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
[Continuité en norme

H^{1}

de surfaces en terme des normes

L^{1}

de leurs formes fondamentales]

Présenté par : Robert Dautray

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

Comptes Rendus. Mathématique, Volume 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 :

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

@article{CRMATH_2005__341_3_201_0,
     author = {Philippe G. Ciarlet and Liliana Gratie and Cristinel Mardare},
     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},
     publisher = {Elsevier},
     volume = {341},
     number = {3},
     year = {2005},
     doi = {10.1016/j.crma.2005.06.031},
     language = {en},
}

TY  - JOUR
AU  - Philippe G. Ciarlet
AU  - Liliana Gratie
AU  - Cristinel Mardare
TI  - Continuity in $ {H}^{1}$-norms of surfaces in terms of the $ {L}^{1}$-norms of their fundamental forms
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 201
EP  - 206
VL  - 341
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2005.06.031
LA  - en
ID  - CRMATH_2005__341_3_201_0
ER  -

%0 Journal Article
%A Philippe G. Ciarlet
%A Liliana Gratie
%A Cristinel Mardare
%T Continuity in $ {H}^{1}$-norms of surfaces in terms of the $ {L}^{1}$-norms of their fundamental forms
%J Comptes Rendus. Mathématique
%D 2005
%P 201-206
%V 341
%N 3
%I Elsevier
%R 10.1016/j.crma.2005.06.031
%G en
%F CRMATH_2005__341_3_201_0

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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.06.031/

Bibliographie
Cité par

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

[2] P.G. Ciarlet Continuity of a surface as a function of its two fundamental forms, J. Math. Pures Appl., Volume 82 (2002), pp. 253-274

[3] P.G. Ciarlet, L. Gratie, C. Mardare, A nonlinear Korn inequality on a surface, J. Math. Pures Appl., in press

[4] P.G. Ciarlet; F. Laurent Continuity of a deformation as a function of its Cauchy–Green tensor, Arch. Rational Mech. Anal., Volume 167 (2003), pp. 255-269

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

[6] P.G. Ciarlet; C. Mardare Continuity of a deformation in $H^{1}$ as a function of its Cauchy–Green tensor in $L^{1}$ , J. Nonlinear Sci., Volume 14 (2004), pp. 415-427

[7] P.G. Ciarlet; C. Mardare Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., Volume 83 (2004), pp. 811-843

[8] P.G. Ciarlet; C. Mardare Recovery of a surface with boundary and its continuity as a function of its two fundamental forms, Anal. Appl., Volume 3 (2005), pp. 99-117

[9] G. Friesecke; R.D. James; S. Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math., Volume 55 (2002), pp. 1461-1506

[10] J. Nečas Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967

[11] M. Szopos On the recovery and continuity of a submanifold with boundary, Anal. Appl., Volume 3 (2005), pp. 119-143

[12] Y.G. Reshetnyak Mappings of domains in $R^{n}$ and their metric tensors, Siberian Math. J., Volume 44 (2003), pp. 332-345

[13] H. Whitney Functions differentiable on boundaries of regions, Ann. of Math., Volume 34 (1934), pp. 482-485

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

An estimate of the H¹-norm of deformations in terms of the L¹-norm of their Cauchy–Green tensors

Philippe G. Ciarlet; Cristinel Mardare

C. R. Math (2004)

New compatibility conditions for the fundamental theorem of surface theory

Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare

C. R. Math (2007)

On rigid displacements and their relation to the infinitesimal rigid displacement lemma in shell theory

Philippe G. Ciarlet; Cristinel Mardare

C. R. Math (2003)