Uniqueness of the Inverse Conductive Scattering Problem

1 . I N T R O D U C T I O N During the last two decades or so, inverse scattering problems for the Helmholtz equation have enjoyed a remarkable degree of popularity, both in pure and applied contexts (see the monograph [1] and the references therein). One of the most important theoretical considerations in inverse scattering problems is uniqueness. Different approaches have been proposed [2-10]. Many of them are based on the finite dimensionality of the eigenspaces of the negative Laplacian in bounded domains. In [5], Isakov proposed a variational approach which was later extended and simplified by Kirsch and Kress [7] by means of boundary integral equation methods. The purpose of this paper is to adapt the method of [7] to the inverse conductive problem [8,9,11,12]. This problem, which generalizes the more classical transmission problem [13], arises in geophysical models in which an obstacle is covered by a thin layer of high conductivity [14]. The scattering of acoustic time-harmonic waves by a penetrable bounded conductive obstacle D, which is assumed to be an open and bounded region in R 3 with R3 \D connected, can be modelled by a boundary value problem as follows. We look for a pair of functions U E C 2 ( R 3 ~ ) CI CI(RZ\D) and v E C2(D) N CI(D) satisfying the Helmholtz equations A u + k2u = O, in RZ\D, (1.1) Av + kgv = O, in D, with the boundary conditions u # v = f , Ou Ov on OD, ( 1 . 2 ) Ou Ov = Au + g, where u is the superposition of the given incident plane wave ui(x) = e ik='d and scattered wave u s, i.e., u(x) = ui(x) + uS(x), and the scattered wave u a is required to satisfy the Sommerfeld radiation condition ( Ou" i k u ' ) = 0 (1.3 / i li-,m Ixl k av