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

TL;DR

This paper investigates the detailed geometric structure of the free boundary in the classical obstacle problem, establishing regularity results for singular points in two and higher dimensions, and providing optimal density decay estimates and examples of anomalous points.

Contribution

It proves that in two dimensions singular points lie on a $C^2$ curve, and in higher dimensions they lie on $C^{1,1}$ or $C^2$ manifolds, extending regularity results and constructing examples of anomalous points.

Findings

Singular points in 2D are contained in a $C^2$ curve.

In higher dimensions, singular points lie in $C^{1,1}$ or countably many $C^2$ manifolds.

The Hausdorff dimension of anomalous points matches the proven bounds, showing sharpness.

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 \cite{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 $C^{1,\alpha}$ by Weiss \cite{W99}. 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 in \cite{W99}. 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.