M. Mrevlishvili and D. Natroshvili

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

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