Transmission eigenvalue problem for inhomogeneous absorbing media with mixed boundary condition

Consider the transmission eigenvalue problem for the wave scattering by a dielectric inhomogeneous absorbing obstacle lying on a perfect conducting surface. After excluding the purely imaginary transmission eigenvalues, we prove that the transmission eigenvalues exist and form a discrete set for inhomogeneous non-absorbing media, by using analytic Fredholm theory. Moreover, we derive the Faber-Krahn type inequalities revealing the lower bounds on real transmission eigenvalues in terms of the media parameters. Then, for inhomogeneous media with small absorption, we prove that the transmission eigenvalues also exist and form a discrete set by using perturbation theory. Finally, for homogeneous media, we present possible components of the eigenvalue-free zone quantitatively, giving the geometric understanding on this problem.