Fourth order geometric evolution equations with nonlinear boundary conditions

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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.