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

We prove the local Hölder continuity of strong local minimizers of the stored energy functional E(u) = ∫ Ω λ|∇u|2 + h(det∇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 → 0+, linearly as s → +∞, and satisfies h(s) = +∞ if s ≤ 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 Hölder continuity then follows by adapting some well-known elliptic regularity theory.