KARP THEOREM IN ELASTODYNAMIC INVERSE SCATTERING

Time-harmonic elastic waves are incident upon a bounded obstacle in three dimensions. The resulting scattered waves are characterized by their far-field patterns. We consider some simple questions concerning the determination of the shape of the obstacle from information-on the far-field patterns. Specifically, we refine and extend some results of Wall, allowing mixed boundary conditions on piecewise-smooth surfaces. We also prove two elastodynamic analogues of a theorem due to Karp, giving sufficient conditions on the far-field patterns for the obstacle to be spherical. The proofs are indirect and are based-on symmetry arguments, as used for scalar problems by Ramm. The possibility of a direct proof is also explored.