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Quadrature surfaces as free boundaries. (English) Zbl 0827.31004

The author formulates several results on the direct scattering by a plane domain bounded by the graph of a periodic function. Some of these results are known (and then there are detailed references) and some are quite recent (then there are outlines of proofs). In particular he makes use of a variational formulation of the scattering problem to derive the linearization with respect to the domain and to study analytic dependence on the wave number and on the incident direction (both turn out to have square root type singularities). This is a nice overview of an important applied problem. The author intends to apply the linearization to solve the inverse problem of finding a domain. Certain uniqueness results in this direction have been recently obtained by A. Kirsch [Inverse Probl. 10, 145-152 (1994; Zbl 0805.35155)] by G. Bao [Inverse Probl. 10, 335-340 (1994; Zbl 0805.35144)] and by H. Ammari [Inverse Probl. 11, 823-833 (1995)].
Reviewer: V.Isakov (Wichita)

MSC:

31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35R35 Free boundary problems for PDEs
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