Three-dimensional basic problems of statics, pseudo-oscillations, general dynamics and steady state oscillations of the thermoelasticity of isotropic bodies have been completely investigated by many authors. In particular, exterior steady state oscillation problems have been studied on the basis of Sommerfeld-Kupradze radiation conditions in the thermoelasticity, and the uniqueness theorems were proved with the help of the well-known Rellich's lemma, since the components of the displacement vector and the temperature in the isotropic case can be represented as a sum of metaharmonic functions . Unfortunately, the methods of investigation of thermoelastic steady state oscillation problems developed for the isotropic case are not applicable in the case of general anisotropy. This is stipulated by a very complicated form of the corresponding characteristic equation which plays a significant role in the study of far field behaviour of solutions to the oscillation equa- tions. We note that the basic and crack type boundary value problems (BVPs) for the pseudo-oscillation equations of the thermoelasticity theory in the anisotropic case are considered in [3,14]. To the best of the authors' knowledge the problems of thermoelastic steady oscillations for anisotropic bodies have not been treated in the scientific literature. In the present paper we will consider a wide class of basic and mixed type BVPs for the equations of thermoelastic steady state oscillations. We will formulate thermoelastic radiation conditions for an anisotropic medium (the generalized Sommerfeld-Kupradze type radiation conditions) and prove the uniqueness theorems in corresponding spaces. To derive these conditions we have essentially applied results of Vainberg. Further, using the potential method and the theory of pseudodifferential equations on manifolds we will prove existence theorems in various functional spaces and establish the smoothness properties of solutions.

The two-dimensional interface crack problem is investigated for anisotropic bodies in the Comninou formulation. It is established that, as in the isotropic case, properly incorporating contact zones at the crack tips avoids contradictions connected with the oscillating asymptotic behaviour of physical and mechanical characteristics leading to the overlapping of material. Applying the special integral representation formulae for the displacement field the problem in question is reduced to the scalar singular integral equation with the index equal to -1. The analysis of this equation is given. The comparison with the results of previous authors shows that the integral equations corresponding to the interface crack problems in the anisotropic and isotropic cases are actually the same from the point of view of the theoretical and numerical analysis.

Three-dimensional basic problems of statics, pseudo-oscillations, general dynamics and steady state oscillations of the thermoelasticity of isotropic bodies have been completely investigated by many authors. In particular, exterior steady state oscillation problems have been studied on the basis of Sommerfeld-Kupradze radiation conditions in the thermoelasticity, and the uniqueness theorems were proved with the help of the well-known Rellich's lemma, since the components of the displacement vector and the temperature in the isotropic case can be represented as a sum of metaharmonic functions . Unfortunately, the methods of investigation of thermoelastic steady state oscillation problems developed for the isotropic case are not applicable in the case of general anisotropy. This is stipulated by a very complicated form of the corresponding characteristic equation which plays a significant role in the study of far field behaviour of solutions to the oscillation equa- tions. We note that the basic and crack type boundary value problems (BVPs) for the pseudo-oscillation equations of the thermoelasticity theory in the anisotropic case are considered in [3,14]. To the best of the authors' knowledge the problems of thermoelastic steady oscillations for anisotropic bodies have not been treated in the scientific literature. In the present paper we will consider a wide class of basic and mixed type BVPs for the equations of thermoelastic steady state oscillations. We will formulate thermoelastic radiation conditions for an anisotropic medium (the generalized Sommerfeld-Kupradze type radiation conditions) and prove the uniqueness theorems in corresponding spaces. To derive these conditions we have essentially applied results of Vainberg. Further, using the potential method and the theory of pseudodifferential equations on manifolds we will prove existence theorems in various functional spaces and establish the smoothness properties of solutions.

Three-dimensional basic and mixed interface problems of the mathematical theory of thermoelastic pseudo-oscillations are considered for piecewise homogeneous anisotropic bodies. Applying the method of boundary potentials and the theory of pseudodifferential equations existence and uniqueness theorems of solutions are proved in the space of regular functions C^(k+ alpha) and in the Bessel-potential (H^(s)_(p)) and Besov (B^(s)_(p,q)) spaces. In addition to the classical regularity results for solutions to the basic interface problems, it is shown that in the mixed interface problems the displacement vector and the temperature are Hölder continuous with exponent 0<alpha<1/2.

Three-dimensional mathematical problems of the interaction between thermoelastic and scalar oscillation fields are considered in a general anisotropic case. An elastic structure is assumed to be a bounded homogeneous anisortopic body occupying domain $\Omega^+\sub\R^3$ , where the thermoelastic field is defined, while in the physically anisotropic unbounded exterior domain $\Omega^-=\R^3\\ \overline{\Omega^+}$ there is defined the scalar field. These two fields satisfy the differential equations of steady state oscillations in the corresponding domains along with the transmission conditions of special type on the interface $\delta\Omega^{+-}$. Uniqueness and existence theorems, for the non-resonance case, are proved by the reduction of the original interface problems to equivalent systems of boundary pseudodifferential equations ($\Psi DEs$) . The invertibility of the corresponding matrix pseudodifferential operators ($\Psi DO$) in appropriate functional spaces is shown on the basis of generalized Sommerfeld-Kupradze type thermoradiation conditions for anisotropic bodies. In the resonance case, the co-kernels of the $\Psi DOs$ are analysed and the efficent conditions of solvability of the transmission problems are established.

The generalized radiation conditions at infinity of Sommerfeld-Kupradze type are established in the theory of thermoelasticity of anisotropic bodies. Applying the potential method and the theory of pseudodifferential equations on manifolds the uniqueness and existence theorems of solutions to the basic three-dimensional exterior boundary value problems are proved and representation formulas of solutions by potential type integrals are obtained.

The two-dimensional interface crack problem is investigated for anisotropic bodies in the Comninou formulation. It is established that, as in the isotropic case, properly incorporating contact zones at the crack tips avoids contradictions connected with the oscillating asymptotic behaviour of physical and mechanical characteristics leading to the overlapping of material. Applying the special integral representation formulae for the displacement field the problem in question is reduced to the scalar singular integral equation with the index equal to -1. The analysis of this equation is given. The comparison with the results of previous authors shows that the integral equations corresponding to the interface crack problems in the anisotropic and isotropic cases are actually the same from the point of view of the theoretical and numerical analysis.

CHAPTER I. Basic Equations. Fundamental Matrices. Thermo-Radiation Conditions 1. Basic differential equations of thermoelasticity theory 2. Fundamental matrices 3. Thermo-radiating conditions. Somigliana type integral representations CHAPTER II. Formulation of Boundary Value and Interface Problems 4. Functional spaces 5. Formulation of basic and mixed BVPs 6. Formulation of crack type problems 7. Formulation of basic and mixed interface problems CHAPTER III. Uniqueness Theorems 8. Uniqueness theorems in pseudo-oscillation problems 9. Uniqueness theorems in steady state oscillation problems CHAPTER IV. Potentials and Boundary Integral Operators 10. Thermoelastic steady state oscillation potentials 11. Pseudo-oscillation potentials CHAPTER V. Regular Boundary Value and Interface Problems 12. Basic BVPs of pseudo-oscillations 13. Basic exterior BVPs of steady state oscillations 14. Basic interface problems of pseudo-oscillations 15. Basic interface problems of steady state oscillations CHAPTER VI. Mixed and Crack Type Problems 16. Basic mixed BVPs 17. Crack type problems 18. Mixed interface problems of steady state oscillations 19. Mixed interface problems of pseudo-oscillations

The generalized radiation conditions at infinity of Sommerfeld-Kupradze type are established in the theory of thermoelasticity of anisotropic bodies. Applying the potential method and the theory of pseudodifferential equations on manifolds the uniqueness and existence theorems of solutions to the basic three-dimensional exterior boundary value problems are proved and representation formulas of solutions by potential type integrals are obtained.

Three-dimensional basic and mixed interface problems of the mathematical theory of thermoelastic pseudo-oscillations are considered for piecewise homogeneous anisotropic bodies. Applying the method of boundary potentials and the theory of pseudodifferential equations existence and uniqueness theorems of solutions are proved in the space of regular functions C^(k+ alpha) and in the Bessel-potential (H^(s)_(p)) and Besov (B^(s)_(p,q)) spaces. In addition to the classical regularity results for solutions to the basic interface problems, it is shown that in the mixed interface problems the displacement vector and the temperature are Hölder continuous with exponent 0<alpha<1/2.