On properties of the free boundary in the neighborhood of contact with the given boundary

For the simpliest elliptic obstacle problem, the behavior of the free boundary in the vicinity of the points where it contacts the prescribed boundary of the domain is studied. The eairlier result concerning the $C^1$ regularity of the boundary $\partial\mathscr N$ of the noncoincidence set is strengthened. It is proved that the assumed earlier Lipschitz condition on $\partial\mathscr N$ can be omitted.