具有旋转对称性的薄板类圆锥形变形
偏微分方程分析
2013-07-25 v2
摘要
数值实验发现，当从弹性薄板中移除一个扇区并将该扇区的边缘重新粘合时，所得构型呈径向对称且近乎圆锥形。我们在两个简化假设下对此情形进行了严格分析：首先，我们仅考虑径向对称构型；其次，我们考虑所谓的 von-Kármán 极限，其中移除区域的大小以及变形量均很小。我们对无限大薄板选择了自由边界条件。我们证明了适当重整化的自由能泛函极小值的存在性。作为副产品，我们获得了在与 d-圆锥相关背景下曾被猜想的弹性能量下界。此外，我们确定了极小值在无穷远处的形状（精确到指数衰减项）。
引用
@article{arxiv.1212.0705,
title = {Almost conical deformations of thin sheets with rotational symmetry},
author = {Stefan Müller and Heiner Olbermann},
journal= {arXiv preprint arXiv:1212.0705},
year = {2013}
}
备注
18 pages. Revised and streamlined version. Section 1: Comment to relation to d-cones added. Section 2: Boundary conditions at 0 no longer imposed, but as consequence of energy estimates (Proposition 1). Section 3: Reorganized and shortened. Explanation of the strategy of proof added. All interpolation estimates now based on Lemma 2. Section 4: Shortened. Weighted energy estimates no longer needed