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Generic regularity of free boundaries for the obstacle problem

2020; Springer Nature; Volume: 132; Issue: 1 Linguagem: Inglês

10.1007/s10240-020-00119-9

1618-1913

Alessio Figalli, Xavier Ros‐Oton, Joaquim Serra,

Advanced Mathematical Modeling in Engineering

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 $n\leq 4$ .