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Using the notion of Gamma-convergence, we discuss the limiting behavior ofthe 3d nonlinear elastic energy for thin elliptic shells, as their thickness hconverges to zero, under the assumption that the elastic energy of deformationsscales like $h^\beta$ with $2<\beta<4$. We establish that, for the givenscaling regime, the limiting theory reduces to the linear pure bending. Twomajor ingredients of the proofs are: the density of smooth infinitesimalisometries in the space of $W^{2,2}$ first order infinitesimal isometries, anda result on matching smooth infinitesimal isometries with exact isometricimmersions on smooth elliptic surfaces.

The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells