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

TL;DR

This paper demonstrates the flexibility of the four-state problem for general linear differential operators by extending convex integration techniques, broadening understanding of microstructure formation in PDEs.

Contribution

It extends convex integration methods to linear operators with potentials, establishing the flexibility of the four-state problem beyond previously known cases.

Findings

Four-state problem for linear operators is flexible.

Extended convex integration to operators with potentials.

Counterexamples for four-state problem constructed.

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].