The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary

Abstract

In this paper, we investigate the interior transmission eigenvalue problem for an inhomogeneous media with conductive boundary conditions. We prove the discreteness and existence of the transmission eigenvalues. We also investigate the inverse spectral problem of gaining information about the material properties from the transmission eigenvalues. In particular, we prove that the first transmission eigenvalue is a monotonic function of the refractive index n and boundary conductivity parameter , and obtain a uniqueness result for constant coefficients. We provide some numerical examples to demonstrate the theoretical results in three dimensions.

Keywords:

• Inverse scattering
• inhomogeneous medium
• transmission eigenvalues
• inverse spectral problem
• conductive boundary condition

AMS Subject Classifications:

• 35P25
• 35P30
• 35J30
• 78A46

Acknowledgements

Some of the results of this work were carried out during a research stay of Oleksandr Bondarenko at the University of Delaware in Summer 2015. He greatly acknowledges the hospitality of Fioralba Cakoni during the stay and the financial support from the Karlsruhe House of Young Scientists (KHYS). The authors would like to thank Fioralba Cakoni for valuable advice and the fruitful discussions.

Notes

No potential conflict of interest was reported by the authors.