Generic regularity of free boundaries for the obstacle problem

By Alessio Figalli, Xavier Ros-Oton, J. Serra. Publications mathématiques (Bures-sur-Yvette), 2020

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The goal of this paper is to establish generic regularity of free boundaries for the obstacle problem in R n $\mathbf {R}^{n}$ . By classical results of Caffarelli, the free boundary is C ∞ $C^{\infty }$ outside a set of singular points. Explicit examples show that the singular set could be in general ( n − 1 ) $(n-1)$ -dimensional—that is, as large as the regular set. Our main result establishes that, generically, the singular set has zero H n − 4 $\mathcal{H}^{n-4}$ measure (in particular, it has codimension 3 inside the free boundary). Thus, for n ≤ 4 $n\leq 4$ , the free boundary is generically a C ∞ $C^{\infty }$ manifold. This solves a conjecture of Schaeffer (dating back to 1974) on the generic regularity of free boundaries in dimensions n ≤ 4 $n\leq 4$ .

Subfields: Applied Mathematics

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