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Discreteness of interior transmission eigenvalues revisited

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Abstract

This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary.

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Notes

  1. In this paper, this means that F is bijective and \(F, F^{-1} \in C^1 ({\bar{\Omega }})\).

  2. One can use two diffeomorphisms \(F_1, \, F_2\) and require the corresponding conditions on \(({F_1}_*A_1, {F_1}_*\Sigma _1)\) and \(({F_2}_*A_2, {F_2}_*\Sigma _2)\) to obtain the discreteness of the ITE problem. However, the same conditions hold by using the diffeomorphisms \(F_1\circ F_2^{-1}, \, I\).

  3. In fact, [23, Lemma 7] is stated for \((u_n) \subset H^1(\Omega )\), however the result also holds for \((u_n) \subset H^1_{_{loc}}(\Omega )\) and the proof is almost unchanged.

  4. The goal is to eliminate \(\Sigma _1 u_2\) from the equation of \({\hat{w}}\).

  5. In Proposition 4, \(G_1 = 0\); nevertheless, the same proof gives the same result in the case \(G_1 \in L^2(\Omega )\) with \({\text {supp}}G_1 \subset \subset \Omega \).

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Authors and Affiliations

  1. École Polytechnique Fédérale de Lausanne, SB MATHAA CAMA, Station 8, 1015, Lausanne, Switzerland

    Hoai-Minh Nguyen & Quoc-Hung Nguyen

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  1. Hoai-Minh Nguyen
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  2. Quoc-Hung Nguyen
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Correspondence to Hoai-Minh Nguyen.

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Communicated by P. Rabinowitz.

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Nguyen, HM., Nguyen, QH. Discreteness of interior transmission eigenvalues revisited. Calc. Var. 56, 51 (2017). https://doi.org/10.1007/s00526-017-1143-7

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