Mathematical Problems in Mechanics/Differential Geometry
An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors
Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 505-510.

Let $\Omega$ be a bounded open connected subset of ${ℝ}^{n}$ with a Lipschitz-continuous boundary and let $\Theta \in {𝒞}^{1}\left(\overline{\Omega };{ℝ}^{n}\right)$ be a deformation of the set $\overline{\Omega }$ satisfying $\mathrm{det}\nabla \Theta >0$ in $\overline{\Omega }$. It is established that there exists a constant $\mathrm{C}\left(\Theta \right)$ with the following property: for each deformation $\Phi \in {\mathrm{H}}^{1}\left(\Omega ;{ℝ}^{n}\right)$ satisfying $\mathrm{det}\nabla \Phi >0$ a.e. in $\Omega$, there exist an n×n rotation matrix $𝐑=𝐑\left(\Phi ,\Theta \right)$ and a vector $𝐛=𝐛\left(\Phi ,\Theta \right)$ in ${ℝ}^{n}$ such that

 ${\parallel \Phi -\left(𝐛+𝐑\Theta \right)\parallel }_{{𝐇}^{1}\left(\Omega \right)}⩽C\left(\Theta \right){\parallel \nabla {\Phi }^{\mathrm{T}}\nabla \Phi -\nabla {\Theta }^{\mathrm{T}}\nabla \Theta \parallel }_{{𝐋}^{1}\left(\Omega \right)}^{1/2}.$
The proof relies in particular on a fundamental ‘geometric rigidity lemma’, recently proved by G. Friesecke, R.D. James, and S. Müller.

Soit $\Omega$ un ouvert borné connexe de ${ℝ}^{n}$ à frontière lipschitzienne et soit $\Theta \in {𝒞}^{1}\left(\overline{\Omega };{ℝ}^{n}\right)$ une déformation de l'ensemble $\overline{\Omega }$ satisfaisant $\text{dét}\phantom{\rule{1.69998pt}{0ex}}\nabla \Theta >0$ dans $\overline{\Omega }$. On établit l'existence d'une constante $\mathrm{C}\left(\Theta \right)$ ayant la propriété suivante : quelle que soit la déformation $\Phi \in {\mathrm{H}}^{1}\left(\Omega ;{ℝ}^{n}\right)$ satisfaisant $\text{dét}\phantom{\rule{1.69998pt}{0ex}}\nabla \Phi >0$ p.p. dans $\Omega$, il existe une matrice n×n de rotation $𝐑$ et un vecteur $𝐛\in {ℝ}^{n}$ tels que

 ${\parallel \Phi -\left(𝐛+𝐑\Theta \right)\parallel }_{{𝐇}^{1}\left(\Omega \right)}⩽C\left(\Theta \right){\parallel \nabla {\Phi }^{\mathrm{T}}\nabla \Phi -\nabla {\Theta }^{\mathrm{T}}\nabla \Theta \parallel }_{{𝐋}^{1}\left(\Omega \right)}^{1/2}.$
La démonstration repose en particulier sur un « lemme de rigidité géométrique » fondamental, récemmment établi par G. Friesecke, R.D. James, et S. Müller.

Accepted:
Published online:
DOI: 10.1016/j.crma.2004.01.014

Philippe G. Ciarlet 1; Cristinel Mardare 2

1 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
2 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France
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Philippe G. Ciarlet; Cristinel Mardare. An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors. Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 505-510. doi : 10.1016/j.crma.2004.01.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.014/

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