On the fine structure of the free boundary for the classical obstacle problem

Abstract

In the classical obstacle problem, the free boundary can be decomposed into "regular" and "singular" points. As shown by Caffarelli in his seminal papers [C77, C98], regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of C 1 manifolds of varying dimension. In two dimensions, this C 1 result has been improved to C 1,α by Weiss [W99].In this paper we prove that, for n = 2 singular points are locally contained in a C 2 curve. In higher dimension n ≥ 3, we show that the same result holds with C 1,1 manifolds (or with countably many C 2 manifolds), up to the presence of some "anomalous" points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a C 1,α manifold, thus extending to every dimension the result in [W99].We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for n ≥ 3 we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.

Regularity of Interfaces in Phase Transitions via Obstacle Problems - Fields Medal Lecture

“…(4) The last part of the statement in the case (n ≥ 3)-(b) corresponds to Theorem 2.13. In [17] we obtain the same result as a simple byproduct of our analysis. In addition, our result on the existence of anomalous points shows that Theorem 2.13 is essentially optimal.…”

“…In a recent paper with Serra [17] we showed that, at most points, (2.7) holds with α = 1. However, there exist some "anomalous" points of higher codimension where not only (2.7) does not hold with α = 1, but actually (2.7) is false for any α > 0.…”

“…Here and in the sequel, dim H (E) denotes the Hausdorff dimension of a set E. The main result in [17] the following: Theorem 2.14. Let Σ := ∪ n−1 m=0 Σ m denote the set of singular points.…”

On the asymptotic behavior of the solutions to parabolic variational inequalities

“…Finally, we recall that the structure of the singular part of the free boundary of minimizers of the obstacle and thin-obstacle problems was studied by several authors; we refer to [5], [6], [23], [16] and [9], for the obstacle problem, and [19], [10] for the thin-obstacle problem. We also note that, the uniqueness of the blow-up and the logarithmic modulus of continuity follow directly by the lograithmic epiperimetric inequality (Theorem 1.10), exactly as in [9,10].…”

Free boundary regularity in obstacle problems

“…In a recent paper with Serra [15] we showed that, up to the presence of some "anomalous" points of higher codimension where (30) is false for any α > 0, one can actually prove that (30) holds with α = 1 at most points (these points will be called "generic"). In particular, up to a small set, singular points can be covered by C 1,1 (and in some cases C 2 ) manifolds.…”