Calculus of Variations and Geometric Measure Theory

M. Sorella - R. Tione

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

created by tione on 23 Jul 2021
modified on 31 Aug 2023


Published Paper

Inserted: 23 jul 2021
Last Updated: 31 aug 2023

Journal: Journal of Functional Analysis
Year: 2021


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. Müller and V. Šverák in (Annals of Mathematics, 2003) to linear operators that admit a potential, and we exploit the notion of large TN configuration introduced by C. Förster and L. Székelyhidi in (Calculus of Variations and Partial Differential Equations, 2017).