Asymptotic expansion of eigenelements of the Laplace operator in a domain with a large number of ‘light’ concentrated masses sparsely situated on the boundary. Two-dimensional case

This paper looks at eigenoscillations of a membrane containing a large number of concentrated masses on the boundary. The asymptotic behaviour of the frequencies of eigenoscillations is studied when a small parameter characterizing the diameter and density of the concentrated masses tends to zero. Asymptotic expansions of eigenelements of the corresponding problems are constructed and the expansions are accurately substantiated. The case where the diameter of the masses is much smaller than the distance between them is investigated under the assumption that the limit boundary condition is still a Dirichlet condition. Introduction The behaviour of bodies with singular density has attracted the attention of scientists for many years. They have studied the influence of concentrated masses (singular lumps) on the variation of the frequencies of eigenoscillations of such bodies. This question proved to be rather too difficult for the mathematics available at the end of the 19th century. We must give due credit to a pioneering paper from the beginning of the 20th century [1]; its author investigated the problem of the oscillation of a string loaded with concentrated masses. This paper was far ahead of its time and was forgotten for some years. It was only after asymptotic methods appeared that the interest of researchers returned to problems with concentrated masses and it became possible to investigate problems with singular density adequately. The author of [2] considered the problem for the Laplace operator with Dirichlet boundary conditions in the three-dimensional case where the mass attached to the system is concentrated in an e-neighbourhood of an interior point, e being a small parameter describing the concentration and size of the mass. In that paper the methods of spectral perturbation theory were used. Another approach was proposed in [3, 4, 5, 6, 7]. As has already frequently been pointed out, a new basic parameter for oscillatory systems with locally attached masses was introduced in these papers, namely, the ratio of the attached mass to the mass of the whole system. This approach has long been firmly established in research papers. It turns out that the introduction of this parameter made it possible not only to analyse the 2000 Mathematics Subject Classification. Primary 35J25; Secondary 35B25, 35B27, 35B40.