changing coefficients in the equations modeling the phenomena. In this paper, we study the
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