Journal of Mathematical Sciences, Vol. 235, No. 5, December, 202018 INTERACTION OF ELASTIC AND SCALAR FIELDS L. Giorgashvili, G. Karseladze, and G. Sadunishvili UDC 517.958 Abstract. In this work, we consider the problem of interaction of elastic body with scalar field. The general solution of a uniform system of equations (of elasticity theory) for the static case is solved by using the Papkovich representation method. The contact problem is solved by using a special boundary-contact condition, in the case where the contact surface is a stretched spheroid. The uniqueness theorem for the solution is also proved. Solutions are obtained in the form of absolutely and uniformly convergent series. CONTENTS 1. Introduction ............................................ 604 2. Some Auxiliary Formulas and Theorems ............................ 604 3. Statement of the Boundary-Value Problems. Uniqueness Theorems ............. 609 4. Solution of the Problem (C) ................................... 611 References ............................................. 621 1. Introduction One of the fundamental methods of solution of spatial problems of elasticity theory is the Fourier method based on the application of a system of curvilinear components and the subsequent separation of variables in the corresponding differential equations. When solving problems by the Fourier method, we use various representations of equilibrium equa- tions through harmonic, biharmonic, and metaharmonic functions. When solving problems for a spheroid (stretched or compressed), a solution in the Papkovich–Neyber sense seems to be the most convenient one. A problem of elasticity theory for an isotropic ellipsoid under the action of arbitrary axially symmetric forces is solved in [7]. The solutions of the first and second basic problems for an ellipsoid of rotation are obtained in [8, 11]. An axially symmetric problem for a hollow ellipsoid of rotation is solved in [5], while a solution of a contact problem for a stretched spheroid is given in [1, 2]. 2. Some Auxiliary Formulas and Theorems The degenerate ellipsoidal coordinates for a stretched ellipsoid of rotation are defined by the equal- ities x 1 = c sinh η sin ϑ cos ϕ, x 2 = c sinh η sin ϑ sin ϕ, x 3 = c cosh η cos ϑ (see [3]), where c is a constant coefficient, 0 ≤ η< +∞,0 ≤ ϑ ≤ π,0 ≤ ϕ< 2π, and x 1 , x 2 , and x 3 are the Cartesian coordinates of the point x. The coordinate surfaces are stretched ellipsoids of rotation η = const, two-sheet hyperboloids of rotation ϑ = const, and cavities ϕ = const. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applica- tions), Vol. 101, Mathematical Analysis and Mathematical Physics, 2016. 604 1072–3374/2018/2355–0604 c  2018 Springer Science+Business Media New York DOI 10.1007/s10958-018-4086-4