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Transmission eigenvalue-free regions near the real axis

Georgi Vodev

Vol. 6 (2024), No. 2, 611–632
Abstract

We consider the interior transmission problem with one complex-valued refraction index, that is, with a damping term which does not vanish on the boundary. Under the condition that all geodesics reach the boundary, for a class of strictly concave domains, we obtain a transmission eigenvalue-free region of the form {λ : CN(|λ| + 1)N |Im λ| C(|λ| + 1)1}, where N > 1 is arbitrary. Under extra conditions on the coefficients we get a larger transmission eigenvalue-free region of the form {λ : CN(|λ| + 1)N |Im λ| C}.

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Keywords
transmission eigenvalues, interior transmission problems
Mathematical Subject Classification
Primary: 35P15
Secondary: 35P20
Milestones
Received: 25 October 2023
Revised: 3 March 2024
Accepted: 25 April 2024
Published: 16 May 2024
Authors
Georgi Vodev
Laboratoire de Mathématiques Jean Leray
Université de Nantes
Nantes
France