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Radiation conditions at the top of a rotational cusp in the theory of water-waves. (English) Zbl 1267.76013
Summary: We study the linearized water-wave problem in a bounded domain (e.g. a finite pond of water) of \({\mathbb R}^3\), having a cuspidal boundary irregularity created by a submerged body. In earlier publications the authors discovered that in this situation the spectrum of the problem may contain a continuous component in spite of the boundedness of the domain. Here, we proceed to impose and study radiation conditions at a point \({\mathcal O}\) of the water surface, where a submerged body touches the surface (see Fig. 1). The radiation conditions emerge from the requirement that the linear operator associated to the problem be Fredholm of index zero in relevant weighted function spaces with separated asymptotics. The classification of incoming and outgoing (seen from \({\mathcal O}\)) waves and the unitary scattering matrix are introduced.

MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35J25 Boundary value problems for second-order elliptic equations
35P99 Spectral theory and eigenvalue problems for partial differential equations
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