The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells.

The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of W 2 , 2 solutions of the Monge–Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometric embeddings of 2d surface in R 3 . In the second part, we rigorously derive the Γ-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness h , where the depth of the shell scales like h α and the applied forces scale like h α + 2 , in the limit when h → 0 . We offer a full analysis of the problem in the parameter range α ∈ ( 1 / 2 , 1 ) . We also complete the analysis in some specific cases for the full range α ∈ ( 0 , 1 ) , applying the results of the first part of the paper. [ABSTRACT FROM AUTHOR]