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

Lewintan, Peter; Neff, Patrizio

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

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

$${∥P∥}_{L^p\left(\Omega ,{ℝ}^{n\times n}\right)}\le c\left({∥symP∥}_{L^p\left(\Omega ,{ℝ}^{n\times n}\right)}+{∥\underline{Curl}P∥}_{L^p\left(\Omega ,{\left(\mathrm{𝔰𝔬}\left(n\right)\right)}^n\right)}\right),$$

où les composantes du rotationnel généralisé s’écrivent ${\left(\underline{Curl}P\right)}_{ijk}:={\partial }_i{P}_{kj}-{\partial }_j{P}_{ki}$ et $c=c(n,p,\Omega )>0$.

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

$${∥P∥}_{L^p\left(\Omega ,{ℝ}^{n\times n}\right)}\le c\left({∥symP∥}_{L^p\left(\Omega ,{ℝ}^{n\times n}\right)}+{∥\underline{Curl}P∥}_{L^p\left(\Omega ,{\left(\mathrm{𝔰𝔬}\left(n\right)\right)}^n\right)}\right),$$

where the generalized $\underline{Curl}$ is given by ${\left(\underline{Curl}P\right)}_{ijk}:={\partial }_i{P}_{kj}-{\partial }_j{P}_{ki}$ and $c=c(n,p,\Omega )>0$

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

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