Free boundary regularity in obstacle problems with a degenerate forcing term

In this paper, we consider the properties of a special free boundary point in the following obstacle problem: The Laplacian of u equals f(x) multiplied by the characteristic function of the set where u is positive within the two-dimensional unit ball, where $f(x)=|x|$ is a degenerate forcing term. The key challenge stems from the degeneracy of $f(x)$, which leads to a more complex structure of the free boundary compared to the classical setting. To analyze it, we introduce the epiperimetric inequality developed by Weiss (Invent Math 138:23-50, 1999). Although this powerful tool was firstly introduced for the classical obstacle problem characterized by $f(x)>0$ in B_1, it also proves effective in our degenerate setting. This allows us to first obtain the decay rate of the Weiss energy for all blow-ups at the origin, which in turn implies the uniqueness of the blow-up profiles. With this uniqueness established, we then prove a very weak directional monotonicity properties satisfied by the solutions. This finally yields the regularity of the free boundary at the origin if the origin is a regular point.