My research interests are partial differential equations and calculus of variations where I mostly focus my attention on problems with a geometric background.
Gradient flows of Curvature energies
The Willmore flow and related topics
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 develop singularities in finite time. Also how to extend the flow after such a singularity. That there are topological restrictions that force a singularity to form was shown in (Blatt, 2009)
If one adds positive multiples of the volume of the area to the Willmore energy, one can in certain cases show that singularities must form in finite time using that these terms are not scaling invariant (Blatt, 2019).
Gradient flows of 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 2 and 4, many of them are degenerate and remind vaguely of p-Laplace operator, and they contain critical operators. These interconnections to very active fields in currently mathematical research but also to applications in harmonic analysis, low-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 modeling 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 enhances the known formulas by Giga and Kohn.\[a^2 +b^2 =c^2\]
Regularity of non-local partial differential equations
Non-local partial differential equations pop up naturally in very different circumstances. Boltzmann’s equation, models for US option prices, fractional minimal surfaces and the knot energies above are examples for this.
- Blatt, S. (2009). A singular example for the Willmore flow. Analysis, 29(4), 407–430. https://doi.org/10.1524/anly.2009.1017
- Blatt, S. (2019). A note on singularities in finite time for the \protectL^2 gradient flow of the Helfrich functional. Journal of Evolution Equations. https://doi.org/10.1007/s00028-019-00483-y