New nonlinear estimates for surfaces in terms of their fundamental forms

Philippe G. Ciarlet; Maria Malin; Cristinel Mardare

doi:10.1016/j.crma.2017.01.002

Mathematical problems in mechanics/Differential geometry

New nonlinear estimates for surfaces in terms of their fundamental forms

Presented by: Philippe G. Ciarlet

Philippe G. Ciarlet ¹; Maria Malin ¹; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Sorbonne Universités, Université Pierre-et-Marie-Curie, Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 226-231.

Abstract
Résumé

We establish several estimates of the distance between two surfaces immersed in the three-dimensional Euclidean space in terms of the distance between their fundamental forms, measured in various Sobolev norms. These estimates, which can be seen as nonlinear versions of linear Korn inequalities on a surface appearing in the theory of linearly elastic shells, generalize in particular the nonlinear Korn inequality established in 2005 by P. G. Ciarlet, L. Gratie, and C. Mardare.

Nous établissons plusieurs majorations de la distance entre deux surfaces immergées dans l'espace euclidien tridimensionnel en fonction de la distance entre leurs formes fondamentales, mesurée à l'aide de diverses normes de Sobolev. Ces estimations, qui peuvent être vues comme des versions non linéaires des inégalités de Korn linéaires sur une surface apparaissant dans la théorie de coques linéairement élastiques, généralisent en particulier l'inégalité de Korn non linéaire sur une surface établie en 2005 par P. G. Ciarlet, L. Gratie et C. Mardare.

Received: 2017-01-06
Accepted: 2017-01-06
Published online: 2017-02-01

DOI: 10.1016/j.crma.2017.01.002

Author's affiliations:

Philippe G. Ciarlet ¹; Maria Malin ¹; Cristinel Mardare ²

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Sorbonne Universités, Université Pierre-et-Marie-Curie, Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France

@article{CRMATH_2017__355_2_226_0,
     author = {Philippe G. Ciarlet and Maria Malin and Cristinel Mardare},
     title = {New nonlinear estimates for surfaces in terms of their fundamental forms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {226--231},
     publisher = {Elsevier},
     volume = {355},
     number = {2},
     year = {2017},
     doi = {10.1016/j.crma.2017.01.002},
     language = {en},
}

TY  - JOUR
AU  - Philippe G. Ciarlet
AU  - Maria Malin
AU  - Cristinel Mardare
TI  - New nonlinear estimates for surfaces in terms of their fundamental forms
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 226
EP  - 231
VL  - 355
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2017.01.002
LA  - en
ID  - CRMATH_2017__355_2_226_0
ER  -

%0 Journal Article
%A Philippe G. Ciarlet
%A Maria Malin
%A Cristinel Mardare
%T New nonlinear estimates for surfaces in terms of their fundamental forms
%J Comptes Rendus. Mathématique
%D 2017
%P 226-231
%V 355
%N 2
%I Elsevier
%R 10.1016/j.crma.2017.01.002
%G en
%F CRMATH_2017__355_2_226_0

Philippe G. Ciarlet; Maria Malin; Cristinel Mardare. New nonlinear estimates for surfaces in terms of their fundamental forms. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 226-231. doi : 10.1016/j.crma.2017.01.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.002/

References
Cited by

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

[2] M. Bernadou; P.G. Ciarlet Sur l'ellipticité du modèle linéaire de coques de W.T. Koiter (R. Glowinski; J.-L. Lions, eds.), Computing Methods in Applied Sciences and Engineering, Lect. Notes Econ. Math. Syst., vol. 134, Springer-Verlag, Heidelberg, 1976, pp. 89-136

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

[4] A. Blouza; H. Le Dret Existence and uniqueness for the linear Koiter model for shells with little regularity, Q. Appl. Math., Volume 57 (1999), pp. 317-337

[5] P.G. Ciarlet An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, The Netherlands, 2005

[6] P.G. Ciarlet; L. Gratie; C. Mardare A nonlinear Korn inequality on a surface, J. Math. Pures Appl., Volume 85 (2006), pp. 2-16

[7] P.G. Ciarlet; C. Mardare On rigid and infinitesimal rigid displacements in shell theory, J. Math. Pures Appl., Volume 83 (2004), pp. 1-15

[8] P.G. Ciarlet; C. Mardare Nonlinear Korn inequalities, J. Math. Pures Appl., Volume 104 (2015), pp. 1119-1134

[9] P.G. Ciarlet, M. Malin, C. Mardare, New estimates of the distance between two surfaces in terms of the distance between their fundamental forms, in preparation.

[10] W. Klingenberg A Course in Differential Geometry, Springer, Berlin, 1978

[11] W.T. Koiter On the nonlinear theory of thin elastic shells, Proc. K. Ned. Akad. Wet., Volume B69 (1966), pp. 1-54

[12] J. Nečas Les méthodes directes en théorie des équations elliptiques, Masson, Paris, 1967

Cited by Sources:

Comments - Policy