Strain tensors on hyperbolic surfaces and their applications

We perform a detailed analysis of the solvability of linear strain equations on several non-characteristic regions to obtain L 2 regularity solutions. As an application, the rigidity results on the strain tensor of the middle surface are implied by the L 2 regularity for non-characteristic regions. Finally, we obtain the optimal constant in the first Korn inequality scales like h 4 / 3 for hyperbolic shells, generalizing the assumption that the middle surface of the shell is given by a single principal system in the literature.