Generic regularity of free boundaries for the obstacle problem

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 $n \leq 4$, 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 $n \leq 4$