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 Rn. By classical results of Caffarelli, the free boundary is C∞ 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 Hn−4 measure (in particular, it has codimension 3 inside the free boundary). Thus, for n ≤ 4, the free boundary is generically a C∞ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions n ≤ 4.