  This paper is concerned with the unique identification of the shape of a
scatterer through a single far-field pattern in an inverse elastic medium
scattering problem with a generalized transmission boundary condition. The
uniqueness issue by a single far-field measurement is a challenging problem in
inverse scattering theory, which has a long and colorful history. In this
paper, we demonstrate the well-posedness of the direct problem by the
variational approach. We establish the uniqueness results by a single far-field
measurement under a generic scenario when dealing with underlying elastic
scatterers exhibiting polygonal-nest or polygonal-cell structures. Furthermore,
for a polygonal-nest or polygonal-cell structure scatterer associated with
density and boundary impedance parameters as piecewise constants, we show that
these physical quantities can be uniquely determined simultaneously by a single
far-field measurement. The corresponding proof relies heavily on examining the
singular behaviour of a coupled PDE system near a corner in a microlocal
manner.

Analysis of PDEs