$L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative

Peter Lewintan; Patrizio Neff

doi:10.5802/crmath.216

Équations aux dérivées partielles

L^{p}

-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with

p

-integrable exterior derivative
[Versions

L^{p}

des inégalités généralisées de Korn pour les champs de tenseurs incompatibles de dimension quelconque avec dérivée extérieure

p

-intégrable]

Peter Lewintan ¹ ; Patrizio Neff ¹

¹ Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany

Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 749-755.

Résumé
Abstract

On montre pour $n \geq 2$ et $1 < p < \infty$ une version $L^{p}$ de l’inégalité généralisée de Korn pour tous les champs de tenseurs incompatibles et $p$ -intégrables $P : Ω \to ℝ^{n \times n}$ , avec rotationnel généralisé $p$ -intégrable et avec zéro trace tangentielle $P τ_{l} = 0$ sur $\partial Ω$ , où ${τ_{l}}_{l = 1, ..., n - 1}$ est un repère tangent sur $\partial Ω$ . Plus précisément on a :

{∥P∥}_{L^{p} (Ω, ℝ^{n \times n})} \leq c ({∥sym P∥}_{L^{p} (Ω, ℝ^{n \times n})} + {∥\underset{̲}{Curl} P∥}_{L^{p} (Ω, {(𝔰𝔬 (n))}^{n})}),

où les composantes du rotationnel généralisé s’écrivent ${(\underset{̲}{Curl} P)}_{i j k} : = \partial_{i} P_{k j} - \partial_{j} P_{k i}$ et $c = c (n, p, Ω) > 0$ .

For $n \geq 2$ and $1 < p < \infty$ we prove an $L^{p}$ -version of the generalized Korn-type inequality for incompatible, $p$ -integrable tensor fields $P : Ω \to ℝ^{n \times n}$ having $p$ -integrable generalized $\underset{̲}{Curl}$ and generalized vanishing tangential trace $P τ_{l} = 0$ on $\partial Ω$ , denoting by ${τ_{l}}_{l = 1, ..., n - 1}$ a moving tangent frame on $\partial Ω$ , more precisely we have:

{∥P∥}_{L^{p} (Ω, ℝ^{n \times n})} \leq c ({∥sym P∥}_{L^{p} (Ω, ℝ^{n \times n})} + {∥\underset{̲}{Curl} P∥}_{L^{p} (Ω, {(𝔰𝔬 (n))}^{n})}),

where the generalized $\underset{̲}{Curl}$ is given by ${(\underset{̲}{Curl} P)}_{i j k} : = \partial_{i} P_{k j} - \partial_{j} P_{k i}$ and $c = c (n, p, Ω) > 0$

Reçu le : 2019-12-18
Révisé le : 2020-07-08
Accepté le : 2021-04-20
Publié le : 2021-09-02

Zbl

DOI : 10.5802/crmath.216

Classification : 35A23, 35B45, 35Q74, 46E35

Affiliations des auteurs :

Peter Lewintan ¹ ; Patrizio Neff ¹

¹ Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Peter Lewintan and Patrizio Neff},
     title = {$L^p$-versions of generalized {Korn} inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {749--755},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {6},
     year = {2021},
     doi = {10.5802/crmath.216},
     zbl = {07390657},
     language = {en},
}

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Peter Lewintan; Patrizio Neff. $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 749-755. doi : 10.5802/crmath.216. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.216/

Bibliographie
Cité par

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[4] Philippe G. Ciarlet; Maria Malin; Cristinel Mardare On a vector version of a fundamental lemma of J. L. Lions, Chin. Ann. Math., Volume 39 (2018) no. 1, pp. 33-46 | DOI | MR | Zbl

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[7] Patrizio Neff; Ingo Münch $Curl$ bounds $Grad$ on $SO (3)$ , ESAIM, Control Optim. Calc. Var., Volume 14 (2008) no. 1, pp. 148-159 | DOI | Numdam | MR | Zbl

[8] Patrizio Neff; Dirk Pauly; Karl-Josef Witsch Maxwell meets Korn: A new coercive inequality for tensor fields in $ℝ^{n \times n}$ with square-integrable exterior derivative, Math. Methods Appl. Sci., Volume 35 (2012) no. 1, pp. 65-71 | DOI | MR | Zbl

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