Vol. 11, No. 1, 2018

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues

Georgi Vodev

Vol. 11 (2018), No. 1, 213–236
Abstract

We study the high-frequency behaviour of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We show that far from the real axis it can be approximated by a simpler operator. We use this fact to get new results concerning the location of the transmission eigenvalues on the complex plane. In some cases we obtain optimal transmission eigenvalue-free regions.

Keywords
Dirichlet-to-Neumann map, transmission eigenvalues
Mathematical Subject Classification 2010
Primary: 35P15
Milestones
Received: 17 January 2017
Revised: 21 June 2017
Accepted: 10 August 2017
Published: 17 September 2017
Authors
Georgi Vodev
Université de Nantes
Laboratoire de Mathématiques Jean Leray
Nantes
France