Uniqueness and factorization method for inverse elastic scattering with a single incoming wave

Abstract

The first part of this paper is concerned with the uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single incident plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.

Uniqueness to inverse acoustic scattering from coated polygonal obstacles with a single incoming wave

“…(ii) A novel path argument for applying 'non-point-to-point' reflection principles to prove the analytical extension of wave fields in polygonal domains. Our method for proving Theorem 1.1 is inspired by the uniqueness proof in inverse conductivity problems with a single measurement [17] and the path argument proposed in [14] where the 'non-point-to-point' reflection principle for the Navier equation was applied. This paper also provides new uniqueness proofs in determining sound-soft/sound-hard polygonal obstacles with a single incoming wave.…”

“…To prove Theorem 1.1, we shall combine the arguments in [12] for treating the Neumann boundary condition and those in elasticity [14] where a 'non-point-to-point' reflection principle for the Navier equation was applied to determine a connected rigid polygon. In contrast with the Dirichlet case [1,4,28,14], the Robin level set (a curve on which the Robin boundary condition is satisfied) of the total field can be unbounded. We will adapt the arguments of [12] by investigating two Robin half-lines starting from a corner point in our uniqueness proof.…”

“…In fact, for this purpose it is only required to redefine the function V appearing in (8) subject to the Helmholtz equation. However, in this paper we prefer an alternative method presented in our previous paper [14], where a non-local extension formula for the Navier equation with the Dirichlet boundary condition was established in 2D. By [14], the existence of an extension operator for the Helmholtz equation relies on the case of the harmonic function.…”

Inverse source problems in an inhomogeneous medium with a single far-field pattern

“…This excludes the possibility of analytical extension in a corner domain for solutions of source problems, which is important in designing inversion algorithms with a single measurement data; see e.g. the enclosure method [19], the range test approach [22,23] as well as [26,Chapter 5] and [8]. In other words, u cannot be analytic at O.…”

The factorization method for recovering cavities in a heat conductor

“…The factorization method was originated by Kirsch [25] and then widely used in various inverse scattering problems and electrical impedance tomography problems; see, for instance, [1,11,18,20,26,29,31,35,37]. The equations for these forward problems are self-adjoint elliptic equations.…”