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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 3 with DC 2 from its scattering amplitude 𝒜(σ,ω) corresponding to the Helmholtz equation Δu+k 2 u=0 outside a soft (u=0 on D) or hard (u/ν=0 on D) impenetrable obstacle D or to the equation div(au)+k 2 u=0 in 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:
35R30Inverse problems for PDE
35P25Scattering theory (PDE)
78A25General electromagnetic theory