Analysis of the transmission eigenvalue problem with two conductivity parameters

arXiv (Cornell University)

Abstract

In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer. In previous studies, this eigenvalue problem was analyzed with one conductive boundary parameter whereas we will consider the case of two parameters. We will prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the physical parameters. We are able to prove monotonicity of the first transmission eigenvalue with respect to the parameters and consider the limiting procedure as the second boundary parameter vanishes. Lastly, we provide extensive numerical experiments to validate the theoretical work.

Authors 4

1. Rafael Ceja Ayala

   Purdue University West Lafayette

   Affiliation as printed

   Department of Mathematics , Purdue University , West Lafayette , IN 47907

2. Isaac Harris Aachen Faculty of Medical Engineering and Technomathematics

   Purdue University West Lafayette · FH Aachen

   Affiliation as printed

   Department of Mathematics , Purdue University , West Lafayette , IN 47907

   University of Applied Sciences Aachen , Faculty of Medical Engineering and Technomathematics , Heinrich-Mußmann-Str. 1 , 52428 Jülich , Germany

3. Andreas Kleefeld

   Forschungszentrum Jülich · Jülich Supercomputing Centre

   Affiliation as printed

   Forschungszentrum Jülich GmbH , Jülich Supercomputing Centre , Wilhelm-Johnen-Straße , 52425 Jülich , Germany

4. Nikolaos Pallikarakis

   National Technical University of Athens

   Affiliation as printed

   Department of Mathematics , National Technical University of Athens , 15780 Athens , Greece