The quasiconvex hull for the five-gradient problem

In 8 Chapter 4.3] Kirchheim and Preiss gave an example of a set K consisting of five 2 × 2 symmetric matrices without rank-one connections, for which there exists a Lipschitz mapping u satisfying ${Du \in K}$ . In the present paper we construct the rank-one convex hull of K. As a corollary we obtain that for each ${F \in {\rm int}\,K^{rc}}$ there exists a Lipschitz mapping u satisfying $$Du \in K\quad{\rm and}\quad u(x) = Fx\,{\rm for}\,x\,\in\,{\partial} \Omega \,.$$ Moreover, we show that the rank-one convex hull of K and the quasiconvex hull of K are equal.