The basic boundary-contact problems of oscillation are considered for a two-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed curves. Asymptotic formulas for the distribution of eigenfunctions and eigenvalues of the considered problems are derived using the correlation method.

Abstract. The basic three-dimensonal boundary-contact dynamic problems are considered for a piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using the Fourier method, the considered problems are proved to be solvable under much weaker restrictions on the initial data of the problems as compared with other methods.1. Two well-known methods -the Laplace transform and the Fourier method -are widely used in investigating dynamic problems. In the works by V. Kupradze and his pupils the Laplace transform method was used to prove the existence of classical solutions of the basic three-dimensional boundary and boundary-contact dynamic problems of elasticity. Based on some results from these works, in this paper we use the Fourier method to show that the basic three-dimensional boundary-contact dynamic problems of elasticity are solvable in the classical sense. We have succeeded in weakening considerably the restrictions imposed on the data of the problems as compared with the Laplace transform method. Detailed consideration is given to the second basic problem. The other problems are treated similarly.2. Throughout the paper we shall use the following notation:2 ) 1/2 is the distance between the points x and y; D 0 ⊂ R 3 is a finite domain bounded by closed surfaces S 0 , S 1 , . . . , S m of the class Λ 2 (α), 0 < α ≤ 1, [1]; note that S 0 covers all other S k , while these latter surfaces do not cover each other and1991 Mathematics Subject Classification. 73B30, 73C25. Key words and phrases. Fourier method, boundary-contact dynamic problems of elasticity, classical solutions, Green's tensor, matrix of fundamential solutions. 559

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We consider the first main dynamic boundary value problem for a three-dimensional piecewise homogeneous hemitropic micropolar medium. By using the Fourier method under sufficiently general assumptions, we prove the solvability of the problem in the classical sense.