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

A nonlinear Korn inequality on a surface estimates a distance between a surface $\theta (\omega )$ and another surface $\phi (\omega )$ in terms of distances between their fundamental forms in the space $L^p(\omega )$, $1.We establish a new inequality of this type. The novelty is that the immersion $\theta $ belongs to a specific set of mappings of class $\mathcal{C}^1$ from $\overline{\omega }$ into $\mathbb{R}^3$ with a unit vector field also of class $\mathcal{C}^1$ over $\overline{\omega }$.