Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics
Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics
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非欧几里得板的缩放定律和黎曼度量的 W^{2,2} 等距浸没
DOI：
10.1051/cocv/2010039
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
R. Pakzad
中科院分区：
文献类型：
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作者：
M. Lewicka;R. Pakzad
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必要和充分条件存在的W2，2等距浸入到R3的一个给定的二维度量。
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 R 3 .