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St. Petersburg Mathematical Journal

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

Author: N. Kuznetsov
Original publication: Algebra i Analiz, tom 31 (2019), nomer 3.
Journal: St. Petersburg Math. J. 31 (2020), 521-531
MSC (2010): Primary 31B10; Secondary 76B15, 35Q35
Published electronically: April 30, 2020
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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 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.

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Additional Information

N. Kuznetsov
Affiliation: Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol′shoy pr. 61, 199178 St. Petersburg, Russian Federation

Keywords: Potential representations, integral operators, integro-differential equation
Received by editor(s): August 20, 2018
Published electronically: April 30, 2020
Dedicated: In memoriam of my mentor S. G. Mikhlin
Article copyright: © Copyright 2020 American Mathematical Society