On the existence of transmission eigenvalues in an inhomogeneous medium

Abstract

We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both the scalar problem and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [Transmission eigenvalues, SIAM J. Math. Anal. 40, (2008) pp. 783–753] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that the contrasts are large enough.

Keywords:

• interior transmission problem
• transmission eigenvalues
• inhomogeneous medium
• inverse scattering

AMS Subject Classifications::

• 35R30
• 35Q60
• 35J40
• 78A25

Acknowledgements

This research was initiated while H. Haddar was visiting the University of Delaware and was finished while F. Cakoni was visiting Ecole Polytechique. Both visits were supported by the associate team ISIP of INRIA-UDEL. The hospitality of both institutions is gratefully acknowledged. The research of F. Cakoni was supported in part by the US Air Force Office of Scientific Research under Grant FA9550-05-1-0127.