$L^p$-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

Partial differential equations

L^{p}

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

p

-integrable exterior derivative

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.

Abstract
Résumé

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 $c = c (n, p, Ω) > 0$

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 $c = c (n, p, Ω) > 0$ .

Received: 2019-12-18
Revised: 2020-07-08
Accepted: 2021-04-20
Published online: 2021-09-02

Zbl

DOI: 10.5802/crmath.216

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

Author's affiliations:

Peter Lewintan ¹; Patrizio Neff ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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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/

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