Asymptotic distribution of eigenfunctions and eigenvalues of the basic boundary-contact oscillation problems of the classical theory of elasticity.

Abstract

The basic boundary-contact oscillation problems are con- sidered for a three-dimensional piecewise-homogeneous isotropic elas- tic medium bounded by several closed surfaces. Using Carleman's method, the asymptotic formulas for the distribution of eigenfunc- tions and eigenvalues are obtained. 1. After the remarkable papers of T. Carleman (1-2) the method based on the asymptotic investigation of the resolvent kernel (or of any other function of the considered operator) with a subsequenet use of Tauberian theorems has become quite popular. By generalizing Carleman's method (and combining it with the variational one) A. Plejel (3) derived the asymp- totic formulas for the distribution of eigenfunctions and eigenvalues of the boundary value oscillation problems of classical elasticity. Mention should also be made of T. Burchuladze's papers (4-5), where the asymptotic formu- las for the distribution of eigenfunctions of the boundary value oscillation problems are obtained for isotropic and anisotropic elastic bodies using in- tegral equations and Carleman's method. Further progress in this direction was made by R. Dikhamindzhia (6). He obtained the asymptotic formulas for the distribution of eigenfunctions and eigenvalues for two- and three- dimensional boundary value oscillation problems of couple-stress elasticity which generalize analogous formulas of classical elasticity. In his recent work M. Svanadze (7) derived the asymptotic formulas for oscillation boundary value problems of the linear theory of mixtures of two homogeneous isotropic elastic materials.