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 $\det (Du)\ne 0$ 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.