Partial differential equations
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.

For n2 and 1<p< we prove an L p -version of the generalized Korn-type inequality for incompatible, p-integrable tensor fields P:Ω 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, n×n csymP L p Ω, 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

On montre pour n2 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 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, n×n csymP L p Ω, 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.216
Classification: 35A23, 35B45, 35Q74, 46E35
Lewintan, Peter 1; Neff, Patrizio 1

1 Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
@article{CRMATH_2021__359_6_749_0,
     author = {Lewintan, Peter and Neff, Patrizio},
     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},
     url = {http://www.numdam.org/articles/10.5802/crmath.216/}
}
TY  - JOUR
AU  - Lewintan, Peter
AU  - Neff, Patrizio
TI  - $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 749
EP  - 755
VL  - 359
IS  - 6
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.216/
DO  - 10.5802/crmath.216
LA  - en
ID  - CRMATH_2021__359_6_749_0
ER  - 
%0 Journal Article
%A Lewintan, Peter
%A Neff, Patrizio
%T $L^p$-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with $p$-integrable exterior derivative
%J Comptes Rendus. Mathématique
%D 2021
%P 749-755
%V 359
%N 6
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.216/
%R 10.5802/crmath.216
%G en
%F CRMATH_2021__359_6_749_0
Lewintan, Peter; Neff, Patrizio. $L^p$-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. http://www.numdam.org/articles/10.5802/crmath.216/

[1] Ciarlet, Philippe G. On Korn’s inequality, Chin. Ann. Math., Volume 31 (2010) no. 5, pp. 607-618 | DOI | MR | Zbl

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

[3] Ciarlet, Philippe G.; Ciarlet, Patrick jun. Another approach to linearized elasticity and a new proof of Korn’s inequality, Math. Models Methods Appl. Sci., Volume 15 (2005) no. 2, pp. 259-271 | DOI | MR | Zbl

[4] Ciarlet, Philippe G.; Malin, Maria; Mardare, Cristinel 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

[5] Geymonat, Giuseppe; Suquet, Pierer-Marie Functional spaces for Norton–Hoff materials, Math. Methods Appl. Sci., Volume 8 (1986) no. 2, pp. 206-222 | DOI | MR | Zbl

[6] Lewintan, Peter; Neff, Patrizio Nečas–Lions lemma revisited: An L p -version of the generalized Korn inequality for incompatible tensor fields, Math. Methods Appl. Sci. (2021), pp. 1-12 | DOI | Zbl

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

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

[9] van der Waerden, Bartel L. Algebra. Volume II, Springer, 2003 (Based in part on lectures by E. Artin and E. Noether. Transl. from the German 5th ed. by John R. Schulenberger. 1st paperback ed.) | Zbl

Cited by Sources: