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

Bevan, JJ (2015) A condition for the Holder regularity of strong local minimizers of a nonlinear elastic energy in two dimensions arxiv, 1509.0.

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Abstract

}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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
AuthorsEmailORCID
Bevan, JJUNSPECIFIEDUNSPECIFIED
Date : 30 September 2015
Additional Information : Copyright 2015 Jonathan J Bevan. This is an arXiv publication.
Depositing User : Symplectic Elements
Date Deposited : 30 Oct 2015 09:25
Last Modified : 30 Oct 2015 09:25
URI: http://epubs.surrey.ac.uk/id/eprint/808817

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