A stationary isotropic thermoelastic field is described by a set of defining relations: 1) the Poisson equation relating the temperature field to volumetric heat sources; 2) Cauchy formulas; 3) Duhamel-Neumann law; 4) equilibrium equations. The linear resolving equations of Lame and Poisson allow us to decompose the solution into the sum of a partial solution and a solution to a problem with adjusted boundary conditions. In the case of volumetric factors of the polynomial type, the partial solution is written out exactly. The interest lies in solving a homogeneous problem. A convenient classification of problems is proposed for linear and piecewise linear boundary conditions in which temperature and mechanical characteristics are generally related. The procedures for forming separable bases of spaces of internal and boundary states in the problems of thermostatics, elastostatics, and thermoelastostatics for regions of arbitrary geometric configuration are strictly prescribed. In isomorphic Hilbert spaces of internal and boundary states the values of the scalar products coincide for isomorphic pairs of elements. The method of boundary states forms an infinite system of linear equations with respect to the coefficients of the Fourier series expansion of the solution. The choice of the size of the truncated basis is based on the Bessel inequality. The accuracy of the solution is estimated by the inconsistency of the constructed boundary state with boundary conditions. As an example, a problem of class "1N"is considered, combining the boundary conditions of the first basic elasticity problem (surface forces) and the Neumann conditions for a temperature field, formulated for a spherical layer weakened by a spherical cavity. Various combinations of geometric parameters of the outer boundary, the central cavity, and the perturbing cavity are considered. The results are analyzed and conclusions are done. 

Victor B. Penkov, Doctor of Physical and Mathematical Sciences, Professor, Professor of the Department of General Mechanics; e-mail: vbpenkov@mail.ru;

Lyubov V. Levina, Candidate of Physical and Mathematical Sciences, Associate Professor, Associate Professor of the Department of Applied Mathematics; e-mail: satalkina_lyubov@mail.ru;

Maxim Y. Levin, Doctor of Technical Sciences, Professor of the Department of Physics and Biomedical Engineering; e-mail: lmu@list.ru;

Valentina V. Zatonskaya, Student of the Department of Applied Mathematics; e-mail: zatonskaya.valentina@yandex.ru