Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality

In this paper, the problem of uniqueness concerning the inverse scattering problem in two-dimensional acoustics for one incident plane wave and one wavenumber is considered. Using the fact that the optimal lower estimate for the eigenvalues of the Laplacian for a domain is given by the Faber-Krahn inequality, which relates the area of the domain to the first eigenvalue of a disc of equal area, it is proved that the uniqueness holds under the restriction that the possible scatterers do not deviate 'too much' in area. Also an improvement of the results due to Colton and Sleeman (1983 IMA J. Appl. Math. 31253-9) is presented, based on the a priori information that the unknown scatterers lie inside a given ball and that the far field is known for a finite number of incident plane waves. The main advantage of this work is that it provides uniqueness for the half number of the needed incoming waves in Colton and Sleeman (1983 IMA J. Appl. Math. 31 253-9). For the case of one incoming plane wave uniqueness is satisfied if the scatterers are contained in a ball of radius R such that kR < t(10) similar or equal to 4.4939, where t(10) is the first root of the spherical Bessel function of first order j(1) (x). The result of local uniqueness is applied to a class of star-shaped scatterers which are smooth perturbations of discs with common centre in R-2 for one incident plane-wave direction. Numerical implementations are presented for smooth perturbations of discs.