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Uniqueness in inverse obstacle scattering. (English) Zbl 0787.35119

The authors study uniqueness of recovery of a bounded simply connected domain \(D \subset \mathbb{R}^ 3\) with \(\partial D \in C^ 2\) from its scattering amplitude \({\mathcal A} (\sigma,\omega)\) corresponding to the Helmholtz equation \(\Delta u+k^ 2u=0\) outside a soft \((u=0\) on \(\partial D)\) or hard \((\partial u/ \partial \nu=0\) on \(\partial D)\) impenetrable obstacle \(D\) or to the equation \(\text{div} (a \nabla u)+ k^ 2 u=0\) in \(\mathbb{R}^ 3\) describing a penetrable obstacle \(D\).
They correct the Schiffer’s proof of uniqueness for soft \(D\) and give a first uniqueness proof for hard \(D\) by using the reviewer’s idea of exploiting singular solutions suggested in the paper [V. Isakov, Commun. Pure Appl. Math. 41, No. 7, 865-877 (1988; Zbl 0676.35082)]. Also they give a simpler proof of uniqueness of a penetrable scatterer than in the reviewer’s previous paper [Commun. Partial Differ. Equations 15, No. 11, 1565-1587 (1990; Zbl 0728.35148)].
For other uniqueness results in the inverse scattering we refer to the book [D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (1992; Zbl 0760.35053)], and to the review paper [V. Isakov, Uniqueness and stability in multidimensional inverse problems, Inverse Problems 9, 579-621 (1993)].
Reviewer: V.Isakov (Wichita)

MSC:

35R30 Inverse problems for PDEs
35P25 Scattering theory for PDEs
78A25 Electromagnetic theory (general)
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