  We consider shells with zero Gaussian curvature, namely shells with one
principal curvature zero and the other one having a constant sign. Our
particular interests are shells that are diffeomorphic to a circular
cylindrical shell with zero principal longitudinal curvature and positive
circumferential curvature, including, for example, cylindrical and conical
shells with arbitrary convex cross sections. We prove that the best constant in
the first Korn inequality scales like thickness to the power 3/2 for a wide
range of boundary conditions at the thin edges of the shell. Our methodology is
to prove, for each of the three mutually orthogonal two-dimensional
cross-sections of the shell, a "first-and-a-half Korn inequality" - a hybrid
between the classical first and second Korn inequalities. These three
two-dimensional inequalities assemble into a three-dimensional one, which, in
turn, implies the asymptotically sharp first Korn inequality for the shell.
This work is a part of mathematically rigorous analysis of extreme sensitivity
of the buckling load of axially compressed cylindrical shells to shape
imperfections.
