Higher order problems with free boundary
In this project, I consider variational problems of higher order whose solutions have a free boundary.
In recent years, a rich theory has been developed for free boundary problems. But the research relies heavily on the maximum principle. Since the maximum principle is unavailable for problems of higher order, new methods need to be developed.
Our work focuses maily on two concrete problems. The first one is the biharmonic AltCaffarelli problem, where we seek to understand the effect of "adhesion terms" on higher order energies. The second one is the obstacle problem for elastic curves, where we minimize a geometric energy with an "obstacle constraint". We are interested in both elliptic and parabolic problems. The study of such free boundary problems leads inevitably to the study of PDE's invovling measures  another vivid field of research.
Here I mention (an incomplete list of) some interesting works of other researchers in this field:

A. Dall'Acqua, K. Deckelnick. An obstacle problem for elastic graphs (2018)

T. Miura. Polar tangential angles and free elasticae (2020)

S. Okabe, K. Yoshizawa. A dynamical approach to the variational inequality on modified elastic graphs (2020)

L. Caffarelli, A. Friedman. The obstacle problem for the biharmonic operator (1979)

S. Dipierro, A. Karakhanyan, E. Valdinoci. A free boundary problem driven by the biharmonic operator (Preprint, 2018)
Longtime behavior of higher order geometric flows
In this project, I look at gradient flows of higher order geometric energies, such as the Willmore energy or Euler's elastic energy. To study asymptotics of those flows, an explicit understanding of the geometry of critical points is required. At times this understanding involves a lot of geometry.
While research about those critical points is not new, the questions arising from the field of geometric flows are increasingly complex and many methods have to be improved to be applicable. In the past years, there have been discovered many singular phenomena, whose nature is not yet understood.
Some (definitely not all) interesting articles of other researchers:

J. Langer, D. Singer. The total squared curvature of closed curves (1984)

M. Pozzetta. Convergence of the elastic flow into manifolds (Preprint, 2020)

E. Kuwert, R. Schätzle. Removability of point sigularities of Willmore surfaces (2004)

S. Blatt. A singular example for the Willmore Flow (2009)
Further interests
Here I mention some further fields I would like to learn more about in the future.

Geometric variational problems on curves

Higher oder curvature energies

Energies involving anisotropic curvature

Networks of curves


PDE's involving measures

Regularity theory for measures arising in free boundary problems

(weak solutions of) geometric equations
