We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the Föppl–von Kármán equations, we find the Γ-limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff–Love extensions of the highly perturbative, Hölder-regular solutions to the Monge–Ampere equation obtained by means of convex integration.

我们关注三维薄弹性薄膜的尺寸缩减分析，该薄膜通过具有弱曲率的黎曼度量进行了预应变。对于推导Föppl–vonKármán方程的不相容版本的预应力，我们发现Γ-重新缩放的能量的极限，确定最佳的能量缩放定律，并根据预应力分量和相关黎曼度量的曲率显示最优性的等效条件。当引起拉伸的预应变不包含平面模式时，我们发现与先前描述的浅壳模型相似。在较高的预应力状态下，我们通过构造变形来证明新的能量上限，这些变形是通过凸积分获得的Monge-Ampere方程的高扰动，Hölder正则解的Kirchhoff-Love扩展。