Vibrations of a Membrane With Many Concentrated Masses Near the Boundary

Abstract

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending on the value of the parameter m(m>2, m=2 or m<2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.

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 function is π-periodic and even in ξ 1 . By (28) and (31), as ξ 2 → +∞, the asymptotics of the function v 1 (ξ; x 1 ) at infinity are as follows:…”

“…By (28), giving the definition of the function X(ξ) and the boundary-value problem (27), it follows that the function…”

“…By contrast with the papers [47,48,49,50,51,52], it is assumed that the masses are situated on the boundary rather sparsely, as was assumed in [25,26,27,28,29,30,31,32], when the distance between the masses is substantially greater than their diameter. The distance between the masses is assumed to be equal to ε, the diameter of the masses equal to a ε, where a = a(ε) → 0 as ε → 0, and the density is assumed to be equal to ε −m , m < 2.…”

On the vibrations of a plate with a concentrated mass and very small thickness

“…Many papers have been devoted to the study of vibrating systems with one single concentrated mass using di erent techniques: let us mention References [1][2][3][4][5][6][7][8][9] for the Laplace operator, Reference [10] for the three-dimensional elasticity operator, and Reference [11] for a Kirchho plate model. We refer to Reference [12] for several concentrated masses and di erent boundary conditions, and, to References [13][14][15][16][17] for many concentrated masses in note that we obtain a di erent behaviour for the eigenelements ( ; u ) of (1) depending on whether m is m¡2, 26m64 or m¿4 (see (63)). It should be mentioned that these intervals for m, have also been found in Reference [11] for the Kirchho -Love plate model with a concentrated mass; that is, for the fourth-order operator arising in this theory (see Remark 7.3); even though the technique used and results obtained in this paper are very di erent from those in Reference [11].…”

The best Sobolev trace constant in a domain with oscillating boundary

“…We prove some homogenization results when the holes are located on the boundary of the domain. This problem is related with the study of the behavior of solutions of periodic mixed conditions on the boundary, Dirichlet and Neumann (see for example [9,20]), and also, with vibration problems of systems with concentrated masses on the boundary (see for example [18,19]). …”