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Extending the Knops-Stuart-Taheri technique to $ C^{1}$ weak local minimizers in nonlinear elasticity


Author: J. J. Bevan
Journal: Proc. Amer. Math. Soc. 139 (2011), 1667-1679
MSC (2010): Primary 49J40; Secondary 49N60, 74G30
DOI: https://doi.org/10.1090/S0002-9939-2010-10637-8
Published electronically: October 8, 2010
MathSciNet review: 2763756
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Abstract: We prove that any $ C^{1}$ weak local minimizer of a certain class of elastic stored-energy functionals $ I(u) = \int_{\Omega} f(\nabla u) dx$ subject to a linear boundary displacement $ u_{0}(x)=\xi x$ on a star-shaped domain $ \Omega$ with $ C^{1}$ boundary is necessarily affine provided $ f$ is strictly quasiconvex at $ \xi$. This is done without assuming that the local minimizer satisfies the Euler-Lagrange equations, and therefore extends in a certain sense the results of Knops and Stuart, and those of Taheri, to a class of functionals whose integrands take the value $ +\infty$ in an essential way.


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Additional Information

J. J. Bevan
Affiliation: Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom
Email: j.bevan@surrey.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2010-10637-8
Keywords: Stored-energy function, local minimizer, uniqueness
Received by editor(s): September 15, 2009
Received by editor(s) in revised form: May 18, 2010
Published electronically: October 8, 2010
Additional Notes: The author gratefully acknowledges the support of an RCUK Academic Fellowship
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society