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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 $D\subset {ℝ}^{3}$ with $\partial D\in {C}^{2}$ from its scattering amplitude $𝒜\left(\sigma ,\omega \right)$ corresponding to the Helmholtz equation ${\Delta }u+{k}^{2}u=0$ outside a soft $\left(u=0$ on $\partial D\right)$ or hard $\left(\partial u/\partial \nu =0$ on $\partial D\right)$ impenetrable obstacle $D$ or to the equation $\text{div}\left(a\nabla u\right)+{k}^{2}u=0$ in ${ℝ}^{3}$ 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)].

##### MSC:
 35R30 Inverse problems for PDE 35P25 Scattering theory (PDE) 78A25 General electromagnetic theory