  Using the notion of Gamma-convergence, we discuss the limiting behavior of
the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h
converges to zero, under the assumption that the elastic energy of deformations
scales like $h^\beta$ with $2<\beta<4$. We establish that, for the given
scaling regime, the limiting theory reduces to the linear pure bending. Two
major ingredients of the proofs are: the density of smooth infinitesimal
isometries in the space of $W^{2,2}$ first order infinitesimal isometries, and
a result on matching smooth infinitesimal isometries with exact isometric
immersions on smooth elliptic surfaces.
