The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in \(\mathbf {R}^{n}\). 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 \(n\leq 4\).

本文的目标是为\(\mathbf {R}^{n}\)中的障碍问题建立自由边界的通用正则性。根据 Caffarelli 的经典结果，自由边界是在一组奇异点之外的\(C^{\infty }\) 。明确的例子表明，奇异集一般可以是\((n-1)\)维的，即与正则集一样大。我们的主要结果表明，一般来说，奇异集的测度为零\(\mathcal{H}^{n-4}\)（特别是，它在自由边界内的余维为 3）。因此，对于\(n\leq 4\)，自由边界通常是\(C^{\infty }\)流形。这解决了 Schaeffer（可追溯到 1974 年）关于维度\(n\leq 4\)中自由边界的一般规律性的猜想。