We look for a body with minimal drag coefficient in a rarefied particle stream. Modelling this problem leads to a nonconvex, noncoercive functional, which is investigated on suitable classes of bodies. It is interesting to note that optimal bodies with circular maximal cross section are not rotational bodies. This phenomenon disappears if frictional effects are added to the model. (B.Kawohl et al.)

Equations of this nature occur in connection with free boundary problems for fluids under microgravity or zero gravity. Their solutions violate classical Dirichlet boundary conditions, but they can be interpreted as viscosity solutions in the sense of Crandall/Lions. (M. Kocan, N.Kutev, B.Kawohl)

Local minimizers of many variational problems in mathematical physics display simple qualitative properties such as monotonicity or radial symmetry. Continuous rearrangements can be used to derive these kinds of results. A typical theorem is the following: Any bounded equilibrium configuration of a liquid, which rotates uniformly about an axis and is subject only to self gravitation, has a plane of symmetry orthogonal to the axis. (F. Brock.)

How should a load be distributed on an elastic membrane so that its deformation becomes minimal ?

How should one distribute a given amount of insulating material on the boundary of a body so as to optimally protect it from losing heat? (S. Cox, B. Kawohl)

For these problems the geometry of the domain is a further unknown. Take for instance the interface between two liquids with surface tension. Their pressure difference determines the mean curvature of the interface. In the time dependent case the interface can move in space. Another free boundary problem is to determine the shape of a body of minimal resistance which is immersed in a liquid, by minimizing a suitable functional. Depending on the physical model this functional may take the surrounding fluid flow into consideration. (A. Wagner)

It has been observed that in the life cycles of cellular slime molds aggregation takes place after a phase of propagation. It is known that for some species this aggregation is caused by deliveration of a chemical substrate by a so called founder cell. After that the other amoebae start moving to this founder cell. It has been shown that for some special initial data the solutions of a system modeling this process blow up. Some of the questions which arise are: How does the blow up take place? What assumptions influence the blow-up? (D.Horstmann)

There are not always classical nonparametric solutions to these equations on nonconvex domains or for incompatible boundary data. There are however generalized solutions, which shall now be identified as viscosity solutions, calculated and plotted. We expect boundary layer phenomena. (H.G.Reschke)

Let g(x) be the brightness distribution of a blurred image. Numerical experiments suggest that there are appropriate diffusion equations which turn the initial value g(x) after some time into a sharper image. This phenomenon is investigated with analytical tools. The diffusion equations are highly nonlinear and contain very difficult subproblems as special cases, such as the Tricomi problem, forward-backward diffusion etc. (B. Kawohl, N. Kutev, M. Mester)

The deformation of elastic bodies is usually restricted by the presence of other bodies and touching can occur. For a long time the investigation of such contact problems was restricted to simple elastic models leading to variational inequalities due to convexity. This classical approach turns out to be too restrictive for modern applications where large deformations and nonlinear constitutive equations are considered. Hence new methods for the treatment of nonlinear contact problems have to be developed. Here the tools of nonsmooth analysis, worked out during the last 20-30 years, are very useful. (F. Schuricht)

Besides bearing the genetic code the mechanical deformation of DNA molecules in the cell has biological relevance too. Due to their special structure the molecules behave like elastic strings. Torsional stresses may cause that the molecule wraps around itself (supercoiling) and one observes the phenomenon of self-touching, which is hard to treat. The analytical treatment of the problem and the improvement of numerical schemes, which still fail, is claimed. (O. Gonzalez, J. Maddocks, F. Schuricht, H. v.d. Mosel)