"Almost positivity'' in the fourth order clamped plate equation
A classical example for a fourth order problem in mechanics is the linear clamped plate boundary value problem. "Linear questions'' may be considered as well understood. This changes completely as soon as one poses the simplest ``nonlinear question'': What can be said about positivity preserving? Does a clamped plate bend upwards when being pushed upwards? It is known that the answer is ''no'' in general. However, there are positivity issues as e.g. "almost positivity'' to be discussed.
The lecture is based on joint work with F. Robert (Nice) and G. Sweers (Cologne).

All constant mean curvature surfaces with three ends
In work joint with Kusner and Sullivan we classify and prove existence of all embedded constant mean curvature surfaces which have genus zero and three ends.  There is a similar theorem for the case of more ends, but only for the case the surfaces have a mirror symmetry.  The proof is based on conjugate surface techniques.

Boundary regularity via Uhlenbeck-Rivière decomposition
Applying a technique due to T. Rivière we prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the $H$-surface system $\triangle u = 2H(u)\partial_{x^1}u\wedge\partial_{x^2} u$ with bounded $H$.
(Joint work with F. Müller)

 

Gauß and mean curvature flow near cones
We present some "Australian souvenirs" concerning the evolution of entire graphs. For mean curvature flow we obtain a stability result, especially near mean convex cones. We show longtime existence for Gauß curvature flow and investigate the stability of solutions.

Self-similar expanding solutions  for the planar network flow

Stability and prescription of curvature for conformal invariant operators