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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, Tome 359 (2021) no. 6, pp. 749-755.

On montre pour n≥2 et 1<p<∞ une version L p de l’inégalité généralisée de Korn pour tous les champs de tenseurs incompatibles et p-intégrables P:Ω→ℝ n×n , avec rotationnel généralisé p-intégrable et avec zéro trace tangentielle Pτ l =0 sur ∂Ω, où {τ l } l=1,...,n-1 est un repère tangent sur ∂Ω. Plus précisément on a :

P L p Ω,ℝ n×n ≤csymP L p Ω,ℝ n×n +Curl ̲P L p Ω,𝔰𝔬(n) n ,

où les composantes du rotationnel généralisé s’écrivent (Curl ̲P) ijk :=∂ i P kj -∂ j P ki et c=c(n,p,Ω)>0.

For n≥2 and 1<p<∞ we prove an L p -version of the generalized Korn-type inequality for incompatible, p-integrable tensor fields P:Ω→ℝ n×n having p-integrable generalized Curl ̲ and generalized vanishing tangential trace Pτ l =0 on ∂Ω, denoting by {τ l } l=1,...,n-1 a moving tangent frame on ∂Ω, more precisely we have:

P L p Ω,ℝ n×n ≤csymP L p Ω,ℝ n×n +Curl ̲P L p Ω,𝔰𝔬(n) n ,

where the generalized Curl ̲ is given by (Curl ̲P) ijk :=∂ i P kj -∂ j P ki and c=c(n,p,Ω)>0

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/crmath.216
Classification : 35A23,  35B45,  35Q74,  46E35
Peter Lewintan 1 ; Patrizio Neff 1

1. Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany 
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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},
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     year = {2021},
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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, Tome 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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