The interior transmission eigenvalue problem for an anisotropic medium by a partially coated boundary

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary ∂Ω is split into two disjoint parts and possesses different transmission conditions. Using the variational method, we obtain the well posedness of the interior transmission problem, which plays an important role in the proof of the discreteness of eigenvalues. Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n ≡ 1, where a fourth order differential operator is applied. In the case of n ≢ 1, we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.