Global solutions to the fully nonlinear obstacle problem

In this paper, we investigate global solutions to the fully nonlinear obstacle problem. We prove that, after factoring out invariant directions, the coincidence set of any global solution belongs to one of the following four types: a single point, a compact convex set with nonempty interior, a ray for $n=1$, or a slender body for $n\ge 2$, where $n$ denotes the spatial dimension. Furthermore, we establish a bijection between the collection of all global solutions with compact coincidence sets and a family of functions describing the asymptotic behavior of such solutions at infinity.