Radiation Condition for a Non-Smooth Interface Between a Dielectric and a Metamaterial

Dhia, Anne‐Sophie Bonnet‐Ben; Chesnel, Lucas; Claeys, Xavier

doi:10.1142/s0218202513500188

Cited by 43 publications

(83 citation statements)

References 27 publications

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“…His result was extended for the case, where Γ has two strictly convex connected components by Kettunen, Lassas, and Ola in [16]. Recently, the well-posedness was extensively studied by Bonnet-Ben Dhia, Ciarlet, and their coauthors in [5][6][7][8][9][10]13] by T-coercivity approach. This approach was introduced by Bonnet-Ben Dhia, Ciarlet, and Zwölf in [9] and is related to the (Banach-Necas-Babuska) inf-sup condition.…”

Section: Introductionmentioning

confidence: 99%

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

Nguyên

2016

Journal de Mathématiques Pures et Appliquées

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In this paper, we investigate the limiting absorption principle associated to and the well-posedness of the Helmholtz equations with sign changing coefficients which are used to model negative index materials. Using the reflecting technique introduced in [26], we first derive Cauchy problems from these equations. The limiting absorption principle and the well-posedness are then obtained via various a priori estimates for these Cauchy problems. Three approaches are proposed to obtain the a priori estimates. The first one follows from a priori estimates of elliptic systems equipped with complementing boundary conditions due to Agmon, Douglis, and Nirenberg in their classic work [1]. The second approach, which complements the first one, is variational and based on the Dirichlet principle. The last approach, which complements the second one, is also variational and uses the multiplier technique. Using these approaches, we are able to obtain new results on the well-posedness of these equations for which the conditions on the coefficients are imposed "partially" or "not strictly" on the interfaces of sign changing coefficients. This allows us to rediscover and extend known results obtained by the integral method, the pseudo differential operator theory, and the T-coercivity approach. The unique solution, obtained by the limiting absorption principle, is not in H

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Section: Introductionmentioning

confidence: 99%

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

Nguyên

2016

Journal de Mathématiques Pures et Appliquées

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“…6.4 below). In this latter case, let us mention that Fredholm well-posedness can be recovered in another functional framework [1]. More exotic situations are investigated in Section 6.4.…”

Section: Discussion On the Assumptions On σmentioning

confidence: 99%

T-coercivity for scalar interface problems between dielectrics and metamaterials

2012

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Abstract. Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of R d , with d = 2, 3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T -coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria.Mathematics Subject Classification. 35Q60, 35Q61, 35J20.

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“…One already sees that the additional constraint will have to depend on the side from which the limit ν → 0 is taken. Note that such results can also be expressed in terms of Fredholm indices [4].…”

Section: An Analytical Solutionmentioning

confidence: 99%

“…Resonant time-harmonic wave equations are found in the modeling of electromagnetic waves in magnetized plasmas [21,13], in the modeling of metamaterials [4] and in aeroacoustics in recirculating flows [3]. The list is non exhaustive.…”

Section: Introductionmentioning

confidence: 99%

Constructive Formulations of Resonant Maxwell's Equations

Pinto¹,

Després²

2017

SIAM J. Math. Anal.

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In this article we propose new constructive weak formulations for resonant time-harmonic wave equations with singular solutions. Our approach follows the limiting absorption principle and combines standard weak formulations of PDEs with properties of elementary special functions adapted to the singularity of the solutions, called manufactured solutions. We show the well-posedness of several formulations obtained by these means for the limit problem in dimension one, and propose a generalization in dimension two.

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Radiation Condition for a Non-Smooth Interface Between a Dielectric and a Metamaterial

Cited by 43 publications

References 27 publications

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients

T-coercivity for scalar interface problems between dielectrics and metamaterials

Constructive Formulations of Resonant Maxwell's Equations

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