Transmission eigenvalues and the nondestructive testing of dielectrics

In this paper we would like to suggest a new method for nondestructive testing of dielectrics. This method is based on determining transmission eigenvalues from electromagnetic scattering data and deducing the presence of cavities in the dielectric from the location of the eigenvalues. The model problem we have in mind (although many others are possible!) is the location of cavities in tree trunks, which for E-polarized electromagnetic waves polarized parallel to the axis of the tree trunk, the scattering problem can be modeled by the Helmholtz equation with variable permittivity (note that, since wood is orthotropic, H-polarized waves would lead to a different equation to the Helmholtz equation). Transmission eigenvalues have appeared only recently in the world of scattering theory and for an introduction to this unusual spectral problem in partial differential equations we refer the reader to sections 8.4 and 8.6 of [6] and the survey paper [7]. In [1] it was shown that, under the assumption that the relative permittivity r was greater than one, transmission eigenvalues could be determined from scattering data and used to give a lower bound for r . In the present paper we will encounter a transmission eigenvalue problem that has not been considered before, i.e. the case when r = 1 in a portion D0 of the scattering obstacle D. In particular D0 corresponds to the location of cavities in the obstacle. We will provide numerical evidence showing that the presence of a nonempty cavity D0 ⊂ D causes the transmission eigenvalues to increase with the magnitude of the increase depending on the size of D0. Although we have been unable to prove this fact analytically, we will establish in