A nonlinear Korn inequality in ${ℝ}^n$ with an explicitly bounded constant

Malin, Maria; Mardare, Cristinel

Comptes Rendus. Mathématique, Tome 358 (2020) no. 5, pp. 621-626.

Il est connu que la distance dans $W^{1,p}$ entre une application dans $W^{1,p}(\Omega ;{ℝ}^n)$ préservant l’orientation et une autre application $\Theta \in C^1(\overline{\Omega };{ℝ}^n)$ préservant l’orientation, où $\Omega$ est un domain de ${ℝ}^n$, $n⩾2$, et $p>1$ est un nombre réel, est majorée par la distance dans $L^p$ entre les racines carrées des champs de tenseurs métriques induits par ces applications, multipliée par une constante dépendant uniquement de $p$, $\Omega$, et $\Theta$.

L’objet de cette Note est d’établir une meilleure inégalité de ce type, et de fournir en plus une borne supérieure explicitement calculable de la constante qui y apparaît. Un rôle essentiel est joué dans nos preuves par la notion de distance géodésique dans un ouvert de ${ℝ}^n$.

It is known that the $W^{1,p}$-distance between an orientation-preserving mapping in $W^{1,p}(\Omega ;{ℝ}^n)$ and another orientation-preserving mapping $\Theta \in C^1(\overline{\Omega };{ℝ}^n)$, where $\Omega$ is a domain in ${ℝ}^n$, $n⩾2$, and $p>1$ is a real number, is bounded above by the $L^p$-distance between the square roots of the metric tensor fields induced by these mappings, multiplied by a constant depending only on $p$, $\Omega$, and $\Theta$.

The object of this Note is to establish a better inequality of this type, and to provide in addition an explicitly computable upper bound on the constant appearing in it. An essential role is played in our proofs by the notion of geodesic distance inside an open subset of ${ℝ}^n$.

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