Critical points of degenerate polyconvex energies

TL;DR

This paper investigates the properties of critical and stationary points of polyconvex energies, showing that such points have nearly constant determinants and establishing rigidity results for related differential inclusions.

Contribution

It provides new insights into the structure of critical points of polyconvex functionals, including their determinant behavior and the rigidity of associated differential inclusions.

Findings

Critical points have locally constant determinants except on measure-zero sets.

Stationary points have constant determinants almost everywhere.

The associated differential inclusion is rigid.

Abstract

We study critical and stationary, i.e. critical with respect to both inner and outer variations, points of polyconvex functionals of the form $f(X) = g(\det(X))$, for $X \in \mathbb{R}^{2\times 2}$. In particular, we show that critical points $u \in Lip(\Omega,\mathbb{R}^2)$ with $\det(Du) \neq 0$ a.e. have locally constant determinant except in a relatively closed set of measure zero, and that stationary points have constant determinant almost everywhere. This is deduced from a more general result concerning solutions $u \in Lip(\Omega,\mathbb{R}^n)$, $\Omega \subset \mathbb{R}^n$ to the linearized problem $curl(\beta Du) = 0$. We also present some generalization of the original result to higher dimensions and assuming further regularity on solutions $u$. Finally, we show that the differential inclusion associated to stationarity with respect to polyconvex energies as above is rigid.