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Transmission eigenvalues for strictly concave domains

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Abstract

We show that for strictly concave domains there are no interior transmission eigenvalues in a region of the form \(\left\{ \lambda \in \mathbf{C}:\mathrm{Re}\,\lambda \ge 0,\,\,|\mathrm{Im}\,\lambda |\ge C_\varepsilon \left( \mathrm{Re}\,\lambda +1\right) ^{\frac{1}{2}+\varepsilon }\right\} \), \(C_\varepsilon >0\), for every \(0<\varepsilon \ll 1\). As a consequence, we obtain Weyl asymptotics for the number of the transmission eigenvalues with an almost optimal remainder term.

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Acknowledgments

I would like to thank Vesselin Petkov for some very usefull discussions and suggestions.

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  1. Département de Mathématiques, UMR 6629 du CNRS, Université de Nantes, 2, rue de la Houssinière, BP 92208, 44332, Nantes Cedex 03, France

    Georgi Vodev

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  1. Georgi Vodev
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Correspondence to Georgi Vodev.

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Vodev, G. Transmission eigenvalues for strictly concave domains. Math. Ann. 366, 301–336 (2016). https://doi.org/10.1007/s00208-015-1329-2

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  • DOI: https://doi.org/10.1007/s00208-015-1329-2

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