The four-state problem and convex integration for linear differential operators

Abstract: We show that the four-state problem for general linear differential operators is flexible. The only flexibility result available in this context is the one for the five-state problem for the curl operator due to B. Kirchheim and D. Preiss, [Section 4.3, Rigidity and Geometry of Microstructures, 2003], and its generalization [Calculus of Variations and Partial Differential Equations, 2017]. To build our counterexample, we extend the convex integration method introduced by S. M\"uller and V. \v Sver\'ak in [Annals of Mathematics, 2003] to linear operators that admit a potential, and we exploit the notion of \emph{large} $T_N$ configuration introduced by C. F\"orster and L. Sz{\'{e}}kelyhidi in [Calculus of Variations and Partial Differential Equations, 2017].