Extending the Knops-Stuart-Taheri technique to 𝐶¹ weak local minimizers in nonlinear elasticity
Jonathan J. Bevan · Proceedings of the American Mathematical Society · 2010
We prove that any C 1 C^{1} weak local minimizer of a certain class of elastic stored-energy functionals I ( u ) = ∫ Ω f ( ∇ u ) d x I(u) = \int _{\Omega } f( abla u)\,dx subject to a linear boundary displacement u 0 ( x ) = ξ x u_{0}(x)=\xi x on a star-shaped domain Ω \Omega with C 1 C^{1} boundary is necessarily affine provided f f is strictly quasiconvex at ξ \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.