Generic regularity of free boundaries for the obstacle problem

TL;DR

This paper proves that, generically, the singular set of free boundaries in the obstacle problem has measure zero in certain dimensions, implying that for dimensions up to four, the free boundary is typically a smooth manifold.

Contribution

It establishes the generic regularity of free boundaries, confirming that singular sets are negligible in measure for most cases, and solves a longstanding conjecture for dimensions up to four.

Findings

Singular set has zero 6; measure in generic cases.

For n  4, free boundary is a smooth manifold.

Confirms Schaeffer's conjecture for n  4.

Abstract

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\mathbb R^n$. By classical results of Caffarelli, the free boundary is $C^\infty$ outside a set of singular points. Explicit examples show that the singular set could be in general $(n-1)$-dimensional ---that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero $\mathcal H^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). In particular, for $n\leq4$, the free boundary is generically a $C^\infty$ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions $n\leq4$.