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Comptes Rendus. Mathématique
Problèmes mathématiques en mécanique, Géométrie différentielle
A nonlinear Korn inequality on a surface with an explicit estimate of the constant
[Une inégalité de Korn non linéaire sur une surface avec une majoration explicite de la constante]
Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 105-111.

Une inégalité de Korn non linéaire sur une surface estime une distance entre une surface θ(ω) et une autre surface ϕ(ω) en fonction des distances entre leur formes fondamentales dans l’espace L p (ω), 1<p<.

Nous établissons une nouvelle inégalité de ce type. La nouveauté réside dans l’appartenance de l’immersion θ à un ensemble particulier d’applications de classe 𝒞 1 de ω ¯ dans 3 avec un champ de vecteurs normaux unitaires aussi de classe 𝒞 1 dans ω ¯.

A nonlinear Korn inequality on a surface estimates a distance between a surface θ(ω) and another surface ϕ(ω) in terms of distances between their fundamental forms in the space L p (ω), 1<p<.

We establish a new inequality of this type. The novelty is that the immersion θ belongs to a specific set of mappings of class 𝒞 1 from ω ¯ into 3 with a unit vector field also of class 𝒞 1 over ω ¯.

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DOI : https://doi.org/10.5802/crmath.122
Maria Malin 1 ; Cristinel Mardare 2

1. Department of Mathematics, University of Craiova, Craiova, Romania
2. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
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     title = {A nonlinear {Korn} inequality on a surface with an explicit estimate of the constant},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {105--111},
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     year = {2021},
     doi = {10.5802/crmath.122},
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Maria Malin; Cristinel Mardare. A nonlinear Korn inequality on a surface with an explicit estimate of the constant. Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 105-111. doi : 10.5802/crmath.122. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.122/

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[3] Philippe G. Ciarlet; Liliana Gratie; Cristinel Mardare A nonlinear Korn inequality on a surface, J. Math. Pures Appl., Volume 85 (2006) no. 1, pp. 2-16 | Article | MR 2200588 | Zbl 1094.53001

[4] Philippe G. Ciarlet; Maria Malin; Cristinel Mardare New nonlinear estimates for surfaces in terms of their fundamental forms, C. R. Acad. Sci. Paris, Volume 355 (2017) no. 2, pp. 226-231 | Article | MR 3612714 | Zbl 1361.53004

[5] Philippe G. Ciarlet; Cristinel Mardare Recovery of a manifold with boundary and its continuity as a function of its metric tensor, J. Math. Pures Appl., Volume 83 (2004) no. 7, pp. 811-843 | Article | MR 2074679 | Zbl 1088.74014

[6] Maria Malin; Cristinel Mardare Nonlinear estimates for hypersurfaces in terms of their fundamental forms, C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017) no. 11, pp. 1196-1200 | Article | MR 3724885 | Zbl 1388.53009

[7] Maria Malin; Cristinel Mardare Nonlinear Korn inequalities on a hypersurface, Chin. Ann. Math., Volume 39 (2018) no. 3, pp. 513-534 | Article | MR 3795281 | Zbl 1396.53009

[8] Maria Malin; Cristinel Mardare A nonlinear Korn inequality in n with an explicitly bounded constant, C. R., Math., Acad. Sci. Paris, Volume 358 (2020) no. 5, pp. 621-626 | MR 4149862 | Zbl 07267923

[9] Maria Malin; Cristinel Mardare Estimates for the constant in two nonlinear Korn inequalities (2021) (in press, to appear in Mathematics and Mechanics of Solids, https://doi.org/10.1177/1081286520957705)

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