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

TL;DR

This paper establishes the Holder continuity of strong local minimizers in a nonlinear elastic energy model in two dimensions, under a positive twist condition ensuring non-interpenetration of material during deformation.

Contribution

It introduces a novel approach linking positive twist to Euler-Lagrange inequalities, enabling regularity proofs for minimizers in nonlinear elasticity.

Findings

Local minimizers are Holder continuous under positive twist.

The positive twist condition is equivalent to mapping circles to star-shaped sets.

The regularizing effect of the energy term is demonstrated in shear maps.

Abstract

We prove the local H\"{o}lder continuity of strong local minimizers of the stored energy functional \[E(u)=\int_{\om}\lambda |\nabla u|^{2}+h(\det \nabla u) \,dx\] 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 local minimizer has positive twist a.e\frenchspacing. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed H\"{o}lder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term $\int_{\om} h(\det \nabla u)\,dx$ can have by analysing the regularity of local minimizers in the class of `shear maps'. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are H\"{o}lder continuous.