A nonlinear Korn inequality on a surface with an explicit estimate of the constant

Malin, Maria; Mardare, Cristinel

Comptes Rendus. Mathématique, Tome 359 (2021) no. 2, pp. 105-111.

Une inégalité de Korn non linéaire sur une surface estime une distance entre une surface $\theta \left(\omega \right)$ et une autre surface $\varphi \left(\omega \right)$ en fonction des distances entre leur formes fondamentales dans l’espace $L^p\left(\omega \right)$, $1<p<\infty$.

Nous établissons une nouvelle inégalité de ce type. La nouveauté réside dans l’appartenance de l’immersion $\theta$ à un ensemble particulier d’applications de classe ${𝒞}^1$ de $\overline{\omega }$ dans ${ℝ}^3$ avec un champ de vecteurs normaux unitaires aussi de classe ${𝒞}^1$ dans $\overline{\omega }$.

A nonlinear Korn inequality on a surface estimates a distance between a surface $\theta \left(\omega \right)$ and another surface $\varphi \left(\omega \right)$ in terms of distances between their fundamental forms in the space $L^p\left(\omega \right)$, $1<p<\infty$.

We establish a new inequality of this type. The novelty is that the immersion $\theta$ belongs to a specific set of mappings of class ${𝒞}^1$ from $\overline{\omega }$ into ${ℝ}^3$ with a unit vector field also of class ${𝒞}^1$ over $\overline{\omega }$.

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