Critical points of degenerate polyconvex energies

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.