The electromagnetic interior transmission problem for regions with cavities

We consider the electromagnetic interior transmission problem in the case when the medium has cavities, i.e., regions in which the index of refraction is the same as in the host medium. We address the configuration where the electromagnetic permeability is constant while the electric permittivity is variable and may be anisotropic. In this case, using appropriate reformulation of the problem into a fourth order PDE, we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. Monotonicity properties of the first eigenvalue in terms of the permittivity and the size of the cavity are established.