On the uniqueness of the shape of a penetrable, anisotropic obstacle

We consider the scattering of time-harmonic waves in an isotropic, homogeneous medium which contains a penetrable, anisotropic, inhomogeneous obstacle. After reviewing existence, uniqueness and regularity of a (variational) solution we prove that a knowledge of the scattered near fields for all incident plane waves at a fixed wave number uniquely determines the obstacle provided the matrix characterizing the anisotropic medium satisfies a suitable condition. To this end we examine the corresponding interior transmission problem and prove that it is a compact perturbation of an interior transmission problem which is well-posed.