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On essential and continuous spectra of the linearized water-wave problem in a finite pond. (English) Zbl 1191.35016
The authors show that the spectrum of the Laplace equation with the Steklov spectral boundary condition, in the connection of the linearized theory of water-waves, can have a nontrivial essential component even in case of a bounded basin with a horizontal water surface. The appearance of the essential spectrum is caused by the boundary irregularities of the type of a rotational cusp or a cuspidal edge. In a previous paper the authors have proven a similar result for the Steklov spectral problem in a bounded domain with a sharp peak.

MSC:
35A18 Wave front sets in context of PDEs
37A30 Ergodic theorems, spectral theory, Markov operators
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35P05 General topics in linear spectral theory for PDEs
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