An inverse cavity problem for Maxwellʼs equations

Consider the scattering of a time-harmonic electromagnetic plane wave by an open cavity embedded in a perfect electrically conducting infinite ground plane, where the electromagnetic wave propagation is governed by the Maxwell equations. The upper half-space is filled with a lossless homogeneous medium above the flat ground surface; while the interior of the cavity is assumed to be filled with a lossy homogeneous medium accounting for the energy absorption. The inverse problem is to determine the cavity structure or the shape of the cavity from the tangential trace of the electric field measured on the aperture of the cavity. In this paper, results on a global uniqueness and a local stability are established for the inverse problem. A crucial step in the proof of the stability is to obtain the existence and characterization of the domain derivative of the electric field with respect to the shape of the cavity.