Interior Transmission Problem for Anisotropic Media

In recent years there has been considerable interest in the inverse scattering problem for anisotropic medium, particularly in the case of acoustic waves and electromagnetic waves. Due to the lack of uniqueness in determining the constitutive parameters, traditional methods for solving the inverse scattering problems based on the use of weak scattering approximations or nonlinear optimization techniques are problematic. On the other hand, since the support is uniquely determined [4], the recently developed linear sampling method [1], [2] for determining the support of the scatterer from the knowledge of the far field pattern at a fixed frequency is ideally suited to solving the inverse scattering problem for anisotropic medium. The unique determination of the support and the linear sampling method for the scalar anisotropic scattering problem were first considered in [4] and [1] respectively. The techniques used in the above papers are based on an analysis of a boundary value problem called the interior transmission problem. This analysis is performed only in the case where the norm of the real part of the matrix A that describes the physical properties of the medium is greater than one. The purpose of this paper is to complete the study of the interior transmission problem in the case where the norm of Re(A) is less then one. Our results imply that the uniqueness result of [4] and the validity of the linear sampling method [1] are also valid for the case ∥Re(A)∥ ≤ 1.