Sharp Results for the Regularity and Stability of the Free Boundary in the Obstacle Problem

In this paper, we study the obstacle problem with obstacles whose Laplacians are not necessarily Hölder continuous. We show that the free boundary at a regular point is C if the Laplacian of the obstacle is negative and Dini continuous. We also show that this condition is sharp by giving a method to construct a counter-example when we weaken the requirement on the Laplacian of the obstacle by allowing it to have any modulus of continuity which is not Dini. In the course of proving optimal regularity we also improve some of the perturbation theory due to Caffarelli (1981). Since our methods depend on comparison principles and regularity theory, and not on linearity, our stability results apply to a large class of obstacle problems with nonlinear elliptic operators. In the case of obstacles where the Laplacian is negative and has sufficiently small oscillation, we establishmeasure-theoretic analogues of the alternative proven by Caffarelli (1977). Specifically, if the Laplacian is continuous, then at a free boundary point either the contact set has density zero, or the free boundary is a Reifenberg vanishing set and the contact set has density equal to one half in a neighborhood of the point. If the Laplacian is not necessarily continuous, but has sufficiently small oscillation, then at a free boundary point either the contact set has density close to zero, or the free boundary is a δ-Reifenberg set and the contact set has density close to one half in a neighborhood of the point.