Inclusion Pairs Satisfying Eshelby's Uniformity Property

Eshelby conjectured that if for a given uniform loading the field inside an inclusion is uniform, then the inclusion must be an ellipse or an ellipsoid. This conjecture has been proved to be true in two and three dimensions provided that the inclusion is simply connected. In this paper we provide an alternative proof of Cherepanov’s result that an inclusion with two components can be constructed inside which the field is uniform for any given uniform loading for two-dimensional conductivity or for anti-plane elasticity. For planar elasticity, we show that the field inside the inclusion pair is uniform for certain loadings and not for others. We also show that the polarization tensor associated with the inclusion pair lies on the lower Hashin-Shtrikman bound, and hence the conjecture of Pólya-Szegö is not true among non-simply connected inclusions. As a consequence, we construct a simply connected inclusion, which is nothing close to an ellipse, but in which the field is almost uniform.