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.
Figalli, Alessio