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The general theory of slender structure buckling by Grabovsky andTruskinovsky [\textit{Cont. Mech. Thermodyn.,} 19(3-4):211-243, 2007], (laterextended in [\textit{Journal of Nonlinear Science.,} Vol. 26, Iss. 1, pp.83--119, 2016] by Grabovsky and the author), predicts that the criticalbuckling load of a thin shell under dead loading is closely related to theKorn's constant (in Korn's first inequality) of the shell under the Dirichletboundary conditions resulting from the loading program. It is known that underzero Dirichlet boundary conditions on the thin part of the boundary ofpositive, negative, and zero (one principal curvature vanishing, and one apartfrom zero) Gaussian curvature shells, the optimal Korn constant in Korn's firstinequality 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 inKorn's first inequality for elliptic shells that contain a finite number ofpoints where both principal curvatures vanish. We prove that the presence of atleast one such point on the shell leads to the scaling drop from the thicknessto the power of $-1$ to the thickness to the power of $-3/2.$ To our bestknowledge, this is the first result in the direction for constant-signcurvature shells, that do not contain a developable region. In addition, underthe assumption that a suitable trivial branch exists, we prove that in fact thebuckling deformation of such shells under dead loading, should be localized atthe vanishing curvature points, as the shell thickness h goes to zero.

A hint on the localization of the buckling deformation at vanishing curvature points on thin elliptic shells