Greenʼs formulas with little regularity on a surface – Application to Donati-like compatibility conditions on a surface

Philippe G. Ciarlet; Oana Iosifescu

doi:10.1016/j.crma.2013.10.016

Mathematical problems in mechanics

Greenʼs formulas with little regularity on a surface – Application to Donati-like compatibility conditions on a surface
[Formules de Green avec peu de régularité sur une surface – Application à des conditions de compatibilité du type de Donati sur une surface]

Présenté par : Philippe G. Ciarlet

Philippe G. Ciarlet ¹ ; Oana Iosifescu ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Départment de mathématiques, Université de Montpellier-2, place Eugène-Bataillon, 34095 Montpellier cedex 5, France

Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 853-858.

Résumé
Abstract

Dans cette Note, on établit deux formules de Green avec peu de régularité sur une surface. Ces formules sont ensuite utilisées pour identifier et justifier des conditions de compatibilité du type de Donati sur une surface, garantissant que les composantes de deux champs de matrices symétriques $(c_{α β})$ et $(r_{α β})$ avec $c_{α β}$ et $r_{α β}$ dans lʼespace $L^{2} (ω)$ , où ω est un domaine ω de $R^{2}$ , sont les composantes covariantes des champs de tenseurs de changement de métrique et de changement de courbure linéarisés associés à un champ de déplacements dʼune surface $θ (\bar{ω})$ , où $θ : \bar{ω} \to R^{3}$ est une immersion régulière.

In this Note, we establish two Greenʼs formulas with little regularity on a surface. These formulas are then used for identifying and justifying Donati-like compatibility conditions on a surface, guaranteeing that the components of two symmetric matrix fields $(c_{α β})$ and $(r_{α β})$ with $c_{α β}$ and $r_{α β}$ in the space $L^{2} (ω)$ , where ω is a domain in $R^{2}$ , are the covariant components of the linearized change of metric and linearized change of curvature tensors associated with a displacement vector field of a surface $θ (\bar{ω})$ , where $θ : \bar{ω} \to R^{3}$ is a smooth immersion.

Reçu le : 2013-10-18
Publié le : 2013-10-30

DOI : 10.1016/j.crma.2013.10.016

Affiliations des auteurs :

Philippe G. Ciarlet ¹ ; Oana Iosifescu ²

@article{CRMATH_2013__351_21-22_853_0,
     author = {Philippe G. Ciarlet and Oana Iosifescu},
     title = {Green's formulas with little regularity on a surface {\textendash} {Application} to {Donati-like} compatibility conditions on a surface},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {853--858},
     publisher = {Elsevier},
     volume = {351},
     number = {21-22},
     year = {2013},
     doi = {10.1016/j.crma.2013.10.016},
     language = {en},
}

TY  - JOUR
AU  - Philippe G. Ciarlet
AU  - Oana Iosifescu
TI  - Greenʼs formulas with little regularity on a surface – Application to Donati-like compatibility conditions on a surface
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 853
EP  - 858
VL  - 351
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2013.10.016
LA  - en
ID  - CRMATH_2013__351_21-22_853_0
ER  -

%0 Journal Article
%A Philippe G. Ciarlet
%A Oana Iosifescu
%T Greenʼs formulas with little regularity on a surface – Application to Donati-like compatibility conditions on a surface
%J Comptes Rendus. Mathématique
%D 2013
%P 853-858
%V 351
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2013.10.016
%G en
%F CRMATH_2013__351_21-22_853_0

Philippe G. Ciarlet; Oana Iosifescu. Greenʼs formulas with little regularity on a surface – Application to Donati-like compatibility conditions on a surface. Comptes Rendus. Mathématique, Volume 351 (2013) no. 21-22, pp. 853-858. doi : 10.1016/j.crma.2013.10.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.10.016/

Bibliographie
Cité par

[1] C. Amrouche; P.G. Ciarlet; L. Gratie; S. Kesavan On the characterizations of matrix fields as linearized strain tensor fields, J. Math. Pures Appl., Volume 86 (2006), pp. 116-132

[2] M. Bernadou; P.G. Ciarlet Sur lʼellipticité du modèle linéaire de coques de W.T. Koiter (R. Glowinski; J.-L. Lions, eds.), Computing Methods in Applied Sciences and Engineering, Springer, 1976, pp. 89-136

[3] M. Bernadou; P.G. Ciarlet; B. Miara Existence theorems for two-dimensional linear shell theories, J. Elast., Volume 34 (1994), pp. 111-138

[4] P.G. Ciarlet Mathematical Elasticity, vol. III: Theory of Shells, Studies in Mathematics and its Applications, North-Holland, Amsterdam, 2000

[5] P.G. Ciarlet An Introduction to Differential Geometry, with Applications to Elasticity, Springer-Verlag, Heidelberg, 2005

[6] P.G. Ciarlet, O. Iosifescu, Donati compatibility conditions on a surface – Application to shell theory, J. Math. Pures Appl., in press.

[7] P.G. Ciarlet; O. Iosifescu The space $H (div, \cdot)$ on a surface – Application to Donati-like compatibility conditions on a surface, C. R. Acad. Sci. Paris, Ser. I (2013) (in press) | DOI

[8] P.G. Ciarlet; C. Mardare Intrinsic formulation of the displacement-traction problem in linearized elasticity, Math. Models Methods Appl. Sci. (2014) (in press)

[9] G. Duvaut; J.-L. Lions Les équations en mécanique et en physique, Dunod, Paris, 1972

[10] G. Geymonat; F. Krasucki Some remarks on the compatibility conditions in elasticity, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., Volume 29 (2005), pp. 175-182

[11] G. Geymonat; P. Suquet Functional spaces for Norton–Hoff materials, Math. Methods Appl. Sci., Volume 8 (1986), pp. 206-222

Cité par Sources :

Commentaires - Politique