2018/10/09 by Н. В. Кузнецов, Kuznetsov, Nikolay
Mathematics · Earth and Planetary Sciences · #Differential Equations and Numerical Methods #Numerical methods in inverse problems #Aquatic and Environmental Studies
paper · pdf · doi:10.48550/arxiv.1810.03998
The linear boundary value problem under consideration describes time-harmonic\nmotion of water in a horizontal three-dimensional layer of constant depth in\nthe presence of an obstacle adjacent to the upper side of the layer (floating\nbody). This problem for a complex-valued harmonic function involves mixed\nboundary conditions and a radiation condition at infinity. Under rather general\ngeometric assumptions the existence of a unique solution is proved for all\nvalues of the nonnegative problem's parameter related to the frequency of\noscillations. The proof is based on the representation of solution as a sum of\nsimple- and double-layer potentials with densities distributed over the\nobstacle's surface, thus reducing the problem to an indefinite\nintegro-differential equation. The latter is shown to be soluble for all\ncontinuous right-hand side terms for which purpose S.~G. Krein's theorem about\nindefinite equations is used.\n