  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.
