This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood of regular enough two dimensional compact surface. We prove an asymptotically sharp geometric rigidity interpolation inequality for thin domains with little regularity. In contrast to that celebrated Friesecke-James-M\"uller rigidity estimate [\textit{Comm. Pure Appl. Math.,} 55(11):1461-1506, 2002] for plates, our estimate holds for any proper rotations $\BR\in SO(3).$ Namely, the estimate bounds the $L^p$ distance of the gradient of any $\By\in W^{1,p}(\Omega,\mathbb R^3)$ field from any constant proper rotation $\BR\in SO(3)$, in terms of the average $L^p$ distance (nonlinear strain) of the gradient $\nabla\By$ from the rotation group $SO(3)$, and the average $L^p$ distance of the field itself from the set of rigid motions corresponding to the rotation $\BR$. There are several remarkable facts about the estimate: 1. The constants in the estimate are sharp in terms of the domain thickness scaling for any thin domains with the required regularity. 2. In the special cases when the domain has positive or negative Gaussian curvature, the inequality reduces the problem of estimating the gradient $\nabla\By$ in terms of the prototypical nonlinear strain $\int_\Omega\mathrm{dist}^p(\nabla\By(x),SO(3))dx$ to the easier problem of estimating only the vector field $\By$ in terms of the nonlinear strain without any loss in the constant scalings as the Ans\"atze suggest. The later will be a geometric rigidity Korn-Poincar\'e type estimate. This passage is major progress in the thin domain rigidity problem. 3. For the borderline energy scaling (bending-to-stretching), the estimate implies improved strong compactness on the vector fields for free.

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薄 Bi-Lipschitz 域中的渐近锐几何刚度插值估计

这项工作是薄域渐近锐几何刚度估计开发计划的一部分。三维欧几里得空间中的薄域大致是足够规则的二维紧凑表面的小邻域。我们证明了一个几乎没有规律性的薄域的渐近尖锐的几何刚度插值不等式。与著名的 Friesecke-James-M\"uller 刚度估计 [\textit{Comm. Pure Appl. Math.,} 55(11):1461-1506, 2002] 相比，我们的估计适用于任何适当的旋转 $ \BR\in SO(3).$ 即估计值限制了任何 $\By\in W^{1,p}(\Omega,\mathbb R^3)$ 的梯度的 $L^p$ 距离来自任何恒定适当旋转 $\BR\in SO(3)$ 的场，根据来自旋转组 $SO(3) 的梯度 $\nabla\By$ 的平均 $L^p$ 距离（非线性应变） $, 以及场本身与对应于旋转 $\BR$ 的一组刚性运动的平均 $L^p$ 距离。关于该估计有几个值得注意的事实： 1. 就具有所需规律性的任何薄域的域厚度缩放而言，估计中的常数是尖锐的。2.在域具有正或负高斯曲率的特殊情况下，不等式减少了从原型非线性应变$\int_\Omega\mathrm{dist}^p估计梯度$\nabla\By$的问题(\nabla\By(x),SO(3))dx$ 到仅根据非线性应变估计矢量场 $\By$ 的更简单的问题，而不会像 Ans\"atze 建议的那样在常数缩放中造成任何损失. 后者将是几何刚度 Korn-Poincar\'e 型估计。这段话是薄域刚性问题的重大进展。3. 对于边界能量缩放（弯曲到拉伸），该估计意味着矢量场上免费的强紧致性得到改善。