Eugenia Pérez

ON VIBRATIONS OF A BODY WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. The density, which depends on a small parameter e, is of order O(1) out of certain regions where it is O(e–m) with m>2. These regions, the concentrated masses with diameter O(e), are located near the boundary, at mutual distances O(η), with η=η(e)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. For the critical size e=O(η2), the asymptotic behavior of the eigenvalues of order O(em−2) is described via a Steklov problem, where the ‘mass’ is localized on the boundary, or through the eigenvalues of a local problem obtained from the micro-structure of the problem. We use the techniques of the formal asymptotic analysis in homogenization to determine both problems. We also use techniques of convergence in homogenization, Semigroups theory, Fourier and Laplace transforms and boundary values of analytic functions to prove spectral convergence. In the same framework we study the case m=2 as well as the case when other boundary conditions are imposed on ∂Ω.

Local problems for vibrating systems with concentrated masses: a review

Abstract In this review we collect certain results obtained in the last decades on vibrating systems with concentrated masses. In particular, we show the connection of the eigenvalues and eigenfunctions of the local problem with the low and high frequency vibrations of the original problem. To cite this article: M. Lobo, E. Perez, C. R. Mecanique 331 (2003).

VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter e, is of order O(1) out of certain regions where it is O(e−m) with m>0. These regions, the concentrated masses with diameter O(e), are located near the boundary, at mutual distances O(η), with η=η(e)→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, 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 e and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.

On the eigenfunctions associated with the high frequencies in systems with a concentrated mass

Abstract We consider the vibrations of a system consisting of the domain Ω of R N , N=2,3 , that contains a small region with diameter depending on a small parameter e . The density is of order O (e −m ) in the small region, the concentrated mass, and it is O (1) outside; m is a parameter, m≥2 . We study the asymptotic behaviour, as e→0 , of the eigenvalues of order O (1) , the high frequencies when m>2 , and the corresponding eigenfunctions of the associated spectral problem. We provide information on the structure of these eigenfunctions. We also check theoretical results with explicit calculations for the dimensions N=1 and N=2 and give correcting terms for the eigenfunctions.

The skin effect in vibrating systems with many concentrated masses

We address the asymptotic behaviour of the vibrations of a body occupying a domain . The density, which depends on a small parameter

On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems

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ON VIBRATING MEMBRANES WITH VERY HEAVY THIN INCLUSIONS

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ON THE WHISPERING GALLERY MODES ON INTERFACES OF MEMBRANES COMPOSED OF TWO MATERIALS WITH VERY DIFFERENT DENSITIES

O(1)

A skin effect for systems with many concentrated masses

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Boundary homogenization in perforated domains for adsorption problems with an advection term

O(\varepsilon^{-m})