The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of W2,2
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 R3.
In the second part, we rigorously derive the -limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness
h, where the depth of the shell scales like hα and the applied forces scale like hα+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.
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