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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 $D \subset \bbfR\sp 3$ with $\partial D \in C\sp 2$ from its scattering amplitude ${\cal A} (\sigma,\omega)$ corresponding to the Helmholtz equation $\Delta u+k\sp 2u=0$ 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 $\text{div} (a \nabla u)+ k\sp 2 u=0$ in $\bbfR\sp 3$ 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 [{\it 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 [{\it D. Colton} and {\it R. Kress}, Inverse acoustic and electromagnetic scattering theory (1992; Zbl 0760.35053)], and to the review paper [{\it V. Isakov}, Uniqueness and stability in multidimensional inverse problems, Inverse Problems 9, 579-621 (1993)].

35R30Inverse problems for PDE
35P25Scattering theory (PDE)
78A25General electromagnetic theory
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