Asymptotic Distribution of Eigenfunctions and Eigenvalues of the Basic Boundary-contact Oscillation Problems of the Classical Theory of Elasticity

The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman’s method, the asymptotic formulas for the distribution of eigenfunctions 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 asymptotic 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 formulas for the distribution of eigenfunctions of the boundary value oscillation problems are obtained for isotropic and anisotropic elastic bodies using integral 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 twoand threedimensional 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. 1991 Mathematics Subject Classification. 73C02.