A central problem in the mathematical theory of elasticity is to predict theories of lower-dimensional objects subject to mechanical deformations, starting from the 3d nonlinear theory. For shells, despite extensive use of the ad-hoc generalizations of plate theories present in the engineering applications, not much is known from the mathematical point of view.

In a recent effort, the limiting behavior (using the notion of Gamma-limit) of the 3d nonlinear elasticity for thin shells has been described, as the shell thickness h converges to 0, and under the assumption that the elastic energy of deformations scales like h^β, with β>2.

In this talk I will also explain the two major ingredients of the proofs, which are: the density of smooth maps in the space of Sobolev first order isometries, and a result on matching smooth infinitesimal isometries to exact isometric immersions.

Another recent related development that I will present concerns the so-called Non-Euclidean plates, which are elastic materials exhibiting the non-zero strain at free equilibria.

This is joint work with Maria Giovanna Mora (SISSA) and Reza Pakzad (University of Pittsburgh).