Main interests

Free boundary value problems

The coupling of PDEs in a volume domain with geometric quantities at interfaces requires the use of weak solution techniques and methods of geometric measure theory. As an exaple, we were thus able to show the existence of weak solutions for a coupled Navier-Stokes Mullins-Sekerka problem for two-phase flows.

Polarization of biological cells

The formation of strongly heterogeneous protein distributions is essential for many cell biological processes. Suitable mathematical descriptions are initially in the form of reaction-diffusion systems, which take into account the reciprocal relationship between processes inside the cell with those on the cell membrane. Rigorous asymptotic simplifications allow the derivation of effective models and the characterization of the qualitative behaviour of such systems.
A particular example is the convergence of certain polarization problems to generalized obstacle problems and the proof of polarization criteria.

Variational analysis of curvature energies

Models for biological membranes are often based on Canham-Helfrich curvature energies (with the Willmore functional as a special case). Compared to phase separation energies, the higher order of such energies poses particular challenges. Typically, compactness statements can only be obtained in spaces of generalized surfaces.
An example of work in this area is the minimization of the Willmore functional under an confinement condition, for example motivated by inner membranes in biological cells.

Phase field models

In many models, the transition between different phases is not described by a low-dimensional interface, but by a thin transition layer. Phase field models from materials science are a classic example, but also so-called diffuse approximations of geometric energies. A crucial question is about asymptotic reductions for vanishing thickness of the transition layer. With variational methods (\(\Gamma\)-convergence) and measure-theoretic formulations (varifolds as a generalized notion of surfaces) we were able to prove the convergence of the De Giorgi approximation of the Willmore functional.

Geometric flows

The evolution by a steepest descent describes a possible dynamics towards low energy states. In the case of phase separation or curvature energies, for example, this leads to mean curvature and Willmore flow. Approximation results for such flows and the behavior under (stochastic) perturbations have been analyzed in several contributions of our group.

Curvature energies determine the character of many variational models of structures and processes in various applications in physics and biology, and are used in image processing and approximation theory.

Examples are Euler's bending energy for curves in the plane or in space and the Willmore functional for surfaces.

A fundamental shape energy for describing the shape of biological cells is given by the Canham-Helfrich energy

\[ \mathcal{E}_{CH}(\Sigma) = \int_\Sigma c_1|H-H_0|^2+c_2K\,d\mathcal{H}^2\]

with the mean curvature \(H\) and Gaussian curvature \(K\) of the surface \(\Sigma\), a given spontaneous curvature \(H_0\) and constants \(c_1,c_2\).

The mathematical treatment of curvature energies poses numerous challenges. Compared to the surface functional, the higher order and the lack of maximum principles are particularly noteworthy.