A condition for the Holder regularity of strong local minimizers of a nonlinear elastic energy in two dimensions

We prove the local Holder continuity of strong local minimizers of the stored energy functional ____[E(u)=____int_____Omega ____lambda|____nabla u|^{2}+h(____det ____nabla u) ____,dx____] subject to a condition of `positive twist'. The latter turns out to be equivalent to requiring that $u$ maps circles to suitably star-shaped sets. The convex function $h(s)$ grows logarithmically as $s____to 0+$, linearly as $s ____to +____infty$, and satisfies $h(s)=+____infty$ if $s ____leq 0$. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a strong local minimizer has positive twist a.e. on a ball then a variational inequality holds and a Caccioppoli inequality can be derived from it. The claimed Holder continuity then follows by adapting some well-known elliptic regularity theory