Localization of the interior transmission eigenvalues for a ball

Vesselin Petkov; Georgi Vodev

doi:10.3934/ipi.2017017

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2017, Volume 11, Issue 2: 355-372. Doi: 10.3934/ipi.2017017

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Localization of the interior transmission eigenvalues for a ball

Vesselin Petkov^1, and
Georgi Vodev^2,

1.
Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France
2.
Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssiniére, BP 92208,44322 Nantes Cedex, France

Received: February 29, 2016

Revised: December 31, 2016

Published: May 2017

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Abstract

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x ∈ \mathbb{R}^d:\: |x| ≤ 1\}, \: d≥ 2,$ and the coefficients $c_j(x), \: j =1,2, $ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$ . We prove that in this case the eigenvalue-free region obtained in [17] for strictly concave domains can be significantly improved. In particular, if $c_j(x), n_j(x), j = 1,2$ are constants for $|x| ≤ 1$ , we show that all (ITEs) lie in a strip $|\operatorname{Im} λ| ≤ C$ .

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Mathematics Subject Classification: Primary: 35P15; Secondary: 35P20, 35P25.

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References

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