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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Extending the Knops-Stuart-Taheri technique to $C^{1}$ weak local minimizers in nonlinear elasticity
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by J. J. Bevan PDF
Proc. Amer. Math. Soc. 139 (2011), 1667-1679 Request permission

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
  • 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
  • © Copyright 2010 American Mathematical Society
  • 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
  • MathSciNet review: 2763756