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 ∈ R 2×2 . In particular, we show that critical points u ∈ Lip(Ω, R 2 ) with det(Du) = 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 ∈ Lip(Ω, R n ), Ω ⊂ R n to the linearized problem curl(βDu) = 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.