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Comptes Rendus Mathématique
Volume 355, n° 2
pages 226-231 (février 2017)
Doi : 10.1016/j.crma.2017.01.002
Received : 6 January 2017 ;  accepted : 6 January 2017
New nonlinear estimates for surfaces in terms of their fundamental forms
Nouvelles estimations pour des surfaces en fonction de leurs formes fondamentales
 

Philippe G. Ciarlet a , Maria Malin a , Cristinel Mardare b
a Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong 
b Sorbonne Universités, Université Pierre-et-Marie-Curie, Laboratoire Jacques-Louis-Lions, 4, place Jussieu, 75005 Paris, France 

Abstract

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.

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Résumé

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.

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