Energy focusing in thin elastic structures and isometric immersions

本项目的目的是促进对薄弹性结构的数学理解。它由两部分组成。在第一部分中，我们考虑了后屈曲状态下的变分问题。这种状态的特点是（自由）弹性能集中在脊和顶点。更好地理解这些现象不仅与应用数学有关，而且与物理和工程有关。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，我们将研究各种设置下弹性能量的最小化，并根据薄片的厚度推导出能量最小值的标度定律，薄片的厚度将是所考虑的模型中的一个小标量参数。我们的目标是获得一个严谨的解释，以这种方式出现后屈曲结构。在考虑的模型中，弹性能量的阶贡献测量了等距浸没造成的弹性变形的偏差。（如果浸入范围内度量的回拉与域内参考度量一致，则浸入是等距的。）通过这种方式，我们自然而然地联想到关于等距沉浸的问题，特别是关于它们的独特性。后者是拟议项目第二部分的重点。等距浸没的独特性在很大程度上取决于所需的规律性。据推测，等距浸没存在一个临界规律，在这个规律之上，等距浸没是唯一的（直到刚性运动，并在适当的进一步假设下）。众所周知，一个合适的外在曲率概念的有限性意味着这种唯一性。因此，我们将研究低规则浸没的外在曲率。在坐标图中，这相当于某个分布雅可比行列式和黑森行列式是否属于合适的函数空间的问题。这些分布决定因素是本建议第二部分关注的中心对象。我们的最大目标是通过将其有效性范围扩展到具有较低正则性的函数来改进具有Hölder连续导数的等距浸入的已知唯一性结果。与第一部分的联系在于外在曲率的研究，它在尺度定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.