Differential inclusions and polycrystals

We study the differential inclusion $Du\in K$, where $K$ is an unbounded and rotationally invariant subset of the real symmetric $3\times 3$ matrices. We exhibit a subset of all possible average fields. The corresponding microgeometries are laminates of infinite rank. The problem originated in the search for the effective conductivity of polycrystalline composites. In the latter context, our result is an improvement of the previously known bounds established by Nesi $\mathrm{\&}$ Milton, hence proving the optimality of a new full-measure class of microgeometries.