On the two-state problem for general differential operators

TL;DR

This paper extends the Ball-James rigidity theorem to general linear differential constraints, demonstrating rigidity for approximate solutions with incompatible states in the linear-growth setting, thus advancing the theory of compensated compactness.

Contribution

It generalizes the rigidity theorem to broader differential operators and establishes results for approximate solutions with linear growth constraints.

Findings

Rigidity holds for approximate solutions with incompatible states.

The theorem applies to sequences bounded in L^1.

Results contribute to the theory of compensated compactness.

Abstract

In this note we generalize the Ball-James rigidity theorem for gradient differential inclusions to the setting of a general linear differential constraint. In particular, we prove the rigidity for approximate solutions to the two-state inclusion with incompatible states for merely $\mathrm{L}^1$-bounded sequences. In this way, our theorem can be seen as a result of compensated compactness in the linear-growth setting.