Transmission eigenvalues and non-destructive testing of anisotropic magnetic materials with voids

In this paper we consider the transmission eigenvalue problem corresponding to the scattering problem for an anisotropic magnetic materials with voids, i.e. subregions with refractive index the same as the background. Here we restrict ourselves to the scalar case of TE-polarization. Under weak assumptions on the material properties, we show that the transmission eigenvalues can be determined from the far field measurements. Then assuming that the contrast on the material properties does not change sign, we prove the existence of at least one transmission eigenvalue for sufficiently small voids. We also show that the first transmission eigenvalue can be used to determining material properties and give qualitative information about the size of the void. Some numerical examples are given to demonstrate the theoretical results.