Solutions to the Eshelby conjectures and construction of special inclusions

We find a way of constructing special inclusions by solving variational inequalities. As a side result, the Eshelby conjectures, which asserts that uniform eigenstress induces uniform elastic strain if and only if the inclusion is an ellipsoid, are solved. In a periodic setting, we can construct optimal ordered structures in the sense of attaining the Hashin-Shtrikman bounds. These works have been submitted and preprints are available at http://www.its.caltech.edu/~liulp/. Examples of multiply-connected inclusion with Eshelby uniformity property are shown below, see the papers for more examples and description of numerical schemes.