We propose results that relate the following two contexts: (i) Given a Riemann metric G on $$\Omega ^1=\omega \times (-\frac{1}{2}, \frac{1}{2})$$ Ω 1 = ω × ( - 1 2 , 1 2 ) , we find the infimum of the averaged pointwise deficit of an immersion from attaining the orientation-preserving isometric immersion of $$G_{\mid \Omega ^h}$$ G ∣ Ω h on $$\Omega ^h=\omega \times (-\frac{h}{2}, \frac{h}{2})$$ Ω h = ω × ( - h 2 , h 2 ) , over all weakly regular immersions. This deficit is measured by the non-Euclidean energies $${\mathcal {E}}^h$$ E h , which can be seen as modifications of the classical nonlinear three-dimensional elasticity. (ii) We complete the scaling analysis of $${\mathcal {E}}^h$$ E h in the context of dimension reduction as $$h\rightarrow 0$$ h → 0 , and the derivation of $$\Gamma $$ Γ -limits of the scaled energies $$h^{-2n}{\mathcal {E}}^h$$ h - 2 n E h for all $$n\geqq 1$$ n ≧ 1 . We show the energy quantisation, in the sense that the even powers 2 n of h are indeed the only possible ones (all of them are also attained). For each n , we identify conditions for the validity of the scaling $$h^{2n}$$ h 2 n , in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $$h\rightarrow 0$$ h → 0 .

中文翻译：

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黎曼度量的定量可浸入性和预应力壳模型的无限层次结构

我们提出了与以下两个上下文相关的结果：(i) 给定 $$\Omega ^1=\omega \times (-\frac{1}{2}, \frac{1}{2}) 上的黎曼度量 G $$ Ω 1 = ω × ( - 1 2 , 1 2 ) ，我们找到了从获得 $$G_{\mid \Omega ^h}$$ G ∣ Ω h on $$\Omega ^h=\omega \times (-\frac{h}{2}, \frac{h}{2})$$ Ω h = ω × ( - h 2 , h 2 ) ，在所有弱规则浸入中。这种赤字是通过非欧式能量 $${\mathcal {E}}^h$$ E h 来衡量的，它可以看作是经典非线性三维弹性的修改。(ii) 我们在降维上下文中完成了 $${\mathcal {E}}^h$$ E h 的缩放分析为 $$h\rightarrow 0$$ h → 0 ，和推导 $$\Gamma $$ Γ - 缩放能量的极限 $$h^{-2n}{\mathcal {E}}^h$$ h - 2 n E h 对于所有 $$n\geqq 1 $$ n ≧ 1 。我们展示了能量量子化，从某种意义上说 h 的偶次幂 2 n 确实是唯一可能的（所有这些都可以实现）。对于每个 n ，我们确定缩放 $$h^{2n}$$h 2 n 的有效性条件，根据 G 的黎曼曲率的消失达到适当的阶数，以及根据匹配的等距扩展。我们还将最小化浸入的渐近行为建立为 $$h\rightarrow 0$$ h → 0 。我们确定缩放 $$h^{2n}$$h 2 n 的有效性条件，根据 G 的黎曼曲率的消失达到适当的阶数，以及匹配的等距扩展。我们还将最小化浸入的渐近行为建立为 $$h\rightarrow 0$$ h → 0 。我们确定缩放 $$h^{2n}$$h 2 n 的有效性条件，根据 G 的黎曼曲率的消失达到适当的阶数，以及匹配的等距扩展。我们还将最小化浸入的渐近行为建立为 $$h\rightarrow 0$$ h → 0 。