Main areas of research are Calculus of Variations and non-linear partial differential equations. Let me present some projects
I am currently working on.

## Knots and Repulsive Energies

During the last 25 years many new geometric energies for curves and surfaces have be introduced
in order to find the nicest shape of a knotted geometric object.
The central idea behind these energies is to punish self-intersections by growing beyond all bounds.

A striking property of these functionals is that there Lagrange operator is a non-local operator
of order between 3 and 4, many of them are degenerate and remind vaguely of p-Laplace operator,
and they contain critical operators.
These interconnections very active fields in currently mathematical research but also to applications in harmonic analysis, l
ow-dimensional topology and last but not least the physics of DNA make this topic such an interesting one.

In my research I concentrate on variational problems, the question of regularity
in this area, and on the heat equation of these energies.

## Supercritical semi-linear equations

Together with Michael Struwe, I have been looking at the Lane-Emden equation which comes
from the modelling of polytropic starts and their gradient flow in the supercritical regime.
One of our basic tools in the analysis is a new monotonicity formula that enhences the known formulas by Giga and Kohn.
We have used this monotonicity formula to analysis singularities in the interior and at the boundary.
Motivated by these findings we derived new short- and long-time existence results the parabolic equation related to the
Lane-Emden equation in critical Morrey spaces.

## The Willmore flow and related geometric evolution equations

Although in the last years a lot of research has been done in the area of geometric evolution equations of higher order, still even some basic questions
are open. For example, it in not known up to now, whether flows like the Willmore flow
or the surface diffusion flow can develope singularities in finite time.
Also how to extend the flow after such a singularity is not known up to now.
In ... we study the gradient flow of the Hefrich functional with vanishing spontaneous curvature.
We show that under very natural and mild assumptions the resulting evolution equations develop singularities in finite time.

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