  The general theory of slender structure buckling by Grabovsky and
Truskinovsky [\textit{Cont. Mech. Thermodyn.,} 19(3-4):211-243, 2007], (later
extended in [\textit{Journal of Nonlinear Science.,} Vol. 26, Iss. 1, pp.
83--119, 2016] by Grabovsky and the author), predicts that the critical
buckling load of a thin shell under dead loading is closely related to the
Korn's constant (in Korn's first inequality) of the shell under the Dirichlet
boundary conditions resulting from the loading program. It is known that under
zero Dirichlet boundary conditions on the thin part of the boundary of
positive, negative, and zero (one principal curvature vanishing, and one apart
from zero) Gaussian curvature shells, the optimal Korn constant in Korn's first
inequality scales like the thickness to the power of $-1, -4/3,$ and $-3/2$
respectively. In this work we analyse the scaling of the optimal constant in
Korn's first inequality for elliptic shells that contain a finite number of
points where both principal curvatures vanish. We prove that the presence of at
least one such point on the shell leads to the scaling drop from the thickness
to the power of $-1$ to the thickness to the power of $-3/2.$ To our best
knowledge, this is the first result in the direction for constant-sign
curvature shells, that do not contain a developable region. In addition, under
the assumption that a suitable trivial branch exists, we prove that in fact the
buckling deformation of such shells under dead loading, should be localized at
the vanishing curvature points, as the shell thickness h goes to zero.
