Neutral coated inclusions of finite conductivity

Abstract

We discuss the conductivity of two-dimensional media with coated neutral inclusions of finite conductivity. Such an inclusion, when inserted in a matrix, does not disturb the uniform external field. We are looking for shapes of the core and coating in terms of the conformal mapping ω ( z ) of the unit disc onto coated inclusions. The considered inverse problem is reduced to an eigenvalue problem for an integral equation containing singular integrals over a closed curve L 1 on the transformed complex plane. The conformal mapping ω ( z ) is constructed via eigenfunctions of the integral equation. For each fixed curve L 1 , the boundary of the core is given by the curve ω ( L 1 ). The boundary of the coating is obtained by the mapping of the unit circle. It is justified that any shaped inclusion with a smooth boundary can be made neutral by surrounding it with an appropriate coating. Shapes of the neutral inclusions are obtained in analytical form when L 1 is an ellipse.