Inverse scattering for shape and impedance

We consider the inverse problem of determining both the shape and the impedance of a two-dimensional scatterer from a knowledge of the far-field pattern of the scattering of time-harmonic acoustic or electromagnetic waves by solving the ill posed nonlinear equation for the operator that maps the boundary and the boundary impedance of the scatterer onto the far-field pattern. We establish results on the injectivity of the linearized map and obtain satisfactory reconstructions by a regularized Newton iteration.