We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. 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>2. 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. For the critical size ε=O(η2), the asymptotic behavior of the eigenvalues of order O(εm−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.

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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.

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Using the unfolding technique, developed in reactor physics, we have developed a method for calculating the frequency distribution function for materials starting from the measured data on the temperature variation of specific heat at constant volume. As examples we have considered the cases of the following materials: graphite, selenium, tellurium, polyethylene and SiO, glass. In the case of graphite the frequency distribution function has been used to calculate the low-energy neutron scattering cross section with encouraging results. K X = Y(3)

The stiff problem here considered models the vibrations of a body consisting of two materials, one of them very stiff with respect to the other. We study the asymptotic behavior of the eigenvalues and eigenfunctions of the corresponding spectral problem, when the stiffness constant of only one of the materials tends to 0. We show that the associated operator has a discrete spectrum "converging", in a certain sense, towards a continuous spectrum in [0,∞) corresponding to an operator. We also provide information on the structure of the eigenfunctions associated with the high frequencies.

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