On continuous spectrum of the linearised water-wave problem in bounded domains

Abstract

We study the spectral problem of the linearised theory of water-waves, in a bounded domain with cuspidal edge. We show that the continuous spectrum of the problem is the set of non-negative real numbers, if the sharpness exponent is large. 1. Introduction and the main result 1.1. Preface. In the linearised theory of water-waves the wave solutions are described by velocity potentials which satisfy a mixed boundary value problem for the Laplace equation. In particular Steklov spectral boundary condition is posed on the horizontal water surface (see (1.2)-(1.4) below, and (19, 5, 6) for the physical background), and the spectral parameter is proportional to the frequency of the wave. In unbounded domains the continuous spectrum of this linearised water-wave problem is typically non-empty, a fact which is related to the existence of propagating waves. In this paper we study the continuous spectrum of the linearised water-wave problem in bounded domains. Our domains will have cuspidal edge at the shoreline, which causes the continuous spectrum to appear. We show that if the sharpness exponent m is greater than 2, then the continuous spectrum is maximal in the sense that it consists of the whole set of non-negative real numbers. Previously this problem has been studied in similar domains, but with different values of the sharpness exponent. In (16) it was shown that for m 0. This result was proved in (8) and an analogous result for a domain with submerged object touching the water-surface at one point, was proved in (16) (see also the remarks at the end of this paper for further details). 1.2. Formulation of the problem. In the sequel we denote a point x ∈ R 3 as x = (y1, y2, z) in order to make a distinction between vertical and horizontal