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The Monge-Ampere constrained elastic theories of shallow shells
Analysis of PDEs · Submitted 30 Nov 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Motivated by the degree of smoothness of constrained embeddings of surfaces in $\mathbb{R}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell scales like $h^\alpha$ and the applied forces scale like $h^{\alpha+2}$, in the limit when $h\to 0$.
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Motivated by the degree of smoothness of constrained embeddings of surfaces in $\mathbb{R}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell 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 independent results: a theorem on approximation of $W^{2,2}$ solutions of the Monge-Amp\`ere equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometries.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by the degree of smoothness of constrained embeddings of surfaces in $\mathbb{R}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell scales like $h^\alpha$ and the applied forces scale like $h^{\alpha+2}$, in the limit when $h\to 0$. | contribution 0.87 | 0.96 | 0.52 | asserted 1.00 |
| s1 | The main analytical ingredients are two independent results: a theorem on approximation of $W^{2,2}$ solutions of the Monge-Amp\`ere equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometries. | method 0.77 | 0.88 | 0.08 | asserted 1.00 |
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Original abstract
Motivated by the degree of smoothness of constrained embeddings of surfaces in $\mathbb{R}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell 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 independent results: a theorem on approximation of $W^{2,2}$ solutions of the Monge-Amp\`ere equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometries.
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1 arXiv paper cite this in 5 sentences
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This may be contrasted with a similar problem considered by the authors in [ref] , where the $Γ$-limit is discussed under energy regime of order $h^2α+2$.
Models for elastic shells with incompatible strains · Discussion · 1.00
Cites it as background 1 paper
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To put the last two results in another context, we draw the reader's attention to the forthcoming paper [ref] , where we analyze the $Γ$-limit of the shallow shell energies $1/h^2α+2 I^h^α,h$ on shells with curvature of order $h^α$.
Models for elastic shells with incompatible strains · The prestrained shallow shell with a first-order isometry constraint: $0 <α<1$ · 0.90
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