# The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells - Mathematics > Analysis of PDEs

Abstract: 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.