A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant

Maria Malin; Cristinel Mardare

doi:10.5802/crmath.84

Problèmes mathématiques en mécanique, Géométrie différentielle

A nonlinear Korn inequality in

ℝ^{n}

with an explicitly bounded constant
[Une inégalité de Korn non linéaire dans

ℝ^{n}

avec une constante majorée explicitement]

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 358 (2020) no. 5, pp. 621-626.

Résumé
Abstract

Il est connu que la distance dans $W^{1, p}$ entre une application dans $W^{1, p} (Ω; ℝ^{n})$ préservant l’orientation et une autre application $Θ \in C^{1} (\bar{Ω}; ℝ^{n})$ préservant l’orientation, où $Ω$ est un domain de $ℝ^{n}$ , $n ⩾ 2$ , et $p > 1$ est un nombre réel, est majorée par la distance dans $L^{p}$ entre les racines carrées des champs de tenseurs métriques induits par ces applications, multipliée par une constante dépendant uniquement de $p$ , $Ω$ , et $Θ$ .

L’objet de cette Note est d’établir une meilleure inégalité de ce type, et de fournir en plus une borne supérieure explicitement calculable de la constante qui y apparaît. Un rôle essentiel est joué dans nos preuves par la notion de distance géodésique dans un ouvert de $ℝ^{n}$ .

It is known that the $W^{1, p}$ -distance between an orientation-preserving mapping in $W^{1, p} (Ω; ℝ^{n})$ and another orientation-preserving mapping $Θ \in C^{1} (\bar{Ω}; ℝ^{n})$ , where $Ω$ is a domain in $ℝ^{n}$ , $n ⩾ 2$ , and $p > 1$ is a real number, is bounded above by the $L^{p}$ -distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on $p$ , $Ω$ , and $Θ$ .

The object of this Note is to establish a better inequality of this type, and to provide in addition an explicitly computable upper bound on the constant appearing in it. An essential role is played in our proofs by the notion of geodesic distance inside an open subset of $ℝ^{n}$ .

Reçu le : 2020-06-08
Accepté le : 2020-06-08
Publié le : 2020-09-14

DOI : 10.5802/crmath.84

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Maria Malin and Cristinel Mardare},
     title = {A nonlinear {Korn} inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {621--626},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {5},
     year = {2020},
     doi = {10.5802/crmath.84},
     language = {en},
}

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Maria Malin; Cristinel Mardare. A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant. Comptes Rendus. Mathématique, Volume 358 (2020) no. 5, pp. 621-626. doi : 10.5802/crmath.84. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.84/

Bibliographie
Cité par

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[2] Sylvia Anicic; Hervé Le Dret; Annie Raoult The infinitesimal rigid displacement lemma in Lipschitz coordinates and application to shells with minimal regularity, Math. Methods Appl. Sci., Volume 27 (2004) no. 11, pp. 1283-1299 | DOI | Zbl

[3] Philippe G. Ciarlet Linear and Nonlinear Functional Analysis with Applications, Other Titles in Applied Mathematics, 130, Society for Industrial and Applied Mathematics, 2013 | Zbl

[4] Philippe G. Ciarlet; Cristinel 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) no. 5, pp. 415-427 | DOI | MR | Zbl

[5] Philippe G. Ciarlet; Cristinel Mardare Nonlinear Korn inequalities, J. Math. Pures Appl., Volume 104 (2015) no. 6, pp. 1119-1134 | DOI | MR | Zbl

[6] Sergio Conti Low-energy deformations of thin elastic plates: Isometric embeddings and branching patterns, 2004 (Habilitationsschrift, Universität Leipzig) | Zbl

[7] Gero Friesecke; Richard D. James; Stefan Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Commun. Pure Appl. Math., Volume 55 (2002) no. 11, pp. 1461-1506 | DOI | MR | Zbl

[8] M. Malin; Cristinel Mardare Estimates for the constant in two nonlinear Korn inequalities (in preparation)

[9] Vladimir G. Mazʼya Sobolev Spaces, Springer, 1985

Cité par Sources :

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