A nonlinear Korn inequality on a surface with an explicit estimate of the constant

Maria Malin; Cristinel Mardare

doi:10.5802/crmath.122

Mathematical Problems in Mechanics, Differential Geometry

A nonlinear Korn inequality on a surface with an explicit estimate of the constant

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

Comptes Rendus. Mathématique, Volume 359 (2021) no. 2, pp. 105-111.

Abstract
Résumé

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 $L^{p} (ω)$ , $1 < p < \infty$ .

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 $\bar{ω}$ into $ℝ^{3}$ with a unit vector field also of class $𝒞^{1}$ over $\bar{ω}$ .

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 $L^{p} (ω)$ , $1 < p < \infty$ .

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 $\bar{ω}$ dans $ℝ^{3}$ avec un champ de vecteurs normaux unitaires aussi de classe $𝒞^{1}$ dans $\bar{ω}$ .

Received: 2020-09-18
Accepted: 2020-09-23
Published online: 2021-03-17

DOI: 10.5802/crmath.122

Author's affiliations:

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Maria Malin and Cristinel Mardare},
     title = {A nonlinear {Korn} inequality on a surface with an explicit estimate of the constant},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {105--111},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {2},
     year = {2021},
     doi = {10.5802/crmath.122},
     language = {en},
}

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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, Volume 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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