The inverse scattering problem for a partially coated penetrable cavity with interior measurements

In this paper we consider the inverse scattering problem for a cavity that is bounded by a partially coated penetrable inhomogeneous medium of compact support and recover the shape of the cavity and the surface conductivity from a knowledge of measured scattered waves due to point sources located on a curve or surface inside the cavity. First, we prove that both the shape of the cavity and the surface conductivity on the coated part can be uniquely determined from a knowledge of the measured data. Next, we establish a linear sampling method for determining both the shape of the cavity and the surface conductivity. A central role in our justification is played by an eigenvalue problem which we call the exterior transmission eigenvalue problem. Finally, we present some numerical examples to illustrate the validity of our method.