Continuity in $ {H}^{1}$-norms of surfaces in terms of the $ {L}^{1}$-norms of their fundamental forms

Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare

doi:10.1016/j.crma.2005.06.031

Mathematical Problems in Mechanics/Differential Geometry

Continuity in

H^{1}

-norms of surfaces in terms of the

L^{1}

-norms of their fundamental forms

Robert Dautray

Philippe G. Ciarlet ¹; Liliana Gratie ²; Cristinel Mardare ³

¹Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
²Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
³Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 201-206

Abstract (Original language)
Résumé

The main purpose of this Note is to show how a ‘nonlinear Korn's inequality on a surface’ can be established. This inequality implies in particular the following interesting per se sequential continuity property for a sequence of surfaces. Let ω be a domain in $R^{2}$ , let $θ : \bar{ω} \to R^{3}$ be a smooth immersion, and let $θ^{k} : \bar{ω} \to R^{3}$ , $k ⩾ 1$ , be mappings with the following properties: They belong to the space $H^{1} (ω)$ ; the vector fields normal to the surfaces $θ^{k} (ω)$ , $k ⩾ 1$ , are well defined a.e. in ω and they also belong to the space $H^{1} (ω)$ ; the principal radii of curvature of the surfaces $θ^{k} (ω)$ stay uniformly away from zero; and finally, the three fundamental forms of the surfaces $θ^{k} (ω)$ converge in $L^{1} (ω)$ toward the three fundamental forms of the surface $θ (ω)$ as $k \to \infty$ . Then, up to proper isometries of $R^{3}$ , the surfaces $θ^{k} (ω)$ converge in $H^{1} (ω)$ toward the surface $θ (ω)$ as $k \to \infty$ .

L'objectif principal de cette Note est de montrer comment on peut établir une « inégalité de Korn non linéaire sur une surface ». Cette inégalité implique en particulier la propriété de continuité séquentielle suivante, intéressante par elle-même. Soit ω un domaine de $R^{2}$ , soit $θ : \bar{ω} \to R^{3}$ une immersion régulière, et soit $θ^{k} : \bar{ω} \to R^{3}$ , $k ⩾ 1$ , des applications ayant les propriétés suivantes : Elles appartiennent à l'espace $H^{1} (ω)$ ; les champs de vecteurs normaux aux surfaces $θ^{k} (ω)$ , $k ⩾ 1$ , sont définis presque partout dans ω et appartiennent aussi à l'espace $H^{1} (ω)$ ; les modules des rayons de courbure principaux des surfaces $θ^{k} (ω)$ sont uniformément minorés par une constante strictement positive ; finalement, les trois formes fondamentales des surfaces $θ^{k} (ω)$ convergent dans $L^{1} (ω)$ vers les trois formes fondamentales de la surface $θ (ω)$ lorsque $k \to \infty$ . Alors, à des isométries propres de $R^{3}$ près, les surfaces $θ^{k} (ω)$ convergent dans $H^{1} (ω)$ vers la surface $θ (ω)$ lorsque $k \to \infty$ .

Article information
Cite this article

Received: 2005-06-15
Published online: 2005-08-01

DOI: 10.1016/j.crma.2005.06.031

Author's affiliations:

Philippe G. Ciarlet ¹ ; Liliana Gratie ² ; Cristinel Mardare ³

¹ Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
² Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
³ Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare. Continuity in $ {H}^{1}$-norms of surfaces in terms of the $ {L}^{1}$-norms of their fundamental forms. Comptes Rendus. Mathématique, Volume 341 (2005) no. 3, pp. 201-206. doi: 10.1016/j.crma.2005.06.031

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