# American Institute of Mathematical Sciences

August  2018, 12(4): 971-992. doi: 10.3934/ipi.2018041

## Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast

 1 Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA 2 Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA

Received  August 2017 Published  June 2018

In this paper we revisit the transmission eigenvalue problem for an inhomogeneous media of compact support perturbed by small penetrable homogeneous inclusions. Assuming that the inhomogeneous background media is known and smooth, we investigate how these small volume inclusions affect the transmission eigenvalues. Our perturbation analysis makes use of the formulation of the transmission eigenvalue problem introduced Kirsch in [8], which requires that the contrast of the inhomogeneity is of one-sign only near the boundary. Thus, our approach can handle small perturbations with positive, negative or zero (voids) contrasts. 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 term involves computable information about the known inhomogeneity as well as the location, size and refractive index of small perturbations. Our asymptotic formula has the potential to be used to recover information about small inclusions from knowledge of the real transmission eigenvalues, which can be determined from scattering data.

Citation: Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041
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##### References:
Comparison of perturbed eigenvalues and corrected eigenvalues. The red circles are the perturbed transmission eigenvalues (squared) and the blue stars the corrected approximations for various values of $\epsilon$. The $x$-axis is $\log_{10}{\epsilon}$
Log/log plot of the error $(\tau_\epsilon-(\tau_0+\epsilon \tau^{(1)} ))/\epsilon$
Parameters for Numerical Example
 Domain $D$ $[-1, 1]$ Background Transmission Eigenvalue $k=\sqrt{-\tau}$ 7.12761 Background Coefficient $q_0$ 6.29 Perturbed Coefficient $q_1$ 24 Parameter $\lambda$ 50.72217
 Domain $D$ $[-1, 1]$ Background Transmission Eigenvalue $k=\sqrt{-\tau}$ 7.12761 Background Coefficient $q_0$ 6.29 Perturbed Coefficient $q_1$ 24 Parameter $\lambda$ 50.72217
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