The Monge-Ampere constrained elastic theories of shallow shells

Motivated by the degree of smoothness of constrained embeddings of surfaces in $\mathbb{R}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell scales like $h^\alpha$ and the applied forces scale like $h^{\alpha+2}$, in the limit when $h\to 0$. The main analytical ingredients are two independent results: a theorem on approximation of $W^{2,2}$ solutions of the Monge-Amp\`ere equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometries.