Complete WKB asymptotics of high frequency vibrations in a stiff problem

Asymptotic behaviour of eigenvalues and eigenfunctions of a stiff problem is described in the case of the fourth-order ordinary differential operator. Considering the stiffness coefficient that depends on a small parameter ε and vanishes as ε → 0 on a subinterval, we prove the existence of low and high frequency resonance vibrations. The low frequency vibrations admit the power series expansions on ε but this method is not applicable to the description of high frequency vibrations. However, the nonclassical asymptotics on ε of the high frequency vibrations were constructed using the WKB method. MSC: Primary 34E20; Secondary 74K10 Introduction and main results. Stiff vibrating systems belong to a class of systems with singularly perturbed potential energy. Stiff problems are known in particular as boundary value problems for differential equations with very contrasting values of coefficients in different sub-domains. They relate to modelling vibrations of elastic systems consisting of two (or more) materials with one of them being very stiff with respect to the other. For the first time the stiff problems were investigated by J. L. Lions [1]. However, it is also of interest to describe the asymptotic behaviour of spectral properties for the stiff problems. These system has two types of eigenvibrations, namely low frequency vibrations and high frequency ones. From a physical viewpoint we can postulate that two kinds of eigenvibrations can appear: one for the stiffer structure and the other for the softer structure. Different aspects of the spectral stiff problems are considered in [2]–[10] with the best general reference being [8]. The asymptotic behaviour of the low frequency vibrations has been widely studied with different techniques [2], [3], [7] and [9]. In this paper, following [10] we consider the phenomenon of high frequency vibrations. The leading terms of high frequency vibrations for different problems were constructed in [7-9]. Information on the behaviour of high frequency vibrations was also provided in [10]. Studying the stiff problem for the forth-order differential operator, we construct the complete asymptotic expansions of high frequency vibrations using the WKB technique. Published in Mat. Stud. 14, no.1 (2001): 59–72 English is improved in 2008