Transmission eigenvalues for dielectric objects on a perfect conductor

We present a new interior transmission problem arising when a dielectric structure sits on a perfect conducting plane. The problem has mixed boundary conditions. We discuss the forward problem, and then briefly formulate the standard near field linear sampling method (LSM) for the inverse problem of shape identification. Next we show that the new mixed transmission eigenvalue problem can be analyzed by appropriate modifications to the standard theory of transmission eigenvalues. In particular this involves proving appropriate density and compactness results. We end with some numerical evidence which shows that the LSM can be used for this problem even if limited aperture of data is used. In addition we demonstrate that transmission eigenvalues can be determined from near field scattering data.