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Motivated by the degree of smoothness of constrained embeddings of surfacesin $\mathbb{R}^3$, and by the recent applications to the elasticity of shallowshells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinearelastic energy of a shallow shell of thickness $h$, where the depth of theshell 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 independentresults: a theorem on approximation of $W^{2,2}$ solutions of theMonge-Amp\`ere equation by smooth solutions, and a theorem on the matching (inother words, continuation) of second order isometries to exact isometries.

The Monge-Ampere constrained elastic theories of shallow shells