Published Paper

Inserted: 24 jun 2020
Last Updated: 19 aug 2024

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 \( \mathbb{R}^n \). 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). Thus, 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 \( n \leq 4 \).

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