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 Alt-Caffarelli 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:

Long-time 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:

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