*Accepted Paper*

**Inserted:** 24 jun 2020

**Last Updated:** 24 jun 2020

**Journal:** Publ. Math. Inst. Hautes Études Sci.

**Year:** 2020

**Abstract:**

The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in $\R^n$. 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). Thus, 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 $n\leq4$

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