The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 45-67.

On démontre d'abord deux résultats indépendants, l'un sur la densité des fonctions régulières dans l'ensemble des solutions de l'équation de Monge–Ampère, l'autre sur la construction d'isométries exactes par continuation à partir d'isométries infinitésimales d'ordre 2, pour des surfaces bidimensionelles.

On dérive ensuite un modèle nouveau pour les coques minces peu profondes d'épaisseur h et profondeur de l'ordre de hα départant de la théorie trois-dimensionnelle de l'élasticité nonlinéaire. Le modèle limite obtenu par la Gamma-convergence consiste à minimiser une énergie biharmonique sous une contrainte de type Monge–Ampère. Ce résultat s'applique au cas où les forces sont de l'order de hα+2 et 1/2<α<1. On peut l'étendre pour α∈(0,1) dans certains cas spécifics, utilisant les résultats de la première partie de l'article.

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 Γ-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.

DOI : 10.1016/j.anihpc.2015.08.005
Classification : 35B65, 53C24, 74K25, 74B20
Mots-clés : Monge–Ampère equation, Isometric continuation, Shallow shells, Nonlinear elasticity, Gamma convergence, Calculus of variations 
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     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Lewicka, Marta; Mahadevan, L.; Pakzad, Mohammad Reza. The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 45-67. doi : 10.1016/j.anihpc.2015.08.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.08.005/

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