On linear water wave problem in the presence of a critically submerged body

We study the problem of propagation of linear water waves in a deep water in the presence of a critically submerged body (i.e. the body touching the water surface). Assuming uniqueness of the solution in the energy space, we prove the existence of the solution which satisfies the radiation conditions at infinity as well as, additionally, at the cusp point where the body touches the water surface. This solution is obtained by the limiting absorption procedure. Next we introduce a relevant scattering matrix and analyse its properties. Under a geometric condition introduced by Maz'ya, see \cite{M1}, we show that the method of multipliers applies to cusp singularities, thus proving a new important property of the scattering matrix, which may be interpreted as the absence of a version of "full internal reflection". This property also allows us to prove uniqueness and existence of the solution in the functional spaces $H^2_{loc}\cap L^\infty $ and $H^2_{loc}\cap L^p $, $2<p<6$, provided a spectral parameter in the boundary conditions on the surface of the water is large enough. This description of the solution does not rely on the radiation conditions or the limiting absorption principle. This is the first result of this type known to us in the theory of linear wave problems in unbounded domains.