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 Ω be a bounded open connected subset of n with a Lipschitz-continuous boundary and let Θ𝒞 1 (Ω ¯; n ) be a deformation of the set Ω ¯ satisfying det Θ>0 in Ω ¯. It is established that there exists a constant C(Θ) with the following property: for each deformation ΦH 1 (Ω; n ) satisfying det Φ>0 a.e. in Ω, there exist an n×n rotation matrix 𝐑=𝐑(Φ,Θ) and a vector 𝐛=𝐛(Φ,Θ) in n such that

Φ-(𝐛+𝐑Θ) 𝐇 1 (Ω) C(Θ)Φ T Φ-Θ T Θ 𝐋 1 (Ω) 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 Ω un ouvert borné connexe de n à frontière lipschitzienne et soit Θ𝒞 1 (Ω ¯; n ) une déformation de l'ensemble Ω ¯ satisfaisant détΘ>0 dans Ω ¯. On établit l'existence d'une constante C(Θ) ayant la propriété suivante : quelle que soit la déformation ΦH 1 (Ω; n ) satisfaisant détΦ>0 p.p. dans Ω, il existe une matrice n×n de rotation 𝐑 et un vecteur 𝐛 n tels que

Φ-(𝐛+𝐑Θ) 𝐇 1 (Ω) C(Θ)Φ T Φ-Θ T Θ 𝐋 1 (Ω) 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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.01.014
Ciarlet, Philippe G. 1; Mardare, Cristinel 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
@article{CRMATH_2004__338_6_505_0,
     author = {Ciarlet, Philippe G. and Mardare, Cristinel},
     title = {An estimate of the {\protect\emph{H}\protect\textsuperscript{1}-norm} of deformations in terms of the {\protect\emph{L}\protect\textsuperscript{1}-norm} of their {Cauchy{\textendash}Green} tensors},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {505--510},
     publisher = {Elsevier},
     volume = {338},
     number = {6},
     year = {2004},
     doi = {10.1016/j.crma.2004.01.014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2004.01.014/}
}
TY  - JOUR
AU  - Ciarlet, Philippe G.
AU  - Mardare, Cristinel
TI  - An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 505
EP  - 510
VL  - 338
IS  - 6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2004.01.014/
DO  - 10.1016/j.crma.2004.01.014
LA  - en
ID  - CRMATH_2004__338_6_505_0
ER  - 
%0 Journal Article
%A Ciarlet, Philippe G.
%A Mardare, Cristinel
%T An estimate of the H1-norm of deformations in terms of the L1-norm of their Cauchy–Green tensors
%J Comptes Rendus. Mathématique
%D 2004
%P 505-510
%V 338
%N 6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2004.01.014/
%R 10.1016/j.crma.2004.01.014
%G en
%F CRMATH_2004__338_6_505_0
Ciarlet, Philippe G.; Mardare, Cristinel. 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. http://www.numdam.org/articles/10.1016/j.crma.2004.01.014/

[1] Adams, R.A. Sobolev Spaces, Academic Press, New York, 1975

[2] Antman, S.S. Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of non-linearly elastic rods and shells, Arch. Rational Mech. Anal., Volume 61 (1976), pp. 307-351

[3] Ball, J.M. Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal., Volume 63 (1977), pp. 337-403

[4] Ciarlet, P.G. Continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl., Volume 82 (2002), pp. 253-274

[5] Ciarlet, P.G.; Laurent, F. Continuity of a deformation as a function of its Cauchy–Green tensor, Arch. Rational Mech. Anal., Volume 167 (2003), pp. 255-269

[6] Ciarlet, P.G.; Mardare, C. Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl. (2004) (in press)

[7] P.G. Ciarlet, C. Mardare, A surface as a function of its two fundamental forms, in preparation

[8] P.G. Ciarlet, C. and Mardare, Continuity of a deformation in H1 as a function of its Cauchy–Green tensor in L1, in preparation

[9] Friesecke, G.; James, R.D.; Müller, S. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., Volume 55 (2002), pp. 1461-1506

[10] Grisvard, P. Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985

[11] John, F. Rotation and strain, Comm. Pure Appl. Math., Volume 14 (1961), pp. 391-413

[12] John, F. Bounds for deformations in terms of average strains (Shisha, O., ed.), Inequalities, III, Academic Press, New York, 1972, pp. 129-144

[13] Kohn, R.V. New integral estimates for deformations in terms of their nonlinear strains, Arch. Rational Mech. Anal., Volume 78 (1982), pp. 131-172

[14] Nečas, J. Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967

[15] Reshetnyak, Y.G. Mappings of domains in n and their metric tensors, Siberian Math. J., Volume 44 (2003), pp. 332-345

[16] Whitney, H. Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., Volume 36 (1934), pp. 63-89

Cited by Sources: