Ellipsoidal inclusion with a shell in an anisotropic medium subjected to a uniform electric field

An electrostatic problem has been solved for a dielectric inclusion consisting of an anisotropic core and a shell immersed in a homogeneous anisotropic dielectric medium (matrix) subjected to a uniform electric field. The outer boundaries of the core and shell are assumed to be ellipsoids, which are confocal after a linear nonorthogonal transformation that eliminates the anisotropy of the dielectric properties of the shell. Analytical expressions have been obtained for the potential and the electric field strength in the matrix, in the shell and core, and an expression for the inclusion polarizability tensor. A special case of inclusion with an isotropic shell is considered. The expressions obtained are applied to the case of an anisotropic sphere with an isotropic shell immersed in an anisotropic medium. It is also shown that in the limiting case of a homogeneous ellipsoidal inclusion in an anisotropic medium, the obtained result agrees with known solutions.