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We establish Korn's interpolation inequalities and the rigidity results ofthe strain tensor of the middle surface for the parabolic and elliptic shellsand show that the best constant in Korn's inequalities scales like $h^{3/2}$for the parabolic shell and $h$ for the elliptic shell, removing the mainassumption that the middle surface of the shell is given by one singleprincipal coordinate in the literature and, in particular, including the closedelliptic shell.

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