Abstract
We prove the local Hölder continuity of strong local minimizers of the stored energy functional
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 \({s \leqq 0}\). 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. on a ball then an Euler-Lagrange type 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. We also demonstrate the regularizing effect that the term \({\int_{\Omega} h({\rm det} \nabla u)\,{\rm d}x}\) 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 Hölder continuous.
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Communicated by G. Dal Maso
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Bevan, J.J. A Condition for the Hölder Regularity of Local Minimizers of a Nonlinear Elastic Energy in Two Dimensions. Arch Rational Mech Anal 225, 249–285 (2017). https://doi.org/10.1007/s00205-017-1104-5
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DOI: https://doi.org/10.1007/s00205-017-1104-5