Extending the Knops-Stuart-Taheri technique to 𝐶¹ weak local minimizers in nonlinear elasticity

Bevan, J.

doi:10.1090/S0002-9939-2010-10637-8

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

Proc. Amer. Math. Soc. 139 (2011), 1667-1679

DOI: https://doi.org/10.1090/S0002-9939-2010-10637-8

Published electronically: October 8, 2010

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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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