**M. Mrevlishvili and D. Natroshvili**

##
Investigation of Interior and Exterior Neumann-Type Static Boundary-Value
Problems of Thermo-Electro-Magneto Elasticity Theory

**abstract:**

We investigate the three-dimensional interior and exterior Neumann-type
boundary-value problems of statics of the thermo-electro-magneto-elasticity
theory. We construct explicitly the fundamental matrix of the corresponding
strongly elliptic non-self-adjoint $6\times 6$ matrix differential operator and
study their properties near the origin and at infinity. We apply the potential
method and reduce the corresponding boundary-value problems to the equivalent
system of boundary integral equations. We have found efficient asymptotic
conditions at infinity which ensure the uniqueness of solutions in the space of
bounded vector functions.

We analyze the solvability of the resulting boundary integral equations in the Hölder
and Sobolev-Slobodetski spaces and prove the corresponding existence theorems.
The necessary and sufficient conditions of solvability of the interior
Neumann-type boundary-value problem are written explicitly.

**Mathematics Subject Classification:**
35J57, 74F05, 74F15, 74B05

**Key words and phrases:** Thermo-electro-magneto-elasticity, boundary-value
problem, potential method, boundary integral equations, uniqueness theorems,
existence theorems