Optimal exponentials of thickness in Korn's inequalities for parabolic and elliptic shells

We consider the scaling of the optimal constant in Korn's first inequality for elliptic and parabolic shells which was first given by Grabovsky and Harutyunyan with hints coming from the test functions constructed by Tovstik and Smirnov on the level of formal asymptotic expansions. Here, we employ the Bochner technique in Remannian geometry to remove the assumption that the middle surface of the shell is given by one single principal coordinate, in particularly, including closed elliptic shells.