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 , let be a smooth immersion, and let , , be mappings with the following properties: They belong to the space ; the vector fields normal to the surfaces , , are well defined a.e. in ω and they also belong to the space ; the principal radii of curvature of the surfaces stay uniformly away from zero; and finally, the three fundamental forms of the surfaces converge in toward the three fundamental forms of the surface as . Then, up to proper isometries of , the surfaces converge in toward the surface as .
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 , soit une immersion régulière, et soit , , des applications ayant les propriétés suivantes : Elles appartiennent à l'espace ; les champs de vecteurs normaux aux surfaces , , sont définis presque partout dans ω et appartiennent aussi à l'espace ; les modules des rayons de courbure principaux des surfaces sont uniformément minorés par une constante strictement positive ; finalement, les trois formes fondamentales des surfaces convergent dans vers les trois formes fondamentales de la surface lorsque . Alors, à des isométries propres de près, les surfaces convergent dans vers la surface lorsque .
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@article{CRMATH_2005__341_3_201_0, author = {Ciarlet, Philippe G. and Gratie, Liliana and Mardare, Cristinel}, 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}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.06.031/} }
TY - JOUR AU - Ciarlet, Philippe G. AU - Gratie, Liliana AU - Mardare, Cristinel 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 UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.06.031/ DO - 10.1016/j.crma.2005.06.031 LA - en ID - CRMATH_2005__341_3_201_0 ER -
%0 Journal Article %A Ciarlet, Philippe G. %A Gratie, Liliana %A Mardare, Cristinel %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 %U http://archive.numdam.org/articles/10.1016/j.crma.2005.06.031/ %R 10.1016/j.crma.2005.06.031 %G en %F CRMATH_2005__341_3_201_0
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, Volume 341 (2005) no. 3, pp. 201-206. doi : 10.1016/j.crma.2005.06.031. http://archive.numdam.org/articles/10.1016/j.crma.2005.06.031/
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