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Localization of the interior transmission eigenvalues for a ball

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  • 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 $|\operatorname{Im} λ| ≤ C$ .

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


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