On global solutions of the obstacle problem -- application to the local analysis close to singularities

The objective of this paper is twofold. First we provide the -- to the best knowledge of the authors -- first result on the behavior of the regular part of the free boundary of the obstacle problem close to singularities. We do this using our second result which is the partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets.