A canonical extension of Kornʼs first inequality to $ \mathsf{H}(\mathrm{Curl})$ motivated by gradient plasticity with plastic spin

Neff, Patrizio; Pauly, Dirk; Witsch, Karl-Josef

doi:10.1016/j.crma.2011.10.003

Partial Differential Equations

A canonical extension of Kornʼs first inequality to

H (Curl)

motivated by gradient plasticity with plastic spin
[Une extension canonique de lʼinégalité de Korn à

H (Curl)

motivée par un modèle de plasticité à gradient avec rotation plastique]

Présenté par : Philippe G. Ciarlet

Neff, Patrizio ¹ ; Pauly, Dirk ¹ ; Witsch, Karl-Josef ¹

¹ Universität Duisburg-Essen, Fakultät für Mathematik, Campus Essen Universitätsstr. 2, 45117 Essen, Germany

Comptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1251-1254.

Résumé
Abstract

Nous démontrons une inégalité de type Korn dans $\overset{\circ}{H} (Curl; Ω, R^{3 \times 3})$ pour des champs tensoriels P appliquant Ω dans $R^{3 \times 3}$ . De façon plus précise, soit Ω un domaine borné de $R^{3}$ dont la frontière ∂Ω est Lipschitz continue et connexe. Il existe alors une constante $c > 0$ , telle que

c ‖ P ‖_{L^{2} (Ω, R^{3 \times 3})} ⩽ ‖ sym P ‖_{L^{2} (Ω, R^{3 \times 3})} + ‖ Curl P ‖_{L^{2} (Ω, R^{3 \times 3})}

(1)

est vérifiée pour tous les champs tensoriels

P \in \overset{\circ}{H} (Curl; Ω, R^{3 \times 3})

, i.e., pour tous les

P \in H (Curl; Ω, R^{3 \times 3})

dont la trace tangentielle sʼannule sur ∂Ω. Ici, rotation et trace tangentielle sont définies ligne par ligne. Pour des champs P compatibles, i.e.,

P = \nabla v

, dʼoù

Curl P = 0

, avec

v \in H^{1} (Ω, R^{3})

et de composante

v_{n}

, telle que

\nabla v_{n}

est normal à ∂Ω, lʼestimation (1) se réduit à

c ‖ \nabla v ‖_{L^{2} (Ω, R^{3 \times 3})} ⩽ ‖ sym \nabla v ‖_{L^{2} (Ω, R^{3 \times 3})},

une variante non classique de la première inégalité de Korn. Par ailleurs, pour des P anti-symétriques, (1) se réduit à une variante non classique de lʼinégalité de Poincaré. Il en résulte que puisque (1) est compatible avec les conditions aux limites classiques, cette estimation généralise tout à la fois lʼinégalité de Poincaré et la première inégalité de Korn.

We prove a Korn-type inequality in $\overset{\circ}{H} (Curl; Ω, R^{3 \times 3})$ for tensor fields P mapping Ω to $R^{3 \times 3}$ . More precisely, let $Ω \subset R^{3}$ be a bounded domain with connected Lipschitz boundary ∂Ω. Then, there exists a constant $c > 0$ such that

c ‖ P ‖_{L^{2} (Ω, R^{3 \times 3})} ⩽ ‖ sym P ‖_{L^{2} (Ω, R^{3 \times 3})} + ‖ Curl P ‖_{L^{2} (Ω, R^{3 \times 3})}

(1)

holds for all tensor fields

P \in \overset{\circ}{H} (Curl; Ω, R^{3 \times 3})

, i.e., all

P \in H (Curl; Ω, R^{3 \times 3})

with vanishing tangential trace on ∂Ω. Here, rotation and tangential traces are defined row-wise. For compatible P, i.e.,

P = \nabla v

and thus

Curl P = 0

, where

v \in H^{1} (Ω, R^{3})

are vector fields having components

v_{n}

, for which

\nabla v_{n}

are normal at ∂Ω, the presented estimate (1) reduces to a non-standard variant of Kornʼs first inequality, i.e.,

c ‖ \nabla v ‖_{L^{2} (Ω, R^{3 \times 3})} ⩽ ‖ sym \nabla v ‖_{L^{2} (Ω, R^{3 \times 3})} .

On the other hand, for skew-symmetric P, i.e.,

sym P = 0

, (1) reduces to a non-standard version of Poincaréʼs estimate. Therefore, since (1) admits the classical boundary conditions our result is a common generalization of these two classical estimates, namely Poincaréʼs resp. Kornʼs first inequality.

Reçu le : 2011-07-05
Accepté le : 2011-09-22
Publié le : 2011-11-23

DOI : 10.1016/j.crma.2011.10.003

Affiliations des auteurs :

Neff, Patrizio ¹ ; Pauly, Dirk ¹ ; Witsch, Karl-Josef ¹

¹ Universität Duisburg-Essen, Fakultät für Mathematik, Campus Essen Universitätsstr. 2, 45117 Essen, Germany

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     title = {A canonical extension of {Korn's} first inequality to $ \mathsf{H}(\mathrm{Curl})$ motivated by gradient plasticity with plastic spin},
     journal = {Comptes Rendus. Math\'ematique},
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Neff, Patrizio; Pauly, Dirk; Witsch, Karl-Josef. A canonical extension of Kornʼs first inequality to $ \mathsf{H}(\mathrm{Curl})$ motivated by gradient plasticity with plastic spin. Comptes Rendus. Mathématique, Tome 349 (2011) no. 23-24, pp. 1251-1254. doi : 10.1016/j.crma.2011.10.003. http://www.numdam.org/articles/10.1016/j.crma.2011.10.003/

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