Fourth order geometric evolution equations with nonlinear boundary conditions

具有非线性边界条件的四阶几何演化方程

• 批准号: 442279986
• 负责人: Dr. Michael Gößwein
• 金额: --
• 依托单位: Arbeitsgruppe Angewandte Mathematik, insbesondere Numerische Mathematik
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/442279986?language=en
• 关键词: Fourth; order; geometric; evolution; equations

In this project we are studying the evolution of surfaces with free boundaries or external boundary contact. A typical example for this kind of geometry is the intersection of three surfaces in one common boundary, like, e.g., in a Mercedes-Benz star. One calls such a boundary a triple point resp. a triple line in higher space dimensions. This appears in a lot of applications like, for example, in foams, cell structures and grain boundaries. The motion of the surfaces is in our situation given by variants of the so-called surface diffusion flow, an evolution law based on the curvature of the surfaces. This model was introduced to model heated polycrystals. In the second half of the last century, a lot of research was done on the evolutions of surfaces without boundary. Yet, the situation of surfaces with boundaries is still hardly studied and there are still a lot of open questions.Our projects has four big aims. Firstly, we want to improve a known existence result for surfaces with triple lines. Hereby, the main issue is to weaken the assumptions on the initial data and to derive a uniqueness result for the solution. The latter is for most evolutions of higher dimensional geometric objects still unknown.In the second part, we want to study the long time behavior of the problem. The surface diffusion flow is well-known for developing singularities in finite time. Now, the questions is how the singularities can be characterised and which conditions are sufficient to exclude a singularity.Furthermore, we want to study the existence of self-similar solutions. These are solutions which do not change for suitable scaling in space and time. Physically, one expects their appearance whenever their is no specific space or time scale for the problem.Finally, we want to consider the coupling of the surface diffusion flow with the so-called mean curvature flow. Hereby, the motivation lies in the fact that in many situations only the exterior surfaces move with respect to the surface diffusion flow. In contrast, the interior surfaces move with respect to the mean curvature flow.

在这个项目中，我们正在研究的自由边界或外部边界接触的表面的演变。这种几何形状的典型示例是三个表面在一个公共边界中的相交，例如，一辆奔驰星星。人们称这样的边界为三相点。在更高的空间维度中的三重线。这出现在许多应用中，例如，在泡沫，细胞结构和晶界中。在我们的情况下，表面的运动是由所谓的表面扩散流的变体给出的，表面扩散流是基于表面曲率的演化定律。该模型被引入到加热的多晶体模型中。上个世纪后半叶，人们对无边界曲面的演化做了大量的研究。然而，对于有边界的曲面的情况，研究还很少，仍然有很多悬而未决的问题。首先，我们要改进一个已知的存在性结果的曲面与三重线。因此，主要问题是削弱对初始数据的假设，并推导出解的唯一性结果。后者是针对大多数高维几何对象的演化问题，在第二部分中，我们想研究问题的长时间行为。表面扩散流是众所周知的发展奇性在有限时间。现在的问题是如何刻画奇点，以及哪些条件足以排除奇点。此外，我们还想研究自相似解的存在性。 这些解决方案不会因空间和时间的适当缩放而改变。从物理上讲，当问题没有特定的空间或时间尺度时，人们就期望它们的出现。最后，我们要考虑表面扩散流与所谓的平均曲率流的耦合。因此，动机在于以下事实：在许多情况下，仅外表面相对于表面扩散流移动。相反，内表面相对于平均曲率流移动。