On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data

We show uniqueness of a (time independent) domain $D$ and of an impedance type boundary condition from either dynamical or scattering data at many frequencies. We assume that the additonal boundary (scattering) data are given for one set of boundary data or for one incident direction. In case of general domain and finite (sharp) observation time we assume Neumann boundary condition on $\partial D$ and for polygonal $D$ we can handle more general case. If the data are available for all times we show uniqueness of the most general impedance boundary condition by using continuation of the corresponding scattering data into complex domain and modifying the Schiffer's argument.