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

Ciarlet, Philippe G.; Gratie, Liliana; Mardare, Cristinel

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
[Continuité en norme

H^{1}

de surfaces en terme des normes

L^{1}

de leurs formes fondamentales]

Présenté par : Robert Dautray

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

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

Résumé
Abstract

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 $k \to \infty$ .

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 $k \to \infty$ .

Reçu le : 2005-06-15
Publié le : 2005-08-01

DOI : 10.1016/j.crma.2005.06.031

Affiliations des auteurs :

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

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     title = {Continuity in $ {H}^{1}$-norms of surfaces in terms of the $ {L}^{1}$-norms of their fundamental forms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {201--206},
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Ciarlet, Philippe G.; Gratie, Liliana; Mardare, Cristinel. Continuity in $ {H}^{1}$-norms of surfaces in terms of the $ {L}^{1}$-norms of their fundamental forms. Comptes Rendus. Mathématique, Tome 341 (2005) no. 3, pp. 201-206. doi : 10.1016/j.crma.2005.06.031. http://www.numdam.org/articles/10.1016/j.crma.2005.06.031/

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