On global solutions of the obstacle problem

Recently great progress has been achieved by Alessio Figalli et al. doing a fine analysis of the singular set in the obstacle problem. Much less is known about the behavior of the regular set close to singularities. The structure of this set may be very complicated, for example it may contain infinitely many connected components approaching the singularity, funnel-type singularities etc.

A characterization of global solutions, that is solutions in the entire space, would help but this problem (related to conjectures in potential analysis) is open since the 1980s. In this talk we will discuss progess in the characterization of global solutions.