Analysis of the transmission eigenvalue problem for biharmonic scattering considering penetrable scatterers

arXiv (Cornell University)

Abstract

In this paper, we provide an analytical study of the transmission eigenvalue problem in the context of biharmonic scattering with a penetrable obstacle. We will assume that the underlying physical model is given by an infinite elastic two--dimensional Kirchhoff--Love plate in $\mathbb{R}^2$, where the plate's thickness is small relative to the wavelength of the incident wave. In previous studies, transmission eigenvalues have been studied for acoustic scattering, whereas in this case, we consider biharmonic scattering. We prove the existence and discreteness of the transmission eigenvalues as well as study the dependence on the refractive index. We are able to prove the monotonicity of the first transmission eigenvalue with respect to the refractive index. Lastly, we provide numerical experiments to validate the theoretical work.

Authors 3

1. Rafael Ceja Ayala

   Arizona State University

   Affiliation as printed

   School of Mathematical and Statistical Sciences , Arizona State University , Tempe , AZ 85287

2. Isaac Harris

   Purdue University West Lafayette

   Affiliation as printed

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

3. Andreas Kleefeld Aachen Faculty of Medical Engineering and Technomathematics

   FH Aachen · Forschungszentrum Jülich · Jülich Supercomputing Centre

   Affiliation as printed

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

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