Asymptotic analysis of vibrating system containing stiff-heavy and flexible-light parts.pdf

a r X i v : 0 7 1 2 .0 7 5 4 v 1 [ m a t h .S P ] 5 D e c 2 0 0 7 N. Babych1, Yu. Golovaty2 ASYMPTOTIC ANALYSIS OF VIBRATING SYSTEM CONTAINING STIFF-HEAVY AND FLEXIBLE-LIGHT PARTS Abstract. A model of strongly inhomogeneous medium with simultaneous perturbation of rigidity and mass density is studied. The medium has strongly contrasting physical character- istics in two parts with the ratio of rigidities being proportional to a small parameter ε. Addi- tionally, the ratio of mass densities is of order ε?1. We investigate the asymptotic behaviour of spectrum and eigensubspaces as ε → 0. Complete asymptotic expansions of eigenvalues and eigenfunctions are constructed and justified. We show that the limit operator is nonself-adjoint in general and possesses two-dimensional Jordan cells in spite of the singular perturbed problem is associated with a self-adjoint operator in appropriated Hilbert space Lε. This may happen if the metric in which the problem is self-adjoint depends on small parameter ε in a singular way. In particular, it leads to a loss of completeness for the eigenfunction collection. We describe how root spaces of the limit operator approximate eigenspaces of the perturbed operator. Introduction We consider a model of strongly inhomogeneous medium consisting of two nearly homogeneous components. Assuming a strong contrast of the corresponding stiffness coefficients k1 ? k2, we get that their ratio k1/k2 has a small order, which we denote by ε. In general, the mass densities r1 and r2 in two parts could be quite different as well or could be the same. We model this assuming that the density ratio r1/r2 is proportional to ε ?m. We investigate how the resonance vibrations of the medium change if the parameter ε tends to 0. In the one-dimensional case we consider the spectral problem d dx ( kε(x) duε dx ) + λε rε(x)uε = 0 in (a, b), α1u ′ ε(a) + α0uε(a) = 0, β1u ′ ε(b) + β0uε(b) = 0, where (a, b) is an interval in R containing the origin and kε(x) =