Sharp results for the regularity and stability of the free boundary in the obstacle problem

Abstract

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 1 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 establish measure-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.

Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem

“…Then the result will follow from the corresponding result for the classical obstacle problem. Namely, we will use Blank's sharp form for the C 1 regularity of the free boundary, see [Bla01] and Theorem 4 below.…”

Eliminating Mixed Asymptotics in Obstacle Type Free Boundary Problems

“…(The constant will depend linearly on L, but otherwise it will only depend on the dimension of the space. See Blank, 2001;Brézis andKinderlehrer, 1973/1974;Caffarelli, 1998 or Caffarelli andKinderlehrer, 1980 for a proof.) In particular, this universal bound gives enough compactness to guarantee that the quadratic rescaling w E ðxÞ :¼ E À2 wðExÞ ð1:3Þ…”

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

“…The first claim of this Theorem on the regular part of the free boundary was already proved (with a different proof for the Laplace operator) in Blank [5]. The second claim of the Theorem on the singular part of the free boundary was proved by Caffarelli [8] for Lipschitz coefficients, and by Monneau [32] for Hölder coefficients, including extensions for double Dini coeffifients).…”

“…On the contrary, if we assume moreover that f ≥ δ 0 > 0, then it is classical that 0 can not be a degenerate point (see Caffarelli [7], Blank [5]). …”

“…Concerning regular points, let us mention slightly more precise results on the regularity of the free boundary when the modulus of continuity is not only controled at the origin 0, but controled at any points (see Blank [5] for sharp results). See the previous works of Caffarelli [6,7,8,9], Weiss [38], and Caffarelli, Karp, Shahgholian [13] for Lipschitz coefficients, and also the work of Caffarelli, Kinderlehrer [14] for some related estimates on the modulus of continuity of the solution or of its gradient.…”