Ellipsoids and paraboloids in free boundaries and potential theory

The purpose of this project is to extend the recent success in answering a long standing conjecture on the structure of global solutions for the obstacle problem, the most extensively studied free boundary problem, to the thin obstacle problem.

Such a rigidity of global solutions is not only interesting in itself but allows for a completely new type of free boundary result.

After the obstacle problem the thin obstacle problem is arguably the most prototypical and most well studied free boundary problem, where so far almost nothing is known about the structure of global solutions and so far not even a conjecture exists.

We will raise a conjecture on the structure of global solutions in the regime of subquadratic growth.On the other hand note that the classification of global solutions of the obstacle problem has also an equivalent formulation in terms of a very old problem in potential theory, i.e. Newton´s no gravity in the cavity problem.

Another classical problem in potential theory posed by Newton is the possible shapes of uniformly rotating stars.

This problem may also be formulated as a free boundary problem, however a typically unstable one.In this problem there is a non-uniqueness of possible shapes of stars for a given angular velocity of rotation. However until today not even stability of the `most physical´ solution has been established.

Combining methods from potential theory and free boundaries we aim to change this.