Deformations with finitely many gradients and stability of quasiconvex hulls

We confirm a conjecture by J.M. Ball and R.D. James about the existence of Lipschitz maps using finitely many gradients without any rank-one connection. For this purpose, we derive a new stability result for quasiconvex hulls which answers a question by Kewei Zhang. The final construction of the functions is based on a new argument which reduces the existence of solutions of partial differential inclusions ∇f∈K to a very natural stability property. In this way our argument unifies and explains the power of both the convex integration method and the present Baire category approach to such existence questions.