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

TL;DR

This paper proves the existence and uniqueness of solutions for a complex boundary value problem modeling water motion around a floating obstacle, using integral equations and potential theory, without encountering irregular frequencies.

Contribution

It introduces a novel approach to solving the floating-body problem via indefinite integro-differential equations, extending solution existence to all frequencies.

Findings

Existence and uniqueness of solutions for all frequencies.

Solution representation as layer potentials reduces the problem.

Application of Krein's theorem ensures solvability for continuous data.

Abstract

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 nonnegative problem's parameter related to the frequency of oscillations. The proof is based on the representation of 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.