The floating-body problem: an integro-differential equation without irregular frequencies

N. Kuznetsov

Algebra i analiz, Tome 31 (2019) no. 3, pp. 170-183.

Résumé

The linear boundary value problem under consideration describes time-harmonic motion of water in a horizontal three-dimensional layer of constant depth in the presence of an obstacle adjacent to the upper side of the layer (floating body). This problem for a complex-valued harmonic function involves mixed boundary conditions and a radiation condition at infinity. Under rather general geometric assumptions the existence of a unique solution is proved for all values of the problem's nonnegative parameter related to the frequency of oscillations. The proof is based on the representation of a solution as a sum of simple- and double-layer potentials with densities distributed over the obstacle's surface, thus reducing the problem to an indefinite integro-differential equation. The latter is shown to be soluble for all continuous right-hand side terms, for which purpose S. G. Krein's theorem about indefinite equations is used.

Détail

Keywords: potential representations, integral operators, integro-differential equation.

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