非欧板的标度律和黎曼度量的$W^{2,2}$等距浸入
Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics
摘要：
本文研究在自由平衡状态下应变非零的弹性结构。许多生长组织（叶、花或海洋无脊椎动物）在自由生长期间获得复杂的构型。我们的研究从三维非协调弹性理论出发，试图解释非欧度规自发形成的机制。回想一下，单连通域上的光滑黎曼度量可以实现为方向保持变形的拉回度量，当且仅当相关的黎曼曲率张量相同地为零。当这个条件失败时，人们寻求产生最接近的度量实现的变形。通过引入非欧形式的非线性弹性泛函，建立了该问题的变分形式，并在适当的尺度下证明了其$Γ$ -收敛性。作为推论，我们得到了一个新的充要条件存在的$W^{2,2}$等距浸入到$\mathbb R^3$的二维度量。
英文摘要：
This paper concerns the elastic structures which exhibit non-zero strain at free equilibria. Many growing tissues (leaves, flowers or marine invertebrates) attain complicated configurations during their free growth. Our study departs from the 3d incompatible elasticity theory, conjectured to explain the mechanism for the spontaneous formation of non-Euclidean metrics. Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its $Γ$-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a $W^{2,2}$ isometric immersion of a given 2d metric into $\mathbb R^3$.
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