We investigate the solvability of three-dimensional dynamical mixed boundary value problems of electro-magneto-elasticity theory for homogeneous anisotropic bodies with interior cracks. Using the Laplace transform technique, the potential method, and the theory of pseudodifferential equations, we prove the existence and uniqueness theorems and analyze asymptotic properties of solutions near the crack edges and near the lines where the different boundary conditions collide.

We investigate asymptotic properties of solutions to mixed boundary value problems of thermopiezoelectricity (thermoelectroelasticity) for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. The singularities and asymptotic behaviour of the mechanical, thermal and electric fields are analysed near the crack edges and near the curves, where the types of boundary conditions change. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well. (2000). Primary 35J55; Secondary 74F05, 74F15, 74B05. Mathematics Subject Classification

We investigate three-dimensional transmission problems related to the interaction of metallic and piezoelectric ceramic bodies. We give a mathematical formulation of the physical problem when the metallic and ceramic sub-domains are bonded along some proper parts of their boundaries. The corresponding nonclassical mixed boundary-transmission problem is reduced by the potential method to an equivalent nonselfadjoint strongly elliptic system of pseudo-differential equations on manifolds with boundary. We investigate the solvability of this system in different function spaces. On the basis of these results we prove uniqueness and existence theorems for the original boundary-transmission problem. We study also the regularity of the electrical and mechanical fields near the curves where the boundary conditions change and where the interfaces intersect the exterior boundary. The electrical and mechanical fields can be decomposed into singular and more regular terms near these curves. A power of the distance from a reference point to the corresponding edge-curves occurs in the singular terms and describes the regularity explicitly. We compute these complex-valued exponents and demonstrate their dependence on the material parameters.

Dirichlet- and Neumann-type boundary value problems of statics are considered in three-dimensional domains with cuspidal edges filled with a homogeneous anisotropic medium. Using the method of the theory of a potential and the theory of pseudodifferential equations on manifolds with boundary, we prove the existence and uniqueness theorems in Besov and Bessel-potential spaces, and study the smoothness and a complete asymptotics of solutions near the cuspidal edges.

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