The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions

Fioralba  Cakoni; Shari  Moskow; Scott  Rome

doi:10.3934/ipi.2015.9.725

Article Contents

2015, Volume 9, Issue 3: 725-748. Doi: 10.3934/ipi.2015.9.725

This issue Previous Article Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials Next Article Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results

The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions

1.
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
2.
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104

Received: August 2014

Revised: January 2015

Published: August 2015

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Abstract

This paper concerns the transmission eigenvalue problem for an inhomogeneous media of compact support containing small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the real transmission eigenvalues. Note that for practical applications the real transmission eigenvalues are important since they can be measured from the scattering data. In particular, in addition to proving the convergence rate for the eigenvalues corresponding to the perturbed media as inclusions' volume goes to zero, we also provide the explicit first correction term in the asymptotic expansion for simple eigenvalues. The correction terms involves the eigenvalues and eigenvectors of the unperturbed known background as well as information about the location, size and refractive index of small inhomogeneities. Thus, our asymptotic formula has the potential to be used to recover information about small inclusions from a knowledge of real transmission eigenvalues.

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Mathematics Subject Classification: Primary: 35R30, 35Q60, 35J40, 78A25.

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