Interior transmission eigenvalue problem with refractive index having C²-transition to the background medium

Abstract

The interior transmission eigenvalue problem for scalar acoustics is studied for a new class of refractive index. Existence of an infinite discrete set of transmission eigenvalues in the case that the acoustic properties of a domain D ⊂ ℝ ⁿ are allowed to have a C ²-transition to the homogeneous background medium is established. It is shown that the transmission problem has a weak formulation on certain weighted Sobolev spaces for this class of refractive index. The weak formulation and the discreteness of the spectrum is justified by using the Hardy inequality to prove compact imbedding theorems. Existence of transmission eigenvalues is demonstrated by investigating a generalized eigenvalue problem associated with the weak formulation.

Keywords:

• interior transmission eigenvalue problem
• transmission spectrum
• transmission eigenvalues
• inhomogeneous medium
• acoustic scattering

AMS Subject Classifications::

• 35R30
• 35Q60
• 35J40
• 35J35
• 78A40

Acknowledgements

Results in this article have been included in the authors PhD thesis completed at Oregon State University under the direction of Professor David Finch. The author is much indebted to Dr Finch for his careful comments and suggestions on this article.