High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues

Vodev, Georgi

doi:10.2140/apde.2018.11.213

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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

DOI: 10.2140/apde.2018.11.213

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

Vol. 11, No. 1, 2018