A straightforward treatment of these problems is given which appears to avoid many of the previously encountered difficulties. Admittedly some generality is lost by assuming that the various associated parameters k j are not space dependent within their respective domains of definition, Dj • Nevertheless, by means of the approach offered here, such problems can be analyzed in just one function space; more general existence and uniqueness theorems can be obtained; there is no need to regularize the operators involved; and, above all, the solutions can be expressed in terms of certain boundary integral equations which, computationally, offer good prospects.

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This paper is concerned with various variational formulations for the fluid -solid interaction problems. The basic approach here is a coupling of field and boundary integral equation methods. In particular, Gårding's inequalities are established in appropriate Sobolev spaces for all the formulations. Existence and uniqueness results of the corresponding weak solutions are given under suitable assumptions.

The problem considered is that of determining the shape of a three-dimensional scattering object, illuminated by an acoustic field, from a knowledge of scattered far-field data. The far-field data are the asymptotic form of the solution of an exterior transmission problem for the Helmholtz equation. The problem is reformulated as an optimisation problem, specifically, finding that surface, in a suitably restricted class, which minimises an appropriate functional of the far field generated by the surface through the solution of the direct problem. Through the use of complete families of solutions, the problem is further reduced to finding a surface which minimises error in satisfying the transmission conditions.