Fourth order geometric evolution equations with nonlinear boundary conditions

在这个项目中，我们研究了具有自由边界或外部边界接触的曲面的演化。这类几何体的一个典型例子是三个曲面在一个公共边界上的相交，例如，在梅赛德斯-奔驰的明星中。人们称这样的边界为三点边界。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、泡孔结构和晶界中。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规律。该模型被引入到加热多晶的模型中。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，有边界曲面的情况仍然很少被研究，还有很多有待解决的问题。我们的项目有四大目标。首先，我们想要改进一个已知的三重直线曲面的存在性结果。因此，主要问题是弱化对初始数据的假设，并推导出解的唯一性结果。后者对于大多数高维几何对象的演化是未知的。在第二部分，我们想要研究该问题的长时间行为。表面扩散流以在有限时间内产生奇点而著称。现在，问题是如何刻画奇点，以及什么条件足以排除奇点。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在空间和时间上的适当缩放而改变。在物理上，当问题没有特定的空间或时间尺度时，人们期望它们的出现。最后，我们想要考虑表面扩散流和所谓的平均曲率流的耦合。因此，其动机在于，在许多情况下，只有外表面相对于表面扩散流运动。相反，内表面相对于平均曲率流移动。

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.