Energy focusing in thin elastic structures and isometric immersions

本项目的目的是推进薄弹性结构的数学理解。第一部分，我们考虑板在后屈曲状态下的变分问题。该区域的特征在于（自由）弹性能在脊和顶点中的聚焦。更好地理解这些现象不仅与应用数学有关，而且与物理学和工程学有关。在我们的情况下，可以预期，在自然界中观察到的构型接近于自由弹性能的最小值。因此，我们将研究在各种设置中的弹性能量的最小化，并推导出在片材厚度方面的最小能量的标度律，这将是所考虑的模型中的一个小标量参数。我们的目标是得到一个严格的解释后屈曲结构的出现在这种方式。在所考虑的模型中，领先的顺序的弹性能量的贡献措施的偏差的弹性变形从等距浸入。(An如果范围中的度量的拉回与域中的参考度量一致，则浸入是等距的。通过这种方式，与等距沉浸问题的自然联系出现了，特别是关于它们的独特性。后者是第二部分的重点。等距浸入的唯一性在很大程度上取决于所需的规则性。它已被证实，有一个关键的规律性，等距浸入以上的等距浸入是唯一的（刚性运动，并在适当的进一步假设）。众所周知，一个适当的外曲率概念的有限性意味着这种唯一性。因此我们将研究低正则浸入的外曲率。在坐标图中，这相当于某些分布雅可比行列式和海森行列式是否属于合适的函数空间的问题。这些分布决定因素的中心感兴趣的第二部分的proposal.Our最大的目标是改进已知的唯一性结果等距浸入Hölder连续导数的有效性范围扩展到较低的正则性函数。与第一部分的联系可以在非本征曲率的研究中找到，它在标度律的推导中也起着重要的作用。

The present project's aim is to advance the mathematical understanding of thin elastic structures. It consists of two parts.In the first part, we consider variational problems that model sheets in the post-buckling regime. This regime is characterized by the focusing of (free) elastic energy in ridges and vertices. A better understanding of these phenomena is relevant not only in Applied Mathematics but also in Physics and Engineering.In our situation it is to be expected that the configurations observed in nature are close to minimizers of the free elastic energy. Hence we will investigate the minimization of elastic energy in various settings and derive scaling laws for the minimum of the energy in terms of the thickness of the sheet, which will be a small scalar parameter in the considered models. Our goal is to obtain a rigorous explanation for the emergence of post-buckled structures in this way.In the considered models, the leading order contribution to the elastic energy measures the deviation of the elastic deformation from an isometric immersion. (An immersion is isometric if the pull-back of the metric in the range coincides with the reference metric in the domain.) In this way, a natural connection to questions about isometric immersions appears, in particular about their uniqueness. The latter is the focus of the second part of the proposed project.The uniqueness of isometric immersions depends heavily on the required regularity. It has been conjectured that there is a critical regularity for isometric immersions above which isometric immersions are unique (up to rigid motions, and under suitable further hypotheses). It is known that the finiteness of a suitable notion of extrinsic curvature implies such uniqueness. Hence we will study the extrinsic curvature of immersions with low regularity. In coordinate charts, this amounts to the question whether or not certain distributional Jacobian and Hessian determinants belong to suitable function spaces. These distributional determinants are the central objects of interest in the second part of the proposal.Our maximal goal is to improve the known uniqueness results for isometric immersions with Hölder continuous derivatives by extending their range of validity to functions with lower regularity. The connection to the first part is to be found in the study of extrinsic curvature, which also plays an important role in the derivation of scaling laws.