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

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, [27, Section 4.3], and its generalization [21]. To build our counterexample, we extend the convex integration method introduced by S. Müller and V. Šverák in [33] to linear operators that admit a potential, and we exploit the notion of large TN configuration introduced by C. Förster and L. Székelyhidi in [21].