Discreteness of interior transmission eigenvalues revisited

Nguyên, Hoài-Minh; Nguyen, Quoc Hung

doi:10.1007/s00526-017-1143-7

Cited by 14 publications

(13 citation statements)

References 29 publications

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“…Remark 3.1. In the proof of Lemma 3.2, we used [30,Lemma 18]. This lemma is in the spirit of the results given in [2] due to Agmon, Douglis, and Nirenberg where more regular data are used.…”

Section: Lemma 32mentioning

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: Lemma 32mentioning

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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“…Natural and interesting questions on the inverse scattering theory include: discreteness of the spectrum (see e.g. [7,6,39,19,32]) location of transmission eigenvalues (see [9,22,40,41], and also [10] for the application in time domain), and the Weyl law of transmission eigenvalues and the completeness of the generalized eigenfunctions (see e.g. [19,20,5,21,38]).…”

Section: Introductionmentioning

confidence: 99%

The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions

Nguyên

Nguyen

2021

Journal of Functional Analysis

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“…A numercial algorithm used for NIMs in the spirit [22] is considered in [1]. Various techniques developed to study NIMs were explored in the context of interior transmission eigenvalues in [28]. The study of NIMs in time domain is recently investigated in [10].…”

Section: Introductionmentioning

confidence: 99%

Cloaking using complementary media for electromagnetic waves

Nguyên

2019

ESAIM: COCV

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Negative index materials are artificial structures whose refractive index has negative value over some frequency range. The study of these materials has attracted a lot of attention in the scientific community not only because of their many potential interesting applications but also because of challenges in understanding their intriguing properties due to the signchanging coefficients in equations describing their properties. In this paper, we establish cloaking using complementary media for electromagnetic waves. This confirms and extends the suggestions in two dimensions of Lai et al. in [15] for the full Maxwell equations. The analysis is based on the reflecting and removing localized singularity techniques, three-sphere inequalities, and the fact that the Maxwell equations can be reduced to a weakly coupled second order elliptic equations.

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Discreteness of interior transmission eigenvalues revisited

Cited by 14 publications

References 29 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

The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions

Cloaking using complementary media for electromagnetic waves

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