The Monge-Ampere constrained elastic theories of shallow shells

2013/11/30 by Marta Lewicka, L. Mahadevan, Lewicka, Marta +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1312.0050

openalex publication_date 2013/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Motivated by the degree of smoothness of constrained embeddings of surfaces in ℝ³, and by the recent applications to the elasticity of shallow shells, 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. The main analytical ingredients 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 isometries.

Citations

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