Calculus of Variations and Geometric Measure Theory

A. Figalli - J. Serra

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

created by figalli on 09 Sep 2018
modified on 03 Jun 2019


Accepted Paper

Inserted: 9 sep 2018
Last Updated: 3 jun 2019

Journal: Invent. Math.
Year: 2019


In the classical obstacle problem, the free boundary can be decomposed into "regular'' and "singular'' points. As shown by Caffarelli in his seminal papers, 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 $C^{1,\alpha}$ by Weiss.

In this paper we prove that, for $n=2$ singular points are locally contained in a $C^2$ curve. In higher dimension $n\ge 3$, 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 $C^{1,\alpha}$ manifold, thus extending to every dimension the result of Weiss.

We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for $n\ge3$ we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.