Uniqueness of scatterer in inverse acoustic obstacle scattering with a single incident plane wave

Abstract

In this paper, we give a positive answer to a challenging open problem for recovering unknown obstacle (which is usually referred to as a scatterer) by acoustic wave probe associated to the Helmholtz equation. We show that the acoustic scattering amplitude $A(\beta, {\alpha}_0, k_0)$, known for all ${\beta}\in {\mathbb S}^2$, where ${\mathbb {S}}^2$ is the unit sphere in ${\mathbb{R}}^3$, ${{\alpha}}_0\in {\mathbb{S}}^2$ is fixed, $k_0>0$ is fixed, determines the obstacle $D$ and the boundary condition on $\partial D$ uniquely (The boundary condition on $\partial D$ is either the Dirichlet, or Neumann, or the impedance one).