任意维度和余维下等距浸入系统的灵活性以及预应变薄膜的能量缩放
Flexibility of the isometric immersion system in arbitrary dimension and codimension and the energy scaling of prestrained thin films
浏览论文内容
中文总结 AI 辅助
研究任意维度和余维下等距浸入系统的灵活性，通过证明可由特定正则性的精确等距浸入逼近，得到阈值灵活性指数\(\alpha_0\)，并应用于预应变薄膜能量缩放估计，确定其缩放指数。
中文摘要 AI 辅助
我们证明，对于在\(d\)维区域上给定的\(C^{r,\beta}\)正则黎曼度量，每一个到欧几里得空间\(\mathbb{R}^{d + k}\)的短浸入，对于任意\(\alpha < \alpha_0 = \min\{\frac{r + \beta}{2}, \frac{1}{1 + d(d + 1)/k}\}\)，都可以由\(C^{1,\alpha}\)正则的精确等距浸入一致逼近。我们的定理恢复了几个先前已知的特殊情况。其新颖之处在于为任意维度\(d\)和余维\(k\)提供了统一的灵活性陈述，同时处理了到目前为止未探索的范围\(k\in (1, \frac{d(d + 1)}{2} - d + 1)\setminus \{d\}\)，在此之前没有相应的一般结果。我们的阈值灵活性指数\(\alpha_0\)与先前为密切相关的蒙日 - 安培系统获得的指数一致。作为应用，我们证明了关于预应变薄膜定量可浸入性的一个新估计，确定了在存在任意预应力度量且薄膜厚度趋于零时非欧几里得能量最小值的缩放指数为\(\frac{4\alpha_0}{\alpha_0 + 1}\)。
英文摘要
We prove that, for a given $C^{r,β}$-regular Riemann metric posed on a $d$-dimensional domain, every short immersion into the Euclidean space $\mathbb{R}^{d+k}$, can be uniformly approximated by exact isometric immersions of regularity $C^{1,α}$ for any $α<α_0=\min\{\frac{r+β}{2}, \frac{1}{1+d(d+1)/k}\}$. Our theorem recovers several previously known results as special cases. The novelty thereof lies in providing a unified flexibility statement for arbitrary dimensions $d$ and codimensions $k$, while also treating the so far uncharted range $k\in (1, \frac{d(d+1)}{2}-d+1)\setminus \{d\}$, where no corresponding general result was previously available. Our threshold flexibility exponent $α_0$ agrees with that previously obtained for the closely related Monge-Ampère system. As an application, we prove a new estimate in the quantitative immersability of thin prestrained films, setting the scaling exponent of the infimum of non-Euclidean energies in presence of an arbitrary prestrain metric, and in the limit of the film's vanishing thickness, at $\frac{4α_0}{α_0+1}$.
